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INDEX
INTRODUCTION
CHAPTER 1
LITERATURE OVERVIEW
CHAPTER 2
DATA
2.1 Data sources
2.2 General descriptive statistics
2.3 Time series description
2.4 A comparative analysis of the asset classes returns
2.5 Stationarity and normality tests
CHAPTER 3
MODEL ESTIMATION
3.1 The basic idea behind regime switching models
3.2 Filtered and smoothed regime probabilities
3.3 The estimation process of a Markov regime switching model
3.4 Overview of the most common multivariate Markov regime
switching models
3.4.1
MMSIAH(k,p)
3.4.2
MMSIA(k,0)
3.4.3
MMSIAH(k,0)
3.4.4
MMSIA(k,p)
3.4.5
MMSIH(k,p)
3.4.6
MMSI(k,p)
3.4.7
Multivariate restricted (common underlying Markov
chain) MSVAR(K,1) model
3.5 Choice of model specification
3.5.1 Alternative models estimation results
3.5.2 multivariate restricted MSVAR(k,1) models
estimation results
3.6 Selected model estimates
3.7 Model restriction test
3.8 Model description
3.8.1
Economic interpretation of regime
3.8.2
Regimes and predictability from the dividend yield
3.8.3
A comparative analysis between the estimated
regimes and the
NBER USA recession indicator
3.8.4
A dynamic correlation analysis of asset classes
returns
3.8.5
Regime shifts in the asset classes returns means
and volatilities
3.9 A single regime VAR(1) model estimates
CHAPTER 4 - ASSET ALLOCATION
4.1 Unconditional and state conditional asset classes returns
distributions and efficient frontiers based on a MSVAR(2,1)
model
4.2 In-sample asset allocation exercise
4.2.1 In-sample asset allocation exercise based on a
MSVAR(2,1) model
4.2.2 In-sample asset allocation exercise based on a
VAR(1) model
4.3 Out-of-sample asset allocation exercise
4.3.1 Out-of-sample asset allocation exercise based on a
MSVAR(2,1) model
4.3.2 Out-of-sample asset allocation exercise based on a
VAR(1) model
4.4 Forecasting ability comparison
4.4.1 RMSE
4.4.2 Pearson correlation coefficient
4.4.3 Regression analysis
4.5 Asset allocation exercise conclusions
CHAPTER 5 - ASSET ALLOCATION BASED ON THE
COPULA-OPINION POOLING APPROACH
5.1 The copula-opinion pooling (COP) approach
5.2 Out-of-sample mean-variance asset allocation exercise
based on COP and MSVAR(2,1) and VAR(1)
5.3 Out-of-sample conditional value-at-risk (CVaR) asset
allocation exercise based on COP and MSVAR(2,1) and
VAR(1)
5.4 COP asset allocation exercise conclusions
CHAPTER 6 - CONCLUSIONS
1
INTRODUCTION
Most studies assume that asset returns are generated by a linear process with
stable coefficients so the predictive power of variables such as dividend yield does
not vary over time. However, there is mounting empirical evidence that asset
returns follow a more complicated process with multiple regimes, each of which is
associated with a very different distribution of asset returns. Since when authors
have started to report evidence of regimes in stock or bond return, regime
switching models have been assuming a central role in financial applications
because of their well-known ability to capture the presence of rich non-linear
patterns in the joint distribution of asset returns. A normal distribution is not
sufficient to characterizes asset returns, as historical data showed fat tails,
skewness, and kurtosis. Furthermore, a simple model with IID probability
distributions, ignoring time series structure of economic stages, seems to be
incomplete, as the financial market changes patterns over time. Regime switching
models are designed to capture discrete changes in the economic mechanism that
generate the data. Usually, regime categorization is linked to the market economic
situations. With constant parameters and linear relationships between asset return
and factors, the classic asset pricing models do work when the financial market
operates normally. However, since authors have displayed that financial crises are
characterized by a significant increase in correlations of stock price movements,
the benefits of portfolio diversification may seriously be subjected to a
reconsideration. As in a standard portfolio optimization model, expected asset
returns and their variance and covariance matrix are derived from an asset pricing
model. If the model do not takes into account these returns features they are
prone to fail to predict expected asset returns correctly since they depend on static
parameters. Consequently, the limitations of the classic asset pricing models
cause a challenge to portfolio optimization. In this context traditional portfolio
optimization models have become unreliable, and the financial industry needs a
dynamic portfolio optimization model that is able to characterize different economic
conditions. Since the performance of any investment portfolio depends on the
accuracy of forecast of assets, the first step is to develop an asset pricing model
for the prediction of asset returns which is able to take into account relevant
2
market returns features, while the second step is to develop a dynamic portfolio
model that maximizes the investment profit with a limited risk exposure.
Dynamic asset allocation is a process of selecting instruments and constructing
optimal portfolios over time. In this context my work proposes an investment model
incorporating market regimes which characterizes different patterns of asset
returns in the unobservable economic situations, such as bear and bull markets.
The developed model is a dynamic vector autoregressive regime-switching model
which also incorporates a predictive variable. The model choose is the result of an
extensive model research amongst the Markov Switching Vector Autoregressive
model MSVAR which is a general class of models that nests the standard VAR
model but additionally accounts for nonlinear regime shifts. The model estimated
is employed to model returns of US small stocks, large stock and bonds, and to
identify different market regimes over time. From the application of the model to
the data, two distinct regimes have been identified. Regime 1 is an extremely
persistent bull regime characterized by low realized volatility and positive average
realized excess returns on all assets. Regime 2 is a not-highly persistent bear
regime characterized by high volatility and large and negative average realized
excess returns on small and large stocks while the average realized excess return
on bonds are significantly positive and relatively larger than in regime 1. Regime 2
includes two oil shocks in the 1970s, the recession of the early 1980s, the October
1987 stock market crash, the Kuwait invasion in the early 1990s and the `Asian flu'
(1996-1998), the dot-com market crash of 2000-2001 (the recent bear market of
2002-2002) and the 2008-2009 financial crisis. In order to evaluate the potential
benefit of a multiple regimes model against a more simple single regime model, I
additionally estimated a less sophisticated single state model, namely a first order
vector autoregressive model VAR(1). The regime switching model covers time
variation not only in the conditional means of stock returns and the dividend price
ratio, but also in their volatilities, their correlations, and their predictive relation. At
each time the returns are assumed to be drawn from one of the different
Gaussians distribution underlying the regime switching model. The latent variable
is an unobserved state variable that governs the switches between regimes and is
assumed to follow a Markov chain. In practice, the Markov chain is unobservable
3
and embedded in the observed sample over time. Regimes switching model can
be seen as mixtures of normal distributions which are considered a very flexible
family that can be used to approximate numerous other distributions. Another
attractive feature of regime switching models is that they are able to capture
nonlinear stylized dynamics of asset returns in a framework based on linear
specifications, or conditionally normal distributions, within a regime. This makes
asset pricing under regime switching analytically tractable. In particular, regimes
introduced into linear asset pricing models can often be solved in closed form
because conditional on the underlying regime, normality is recovered. The
expected returns and their variance-covariance matrix used in the objective
function of the optimization have been generated both by the regimes switching
asset pricing model and by the single state asset pricing model.
My work consists of a comparative study of the performances of the multivariate
regime switching model against the single regime model in terms of portfolio
returns in the context of dynamic asset allocation. The study was conducted
through the practical application, both in-sample and out of-sample, of the two
models under various portfolio optimization approaches. In the first part of the
asset allocation exercise I constructed for any asset pricing model, both in-sample
and out-of-sample, two dynamic recursive efficient portfolios that maximize the
Sharpe among portfolios on the efficient frontier; in the first in-sample dynamic
recursive portfolio the budget constraint is opened up to permit between 0% and
100% in the riskless asset while the second one requires fully-invested portfolios
whose weights must sum to 1; in addition short selling, thus negative asset class
weights, is not allowed. The other three dynamic recursive portfolios that I
constructed have been chosen as those that maximize the investor utility function
with three different risk aversion coefficient subject to non negative weights and
opened upper budget constraint. The second part of the asset allocation exercise
focuses only on the out-of-sample period. Here the Copula-Opinion Pooling
approach is applied to implement in the asset pricing model views on the asset
returns produced by both the single regime model and the more sophisticated
regime switching model. The purpose of this section is to investigate and make a
comparison of the behavior of the regime switching model and the single state
4
model in the COP framework in terms of both expected and realized portfolio
returns and Sharpe ratio in the context of mean-variance and conditional value-at-
risk (CVaR) portfolio optimization. Therefore, in addition to the five recursive
optimal portfolios chosen with the same portfolio selection process as in the first
part of the asset allocation problem, here using conditional value-at-risk as the risk
exposure constraint, I derived the dynamic optimal weights of other five different
portfolios equally distributed, in terms of CVaR, along the time dependent efficient
frontier. The exercise has been repeated for different values of the confidence in
the views.
From the evaluation of the portfolio performances in the empirical analysis, I
expect the superiority of the regime switching model to be noticed. The
overperformance can be achieved by the more efficient and desirable risk-reward
combinations on the state-dependent frontier that can be obtained only by
systematically altering portfolio allocations in response to changes in the
investment opportunities as the economy switches back and forth among different
states. An investor who ignores regimes sits on the unconditional frontier, thus an
investor can do better by holding a higher Sharpe ratio portfolio when the low
volatility regime prevails. Conversely, when the bad regime occurs, the investor
who ignores regimes holds too high a risky asset weight. She would have been
better off shifting into the risk-free asset when the bear regime hits. As a
consequence, the presence of two regimes and two frontiers means that the
regime switching investment opportunity set dominates the investment opportunity
set offered by one frontier.
The plan of the paper is structured as follows. Section 1 describes the literature
overview on regime-switching models. Section 2 gives information on the data
employed in the paper. Section 3 provides the theoretical background for the
Markov regime switching models, estimates a range of regime switching models,
select the best one and provides an interpretation of the parameter estimates and
the resulting regimes. Section 4 examines the in-sample and out-of-sample
performances of asset allocation schemes based on the two different asset pricing
models. Section 5 examines the out-of-sample performances of asset allocation
5
schemes based on the two different asset pricing models under the Copula-
Opinion Pooling approach. Section 6 concludes.
6
CHAPTER 1
LITERATURE OVERVIEW
Most studies assume that asset returns are generated by a linear process with
stable coefficients so the predictive power of state variables such as dividend
yield, default and term spreads does not vary over time. However, there is
mounting empirical evidence that asset returns follow a more complicated process
with multiple "regimes", each of which is associated with a very different
distribution of asset returns. Regime switching models were introduced into
economics by Goldfield and Quandt (1973) but their breakthrough in economics
came with the seminal paper of Hamilton (1989). Ang and Bekaert (2002a, 2002b),
Ang and Chen (2002), Garcia and Perron (1996), Gray (1996), Guidolin and
Timmermann (2005a; 2005b; 2006a; 2006b; 2006c), Perez-Quiros and
Timmermann (2000), Turner, Startz and Nelson (1989) and Whitelaw (2001) all
report evidence of regimes in stock or bond returns. Regime switching models
have been assuming a central role in financial applications because of their well-
known ability to capture the presence of rich non-linear patterns in the joint
distribution of asset returns. A normal distribution is not sufficient to characterizes
asset returns, as historical data showed fat tails, skewness, and kurtosis.
Furthermore, a simple model with IID probability distributions, ignoring time series
structure of economic stages, seems to be incomplete, as the financial market
changes patterns over time. There are good economic reasons why the
equilibrium distribution of asset returns must be involved with economic sentiment
regimes. Regime switching models are designed to capture discrete changes in
the economic mechanism that generate the data. Usually, regime categorization is
linked to the market economic situations. In the seminal work by Hamilton (1989)
the regime switching model allows the data to be drawn from two or more possible
distributions, where the transition from one regime to another is driven by the
realization of a discrete variable (the regime), which follows a Markov chain
process. That is, at each point in time, there is a certain probability that the
process will stay in the same regime next period. The transition probability may be
constant or they may depend on the other variables. In regime switching model the
returns can be described with a hidden Markov model (HMM) of Gaussian
mixtures with a number of different regimes. At each time the stock return is
7
assumed to be drawn from one of the different Gaussian distribution. The latent
variable is an unobserved state variable that governs the switches between
regimes and is assumed to follow a first order Markov chain, i.e. only the most
present history of the chain matters and the switching probabilities are constant. In
practice, the Markov chain is unobservable and embedded in the observed sample
over time. As pointed out by Marron and Wand (1992), mixtures of normal
distribution provide a very flexible family that can be used to approximate
numerous other distributions. Mixture of normals can also be viewed as a
nonparametric approach if the number of state, k, is allowed to grow with the
sample size. They can capture skewness and kurtosis in a way that is easily
characterized as a function of the mean, variance and persistence parameters of
the underlying starts. They can also accommodate predictability and serial
correlation in returns and volatility clustering since they allow the first and second
moments to follow a step function driven by shifts in the underlying regime
process, c.f. Timmermann (2000). Recent papers have emphasized the
importance of adopting flexible models capable of capturing even complicated
time-varying forms of heteroskedasticity, fat tails and skews in the underlying
distribution of returns, see Manganelli (2004), Patton (2004) and Timmermann
(2000). The differences in performance measures between the single-state model
and the regime switching model could become larger if higher-order moments are
taken into consideration as well. Perez-Quiros and Timmermann (2001) provide an
excellent study on higher-order moments under regime switching. Any finite state
model is best viewed as an approximation to a more complex and evolving data
generating process with non-recurrent states (see, e.g., Pesaran, Pettenuzzo and
Timmermann (2006)). Other previous researches indicated that a probability
distribution with a structural Markov chain is efficient to describe the dynamics of
the economic regimes. Hamilton (1989) successfully applied a two-regime hidden
Markov model to the U.S. GDP data and characterized changing pattern of the US
economy. Cai (1994), Hamilton and Susmel (1994), and Gray (1996) use
variations of the standard Markov regime switching model to describe the time
series behavior of U.S. short-term interest rates. Bekaert and Hodrick (1993)
document regime shifts in major foreign exchange rates. Schwert (1989)
8
considered that asset returns may be associated with either high or low volatility
which switches over time. Liu, Xu and Zhao (2011) showed that the regime
switching model is an effective way for linking sector ETF returns to style and
macro factors in changing market regimes over time. Whitelaw (2001) constructed
an equilibrium model where growth in consumption follows a regime switching
process so investors' intertemporal marginal rate of substitution also follows a
regime process. Ang and Bekaert (2002) studied an international asset allocation
model with regime shifts and examine portfolio choice for a small number of
countries. Guidolin and Timmermann (2006; 2008) provide important economic
insights on how investments vary across different market regimes. Recently, Jun
Tu (2010) provides a Bayesian framework for making portfolio decisions with
regime switching and asset pricing model uncertainty. The joint distribution of
future economic indicators not only depends on the current observations but also
on the regime switching model parameters. Consequently, asset classes time
series follow a multivariate mixture of normal distribution with time-varying mixing
parameters over time. As regimes are not observable, the probability distribution of
regimes must be dynamically updated with newly observed data and the
unconditional joint probability distribution of the returns is a multivariate mixture of
normals with mixing parameters equal to the prior distribution of the regimes at
time t. Regime switching models typically identify bull and bear regimes with very
different means, variances and correlations across assets, as noticed by Maheu
and McCurdy (2000). As the underlying state probabilities change over time this
leads to time-varying expected returns, volatility persistence and changing
correlations and predictability in higher order moments such as the skew and
kurtosis. The degree of predictability of mean and returns can also vary
significantly over time in regime switching models
a feature that seems present
in stock returns data as noticed by Bossaerts and Hillion (1999). Under the
traditional ARCH and GARCH models of Engle (1982) and Bollerslev (1986),
changes in volatility are sometimes found to be too gradual and unable to capture,
despite the additions of asymmetries and other tweaks to the original GARCH
formulations. Hamilton and Susmel (1994) and Hamilton and Lin (1996) developed
regime switching version of ARCH dynamics applied to equity returns that allowed
9
volatilities to rapidly change to new regimes. A version of a regime switching
GARCH model was proposed by Gray (1996). There have been many versions of
regime switching models applied to vector of asset returns. Ang and Bekaert
(2002a) and Ang and Chen (2002) show that regime switching models provide the
best fit out of many alternatives models to capture the tendency of many assets to
exhibit higher correlations during down markets than in up markets. Ang and Chen
(2002) interestingly find that there is little additional benefit to allowing regime
switching GARCH effects compared to the heteroskedasticity already present in a
standard regime switching model of normals. It has been known for some time that
international equity returns are more highly correlated with each other in bear
markets than in normal times (see Erb, Harvey and Viskanta (1994); Campbell,
Koedijk and Kofman (2002)). Longin and Solnik (2001) recently formally establish
the statistical significance of this asymmetric correlation phenomenon. Whereas
standard models of time varying volatility (such as GARCH models) fail to capture
this salient feature of international equity return data, recent work by Ang and
Bekaert (2002a) shows that asymmetric correlations are well captured by a regime
switching model. Stock and bond returns are - to a limited extent
predictable
(e.g., Campbell (1987), Fama and French (1988; 1989) and Keim and Stambaugh
(1986)), their volatility cluster over time (e.g., Bollerslev, Chou, and Kroner (1992)
and Glosten, Jagannathan, and Runkle (1993)) and correlations are not the same
in bull and bear markets (e.g., Ang and Chen (2002) and Perez-Quiros and
Timmermann (2000)). At shorter horizons stock returns are also far from normally
distributed and affected by occasional outliers. Campbell and Ammer (1993) and
Fama and French (1989) have showed that variables found to forecast stock
returns also predict bond returns. Henkel, Martin, and Nardari (2001) capture the
time-varying nature of return predictability in a regime switching context. They use
a regime switching vector autoregression (VAR) with several predictors, including
dividend yields, and interest rate variables along with stock returns. They find that
predictability is very weak during business cycle expansions but is very strong
during recessions. Thus, most predictability occurs during market downturns, and
the regime switching model captures this countercyclical predictability by exhibiting
significant predictability only in the contraction regime. A branch of the existing
10
regime switching literature concern with the issue of parameter estimation. For an
extensive overview concerning the econometrics issues of regime switching model
and an overview about empirical evidence many authors refer to Kim and Nelson
(1999). One of the first papers in financial econometrics that estimates time-
varying integration of single countries to the world market is Bekaert and Harvey
(1995). Hamilton (1994) and Kim and Nelson (1999) give an overview about the
econometrics of state-space models with regime switching and provide an
overview of possible applications to finance. From an econometric point of view,
the main problem in estimating regime switching models is the unobservability of
the prevailing regime. Two different approaches have been suggested: a classical
maximum likelihood based on the filters such as the Hamilton filter or on the
expectation maximization algorithm and a Bayesian approach based on numerical
Bayesian methods such as the Gibbs sample and Markov Chain Monte Carlo
methods. Regime probabilities play a critical role in the estimation for regime
switching models, which uses maximum likelihood techniques (see Hamilton
(1994), Gray (1996); and Ang and Bekaert (2002b)) or Bayesian techniques (see
Albert and Chib (1993)). It is well known that applications of classical mean-
variance frontier (MVF) technology to dynamic asset allocation problems in which
the MVF is allowed to depend on one more variables capturing the state of market
investment opportunities, suffer from a number of issues (e.g., see Schöttle an
Werner (2006)). For instance, the shape of the MVF together with the location of
the efficient portfolios has been observed to change drastically as market data are
progressively updated and expanded. Moreover, it is typical to observe that MVFs
often occupy rather unrealistic regions of the mean-standard deviation space as a
result of optimization based on error-prone estimations, resulting in large
deviations between the ex-ante, in-sample and the ex-post, out-of-sample Sharpe
ratios. Guidolin and Ria (2010) can be seen as an attempt to produce more robust
estimates of the MVF and hence
after appropriate mean-variance preferences
have been assumed
more robust optimal portfolios not by changing methods of
estimation or by resampling the data, but instead by exploring the implication of a
simple and yet powerful parametric approach that explicitly tracks the time
variation in the features of the investment opportunity sets (means, variances, and
11
correlations) as depending from a latent Markov
state variable.
In the presence of regimes, asset returns may have entirely different relationships
with the predictors in different regimes. One of the key improvements upon a
traditional investment model is that portfolio decisions are based on a Bayesian
type of dynamic updating on the probability distribution of the economic regimes.
Essentially, this modeling approach provides time-varying risk premiums and risk
magnitudes, depending not only on the economic indicators but also the updated
regime distribution at any point in time. Investment securities may exhibit different
risk levels in different economic situations and therefore, different risk premiums.
However, there is no clear determination as to which economic regimes we are in
by directly observing the market data. The key idea for a regime switching model
is to resolve the issue of unobserved economic regimes over time. Regime
switching means that all conditional moments of the asset returns distribution are
time-varying, so it is possible to extend the previous literature on strategic asset
allocation to cover the case where all moments may be appropriate. However,
none of the state can be perfectly anticipated: starting from any one of these the
investor always assigns a positive and non-negligible probability to the possibility
of transitioning to a different state. Regime switching models also nest as a special
case jump models, given that a jump is a regime that is immediately exited next
period and, when the number of regimes is large, the dynamics of a regime
switching model approximates the behavior of time-varying parameter models
where the continuous state space of the parameter is appropriately discretized.
Finally, another attractive feature of regime switching models is that they are able
to capture nonlinear stylized dynamics of asset returns in a framework based on
linear specifications, or conditionally normal log-normal distributions, within a
regime. This makes asset pricing under regime switching analytically tractable. In
particular, regimes introduced into linear asset pricing models can often be solved
in closed form because conditional on the underlying regime, normality (or log-
normality) is recovered. This makes incorporating regime dynamics in affine
models straightforward. Regime shifts continue to have a significant effect on the
optimal asset allocation and expected utility even after accounting for parameter
uncertainty. Guidolin and Timmermann Size and Value Anomalies under Regime
12
Shifts (2005) find that four-state models perform better than single-state
alternatives both in terms of precision of their out-of-sample forecasts and in terms
of sample estimates of mean returns and that accounting for the presence of
regimes lead to higher average realized utility even after accounting for parameter
estimation error. Another branch of literature that concerns with the portfolio
choice and regime switching analyzes the effects of regime switching on asset
allocation. Overall, the main findings are that regime switching induces a change
in the asset allocation depending on the investment horizon and depending on the
current regime. One of the main references for asset allocation in regime switching
framework is Ang and Bekaert (2002). In their paper, they analyze dynamic asset
allocation with regimes shifts in an international context. The starting point of their
paper are time-varying correlation between different equity markets. In bad times
correlation and volatilities increase in comparison to good times and, therefore the
investment opportunity set is stochastic. In the empirical part, they assume two
state model with Markov switching and constant transition probabilities. For
parameter estimation, they use a Bayesian procedure similar to Hamilton (1989)
and Gray (1996). Overall, there are always relatively large benefits of international
diversification, although the optimality of the home-biased portfolios cannot always
be rejected statistically. The cost of ignoring regime switching are very high if the
investor is allowed to switch to cash position. If the investment universe is limited
to equities, costs of ignorance are lower. With respect to hedging demands, they
find that intertemporal hedging demands under regime switching are economically
negligible and statistically insignificant. Similar, Ang and Bekaert (2004) find that
for a global all-equity portfolio, the regime switching strategy dominates static
strategies in an out-of-sample test. In a persistent high-volatility market, the model
tells the investors to switch primarily to cash. Recent contributions include
Graflund and Nilsson (2003), Bauer, Haerden and Molenaar (2003), Ang and
Bekaert (2004), and Guidolin and Timmermann (2005). A number of authors
analyze the implications of regime switching in portfolio selection. The presence of
asymmetric correlations in equity returns has so far primarily raised a debate on
whether they cast doubt on the benefits of international diversification, in that
these benefits are not forthcoming when you need them the most. However, the
13
presence of regimes should be exploitable in an active asset allocation program.
The optimal equity portfolio in the high volatility regimes is likely to be very
different (for example more home biased) than the optimal portfolio in the normal
regime. When bonds and T-bills are considered, optimally exploiting regime
switching may lead to portfolio shifts into bonds or cash when a bear market
regime is expected. In particular, investor often use available realized returns at a
given point in time to determine whether the market is in a bull or a beat state.
Turner, Starts, and Nelson (1989) provide a rigorous econometric model for
analyzing bull and bear markets and find that the S&P500 index displays different
means and variances across these markets. Schwert (1989) and Hamilton and
Susmel (1994) also document regime-dependent market volatility, while Ang and
Bekaert (2002; 2004) and Guidolin and Timmermann (2005; 2007; 2008a; 2008b)
provide important economic insights on how investments vary across different
market regimes. Jun Tu (2010) found that the certainty-equivalent loses
associated with ignoring regime switching are generally above 2% per year, and
can be as high as 10%. Tu and Zhou (2004) find that the certainty-equivalent
losses associated with ignoring fat tails are typically less than 1% for mean-
variance investors facing model and parameter uncertainty. However, the findings
of this study reveal that ignoring regime switching can lead to sizable economic
costs. These findings support the qualitative conclusions of the earlier regime
studies by Ang and Bekaert (2002; 2004) and Guidolin and Timmermann (2005;
2007; 2008a; 2008b), despite their classical framework, which does not
incorporate model or parameter uncertainty. This is because the impact of these
types of uncertainty could be less important than the impact of regimes. Where the
investor is allowed to rebalance his or her portfolios during the investment period,
the corresponding problem is usually discussed in terms of intertemporal hedging,
first mentioned by Merton (1971). The intertemporal hedging is a desire of the
investor to protect him from unfavorable changes in the set of investment
opportunities, or a desire to be able to profit from favorable changes. Ang and
Bekaert (1999) investigates international diversification and intertemporal hedging
demands within a regime switching framework from the perspective of a US
investor who is allowed to buy foreign stocks in addition to US stocks. Nilsson and
14
Graflund (2001) instead assume that the investment opportunity set faced by the
investor is spanned by a well-diversified home stock market portfolio and risk-free
short-term bill. The aim of the authors and contribution of their paper is to study
the importance of regime switching in this classical portfolio selection setup. The
authors investigated if the optimal portfolio differs across different regimes, and if
the intertemporal hedging demand differs across regimes. In the regime switching
model described by the authors the returns are represented as a mixture of
Gaussian distributions. The joint effects of learning about the underlying state
probabilities and predictability of asset returns from the dividend yield give rise to a
non-parametric relationship between the investment horizon and the demand for
stocks. Strategic asset allocation decisions can only be made in the context of a
model for the joint distribution of asset returns. Most studies assume that asset
returns are generated by a liner process with stable coefficients so the predictive
power of state variable such as dividend yields, default and term spreads does not
vary over time. However, there is mounting empirical evidence that asset returns
follow a more complicated process with multiple "regimes", each of
which is
associated with a very different distribution of asset returns. In Guidolin and
Timmermann the authors characterize investors' strategic asset allocation and
consumption decisions under a regime switching model for asset returns with four
states characterized as crash, slow growth, bull and recovery states. A difference
to earlier studies is that the authors allow the underlying states to be unobservable
to the investor who must infer the state probability from the sequence of returns
data. Kao and Shumaker (1999) analyze the opportunities for equity style timing,
based on Fama and French (1993) factors, using recursive partitioning (regression
and classification trees) and macroeconomic factors (term spread, real bond yield,
corporate credit spread, high-yield spread, estimated GDP growth, earnings-yield
gap, CPI), they try to predict future differences in style returns. They find that
timing strategies in the US market based on asset class and size have historically
provided more opportunity for outperformance than a timing strategy based on
value and growth. For example, during the Internet bubble period 1998-2002, the
stock market was extremely volatile, while market volatility was relatively low in the
period of 2003-2006. It is highly probable that market sentiment, market volatility,
15
and the non-smooth asset return processes are regime dependent. Ignoring such
a possibility and simply averaging information across the market regimes may
result in suboptimal investment strategies. If regimes exist and may be identified ,
estimated, and predicted, then it is an open question whether an investor should
take notice of them, and go through the relatively sophisticated econometric
techniques required by her acknowledging this state-dependence. Clarke and de
Silva (1998) note that no static mix to be applied to standard mean-variance
portfolios can be used to achieve a point along a state-dependent efficient frontier.
The more efficient and desirable risk-reward combinations on the state-dependent
frontier may be achieved only by systematically altering portfolio allocations in
response to changes in the investment opportunities as the economy switches
back forth among different states. The reason is that in the presence of state-
dependence (when two states with probabilities p and 1-p are possible), a mixture
of Gaussian (more generally, elliptical) densities is never the same as a Gaussian
density that has means and variances which are probably-weighted (with weights
p and 1-p) averages of the state-dependent means and variances. When
investment opportunities remain constant over time, a power utility investor's
horizon does not affect the optimal asset allocation, c.f. Samuelson (1969). In the
absence of predictor variables, standard models therefore imply constant portfolio
weights. In contrast, using the dividend yield as a predictor, Barberis (2000) finds
that the weight on stocks should increase as a function of the investor's horizon.
Even in the absence of predictor variables, regime switching models imply that
investors' asset allocation varies over time as the underlying states offer different
investment opportunities and investors revise their beliefs about the state
probabilities. Ang and Chen (2002) find that equity correlations that differ across
high/low return states can be successfully captured by a regime switching model.
They note that small firms' returns exhibit relatively strong asymmetries and argue
that such asymmetric correlations may be important for strategic asset allocation
purposes, although they stop short of analyzing this question. Guidolin and
Timmermann (2005) extend the regimes switching model for asset returns to
include predictability from state variables such as dividend yield. Consistent with
earlier findings in the literature (e.g., Campbell, Chan and Viceira (2003)), Guidolin
16
and Timmermann (2005) find that the recursively updated portfolio weights vary
significantly over time as a result of changing investment opportunities and that
optimal asset holdings are sensitive to how predictability is modeled. When
regimes are taken into account, there is evidence that the allocation to stocks and
bonds as well as the division of stock holdings among small and large firms is
quite different from that obtained under linear models of predictability in asset
returns. Furthermore, the authors generally find that the average realized utility is
highest for model that account for regime switching. Huy Thanh Vo and Maurer
(2013) solve the asset allocation problem under stock return predictability based
on the dividend yield for an investor who accounts for both; the uncertainty about
the true underlying predictive power of the dividend yield and changes in the joint
distribution of stocks and predictors due to shifts in regimes. The model proposed
by the authors covers time variation not only in the conditional means of regime
switching and the dividend yield, but also in their volatilities, their correlations, and
their predictive relation. The possibility of switching across regimes, even if it
occurs relatively rarely, induces an important additional source of uncertainty that
investors want to hedge against. It is reasonable to expect that if the market
portfolio exhibits regime switches, then portfolios of stocks would also switch
regimes, and the regimes and behavior within each regimes of the portfolios
should be related across portfolios. This is indeed the case Perez-Quiros and
Timmermann (2000), Gu (2005), and Guidolin and Timmermann (2008b), among
others, fit regime switching models to small cross section of stock portfolios. These
studies show that the magnitude of size and value premiums, among other things,
varies across regimes in the same direction. On the other hand, the dynamics of
certain stock portfolios react differently across regimes, such as small firms
displaying the greatest differences in sensitivities to credit risk across recessions
and expansions compared to large firms. Factor loadings of value and growth
firms also differ significantly across regimes. They exploit the ability of the regime
switching model to capture higher correlation during market downturns and
examine the question of whether such higher correlations during bear markets
negate the benefits of international diversification. They find there are still large
benefits of international diversifications. The cost of ignoring regimes is very large
17
when a risk-free asset can be held; investors need to be compensated
approximately two to three cents per dollar of initial wealth to not take into account
regimes changes. In the regimes switching context the risk-return trade-off can
vary across states in a way that may have strong asset allocation implications. For
example, knowing that the current state is a persistent bull state will make most
risky assets more attractive than in a bear state. Asset allocation under an
unknown number of permanent structural breaks has been studied by Pettenuzzo
and Timmermann (2011), who apply a multiple change point model proposed by
Chib (1998). When dividend yield as a predictor is added to the a regime switching
model the resulting regime switching VAR model nests many of the models in the
existing literature and enables the correlation between the dividend yield and asset
returns to vary across different regimes. The relationship between stock returns
and the dividend yield is linear within a given regimes. However, since the
coefficient on the dividend yield varies across regimes, as the regime probabilities
change the model is capable of tracking a non-liner relationship between asset
returns and the yield. This is important given the evidence of a non-linear
relationship between the yield and stock returns uncovered by Ang and Bekaert
(2004). A large literature in finance has reported evidence that variables related to
the business cycle predicted stock and bond returns. One of the key instruments is
the dividend yield; see, e.g., Campbell and Shiller (1988), Fama and French
(1998; 1989), Ferson and Harvey (1991), Goetzmann and Jorion (1993) and
Kandel and Stambaugh (1996). Due to its high persistence coupled with the strong
negative correlation between shocks to returns and shocks to the dividend yield,
Campbell, Chan, and Viceira (2003) find that the dividend yield generate the
largest hedging demand among a wider set of predictable variables. Empirical
studies on predictive regressions find economic indicators related to the business
cycle slightly effective in predicting stock returns, which contradicts the assumption
of independently and identically distributed stock returns (Campbell, Low, and
MacKinlay (1997)). Against the background of these empirical findings, portfolio
theory reveals that optimal policies under predictability do exhibit horizon effects,
as time variation in the expected returns induce intertemporal hedging demands
against changes in the investment opportunity set (Brennan, Schwartz, and Lynch
18
(1999)). The implications were first formulated by Merton (1973). The lasting stock
bull market during the 80s and 90s reinforced doubts about predictability, as the
dividend yield ability to forecast expected stock returns declined noticeably.
Acknowledging this ongoing dispute, portfolio theory has accounted for uncertainty
in predictive relations by incorporating estimation risk (Kandel and Stambaugh
(1996), Barberis (2000)), model risk (Avramov (2002)), and learning (Xia (2001),
Brand and Santa-Clara (2006)). These studies adopt a Bayesian framework
instead of taking a binary view, either accepting or rejecting predictability solely
based on statistical significance. Nevertheless, a comprehensive re-examination of
predictive regressions using an extended data set and with newly formulated test
statistics has led to some researchers to conclude that stock return predictability,
especially long run predictability, was a statistical fluke and never truly existed
(Ang and Bekaert (2007), Boudoukh, Richardson, and Whitelaw (2008)). Moreover
the out-of-sample performance of several predictors has been reported to be poor
and, in many cases, even worse, than unconditional mean of stock returns
(Bossaerts and Hillion (1999), Welch and Goyal (2008)). This too casts doubt on
the practical relevance of predictability. In contrast to these studies, another strand
of literature argues that the predictive relation itself is subject to time variations
and hence cannot be sufficiently described by a simple linear regression. For
instance, Pesaran and Timmermann (1995), Paye and Timmermann (2006), and
Henkel, Martin, and Nardari (2011) provide evidence for changes in the predictive
relation over time. Lettau and Van Nieuwerburgh (2008) argue that the mean of
dividend yield was affected at least by one or even two structural breaks (in 1994
and possibly in 1951), and that once adjusted for the subsample means, the
predictive power remains stable and significant. Actually standard regime
switching models cannot capture the effects of permanent structural breaks.
Nevertheless, they can approximate permanent breaks to a certain extent by
including extremely persistent regimes. In the long run, however, they imply a
steady state distribution. Furthermore, the combination of regime shifts and stock
predictability also incorporates two aspects of major importance in asset
allocations. Generally, regime shifts generate a momentum effect, which increases
the variance in the long run, whereas stock return predictability generates a mean
19
reversion, or rebound effect, which decreases the variance in the long run (Ang
and Bekaert (2002)).
Samuelson (1991) define a "rebound" process, or mean
-
reverting process, as having a transition matrix which has a higher probability of
transitioning to the alternative state than staying in the current state. Samuelson
shows that with a rebound process, risk-averse investors increase their exposure
to the risky assets as the horizon increases. That is, under rebound, long-horizon
investors are more tolerant of risky assets than short-horizon investors. The
opposite of rebound process is called "momentum" process: it is more likely to
continue in the same state rather than transition to the other state. Under a
momentum process, risk-averse investors want to decrease their exposure to risky
assets as horizon increases. Intuitively, long-run volatility is smaller under a
rebound process that under a momentum process (with the same short-run
volatility).
20
CHAPTER 2
DATA
In this chapter I am going to describe in more detail the data I have used in this
work. In the first paragraph the data are listed and the sources are provided; in the
second paragraph some general descriptive statistics are provided for the risk free
rate, the dividend yield and the three asset classes time series; in the third
paragraph a comment of the data presented in the second paragraph and a
general overview are provided for each time series; in the fourth paragraph, the
three asset classes time series are compared and some further comments are
provided; lastly, in the fifth paragraph the results of the normality and stationarity
tests are provided and described for the three asset classes time series and the
dividend yield.
2.1 Data sources
Table 1
Data Short Name and Description
Variable Definition
lo20
excess value weighted monthly returns of first and second CRSP size decile
US equity portfolios
hi20
excess value weighted monthly returns of ninth and tenth CRSP size decile
US portfolios
tbond
excess monthly returns on a 10 years bond priced using the monthly FED
10-Year Treasury Constant Maturity Rate
risk_free 1-month US T-Bill monthly returns
div_y
S&P 500 Dividend Yield
The time series lo20 and hi20 represent respectively the excess value weighted
monthly returns of the first and second, and the ninth and tenth size deciles US
equity portfolio. The data are made available by Kenneth R. French on his
academic website's data librar
y
1
. According to the authors the portfolios are
constructed at the end of each June using the June market equity and NYSE
breakpoints. The portfolios for July of year t to June of t+1 include all NYSE,
AMEX, and NASDAQ stocks for which market equity data for June of year t were
available.
1
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html and
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_port_form_sz.html
21
The version of the data I have used dates back to March 2015 and the data author
declares the file was created using the January 2015 CRSP (The Center for
Research in Security Prices) database.
The time series tbond represents the excess monthly returns on a US 10-year
constant maturity Treasury Bond. The time series has been realized pricing a 10-
year constant maturity bond using the monthly US 10-year Treasury Constant
Maturity Rate provided by the US Federal Reserve Bank of St. Louis. The pricing
formula applied is the monthly version of the Damodaran formula
2
which models
the bond returns as the monthly fraction of the annual promised coupon at the
start of the year and the bond price change due to interest rate changes.
The risk_free time series represents the US 1-month Treasury Bill rate. The data
are published by Kenneth R. French on his academic website's data library
3
and
are made available by Ibbotson and Associates Inc. The div_y time series
represents the S&P 500 Dividend Yield which consists in the annual dividends
paid by the companies in the index divided by the price index. The data are made
available by Robert Shiller on his academic website
4
in the stock market data
section. The same data have been used by the author in his book Irrational
Exuberance (2005).
2.2 General descriptive statistics
In this paragraph some general descriptive statistics of the four time series are
shown below in Table 2.
Table 2
Monthly Time Series General Descriptive Statistics
lo20
hi20
tbond risk_free div_y
# of observations
540
540
540
540
540
Mean
0.0068
0.0036
0.0016
0.0046
0.0311
Median
0.0103
0.0065
0.0018
0.0043
0.0308
2
http://quant.stackexchange.com/questions/3941/t-note-returns-from-t-note-yields-derivation-of-
damodarans-formula and http://www.stern.nyu.edu/~adamodar/pc/datasets/histretSP.xls
3
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html and
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/det_port_form_sz.html
4
http://www.econ.yale.edu/~shiller/data.htm
22
Minimum
-0.3022
-0.2091
-0.0894
0.0000
0.0111
Maximum
0.2732
0.1761
0.1005
0.0135
0.0624
Standard Dev.
0.0650
0.0437
0.0202
0.0023
0.0120
Variance
0.0042
0.0019
0.0004
0.0000
0.0001
Skewness
-0.2177
-0.3726
0.4497
0.8436
0.2917
Kurtosis
5.2645
4.7089
6.1871
4.5912
2.3683
Sharpe Ratio
0.1051
0.0824
0.0776
Percentile (10%)
-0.0697
-0.0483
-0.0215
0.0015
0.0158
Percentile (25%)
-0.0295
-0.0207
-0.0111
0.0032
0.0203
Percentile (50%)
0.0103
0.0065
0.0018
0.0043
0.0308
Percentile (75%)
0.0442
0.0318
0.0130
0.0057
0.0386
Percentile (90%)
0.0794
0.0536
0.0236
0.0075
0.0489
Mean (normal fit)
0.0068
0.0036
0.0016
0.0046
0.0311
Mean annualized (normal fit)
0.0821
0.0433
0.0188
0.0547
Standard Dev. (normal fit)
0.0650
0.0437
0.0202
0.0023
0.0120
Standard Dev. Annualized
(normal fit)
0.2253
0.1515
0.0699
0.0081
2.3 Time series description
The US small cap excess returns, hereafter called lo20, has yielded on average a
monthly mean return of 0.68%, a standard deviation of 6.5%, which is equivalent
to a variance of 0.42%, and therefore a Sharpe ratio of 0.1051; the median is
equal to 1.03% and the maximum and minimum sample values are respectively
27.32% and -30.22%. In addition a normal distribution has been estimated from
the sample data, the estimation returns an estimate of the mean equal to 0.68%,
which is equivalent to an annualized mean of 8.21%, and an estimate of the
standard deviation of 6.50%, which is equivalent to an annualized standard
deviation of 22.53%. The time series exhibits a negative skewness and kurtosis in
excess of the Gaussian benchmark (equal to three) respectively -0.2177 and
5.2645. The lo20 time series is depicted in the plots below.
23
Figure 1
lo20 time series plot
Figure 2
lo20 cumulative returns time series plot
From the two plots above it can be seen that the US small cap excess returns has
experienced, over the sample period, several bull and bear phases. From a first
glance it can be clearly seen that there is an alternation of high and low volatility
periods. Overally the time series is characterized by an uptrend which becomes
steeper starting from the beginning of the 1990s.The asset class has overally
24
performed very well reaching a cumulative return equal to about 11 (1100%) over
the sample period. The peak in the evolution of a unit of wealth invested in lo20
has been reached around 2007, just before the last financial crisis, at a level of
about 18 (1700% cumulated return).
The US large cap excess returns, hereafter called hi20, has yielded on average a
monthly mean return of 0.36%, a standard deviation of 4.37%, which is equivalent
to a variance of 0.19%, and therefore a Sharpe ratio of 0.0824; the median is
equal to 0.65% and the maximum and minimum sample values are respectively
17.61% and -20.91%. In addition a normal distribution has been estimated from
the sample data, the estimation returns an estimate of the mean equal to 0.36%,
which is equivalent to an annualized mean of 4.33%, and an estimate of the
standard deviation of 4.37%, which is equivalent to an annualized standard
deviation of 15.15%. The time series exhibits a negative skewness and kurtosis in
excess of the Gaussian benchmark (equal to three) respectively -0.3726 and
4.7089. The hi20 time series is depicted in the plot below.
Figure 3
hi20 time series plot
25
Figure 4
hi20 cumulative returns time series plot
From the two plots above it can be seen that the US large cap excess returns has
experienced, over the sample period, several bull and bear phases, similar to what
happened to lo20. From a first glance it can be clearly seen that there is a
succession of high and low volatility periods. Overally the time series is
characterized by an uptrend which becomes steeper starting from the beginning of
the 1990s and much steeper from the second half of the 2000s.The asset class
has overall performed slightly well reaching a cumulative return equal to about 3
(300%) over the sample period. The peak in the evolution of a unit of wealth
invested in hi20 has been reached around January 2000, just before the dot-com
burst, at a level slightly above 6 (500% cumulated return).
The 10-Year US Treasury Bond excess returns, hereafter called tbond, has
yielded on average a monthly mean return of 0.16%, a standard deviation of
2.02%, which is equivalent to a variance of 0.04%, and therefore a Sharpe ratio of
0.0776; the median is equal to 0.28% and the maximum and minimum sample
values are respectively 10.05% and -8.94%. In addition a normal distribution has
been estimated from the sample data, the estimation returns an estimate of the
mean equal to 0.16%, which is equivalent to an annualized mean of 1.88%, and
an estimate of the standard deviation of 2.02%, which is equivalent to an
26
annualized standard deviation of 6.99%. The time series exhibits a positive
skewness and kurtosis in excess of the Gaussian benchmark (equal to three)
respectively 0.4497 and 6.1871. The tbond time series is depicted in the plot
below.
Figure 5
tbond time series plot
Figure 6
tbond cumulative return time series plot
27
From the two plots above it is immediately apparent that the time series shows two
distinct behaviors over the sample period. In the first period, namely between
January 1965 and the first years of the 1980s, an overall slight decrease occurred;
in the second period a consistent uptrend occurred throughout the entire period, it
then reached a peak of about 2 almost by the end of the period.
The 1-month US T-bill rate, hereafter called risk_free, has showed, over the
sample period, an average value equal to 0.46%, a standard deviation of 0.23%,
which is equivalent to a variance of about 0.0001%. Since the risk free rate is not
one of the asset classes the investor is assumed to invest in, the Sharpe ratio has
not been computed. The median is equal to 0.43% and the maximum and
minimum sample values are respectively 1.35% and 0%. In addition a normal
distribution has been estimated from the sample data, the estimation returns an
estimate of the mean equal to 0.46%, which is equivalent to an annualized mean
of 5.47%, and an estimate of the standard deviation of 0.23%, which is equivalent
to an annualized standard deviation of 0.81%. The time series exhibits a
substantial positive skewness equal to 0.8436 and a kurtosis in excess of the
Gaussian benchmark (equal to three) equal to 4.5912. The risk_free time series is
depicted in the plot below.
Figure 7
risk_free time series plot
28
From the risk_free plot it can be seen that the 1-month US T-Bill rate has been
characterized by significant fluctuations; however, two distinct overall trends can
be easily detected. An uptrend occurred in the first period and culminated in the
first years of the 1980s at nearly 1.4% followed by a downtrend which reached its
minimum at nearly 0% by the end of the sample period.
The S&P500 dividend yield, hereafter called div_y, has showed, over the sample
period, an average value equal to 3.11%, a standard deviation of 1.20%, which is
equivalent to a variance of about 0.01%. Since the risk free rate is not one of the
asset classes the investor is assumed to invest in, the Sharpe ratio has not been
computed. The median is equal to 3.08% and the maximum and minimum sample
values are respectively 1.11% and 6,24%. In addition a normal distribution has
been estimated from the sample data, the estimation returns an estimate of the
mean equal to 3.11% and an estimate of the standard deviation equal to 0.23%.
Since the time series already represents an annual measure, i.e. the annual
dividends paid by the companies in the S&P500 index divided by the price index,
the estimated mean and standard deviation are not further annualized. The time
series exhibits a substantial positive skewness equal to 0.8436 and a kurtosis in
excess of the Gaussian benchmark (equal to three) equal to 4.5912. The div_y
time series is depicted in the plot below.
29
Figure 8
div_y time series plot
The figure above shows the evolution of the S&P500 dividend yield over the
sample period. The div_y time series characterized by a first period of relatively
high volatility followed by a less volatile one. Even if in the first period the dividend
yield fluctuates significantly, a general uptrend that culminated just above 6% in
the first years of 1980s can be detected. Since the mid-1980s a steep downtrend
happened, the period ended with a fall to nearly 0% by the end of the 2000s
followed by a considerable rise in the last decade of the sample.
2.4 A comparative analysis of the asset classes returns
In this paragraph a comparison of the three asset class is provided with the aim to
detect the similarities and differences between them, to identify peculiarities that
make each one of the time series unique, and to study how they comove over the
entire sample period. In doing so some figures and plots are provided with the
intent to make the analysis more clear and comprehensible.
30
Figure 9
lo20, hi20 and tbond time series plot
Figure 10
lo20, hi20 and tbond cumulative returns time series plot
31
Figure 11
lo20, hi20 and tbond scatter plots and histogram matrix
Figure 9 shows the three asset class time series of returns in the same plot; from
this figure it can be recognized that the lo20 time series, over the entire sample
period, has experienced wider fluctuations than hi20 and tbond with the latter one
characterized to be the least volatile asset class. It can be also noted that some
volatility clusters occurred in a pretty synchronized way among the asset class, as
shown by the data between 1995 and 2005; this feature of the data may lead to
consider that the three asset class have been positively correlated and perhaps
caused by a common exogenous phenomena. Figure 10 shows the evolution of
the accumulated returns of one unit of wealth invested in each one of the three
asset classes. As widely suggested by the literature the US small cap generally
yields a higher average return and a higher volatility, this peculiarity is clearly
witnessed by the sample data, in fact the lo20 time series reacted significantly
faster and sharper than hi20 and tbond after swing points in the data, leading to a
higher standard deviation. The evolution of the hi20 cumulative returns shows a
trend generally comparable to the one of lo20 with the difference that the former
one has been characterized to be much less volatile with a period of substantial
lack of fluctuations that is extended through the sample data from the 1965 to the
1990. The tbond cumulated returns exhibits a slight and constant uptrend that had
32
reached its highest point just above 2 (100% cumulated returns) by the end of the
sample data period. The almost total absence of downtrend phases, opposed to
the wide fluctuations that have been experienced by the two equity asset classes,
may suggest that the tbond time series is uncorrelated, or at least very slightly
correlated, with the two equity asset classes time series. From the examination of
the two equity time series, especially lo20, is straightforward to identify the
moments in which the financial crises occurred; a first downtrend started at the
end of the 1980s coinciding with the 1987 Black Monday, the largest one-day
percentage decline in stock market history; a second significant downtrend started
around 2000 following the concern about the internet companies (perhaps many of
them were small cap companies and then comprised in the first and second CRSP
size decile US equity portfolios and thereafter represented by the lo20 time
series); a third downtrend started approximately at 2007 coinciding with the Global
Financial Crises. The fact that the sample extends well into 2008-2009 allows me
to reach the conclusion that it is fully affected by the recent turmoil in international
equity markets. Additionally, the bull and bear phases present in the data lead to
the conclusions that the selected sample period is probably representative of
market regimes. All the three asset classes time series are characterized by a
kurtosis in excess of the Gaussian benchmark (three). However, only the tbond
time series has a positive skewness coefficient, while both lo20 and hi20 have
negative skewness coefficients. Figure 11 consists in a matrix of subaxes
containing scatter plots of the asset class in the vertical axis against the asset
class in the horizontal axis, the diagonal are replaced with histogram plots of the
asset class in the corresponding position along the vertical or horizontal axis. The
trend line superimposed on each scatter gives the direction of the correlation
between two time series, while the closer the dots are to this line, the stronger the
correlation between the time series. Although all the linear correlation coefficient
between the three asset classes are positive, some of them are higher than
others. The strength of the linear correlation between lo20 and hi20 is the highest
and is equal to 0.7180, the one between lo20 and tbond is the lowest and is equal
to 0.1210 while the one between hi20 and tbond is equal to 0.2171.
2.5 Stationarity and normality tests
33
In this paragraph the results of the normality and stationarity tests are provided
and described for the three asset classes time series and the dividend yield.
Additional figures are provided with the intent to make the dissertation clearer and
more comprehensive.
Firstly, I started by performing normality test and secondly stationarity test. In the
attempt to determine if each time series was well-modeled by a normal distribution
I proceeded adopting in the very first stage an informal graphical method that
consisted in comparing a histogram of the sample data to a Normal probability
curve. I further examined the time series with the help of the normal probability
plot, the quantile-quantile plot (QQ plot), the kernel smoothing probability density
estimate and the empirical cumulative distribution function for each time series. In
the second stage I performed some common statistical hypothesis tests in which
data are tested against the null hypothesis that they are normally distributed. In the
first part of the next passage the results of the graphical method is provided for
each time series while at the end of the passage the results of the hypothesis test,
contained in a table, are exposed.
The interpretation of the histogram and the superimposed normal density
distribution fitted to the data is the following: if the time series has been drawn
from a Normal distribution then empirical distribution of the data, represented by
the histogram, must be bell-shaped and resemble the normal distribution fitted,
otherwise a suspect that the data are not normally distributed should rise. The
normal probability plot has the sample data displayed with the plot symbol '+'.
Superimposed on the plot is a line joining the first and third quartiles of the sample
data. This line is extrapolated out to the ends of the sample to help evaluate the
linearity of the data. The purpose of a normal probability plot is to graphically
assess whether the sample data could come from a normal distribution. If the data
are normal the plot will be linear. Other distribution types will introduce curvature in
the plot. The QQ displays a quantile-quantile plot of the quantiles of the sample
data versus theoretical quantiles from a normal distribution. If the distribution of the
sample data is normal, the plot will be close to linear. The plot has the sample data
displayed with the plot symbol '+'. Superimposed on the plot is a line joining the
34
first and third quartiles of the sample data. This line is extrapolated out to the ends
of the sample to help evaluate the linearity of the data. The kernel smoothing
probability density estimate returns a probability density estimate for the sample
data. The estimate is based on a normal kernel function, and is evaluated at 100
equally spaced points, that cover the range of the sample data. If the sample data
is normally distributed then the plot of the kernel smoothing probability density
estimate must resemble the shape of a normal density curve, otherwise a suspect
that the data are not normally distributed should rise.
Figure 12
lo20 frequency histogram and normal density function fit
35
Figure 13
lo20 normal probability plot
Figure 14
lo20 QQ plot
36
Figure 15
lo20 kernel smoothing probability density estimate
Figure 16
lo20 empirical cumulative distribution function
From the examination of the five plots above the lo20 time series clearly appeared
to be not normally distributed. Figure 12 shows that the lo20 probability density
distribution presents fat tails and, returns close to zero have a significant higher
probability of happen than in the case of a normal distribution. Figure 13 and 14
both shows the presence of outliers returns that diverge significantly from the
straight line that join the first and the third quantiles. Figure 15 points out the
37
presence of the fat tail characteristic represented by the fact that the kernel
smoothing probability density estimate does not appear to have the shape of a
normal distribution, on the contrary it shows returns far from the mean higher
density than the normal distribution.
Figure 17
hi20 frequency histogram and normal density function fit
Figure 18
hi20 normal probability plot
38
Figure 19
hi20 QQ plot
Figure 20
hi20 kernel smoothing probability density estimate
39
Figure 21
hi20 empirical cumulative distribution function
The hi20 time series at a first glance seems to be very similar to lo20 regarding the
distribution characteristic. From the examination of the five plots above the hi20
time series clearly appeared to be not normally distributed. Figure 17 shows that
the hi20 probability density distribution presents fat tails and, returns close to zero
have a significant higher probability of happen than in the case of a normal
distribution. Figure 18 and 19 both show the presence of outlier returns that
diverge significantly from the straight line that join the first and the third quantiles.
Figure 20 points out the presence of the fat tail characteristic represented by the
fact that the kernel smoothing probability density estimate does not appear to have
the shape of a normal distribution, on the contrary it shows, for returns far from the
mean, higher density than the normal distribution.
40
Figure 22
tbond frequency histogram and normal density function fit
Figure 23
tbond normal probability plot
41
Figure 24
tbond QQ plot
Figure 25
tbond kernel smoothing probability density estimate
42
Figure 26
tbond empirical cumulative distribution function
From the examination of the five tbond plots above emerges that the time series is
characterized to have a significant positive skewness that may prevent it to be
normally distributed. Figure 22 shows that the tbond probability density distribution
presents fat tails, large skewness and, returns close to zero have a slight higher
probability of happen than in the case of a normal distribution. Figure 23 and 24
both shows the presence of outlier returns that diverge significantly from the
straight line that join the first and the third quantiles. Figure 25 points out the
presence of a large positive skewness by the fact that the kernel smoothing
probability density estimate does not appear to have the shape of a normal
distribution, on the contrary it shows a marked asymmetry.
43
Figure 27
div_y frequency histogram and normal density function fit
Figure 28
div_y normal probability plot
44
Figure 29
div_y QQ plot
Figure 30
div_y kernel smoothing probability density estimate
45
Figure 31
div_y empirical cumulative distribution function
From the examination of the five plots above the div_y time series appeared
extremely not normally distributed. In Figure 27 I decided to show, in addition to a
fitted normal distribution (green solid line), a gamma distribution (red solid line)
which has in common with the div_y sample a positive probability density support.
From a graphical comparison of the degree of fit of the two different fitted
distributions to the sample date appears clear that the div_y is not normally
distributed. Figure 28 and 28 both shows the presence of outlier returns that
diverge significantly from the straight line that join the first and the third quantiles.
Figure 30 points out also the fact that the probability density distribution of the
sample data is even not unimodal, which is in turn a large divergence from the
normal distribution case.
Even though at this point of the analysis it appeared quite likely that all the time
series were not normally distributed I further analyzed them performing some
common statistical hypothesis tests in which data are tested against the null
hypothesis that they are normally distributed. In the following table the results are
exhibited.
Table 3
Normality Tests Results
test name
alpha
object
lo20
hi20
div_y
tbond
46
Chi-square goodness-of-fit
10% decision
1
1
1
1
Chi-square goodness-of-fit
10% p-value
0.0014
0.0348
0
0.012
Chi-square goodness-of-fit
5%
decision
1
1
1
1
Chi-square goodness-of-fit
5%
p-value
0.0014
0.0348
0
0.012
Chi-square goodness-of-fit
1%
decision
1
0
1
0
Chi-square goodness-of-fit
1%
p-value
0.0014
0.0348
0
0.012
Anderson-Darling
10% decision
1
1
1
1
Anderson-Darling
10% p-value
0
0
0
0
Anderson-Darling
10% statistic
2.28
1.78
4.81
2.14
Anderson-Darling
10%
cvalue
0.62
0.62
0.61
0.65
Anderson-Darling
5%
decision
1
1
1
1
Anderson-Darling
5%
p-value
0
0
0
0
Anderson-Darling
5%
statistic
2.28
1.78
4.81
2.14
Anderson-Darling
5%
cvalue
0.80
0.74
0.74
0.75
Anderson-Darling
1%
decision
1
1
1
1
Anderson-Darling
1%
p-value
0
0
0
0
Anderson-Darling
1%
statistic
2.28
1.78
4.81
2.14
Anderson-Darling
1%
cvalue
0.94
0.97
1.13
1.04
Jarque-Bera
10% decision
1
1
1
1
Jarque-Bera
10% p-value
0
0
0.0025
0
Jarque-Bera
10% statistic
119.64
78.20
16.63
246.75
Jarque-Bera
10%
cvalue
4.36
4.34
4.35
4.37
Jarque-Bera
5%
decision
1
1
1
1
Jarque-Bera
5%
p-value
0
0
0
0
Jarque-Bera
5%
statistic
119.64
78.20
16.63
246.75
Jarque-Bera
5%
cvalue
5.87
5.89
5.85
5.84
Jarque-Bera
1%
decision
1
1
1
1
Jarque-Bera
1%
p-value
0
0
0
0
Jarque-Bera
1%
statistic
119.64
78.20
16.63
246.75
Jarque-Bera
1%
cvalue
10.65
10.82
10.69
10.66
Three different normality tests have been performed in the attempt to assess how
likely each one of the three asset class time series of returns, individually
considered, could have been drawn from a normal density distribution. The Chi-
square goodness-of-fit is a test that returns a decision for the null hypothesis that
the sample data comes from a normal distribution with a mean and variance
estimated from the sample data, using the chi-square goodness-of-fit test. The
alternative hypothesis is that the data does not come from such a distribution. The
result (the row "decision" in Table 3)
is 1 if the test rejects the null hypothesis at
47
the given significance level, and 0 otherwise. The test groups the data into bins,
calculating the observed and expected counts (based on the hypothesized
distribution) for those bins, and computing the chi-square test statistic. The test
statistic has an approximate chi-square distribution when the counts are
sufficiently large. The Andreson-Darling is a test that returns a decision for the null
hypothesis that the sample data comes from a population with a normal
distribution, using the Anderson-Darling test. The alternative hypothesis is that the
sample data does not come from a population with a normal distribution. The
result (the row "decision" in Table 3)
is 1 if the test rejects the null hypothesis at
the given significance level, or 0 otherwise. The test statistic belongs to the family
of quadratic empirical distribution function statistics, which measure the distance
between the hypothesized distribution and the empirical one. The weight function
for the Anderson-Darling test places greater weight on the observations in the tails
of the distribution, thus making the test more sensitive to outliers and better at
detecting departure from normality in the tails of the distribution. The Jarque-Bera
test returns a test decision for the null hypothesis that the sample data comes from
a normal distribution with an unknown mean and variance. The alternative
hypothesis is that it does not come from such a distribution. The result (the row
"decision" in Table 3)
is 1 if the test rejects the null hypothesis at the given
significance level, and 0 otherwise. The Jarque
Bera test is a goodness-of-fit test
of whether sample data have the skewness and kurtosis matching a normal
distribution.
If
the
data
come
from
a
normal
distribution,
the JB statistic asymptotically has a chi-squared distribution with two degrees of
freedom, so the statistic can be used to test the hypothesis that the data are from
a normal distribution. The null hypothesis, in other words, is a joint hypothesis of
the skewness being zero and the excess kurtosis being zero. The three tests
returns for any significance level the decision that the four time series are
statistically not normally distributed except for the Chi-square goodness-of-fit that
failed to reject the null hypothesis at 1% significance level for ho20 and tbond.
From the point of view of this dissertation, it is important to note that all the time
series displayed significant deviation from the normal distribution benchmark, as
evidenced by the statistically significant normality tests results, and then they can
48
be considered not normally distributed. This is a clear indication that the returns on
the three asset classes cannot be captured by linear models to reinforce the idea
of a regime switching model, which is more flexible in accommodating the mixing
of several empirical distributions. The regime switching model is also supported by
the results of normality tests, as all null hypotheses are strongly rejected. Since
regime switching models account for non-normality by using a mixture-of-normal
distributions approach, they deliver a more accurate way of modeling the
dynamics and the distribution of the three asset classes returns than models using
only one normal distribution.
In the following part of the paragraph stationarity tests results are provided and
commented. These tests can be used to determine if trending data should be first
differenced or regressed on deterministic functions of time to render the data
stationary.
A process is said to be covariance-stationary, or weakly stationary, if its
first and second moments (hence also the covariance) are time invariant. In other
words the structure of the series does not change with the time. A stationary series
is relatively easy to predict since its statistical properties will be the same in the
future as they have been in the past.
Table 4
Stationarity Tests Results
test name
alpha object lo20 hi20 div_y tbond
Augmented Dickey-Fuller TS LAG 0
10% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 0
10% p-value 0.00
0.00
0.60
0.00
Augmented Dickey-Fuller TS LAG 0
10% statistic -18.29 -22.08 -1.98 -16.93
Augmented Dickey-Fuller TS LAG 0
10%
cvalue -3.13 -3.13 -3.13 -3.13
Augmented Dickey-Fuller TS LAG 1
10% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 1
10% p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller TS LAG 1
10% statistic -15.39 -16.37 -2.43 -16.85
Augmented Dickey-Fuller TS LAG 1
10%
cvalue -3.13 -3.13 -3.13 -3.13
Augmented Dickey-Fuller TS LAG 2
10% decision 1.00
1.00
0.00
1.00
Augmented Dickey-Fuller TS LAG 2
10% p-value 0.00
0.00
0.43
0.00
Augmented Dickey-Fuller TS LAG 2
10% statistic -13.09 -12.84 -2.33 -12.47
Augmented Dickey-Fuller TS LAG 2
10%
cvalue -3.13 -3.13 -3.13 -3.13
Augmented Dickey-Fuller TS LAG 0
5% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 0
5%
p-value 0.00
0.00
0.60
0.00
Augmented Dickey-Fuller TS LAG 0
5%
statistic -18.29 -22.08 -1.98 -16.93
Augmented Dickey-Fuller TS LAG 0
5%
cvalue -3.42 -3.42 -3.42 -3.42
Augmented Dickey-Fuller TS LAG 1
5% decision
1
1
0
1
49
Augmented Dickey-Fuller TS LAG 1
5%
p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller TS LAG 1
5%
statistic -15.39 -16.37 -2.43 -16.85
Augmented Dickey-Fuller TS LAG 1
5%
cvalue -3.42 -3.42 -3.42 -3.42
Augmented Dickey-Fuller TS LAG 2
5% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 2
5%
p-value 0.00
0.00
0.43
0.00
Augmented Dickey-Fuller TS LAG 2
5%
statistic -13.09 -12.84 -2.33 -12.47
Augmented Dickey-Fuller TS LAG 2
5%
cvalue -3.42 -3.42 -3.42 -3.42
Augmented Dickey-Fuller TS LAG 0
1% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 0
1%
p-value 0.00
0.00
0.60
0.00
Augmented Dickey-Fuller TS LAG 0
1%
statistic -18.29 -22.08 -1.98 -16.93
Augmented Dickey-Fuller TS LAG 0
1%
cvalue -3.98 -3.98 -3.98 -3.98
Augmented Dickey-Fuller TS LAG 1
1% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 1
1%
p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller TS LAG 1
1%
statistic -15.39 -16.37 -2.43 -16.85
Augmented Dickey-Fuller TS LAG 1
1%
cvalue -3.98 -3.98 -3.98 -3.98
Augmented Dickey-Fuller TS LAG 2
1% decision
1
1
0
1
Augmented Dickey-Fuller TS LAG 2
1%
p-value 0.00
0.00
0.43
0.00
Augmented Dickey-Fuller TS LAG 2
1%
statistic -13.09 -12.84 -2.33 -12.47
Augmented Dickey-Fuller TS LAG 2
1%
cvalue -3.98 -3.98 -3.98 -3.98
Augmented Dickey-Fuller AR LAG 0
10% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 0
10% p-value 0.00
0.00
0.40
0.00
Augmented Dickey-Fuller AR LAG 0
10% statistic -18.16 -21.97 -0.68 -16.80
Augmented Dickey-Fuller AR LAG 0
10%
cvalue -1.62 -1.62 -1.62 -1.62
Augmented Dickey-Fuller AR LAG 1
10% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 1
10% p-value 0.00
0.00
0.37
0.00
Augmented Dickey-Fuller AR LAG 1
10% statistic -15.22 -16.23 -0.77 -16.62
Augmented Dickey-Fuller AR LAG 1
10%
cvalue -1.62 -1.62 -1.62 -1.62
Augmented Dickey-Fuller AR LAG 2
10% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 2
10% p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller AR LAG 2
10% statistic -12.90 -12.69 -0.75 -12.25
Augmented Dickey-Fuller AR LAG 2
10%
cvalue -1.62 -1.62 -1.62 -1.62
Augmented Dickey-Fuller AR LAG 0
5% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 0
5%
p-value 0.00
0.00
0.40
0.00
Augmented Dickey-Fuller AR LAG 0
5%
statistic -18.16 -21.97 -0.68 -16.80
Augmented Dickey-Fuller AR LAG 0
5%
cvalue -1.94 -1.94 -1.94 -1.94
Augmented Dickey-Fuller AR LAG 1
5% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 1
5%
p-value 0.00
0.00
0.37
0.00
Augmented Dickey-Fuller AR LAG 1
5%
statistic -15.22 -16.23 -0.77 -16.62
Augmented Dickey-Fuller AR LAG 1
5%
cvalue -1.94 -1.94 -1.94 -1.94
Augmented Dickey-Fuller AR LAG 2
5% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 2
5%
p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller AR LAG 2
5%
statistic -12.90 -12.69 -0.75 -12.25
Augmented Dickey-Fuller AR LAG 2
5%
cvalue -1.94 -1.94 -1.94 -1.94
Augmented Dickey-Fuller AR LAG 0
1% decision
1
1
0
1
50
Augmented Dickey-Fuller AR LAG 0
1%
p-value 0.00
0.00
0.40
0.00
Augmented Dickey-Fuller AR LAG 0
1%
statistic -18.16 -21.97 -0.68 -16.80
Augmented Dickey-Fuller AR LAG 0
1%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller AR LAG 1
1% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 1
1%
p-value 0.00
0.00
0.37
0.00
Augmented Dickey-Fuller AR LAG 1
1%
statistic -15.22 -16.23 -0.77 -16.62
Augmented Dickey-Fuller AR LAG 1
1%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller AR LAG 2
1% decision
1
1
0
1
Augmented Dickey-Fuller AR LAG 2
1%
p-value 0.00
0.00
0.38
0.00
Augmented Dickey-Fuller AR LAG 2
1%
statistic -12.90 -12.69 -0.75 -12.25
Augmented Dickey-Fuller AR LAG 2
1%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller ARD LAG 0
10% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 0
10% p-value 0.00
0.00
0.70
0.00
Augmented Dickey-Fuller ARD LAG 0
10% statistic -18.30 -22.08 -1.10 -16.86
Augmented Dickey-Fuller ARD LAG 0
10%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller ARD LAG 1
10% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 1
10% p-value 0.00
0.00
0.50
0.00
Augmented Dickey-Fuller ARD LAG 1
10% statistic -15.40 -16.37 -1.55 -16.72
Augmented Dickey-Fuller ARD LAG 1
10%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller ARD LAG 2
10% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 2
10% p-value 0.00
0.00
0.54
0.00
Augmented Dickey-Fuller ARD LAG 2
10% statistic -13.10 -12.84 -1.45 -12.34
Augmented Dickey-Fuller ARD LAG 2
10%
cvalue -2.57 -2.57 -2.57 -2.57
Augmented Dickey-Fuller ARD LAG 0
5% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 0
5%
p-value 0.00
0.00
0.70
0.00
Augmented Dickey-Fuller ARD LAG 0
5%
statistic -18.30 -22.08 -1.10 -16.86
Augmented Dickey-Fuller ARD LAG 0
5%
cvalue -2.87 -2.87 -2.87 -2.87
Augmented Dickey-Fuller ARD LAG 1
5% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 1
5%
p-value 0.00
0.00
0.50
0.00
Augmented Dickey-Fuller ARD LAG 1
5%
statistic -15.40 -16.37 -1.55 -16.72
Augmented Dickey-Fuller ARD LAG 1
5%
cvalue -2.87 -2.87 -2.87 -2.87
Augmented Dickey-Fuller ARD LAG 2
5% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 2
5%
p-value 0.00
0.00
0.54
0.00
Augmented Dickey-Fuller ARD LAG 2
5%
statistic -13.10 -12.84 -1.45 -12.34
Augmented Dickey-Fuller ARD LAG 2
5%
cvalue -2.87 -2.87 -2.87 -2.87
Augmented Dickey-Fuller ARD LAG 0
1% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 0
1%
p-value 0.00
0.00
0.70
0.00
Augmented Dickey-Fuller ARD LAG 0
1%
statistic -18.30 -22.08 -1.10 -16.86
Augmented Dickey-Fuller ARD LAG 0
1%
cvalue -3.44 -3.44 -3.44 -3.44
Augmented Dickey-Fuller ARD LAG 1
1% decision
1
1
0
1
Augmented Dickey-Fuller ARD LAG 1
1%
p-value 0.00
0.00
0.50
0.00
Augmented Dickey-Fuller ARD LAG 1
1%
statistic -15.40 -16.37 -1.55 -16.72
Augmented Dickey-Fuller ARD LAG 1
1%
cvalue -3.44 -3.44 -3.44 -3.44
Augmented Dickey-Fuller ARD LAG 2
1% decision
1
1
0
1
51
Augmented Dickey-Fuller ARD LAG 2
1%
p-value 0.00
0.00
0.54
0.00
Augmented Dickey-Fuller ARD LAG 2
1%
statistic -13.10 -12.84 -1.45 -12.34
Augmented Dickey-Fuller ARD LAG 2
1%
cvalue -3.44 -3.44 -3.44 -3.44
Leybourne-McCabe stationarity
10% decision
0
0
0
0
Leybourne-McCabe stationarity
10% p-value 0.10
0.10
0.10
0.10
Leybourne-McCabe stationarity
10% statistic 0.03
0.11 -1654 0.05
Leybourne-McCabe stationarity
10%
cvalue
0.12
0.12
0.12
0.12
Leybourne-McCabe stationarity
5% decision
0
0
0
0
Leybourne-McCabe stationarity
5%
p-value 0.10
0.10
0.10
0.10
Leybourne-McCabe stationarity
5%
statistic 0.03
0.11 -1654 0.05
Leybourne-McCabe stationarity
5%
cvalue
0.15
0.15
0.15
0.15
Leybourne-McCabe stationarity
1% decision
0
0
0
0
Leybourne-McCabe stationarity
1%
p-value 0.10
0.10
0.10
0.10
Leybourne-McCabe stationarity
1%
statistic 0.03
0.11 -1654 0.05
Leybourne-McCabe stationarity
1%
cvalue
0.22
0.22
0.22
0.22
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
10% decision
0
0
1
0
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
10% p-value 0.10
0.10
0.01
0.10
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
10% statistic 0.04
0.10
0.36
0.07
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
10%
cvalue
0.12
0.12
0.12
0.12
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
5% decision
0
0
1
0
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
5%
p-value 0.10
0.10
0.01
0.10
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
5%
statistic 0.04
0.10
0.36
0.07
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
5%
cvalue
0.15
0.15
0.15
0.15
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
1% decision
0
0
1
0
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
1%
p-value 0.10
0.10
0.01
0.10
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
1%
statistic 0.04
0.10
0.36
0.07
Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS)
1%
cvalue
0.22
0.22
0.22
0.22
Ljung-Box Q- residual autocorrelation
LAG 5
10% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 5
10% p-value 0.00
0.20
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 5
10% statistic 30.02 7.31
2581 56.35
Ljung-Box Q- residual autocorrelation
LAG 5
10%
cvalue
9.24
9.24
9.24
9.24
52
Ljung-Box Q- residual autocorrelation
LAG 10
10% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 10
10% p-value 0.00
0.53
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 10
10% statistic 33.83 8.99
4914 66.37
Ljung-Box Q- residual autocorrelation
LAG 10
10%
cvalue 15.99 15.99 15.99 15.99
Ljung-Box Q- residual autocorrelation
LAG 15
10% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 15
10% p-value 0.00
0.58
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 15
10% statistic 41.56 13.24 7062 80.19
Ljung-Box Q- residual autocorrelation
LAG 15
10%
cvalue 22.31 22.31 22.31 22.31
Ljung-Box Q- residual autocorrelation
LAG 5
5% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 5
5%
p-value 0.00
0.20
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 5
5%
statistic 30.02 7.31
2581 56.35
Ljung-Box Q- residual autocorrelation
LAG 5
5%
cvalue 11.07 11.07 11.07 11.07
Ljung-Box Q- residual autocorrelation
LAG 10
5% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 10
5%
p-value 0.00
0.53
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 10
5%
statistic 33.83 8.99
4914 66.37
Ljung-Box Q- residual autocorrelation
LAG 10
5%
cvalue 18.31 18.31 18.31 18.31
Ljung-Box Q- residual autocorrelation
LAG 15
5% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 15
5%
p-value 0.00
0.58
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 15
5%
statistic 41.56 13.24 7062 80.19
Ljung-Box Q- residual autocorrelation
LAG 15
5%
cvalue 25.00 25.00 25.00 25.00
Ljung-Box Q- residual autocorrelation
LAG 5
1% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 5
1%
p-value 0.00
0.20
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 5
1%
statistic 30.02 7.31
2581 56.35
Ljung-Box Q- residual autocorrelation
LAG 5
1%
cvalue 15.09 15.09 15.09 15.09
53
Ljung-Box Q- residual autocorrelation
LAG 10
1% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 10
1%
p-value 0.00
0.53
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 10
1%
statistic 33.83 8.99
4914 66.37
Ljung-Box Q- residual autocorrelation
LAG 10
1%
cvalue 23.21 23.21 23.21 23.21
Ljung-Box Q- residual autocorrelation
LAG 15
1% decision
1
0
1
1
Ljung-Box Q- residual autocorrelation
LAG 15
1%
p-value 0.00
0.58
0.00
0.00
Ljung-Box Q- residual autocorrelation
LAG 15
1%
statistic 41.56 13.24 7062 80.19
Ljung-Box Q- residual autocorrelation
LAG 15
1%
cvalue 30.58 30.58 30.58 30.58
Phillips-Perron one unit root against
TS LAG 4
10% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 4
10% p-value 0.00
0.00
0.45
0.00
Phillips-Perron one unit root against
TS LAG 4
10% statistic -18.12 -22.08 -2.29 -16.58
Phillips-Perron one unit root against
TS LAG 4
10%
cvalue -3.13 -3.13 -3.13 -3.13
Phillips-Perron one unit root against
TS LAG 5
10% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 5
10% p-value 0.00
0.00
0.42
0.00
Phillips-Perron one unit root against
TS LAG 5
10% statistic -18.09 -22.11 -2.34 -16.56
Phillips-Perron one unit root against
TS LAG 5
10%
cvalue -3.13 -3.13 -3.13 -3.13
Phillips-Perron one unit root against
TS LAG 6
10% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 6
10% p-value 0.00
0.00
0.41
0.00
Phillips-Perron one unit root against
TS LAG 6
10% statistic -18.06 -22.12 -2.37 -16.52
Phillips-Perron one unit root against
TS LAG 6
10%
cvalue -3.13 -3.13 -3.13 -3.13
Phillips-Perron one unit root against
TS LAG 4
5% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 4
5%
p-value 0.00
0.00
0.45
0.00
Phillips-Perron one unit root against
TS LAG 4
5%
statistic -18.12 -22.08 -2.29 -16.58
Phillips-Perron one unit root against
TS LAG 4
5%
cvalue -3.42 -3.42 -3.42 -3.42
54
Phillips-Perron one unit root against
TS LAG 5
5% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 5
5%
p-value 0.00
0.00
0.42
0.00
Phillips-Perron one unit root against
TS LAG 5
5%
statistic -18.09 -22.11 -2.34 -16.56
Phillips-Perron one unit root against
TS LAG 5
5%
cvalue -3.42 -3.42 -3.42 -3.42
Phillips-Perron one unit root against
TS LAG 6
5% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 6
5%
p-value 0.00
0.00
0.41
0.00
Phillips-Perron one unit root against
TS LAG 6
5%
statistic -18.06 -22.12 -2.37 -16.52
Phillips-Perron one unit root against
TS LAG 6
5%
cvalue -3.42 -3.42 -3.42 -3.42
Phillips-Perron one unit root against
TS LAG 4
1% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 4
1%
p-value 0.00
0.00
0.45
0.00
Phillips-Perron one unit root against
TS LAG 4
1%
statistic -18.12 -22.08 -2.29 -16.58
Phillips-Perron one unit root against
TS LAG 4
1%
cvalue -3.98 -3.98 -3.98 -3.98
Phillips-Perron one unit root against
TS LAG 5
1% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 5
1%
p-value 0.00
0.00
0.42
0.00
Phillips-Perron one unit root against
TS LAG 5
1%
statistic -18.09 -22.11 -2.34 -16.56
Phillips-Perron one unit root against
TS LAG 5
1%
cvalue -3.98 -3.98 -3.98 -3.98
Phillips-Perron one unit root against
TS LAG 6
1% decision
1
1
0
1
Phillips-Perron one unit root against
TS LAG 6
1%
p-value 0.00
0.00
0.41
0.00
Phillips-Perron one unit root against
TS LAG 6
1%
statistic -18.06 -22.12 -2.37 -16.52
Phillips-Perron one unit root against
TS LAG 6
1%
cvalue -3.98 -3.98 -3.98 -3.98
Phillips-Perron one unit root against
AR LAG 4
10% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 4
10% p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 4
10% statistic -18.02 -21.98 -0.74 -16.47
Phillips-Perron one unit root against
AR LAG 4
10%
cvalue -1.62 -1.62 -1.62 -1.62
55
Phillips-Perron one unit root against
AR LAG 5
10% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 5
10% p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 5
10% statistic -18.00 -22.01 -0.75 -16.46
Phillips-Perron one unit root against
AR LAG 5
10%
cvalue -1.62 -1.62 -1.62 -1.62
Phillips-Perron one unit root against
AR LAG 6
10% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 6
10% p-value 0.00
0.00
0.37
0.00
Phillips-Perron one unit root against
AR LAG 6
10% statistic -17.98 -22.03 -0.76 -16.44
Phillips-Perron one unit root against
AR LAG 6
10%
cvalue -1.62 -1.62 -1.62 -1.62
Phillips-Perron one unit root against
AR LAG 4
5% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 4
5%
p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 4
5%
statistic -18.02 -21.98 -0.74 -16.47
Phillips-Perron one unit root against
AR LAG 4
5%
cvalue -1.94 -1.94 -1.94 -1.94
Phillips-Perron one unit root against
AR LAG 5
5% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 5
5%
p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 5
5%
statistic -18.00 -22.01 -0.75 -16.46
Phillips-Perron one unit root against
AR LAG 5
5%
cvalue -1.94 -1.94 -1.94 -1.94
Phillips-Perron one unit root against
AR LAG 6
5% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 6
5%
p-value 0.00
0.00
0.37
0.00
Phillips-Perron one unit root against
AR LAG 6
5%
statistic -17.98 -22.03 -0.76 -16.44
Phillips-Perron one unit root against
AR LAG 6
5%
cvalue -1.94 -1.94 -1.94 -1.94
Phillips-Perron one unit root against
AR LAG 4
1% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 4
1%
p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 4
1%
statistic -18.02 -21.98 -0.74 -16.47
Phillips-Perron one unit root against
AR LAG 4
1%
cvalue -2.57 -2.57 -2.57 -2.57
56
Phillips-Perron one unit root against
AR LAG 5
1% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 5
1%
p-value 0.00
0.00
0.38
0.00
Phillips-Perron one unit root against
AR LAG 5
1%
statistic -18.00 -22.01 -0.75 -16.46
Phillips-Perron one unit root against
AR LAG 5
1%
cvalue -2.57 -2.57 -2.57 -2.57
Phillips-Perron one unit root against
AR LAG 6
1% decision
1
1
0
1
Phillips-Perron one unit root against
AR LAG 6
1%
p-value 0.00
0.00
0.37
0.00
Phillips-Perron one unit root against
AR LAG 6
1%
statistic -17.98 -22.03 -0.76 -16.44
Phillips-Perron one unit root against
AR LAG 6
1%
cvalue -2.57 -2.57 -2.57 -2.57
Phillips-Perron one unit root against
ARD LAG 4
10% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 4
10% p-value 0.00
0.00
0.55
0.00
Phillips-Perron one unit root against
ARD LAG 4
10% statistic -18.14 -22.09 -1.43 -16.52
Phillips-Perron one unit root against
ARD LAG 4
10%
cvalue -2.57 -2.57 -2.57 -2.57
Phillips-Perron one unit root against
ARD LAG 5
10% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 5
10% p-value 0.00
0.00
0.53
0.00
Phillips-Perron one unit root against
ARD LAG 5
10% statistic -18.11 -22.12 -1.48 -16.51
Phillips-Perron one unit root against
ARD LAG 5
10%
cvalue -2.57 -2.57 -2.57 -2.57
Phillips-Perron one unit root against
ARD LAG 6
10% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 6
10% p-value 0.00
0.00
0.52
0.00
Phillips-Perron one unit root against
ARD LAG 6
10% statistic -18.08 -22.13 -1.50 -16.47
Phillips-Perron one unit root against
ARD LAG 6
10%
cvalue -2.57 -2.57 -2.57 -2.57
Phillips-Perron one unit root against
ARD LAG 4
5% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 4
5%
p-value 0.00
0.00
0.55
0.00
Phillips-Perron one unit root against
ARD LAG 4
5%
statistic -18.14 -22.09 -1.43 -16.52
Phillips-Perron one unit root against
ARD LAG 4
5%
cvalue -2.87 -2.87 -2.87 -2.87
57
Phillips-Perron one unit root against
ARD LAG 5
5% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 5
5%
p-value 0.00
0.00
0.53
0.00
Phillips-Perron one unit root against
ARD LAG 5
5%
statistic -18.11 -22.12 -1.48 -16.51
Phillips-Perron one unit root against
ARD LAG 5
5%
cvalue -2.87 -2.87 -2.87 -2.87
Phillips-Perron one unit root against
ARD LAG 6
5% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 6
5%
p-value 0.00
0.00
0.52
0.00
Phillips-Perron one unit root against
ARD LAG 6
5%
statistic -18.08 -22.13 -1.50 -16.47
Phillips-Perron one unit root against
ARD LAG 6
5%
cvalue -2.87 -2.87 -2.87 -2.87
Phillips-Perron one unit root against
ARD LAG 4
1% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 4
1%
p-value 0.00
0.00
0.55
0.00
Phillips-Perron one unit root against
ARD LAG 4
1%
statistic -18.14 -22.09 -1.43 -16.52
Phillips-Perron one unit root against
ARD LAG 4
1%
cvalue -3.44 -3.44 -3.44 -3.44
Phillips-Perron one unit root against
ARD LAG 5
1% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 5
1%
p-value 0.00
0.00
0.53
0.00
Phillips-Perron one unit root against
ARD LAG 5
1%
statistic -18.11 -22.12 -1.48 -16.51
Phillips-Perron one unit root against
ARD LAG 5
1%
cvalue -3.44 -3.44 -3.44 -3.44
Phillips-Perron one unit root against
ARD LAG 6
1% decision
1
1
0
1
Phillips-Perron one unit root against
ARD LAG 6
1%
p-value 0.00
0.00
0.52
0.00
Phillips-Perron one unit root against
ARD LAG 6
1%
statistic -18.08 -22.13 -1.50 -16.47
Phillips-Perron one unit root against
ARD LAG 6
1%
cvalue -3.44 -3.44 -3.44 -3.44
Variance ratio random walk
10% decision
1
1
1
1
Variance ratio random walk
10% p-value 0.00
0.00
0.00
0.00
Variance ratio random walk
10% statistic -5.38 -7.50 4.32
-3.14
Variance ratio random walk
10%
cvalue
1.64
1.64
1.64
1.64
Variance ratio random walk
5% decision
1
1
1
1
Variance ratio random walk
5%
p-value 0.00
0.00
0.00
0.00
Variance ratio random walk
5%
statistic -5.38 -7.50 4.32
-3.14
58
Variance ratio random walk
5%
cvalue
1.96
1.96
1.96
1.96
Variance ratio random walk
1% decision
1
1
1
1
Variance ratio random walk
1%
p-value 0.00
0.00
0.00
0.00
Variance ratio random walk
1%
statistic -5.38 -7.50 4.32
-3.14
Variance ratio random walk
1%
cvalue
2.58
2.58
2.58
2.58
Ten different stationarity tests have been performed with combinations of different
lags and significance levels in the attempt to check if each one of the three asset
class time series of returns would be stationary.
The Augmented Dickey-Fuller TS tests the null hypothesis of a unit root against
the trend-stationary alternative, the Augmented Dickey-Fuller AR test the null
hypothesis of a unit root against the autoregressive alternative while the
Augmented Dickey-Fuller ARD tests the null hypothesis of a unit root against the
autoregressive with drift alternative. For all the three different versions of the
Augmented Dickey-Fuller test have been performed three tests at lags 0,1 and 2
and significance levels of 10%,5% and 1%. The null hypothesis of a unit root has
been rejected for the four time series in all the tests except for the div_y time
series which appeared to be not stationary. The Leybourne-McCabe stationarity
test assesses the null hypothesis that a univariate time series is a trend stationary
AR(p) process, against the alternative that it is a nonstationary ARIMA(p,1,1)
process, where p represents the autoregressive order, in this case I chose p equal
to 1. The test has been performed at significance levels of 10%,5% and 1%. The
null hypothesis failed to be rejected for all the time series at any significance level.
The Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) assesses the null hypothesis
that a univariate time series is trend stationary against the alternative that it is a
nonstationary unit root process. The authors of the test suggest that a number of
lags on the order of square root of T, where T is the sample size, is often
satisfactory under both the null and the alternative. Given that the length of the
four time series analyzed is 540 I chose the number of lags equal to 23. The test
has been performed at significance levels of 10%,5% and 1%. The null hypothesis
of a unit root failed to be rejected in all the tests except for the div_y time series
which appeared to be not stationary. The Phillips-Perron TS tests null hypothesis
of a unit root against the trend-stationary alternative, the Phillips-Perron AR tests
59
null hypothesis of a unit root against the autoregressive alternative while the
Phillips-Perron ARD tests null hypothesis of a unit root against the autoregressive
with drift alternative.
The Phillips and Perron's test statistics can be viewed as
Dickey
Fuller statistics that have been made robust to serial correlation by using
the Newey
West (1987) heteroskedasticity and autocorrelation-consistent
covariance matrix estimator. The lags, which is an input of the statistics test,
represent the number of autocovariance lags to include in the Newey-West
estimator of the long-run variance. The Newey-West estimator (1987)
5
is
consistent if the number of lags are O(T
1/4
), where T is the effective sample size; in
the case of this study the numbers of lags have been chosen to be 4,5 and 6. The
test has been performed at significance levels of 10%,5% and 1%. The null
hypothesis of a unit root has been rejected for the four time series in all the tests
except for the div_y time series which appeared to be not stationary. The variance
ratio test assesses the null hypothesis that a univariate time series is a random
walk. The test has been performed at significance levels of 10%,5% and 1% and
then null hypothesis has been rejected at any significance level for all the four time
series.
In the following passage the sample autocorrelation function (ACF) and the
sample partial autocorrelation function (PACF) are plotted for the four time series.
Autocorrelation is the linear dependence of a variable with itself at two points in
time, it measures the correlation between yt and yt + k, where k = 0,...,K and yt is
a stochastic process . Correlation between two variables can result from a mutual
linear dependence on other variables. Partial autocorrelation is the autocorrelation
between yt and yt
h after removing any linear dependence on y1, y2, ..., yt
h+1.
As suggested by Box, Jenkins, and Reinsel (1994)
6
the correlation at each lag is
scaled by the sample variance so that the autocorrelation and the partial
autocorrelation at lag 0 is unity.
5
Whitney K. Newey and Kenneth D. West in their paper "A simple, positive semi-definite,
heteroskedasticity and autocorrelation consistent covariance matrix" (1987)
6
Box, G. E. P., G. M. Jenkins, and G. C. Rei
nsel. "Time Series Analysis: Forecasting and Control.
3rd ed. Englewood Cliffs, NJ: Prentice Hall" (1994)
60
Figure 32
lo20 sample autocorrelation and sample partial autocorrelation
Figure 33
hi20 sample autocorrelation and sample partial autocorrelation
61
Figure 34
tbond sample autocorrelation and sample partial autocorrelation
Figure 35
div_y sample autocorrelation and sample partial autocorrelation
The Ljung-Box Q-test is a quantitative way to test for autocorrelation at multiple
lags jointly. The test assesses whether a series of residuals exhibits no
autocorrelation for a fixed number of lags, against the alternative that some
autocorrelation coefficient is nonzero. The null hypothesis for this test is that the
first m autocorrelations are jointly zero. Under the null hypothesis, the asymptotic
distribution of the test statistics is chi-square with degrees of freedom equal to the
sample size. I performed the test at lags 15,10 and 5 and significance levels of
62
10%,5% and 1%. The null hypothesis failed to be rejected only for hi20 at any lags
value and significance level.
From the examination of the results presented in this paragraph emerges that the
three asset classes time series are stationary and may need to be model by an
autoregressive model while the div_y time series is not stationary.
63
CHAPTER 3
MODEL ESTIMATION
3.1 The basic idea behind regime switching models
The basic idea is simple, financial operators care for detecting and forecasting
instability in statistical relationships. A naïve approach is to model such instability
using dummy variables in
"regression
-
type" analysis, in other words it can be said
that a regime applies before the break and the other after the break. Unfortunately
this approach is feasible only ex-post because the current regime and its duration
are not observable but only forecastable. Econometricians have developed
methods in which instability is stochastic, it has a structure, and it can be
predicted. Suppose that the historical behavior of returns on some asset can be
described with a first-order autoregression process AR(1): the model can be
adequate until a certain date when its forecast accuracy declined, at this point the
idea would be to simply estimate again the parameter of the model if this break
can be observed, otherwise if the break is not observable the operators face
uncertainty and a possible model misspecification which can lead to a poor
forecast accuracy. In this framework the change that occurred at a certain date
was considered "deterministic" b
y operators and anyone would have been able to
predict with certainty looking ahead and anyone would have estimated the
parameter of the model again just after the break occurrence.
Instead in reality there must have been some imperfectly predictable forces that
produced the change. Hence, the econometricians developed a more elegant
approach that assumes there is some larger model encompassing both regimes. If
our model is still the AR(1) now it would have switching parameters whose values
are conditional on the regimes the process is at certain time. This means that the
parameters of the AR(1) are now time-variant parameters which depend on the
value of the current regime/state of the process. A complete description of the
probability law governing the observed data would then require a probabilistic
model of what caused the change from one regime to another. More generally, to
specify and estimate a regime switching model will mean to specify a structural
model for the temporal dependence of the regimes state variable.
At this point, the problem becomes to choose what kind of stochastic process can
be specified for the state variable; three mainly different solutions have been
64
proposed: threshold models in which the state variable can assume k distinct
values in dependence of the values as of time t of some threshold variable x,
smooth transition models in which the state variable comes from some discrete
probability distribution that can take k distinct values and the probability depends
on the values as of time t of some threshold variable x, it is called smooth because
the variable x no longer determistically determines the state, but simply the
cumulative distribution function of the regime; Markov switching models in which
the state variable is unobservable from a discrete, first-order, k-state, irreducible,
ergodic Markov chain. Markov switching models are of intermediate generality
between threshold and smooth transition models. Discrete means that the state
variable may only take k distinct values. First-order Markov chain means that the
probability of the realization of a certain value i of the state variable at time t only
depends on the value of the state variable at time t-1. Irreducible means that there
is no absorbing state, i.e. no special regime i such that
Pr(S
t
= | S
t-1
= j) = 0
for
all
= 1,2, ... ,
in other words no regimes exists such that the chain gets trapped
into it. Ergodic means that the chain has a long-run mean; in fact, defining
as a
(k x 1) vector made of zeros except the j-th element that equals 1 to signal
S
t
=
and 0 otherwise, we have:
+1
= P
+ v
+1
where P is the transition matrix that
collects the quantities:
(1)
Pr(S
t
= | S
t-1
= ) = p
,
and
v
+1
is some error term. Notice that such representation implied that the
forecast is:
E[
+T
|
,
-1
,
-2
, ... ] = (P)
T
. A Markov chain is ergodic if and only
if:
plim E[
+T
|
,
-1
,
-2
, ... ] = plim (P)
T
=
where
plim
means "limit
probability" as
T
and
is called vector of unconditional or ergodic
probabilities (also called
). One useful implication from all of this is that is
possible to forecast the probability of a regime T steps ahead using the following
formula
(P)
T
because
Pr(S
t+T
| S
t
) = E[
+T
|
]
.
3.2 Filtered and smoothed regimes probabilities
The key feature of Markov switching models is that they allow you to
endogenously make inferences on the regimes based on the data, for instance, in
65
the case with two regimes/states, because the regimes are unobservable (also
said, latent) the econometrician observes the returns directly but can only make an
inference about the value of the state variable
S
t
based on what he sees
happening with the returns. With two regimes this inferences will take the form of
two probabilities:
(2)
1t
Pr(S
t
= 1|
t
, )
and
2t
Pr(S
t
= 2|
t
, )
where
t
is all the past information up on time t and
is the vector collecting the
parameters to be estimated.
1t
and
2t
are called "filtered state (regime)
probabilities" because they depend on "real time" information
t
and
1t
+
2t
= 1
.
A filtered probability is the best inference of the current state, based on real time
information.
The Hamilton's filter is used to calculate the filtered probabilities of
each state based on the arrival of new information.
Here is illustrated how to compute filtered probabilities. The following formulas
regard the univariate case but are easily extended to deal with the multivariate
case.
Assume that
1t-1
is already known and that
(
|S
t
= ,
t-1
; )
is the log-
likelihood of the time t observation
conditional on state
S
t
=
. Then the
conditional density of the time t observation is:
(3)
(
|
t-1
; ) = Pr(
|
t-1
; ) =
1t-1
p
11
(
|S
t
= 1,
t-1
; ) +
1t-1
p
21
(
|S
t
= 2,
t-1
; ) + (1 -
1t-1
)p
12
(
|S
t
= 1,
t-1
; ) +
(1 -
1t-1
)p
22
(
|S
t
= 2,
t-1
; )
At this point, by an application of Bayes' rule, can be obtain:
(4)
jt
=
1t-1
p
j1
(
|
S
t
= ,
t-1
;
)+
2t-1
p
j2
(
|
S
t
= ,
t-1
;
)
(
|
t-1
;
)
= 1,2
In other words the probability of being in regime
at time t is the ratio between the
probability of reaching
from
S
t-1
= 1
plus the probability of reaching
from
S
t-1
=
66
2
and the total probability of
given past information. There are various
techniques to initialize
10
in order to compute
11
and
12
: if the Markov chain is
presumed to be ergodic one can use the unconditional probabilities:
(5)
1
=
10
=
1-
22
2-
11
-
22
2
=
20
=
1-
11
2-
11
-
22
Other common choices are
10
=
20
= 0.5
or
10
and therefore
20
= (1 -
10
)
can
be estimated by MLE as if they were themselves parameters.
One is also often interested in forming an inference about what regime the
economy was in at date t based on observation obtained through a later date T:
1t|T
Pr(S
t
= 1|
T
; )
and
2t|T
Pr(S
t
= 2|
T
; )
. These are referred to as
"smoothed probabilities". The obvious difference between smoothed and filtered
probabilities is that the former ones use all the information in the sample and as
such they represent ex-post measures. A filtered probability provides instead a
recursive,
real
time
assessment
on
the
current
state.
The average duration of state i, in a two states model, is equal to:
(6)
1
1-
3.3 The estimation process of a Markov regime switching model
Different econometric methods can be used to estimate regime switching models:
maximum-likelihood and EM algorithms are outlined by Hamilton (1988; 1990).
The maximum likelihood algorithm involves a Bayesian updating procedure, which
infers the probability of being in a regime given all available information up until
that time. An alternative to maximum likelihood estimation is Gibbs sampling,
which was developed for regime switching models by Albert and Chib (1993) and
Kim and Nelson (1999), in this approach both the parameters and the Markov
switching are treated as random variables.
An important issue in estimating regime switching models is specifying the number
of regimes. This is often difficult to determine from data, and as far as possible the
choice should be based on economic arguments. Such decision can be difficult
given that the regimes themselves are often thought of as approximation to
underlying states that are, unobserved. It is not uncommon to simply fix the
67
number of regime at some values, typically two (bear and bull markets), rather
than basing the decision on econometric tests. The reason is that tests for the
number of regimes are typically difficult to implement because they do not follow
standard distributions. Under the null of a single regime, the parameters of the
other regime are not identified, so there are unidentified nuisance parameters.
This means that conventional likelihood ratio tests are not asymptotically Chi-
squared distributed.
The algorithm for the estimation of Markov switching model proposed by Kim and
Nelson, instead of proceeding forward in a recursive fashion, proceeds backwards,
starting from the fact that for T = t, filtered and smoothed probabilities will be
identical. The estimation of Markov switching models performed by MLE
possesses optimal properties. This means that under standard regularity
conditions, the ML estimator of
will be: consistent, asymptotically normal, the
most efficient in a wide class of estimators. The conditional log-likelihood of the
observed data arises rather naturally from the recursive calculation of filtered
probabilities
(7)
ln(
1
,
2
, ... ,
T
|
0
; ) =
ln(
t
|
t-1
; )
T
t=1
.
The estimation procedure by maximum-likelihood is here described assuming the
existence of two regimes. We observe
directly but we can only make an
inference about the value of
S
t
based on what we see happening with
.The
inference will take the form of the two probabilities
jt
= Pr(S
t
= |
t
, )
for
= 1,2
where
t
is the set of observations obtained as of date t and
represents the
vector of population parameters. The inference is performed iteratively for
=
1, 2, ... ,
with step t accepting as input the values
it-1
= Pr(S
t-1
= |
t-1
, )
for
= 1,2
and producing as output
1t
and
2t
. The densities under the two regimes
are denoted as
(8)
= (
|S
t
= ,
t-1
; )
for
= 1,2
.
Given the input
it-1
it is possible to calculate the conditional density of the t-th
observation from:
68
(9)
7
(
|
t-1
; ) =
p
2
=1
2
=1
it-1
and the desired output is then:
(10)
jt
=
p
2
=1
-1
(
|
t-1
;
)
Executing this iteration will lead to evaluate the sample conditional log-likelihood of
the observed data for the specified value of
(7)
ln (
1
,
2
, ... ,
|
0
; ) =
ln (
|
t-1
; )
t=1
An estimation of
can then be obtained by maximizing the log likelihood of the
observed data by numerical optimization. Several options are available for the
value
0
to use to start these iterative process, some of them have been already
described in this paragraph.
The previous estimation steps can be easily generalized to consider a vector of
observation
at time t and then a multivariate model. Let
-
= {
,
-
, ... ,
}
be the observation through date t,
be an (n x n) matrix whose row j, column i
element is the transition probability
p
,
be an (n x 1) vector whose j-th element
(
|S
t
= ,
t-1
; )
is the density in regime j, and
|
an (n x 1) vector whose j-th
element is
Pr(S
t
= |
t
; )
. Then the conditional density of the t-th observation is:
(11)
8
(
|
t-1
; ) =
(
-|-
)
where
|-
=
-|-
and desired output is then:
(12)
|
=
-|-
(
|
t-1
;
)
where
is a (n x 1) vector all of whose elements are unity and
denotes
element-by-element multiplication.
7
James D. Hamilton "Regime
-
Switching Models" (2005)
8
James D. Hamilton "Regime
-Switching Models.
Palgrave Dictionary of Economics
" (2005) and
Birger Nilsson and Andreas Graflund
"Dynamic Portfolio Selection: The Relevance of Switching
Regimes and Investment Horizon" (2001)
69
3.4 Overview of the most common multivariate Markov regime
switching models
In this paragraph
I am going to briefly describe some of the most common and
employed Markov Switching model specifications. The Markov Switching model
allows for a great variety of specifications, especially the MSVAR family which
nests all the models illustrated in this paragraph. According to the most common
notation the presence and the configuration of the regime-dependent parameters
is indicated by the presence of particular capital letters in the name of the model
specification. The general term MMS(M) stands for Multivariate Markov Switching
and the mutable string (M) assumes values in accordance to the configuration of
the regime-dependent parameters; the possible (M) values are here explained:
I
Markov switching intercept term;
A
Markov switching autoregressive parameters;
H
Markov switching heteroskedasticity
For example if a multivariate model specification is characterized to accommodate
heteroskedasticity through the presence of a regime-dependent covariance matrix
and has a regime-dependent intercept term but invariant autoregressive
parameters, the model according to the common notation take the name MMSIH-
VAR(k,p).
The Markov Switching Vector Autoregressive model MSVAR is a general class of
models which nests the standard VAR model but additionally accounts for
nonlinear regime shifts. These models are particularly useful when one extends
them to capture the dynamics not of the returns only, but also of some vector
variables that also collects one or a number of predictors of subsequent asset
returns. Many authors adopted this family of Markov switching model to predict
stocks and bonds returns using a number of macroeconomic variables as
predictors. Many authors have also showed that while a simple VAR(1) model
does not produce useful predictions, models with two or more regimes/states
manage to be rather useful. MSVAR models are also particularly suitable to model
and study contagion dynamics tanks to the presence of the autoregressive
70
coefficient matrix which might also be time-dependent in certain model
configurations.
There are several models configuration nested in the general MSVAR family
model, the most relevant are illustrated in the following subparagraph.
3.4.1 MMSIAH(k,p)
The model, according to different notations, is also known as MMSIAH(k)-VAR(p)
or MSIVARH(k,p). This model configuration can be considered the most general
and complete among all the MSVAR family models; in fact all of its terms
intercept, autoregressive coefficients matrix and covariance matrix
are regime-
dependent. It is able to capture and model in a multivariate framework many of the
financial time series features. In a MMSIAH(k,p) model there are three types of
contagion effects: a simultaneous one through the off-diagonal elements of the
variance and covariance matrix that captures the dynamics across regimes of
correlations, a dynamic and linear one through the VAR coefficients, a dynamic
and nonlinear one through the fact that the regime variables that drives the
process
of
all
variables
are
common
to
all
variables.
When k=1 a MMSIAH(1)-VAR(p) becomes a VAR(p) model with Gaussian shocks.
A VAR model is just a multivariate extension of a standard autoregressive model,
in which lags of variable i may in principle affects the subsequent value of j. The
joint distribution of a vector of n returns
r
t
= [r
1t
r
2t
... r
nt
]
can be modeled as a
multivariate regime switching process driven by a common discrete state
variables,
S
t
, that takes integer values between 1 and k:
(13)
=
+
,
p
=1
-
+
where
= [
1S
t
...
nS
t
]
is a vector of mean returns in state
S
t
,
,
is a (n x n)
matrix of autoregressive and regression coefficients at lag j in state
S
t
and
=
[
1t
...
nt
]~(,
)
is the vector of return innovations that are assumed to be
joint normally distributed with zero mean and state-specific covariance matrix
.
Innovations to returns are thus drawn from a Gaussian mixture distribution that is
known to provide a flexible approximation to a wide class of distributions. The
state dependence of the covariance matrix captures the possibility of
71
heteroskedastic shocks to asset returns and a non diagonal
makes the asset
returns simultaneously cross-correlated.
Each state is the realization of a first-order Markov chain governed by the (k x k)
transition probability matrix,
P
, with generic element
p
defined as
Pr(S
t
=
| S
t-1
= ) = p
, = 1, ... , .
In the framework of this model the estimation
method allows
S
t
to be unobservable and treated as a latent variable.
The version of the model that has been just described above is characterized to
have a full covariance matrix estimated from the data, however a less
sophisticated version in which the structure of the covariance matrix is diagonal is
also possible; in this version the covariance between the residuals of the different
equations are not allowed. In the diagonal covariance matrix version of the model
there is no evidence of a simultaneous contagious effect one through the off-
diagonal elements of the variance and covariance matrix that captures the
dynamics across regimes of correlations because these covariance elements are
equal to 0. The model can be extended to incorporate an (l x 1) vector of predictor
variables such as the dividend yield:
(14)
= (
) +
,
p
=1
-
+ (
)
the same model can be represented alternatively in the following way:
(15)
(
) = (
) +
,
p
=1
(
-
-
) + (
)
where
= (
)
is a (l + n) x 1,
= [
1S
t
...
lS
t
]
is the intercept vector for
in state
S
t
,
{
,
}
=1
p
are (l + n) x (l + n) matrices of autoregressive and regression
coefficients in state
S
t
and
[
t
...
zt
]~(,
)
, where
is an (l + n) x (l + n)
covariance matrix. This model allows for predictability in returns through the
lagged values of
. The relationship between stock returns and the dividend yield
is linear within a given regime but the model is capable of tracking a non-linear
relationship between asset returns and the dividend yield since the coefficient on
the yield varies across regimes and the regime probabilities changes as well. This
family of models even in the absence of autoregressive terms or predictor
72
variables implies time-varying investment opportunities. For example, the
conditional mean of asset returns is an average of the vector of mean returns,
,
weighted by the filtered state probabilities
[Pr(S
t
= 1|
t
) ... Pr(S
t
= |
t
)]
,
conditional on information available at time t,
t
. Since these state probabilities
vary over time, the expected return will also change. These same dynamic is
applicable to higher order moments of the returns distribution.
The model is stationary if the absence of roots outside the unit circle is verified for
the matrices of autoregressive coefficients. Ang and Bekaert (2002) have showed
that formally, it is just sufficient for such a condition to be verified in at least one of
the k regimes.
Likewise the version without any predictor variables, also the less sophisticated
version with predictor variables can be characterized by a diagonal structure of the
covariance matrix. In this version the covariance between the residuals of the
different equations are not allowed thus there is no evidence of a simultaneous
contagious effect through the off-diagonal elements of the variance and
covariance matrix.
3.4.2 MMSIA(k,0)
The model, according to different notations, is also known as MMSI(k), MMSI(k)-
VAR(0) or MSIVAR(k,0).
(16)
=
+
where
= [
1t
...
nt
]~(, )
The only regime-dependent term is the intercept which is driven by a Markov state
variable while the autoregressive coefficients matrix and the covariance matrix are
static. Only two types of contagion effects are present in this model configuration:
a static and simultaneous one through the off-diagonal elements of the variance
and covariance matrix and a dynamic and nonlinear one through the fact that the
regime variables that drives the process of all variables are common to all
variables.
Also for this model exists a diagonal covariance matrix version which is practically
73
equivalent to a model that consists of N (number of asset) independent univariate
homoskedastic Markov regime switching Normal distributions in which the regime
switching dynamic is unique and common for all the asset returns functions. In this
case the total log-likelihood of the diagonal MMSIA(k,0) model is equal to the sum
of the individually (one for each asset) maximized log-likelihood function.
3.4.3 MMSIAH(k,0)
The model, according to different notations, is also known as MMSIH(k),
MMSIH(k)-VAR(0) or MSIVARH(k,0).
(17)
=
+
where
= [
1t
...
nt
]~(,
)
The model differs from the last model configuration described because here a
regime-dependent variance and covariance matrix which add a dynamic and
simultaneous contagious effect through the off-diagonal elements of the variance
and covariance matrix is present. Again, also for this model exists a diagonal
covariance matrix version which is practically equivalent to a model that consists
of N (number of asset) independent univariate Markov regime switching Normal
distributions in which the regime switching dynamic is unique and common for all
the asset returns functions. In this case the total log-likelihood of the diagonal
MMSIA(k,0) model is equal to the sum of the individually (one for each asset)
maximized log-likelihood function.
3.4.4 MMSIA(k,p)
The model, according to different notations, is also known as MMSIA(k,p),
MMSIA(k)-VAR(p) or MSIVAR(k,p).
(18)
=
+
,
p
=1
-
+
where
= [
1t
...
nt
]~(, )
This is a homoskedastic model configuration in which a dynamic and linear
contagious effect through the VAR coefficients, and a dynamic and nonlinear one
through the shared regime driven variables, are generated by the regime-
74
dependent
intercept
and
autoregressive
coefficients
matrix
terms.
Again, equivalently to the first three models illustrated before, for this model exists
a diagonal covariance matrix version. In this version the covariance between the
residuals of the different equations are not allowed thus there is no evidence of a
simultaneous contagious effect through the off-diagonal elements of the variance
and covariance matrix.
3.4.5 MMSIH(k,p)
The model, according to different notations, is also known as MMSIH(k)-VAR(p).
(19)
=
+
p
=1
-
+
where
= [
1t
...
nt
]~(,
)
. The model is a special case of the MSVAR(k,p)
in which while intercepts and the covariance matrix are regime-dependent, the
VAR(p) autoregressive coefficients are not. This model configuration shows a
dynamic and simultaneous contagious effect through the off-diagonal elements of
the variance and covariance matrix, a static and linear one through the VAR
coefficients and a dynamic and nonlinear one through the fact that the regime
variables that drives the process of all variables is common to all variables.
Likewise the last model illustrated for this model exists a diagonal covariance
matrix version characterized by the fact that the covariance between the residuals
of the different equations are not allowed thus there is no evidence of a
simultaneous contagious effect through the off-diagonal elements of the variance
and covariance matrix.
3.4.6 MMSI(k,p)
The model, according to different notations, is also known as MMSI(k)-VAR(p).
(20)
=
+
p
=1
-
+
where
= [
1t
...
nt
]~(, )
. The model is a special case of the MSVAR(k,p) in
which while intercepts are regime-dependent, the VAR(p) autoregressive
coefficients and the covariance matrix are not, then the model is homoskedastic
.
75
This model configuration shows a static and simultaneous contagious effect
through the off-diagonal elements of the variance and covariance matrix, a static
and linear one through the VAR coefficients and a dynamic and nonlinear one
through the fact that the regime variables that drives the process of all variables is
common to all variables. In the same way as for all the model previously illustrated
there is a less sophisticated version of the model characterized by a diagonal
covariance matrix, thus the covariance between the residuals of the different
equations are not allowed and for this reason in this version of the model there is
no evidence of a simultaneous contagious effect through the off-diagonal elements
of the variance and covariance matrix.
3.4.7 Multivariate restricted (common underlying Markov chain)
MSVAR(K,1) model
Guidolin
9
proposed a method to obtain a multivariate MSVAR(k,1) model starting
from the estimation of single univariate models, one for each asset class, and
making assumption on their underlying Markov chains. Each univariate model is a
Markov Switching first-order autoregressive model MSARH(k,1). The intercept, the
autoregressive coefficient, the regression coefficients and the variance of the
shock are regime-dependent. The model takes the following form:
(21)
r
,t
=
S
t
+
S
t
,
r
,t-1
+
S
t
,
=1
r
,t-1
+
,t
where
,t
~(0,
,S
t
2
)
,
r
,t-1
is the return at time t-1 on a generic asset or predictor j,
S
t
,
is the autoregressive coefficient in state
S
t
and
S
t
,
is the regression coefficient
of the return of asset i on the return of asset j in state
S
t
.
Once n univariate MSARH(k,1) models have been estimated, when n represent
the number of assets, is possible to estimate a multivariate model which is a
restricted version of the n univariate models. The restriction imposes
S
t
= S
,t
for
= 1, ... ,
, in words a unique Markov chain is assumed to drive simultaneously the
9
Massimo Guidolin
"Modelling, Estimating and Forecasting Financial Data under Regime (Markov)
Switching"
download at "http://dida
ttica.unibocconi.it/mypage/dwload.php?nomefile=Lecture_7_-
_Markov_Switching_Models20130520235704.pdf"
76
regime switching dynamics of all the asset classes. This model may be obtained
from a set of n univariate models, one for each of the n asset returns, when the
means, the variance parameters, the autoregressive coefficients and the
regression coefficients in the multivariate model are set to be identical to those of
the univariate models. The restriction
S
t
= S
,t
for
= 1, ... ,
implies that exists a
unique transition matrix and therefore the transition probabilities from one regime
to another are identical for all the asset returns processes. For instance if has
been chosen a two states model the restriction implies that
11
=
,11
and
22
=
,22
for
= 1, ... ,
. It is important to notice that the model is a multivariate
restricted version of n univariate models only when the simultaneous covariance
coefficients are restricted to be zero, this impose a diagonal structure of the
covariance matrix, for this reason the total log-likelihood for the multivariate
restricted model is equal to the sum of the log-likelihood of the n univariate models
individually maximized. A likelihood ratio test is proposed by the author to assess
the null hypothesis that the restriction cannot be rejected based on the available
data. The model proposed by Guidolin in this framework, for the reason explained
just before, is characterized by the fact that the covariance coefficients in all
regimes are restricted to be zero, i.e., the only source of correlation in the system
is the fact that the same Markov state variables drive simultaneously the switches
in all the time series. Given that in the multivariate model the parameters of each
asset class are constrained to assume the value estimated from the corresponding
single univariate model, in which context covariance terms cannot be estimated,
thus the covariance matrix of the multivariate model is also constrained to be
diagonal.
The model can be represented in the following way (assuming there are only three
assets
r
i,t
and a predictor
z
t
= r
4,t
, thus n=4):
(22)
r
1,t
=
1S
t
+
S
t
1,1
r
1,t-1
+
S
t
1,
4
=1
r
,t-1
+
1,t
1
77
r
2,t
=
2S
t
+
S
t
2,2
r
2,t-1
+
S
t
2,
4
=1
r
,t-1
+
2,t
2
r
3,t
=
3S
t
+
S
t
3,3
r
3,t-1
+
S
t
3,
4
=1
r
,t-1
+
3,t
3
z
t
= r
4,t
=
4S
t
+
S
t
4,4
r
4,t-1
+
S
t
4,
4
=1
r
,t-1
+
4,t
4
where
[
1,t
2,t
3,t
4,t
] ~ ([
0
0
0
0
] , [
1,1,S
t
2
0
0
0
0
2,2,S
t
2
0
0
0
0
3,3,S
t
2
0
0
0
0
4,4,S
t
2
])
is the multivariate normal distribution of the shocks at time t,
S
t
is the intercept
term of the asset i,
r
,t-1
is the return at time t-1 on a generic asset (
z
t
= r
4,t
is the
predictor),
S
t
,
is the autoregressive coefficient in state
S
t
and
S
t
,
is the regression
coefficient of the return of asset i on the return of asset j in state
S
t
.
The same model can be represented in matricial form in the following way:
(23)
=
+
,
-
+
where
[
1,t
2,t
3,t
4,t
] ~ ([
0
0
0
0
] , [
1,1,S
t
2
0
0
0
0
2,2,S
t
2
0
0
0
0
3,3,S
t
2
0
0
0
0
4,4,S
t
2
])
is the multivariate normal distribution of the shocks at time t,
= (
)
is a (3 +
1) x 1,
= [
1S
t
...
4S
t
]
is the intercept vector for
in state
S
t
and
,
is a (3 +
1) x (3 + 1) matrix of autoregressive and regression coefficients.
The following table provide a recap of the different notation commonly used for
each model configuration previously described in this paragraph.
78
Table 5
Commonly used alternative notation for MSVAR family models
alternative notations
m
o
d
e
l sp
e
cificat
ion
MMSI(k)
MMSIA(k,0) MMSI(k)-VAR(0)
MSIVAR(k,0)
MMSIA(k,p) MMSIA(k)-VAR(p) MSIVAR(k,p)
MMSIH(k,p) MMSIH(k)-VAR(p)
MMSI(k,p)
MMSI(k)-VAR(p)
MMSIH(k) MMSIAH(k,0) MMSIH(k)-VAR(0) MSIVARH(k,0)
MMSIAH(k,p) MMSIAH(k)-VAR(p) MSIVARH(k,p)
3.5 Choice of model specification
The objective of the model estimation procedure is to find the best fitting model
and the number of regimes is part of the fit.
Suppose you have a general model parameterized by
and that you want to test a
set of r restriction that transforms
into a sub-vector
r
of dimension r. Once both
model have been estimated by MLE the maximized log-likelihoods
()
and
(
R
)
are available and the log-likelihood test statistic is
LRT = 2[() - (
R
)]~
2
.
Unfortunately is not possible to use the likelihood ratio test , in fact the usual
regularity conditions fail because under the null hypothesis, some of the
parameters of the model would be unidentified (they are called "nuisances"). For
example, if there is really only one regime, the ML estimate of the transition
probability matrix does not converge to a well-defined value, meaning that the
likelihood ratio test does not have the usual Chi-squared limiting distribution. To
interpret a LRT statistic one instead needs to appeal to a simulation methods.
A practical alternative consists of using information criteria that are essentially
penalized log-likelihood tests in which the trade-off between fit and parsimony is
quantified. The goal is to minimize the value of the information criteria. In general
there are two ways to reduce the value of the information criteria: increase the
maximized log-likelihood or reduce the number of parameters.
79
A range of values for the number of regimes is considered
(k = 1,2,3, ... )
, doing so
very parsimonious as well as heavily parameterized models are covered. To select
among the regimes specifications, Akaike (AIC), Schwartz (SIC) or Bayesian (BIC)
and Hannan-Quinn (HQC) are usually considered. Unlike formal hypothesis tests
which are subject to nuisance parameter problems, these criteria do not, however,
provide rigorous tests for the presence of regimes. The AIC tends to suggest the
selection of overparameterized models according to Fenton and Gallant (1996),
the BIC on the contrary tends to favors small models while Hannan-Quinn is
usually in an intermediate position when compared to BIC and AIC. The three
information criteria formula are here illustrated:
(24)
= 2 - 2 ln(
)
(25)
= ln () - 2 ln(
)
(26)
= 2 ln (ln()) - 2 ln (
)
where k stands for the number of free parameters to be estimated, n stands for the
number of observations while
is the maximized value of the
log-Likelihood for the estimated model
.
3.5.1
Alternative models estimation results
In this subparagraph I am going to describe the method I adopted to estimate the
model and the results of the estimation procedure. Firstly I started by estimating a
range of MSVAR model commonly used by many authors in the relevance
literature, and a less common restricted version, secondly I applied an information
criteria to select the best model according to it. To determine the best model I
undertake an extensive specification search estimating models that span a
number of regimes from 1 to 4 and autoregressive order from 1 to 2. This
approach covers very parsimonious as well as heavily parameterized models. For
the estimation I employed the Marcelo Perlin's MS_Regress, a MATLAB toolbox
10
(October 30, 2014 versions) specially designed for the estimation and simulation
of markow regime switching model. The package allows the user to estimate a
10
Perlin, M. (2014) MS Regress - The MATLAB Package for Markov Regime Switching Models.
Available at SSRN: http://ssrn.com/abstract=1714016 or http://dx.doi.org/10.2139/ssrn.1714016
80
large number of different Markov switching specifications, without any change in
the original code. I wrote my own script to manage the estimation routine of
several models in a row, including the composition of the input structure for each
model
into the package's notation
, the filling of the package fitting function and the
options to feed it, the parsing of the output and the storing of the estimation results
and charts. In this Matlab package all of the models are estimated using maximum
likelihood. Results of the model specification analysis are presented in Table 6.
Table 6
Estimation results
Model
Number of
parameters
(saturation
ratio)
Log-
likelihood
AIC
BIC
Hannan-
Quinn
MMSIAH(k,p)
MMSIAH(2,1)
62 (34.84)
-12383.33 -12383.33 -12031.30 -12279.27
MMSIAH(3,1)
96 (22.5)
-12996.57 -12996.57 -12451.49 -12835.44
MMSIAH(3,2)
144 (15)
-12740.35 -12740.35 -11922.74 -12498.66
MMSIAH(4,1)
132 (16.36) -12575.36 -12575.36 -11825.88 -12353.81
MMSIAH(4,2)
196 (11.02) -12528.85 -12528.85 -11415.99 -12199.88
MMSIAH(k,0)
MMSIAH(2,0)
30 (72)
4796.60
-9533.21 -9362.87 -9482.85
MMSIAH(3,0)
48 (45)
5374.93 -10653.85 -10381.32 -10573.29
MMSIAH(4,0)
68 (31.76)
5373.71 -10611.42 -10225.33 -10497.29
MMSIAH(k,p) diagonal covariance matrix
MMSIAH(2,1) diagonal
covariance matrix
50 (43.2)
-11786.48 -11786.48 -11502.59 -11702.56
MMSIAH(3,1) diagonal
covariance matrix
78 (27.69)
-12139.21 -12139.21 -11696.33 -12008.29
MMSIAH(3,2) diagonal
covariance matrix
126 (17.14)
-12093.47 -12093.47 -11378.05 -11881.98
MMSIAH(4,1) diagonal
covariance matrix
168 (12.86)
-12047.66 -12047.66 -11434.45 -11866.39
MMSIAH(4,2) diagonal
covariance matrix
172 (12.56)
-11957.77 -11957.77 -10981.18 -11669.08
MMSIAH(k,0) diagonal covariance matrix model
MMSIAH(2,0) diagonal 18 (120)
4836.79
-9637.57 -9535.37 -9607.36
81
covariance matrix
MMSIAH(3,0)
diagonal
covariance matrix
30 (72)
4914.18
-9768.36 -9598.02 -9718.00
MMSIAH(4,0)
diagonal
covariance matrix
44 (49.09)
5119.76 -10151.52 -9901.69 -10077.67
The previous table contains the number of estimated parameters, the saturation
ratio value, the maximized value of the log-Likelihood and three information criteria
values for each model configuration estimated. I decided to use the Hannan-Quinn
as model selection method since it holds the middle ground between the Akaike
and Schwartz information criteria. The Hannan-Quinn information criteria supports
the
MMSIAH(3,1)
model; I therefore would have settled on a three states model
with first order autoregressive components.
Estimating a richly parameterized model in which the number of parameters is so
large that the saturation ratio (i.e., the number of observations available to
estimate each parameter, on average) is below 20, leads to encounter difficulties
at obtaining reliable parameters estimates. A common rule of thumb proposes that
nonlinear estimation results based on saturation ratios less than to 20 ought to be
taken with great caution.
3.5.2 Multivariate restricted MSVAR(K,1) models estimation results
Given that the preferred model according to the Hannan-Quinn information ratio is
characterized to have a low saturation ratio equal to 22.5 and that the two next
most supported models, a MMSIAH(3,2) and a MMSIAH(4,1), have saturation
ratios respectively of 15 and 16.36, and given that a high number of estimated
parameters fail to be statistically significant - probably because of the bad quality
of the estimation caused by the high number of free parameters - I preferred to
perform the estimation of an additional class of MSVAR family models, namely the
multivariate restricted MSVAR(k,1) model. This class of model configuration has
been described previously in the subparagraph 3.4.7. I proceeded estimating three
MSARH(k,p) models for each asset class. From the adoption of the Hannan-Quinn
information ratio emerged that, for three out of four asset classes, a 2 states first
order autoregressive model is sufficient to capture and described the asset returns
82
dynamic. The only times series for which there is evidence that a 2 states model is
too parsimonious is the div_y, indeed a 3 states model is supported by the data.
Once the 4 univariate MSARH(k,1) models are estimated, and the free estimated
parameters available, is possible to proceed to estimate the restricted multivariate
MSVAR(k,1) models conditionally to the parameters values previously obtain by
the estimation of the 4 univariate MSARH(k,1). The restriction imposes
S
t
= S
,t
for
= 1, ... ,
, which means that a unique Markov chain is assumed to drive
simultaneously all the regime switching dynamics of all the asset classes time
series which implies that each asset class in a MSVAR(k,1) multivariate model is
assumed to have a regime switching dynamic characterized by a number of states
identical to the ones of all the other asset classes. Lastly, I estimated a
multivariate model in which the restriction imposes all the asset classes time
series to be explained either by a unique 2 states underlying Markov chain,
namely a MSVAR(2,1) or by a unique 3 states underlying Markov chain, namely a
MSVAR(3,1) or by an additional 4 sates model called MSVAR(4,1). All the
estimation results are presented in the following tables, the bold Hannan-Quinn
information ratio value indicates the selected univariate model for each asset class
and for the multivariate restricted model.
Table 7
Univariate models estimation results
Model (k,p)
Number of
parameters
(saturation
ratio)
Log-
likelihood
AIC
BIC
Hannan-
Quinn
variable: lo20
MSARH(2,1) 14 (38.6)
760.92 -1,493.84 -1,433.79 -1,470.35
MSARH(3,1) 24 (22.5)
768.72 -1,489.43 -1,386.48 -1,449.17
MSARH(4,1) 36 (15)
782.98 -1,493.97 -1,339.54 -1,433.57
variable: hi20
MSARH(2,1) 14 (38.6)
953.29 -1,878.58 -1,818.52 -1,855.09
MSARH(3,1) 24 (22.5)
963.31 -1,878.61 -1,775.66 -1,838.35
MSARH(4,1) 36 (15)
968.64 -1,865.29 -1,710.86 -1,804.89
variable: tbond
MSARH(2,1) 14 (38.6)
1,411.50 -2,795.01 -2,734.95 -2,771.52
MSARH(3,1) 24 (22.5)
1,416.00 -2,784.00 -2,681.05 -2,743.73
83
MSARH(4,1) 36 (15)
1,419.09 -2,766.17 -2,611.74 -2,705.77
variable: div_y
MSARH(2,1) 14 (38.6)
2,951.94 -5,875.87 -5,815.82 -5,852.38
MSARH(3,1) 24 (22.5)
3,017.29 -5,986.58 -5,883.63 -5,946.32
MSARH(4,1) 36 (15)
2,979.60 -5,887.19 -5,732.76 -5,826.79
Table 8
Multivariate restricted models estimation results
Model
Number of
parameters
(saturation
ratio)
Log-
likelihood
AIC
BIC
Hannan-
Quinn
MSVAR(2,1)
restricted
4 (135)
6,066.08
-12,116.1
-12,070.7
-12,102.7
MSVAR(3,1)
restricted
9 (60)
6,050.59
-12,075.1
-12,001.3
-12,053.3
MSVAR(4,1)
restricted
16 (33.8)
5343.32
-10654.64 -10563.80 -10627.79
3.6 Selected model estimates
According to the Hannan-Quinn information criteria, the multivariate restricted
model most supported by the data is the 2 regimes restricted first order
autoregressive MSVAR(2,1). The same conclusion would have been reached
even if either AIC or BIC information criteria had been used instead of the Hannan-
Quinn information criteria.
Table 9 shows parameters estimates (pvalues are reported in parentheses) for the
regime
switching
restricted
MSVAR(2,1)
model:
(27)
=
+
,
-
+
where
= (
)
is a (3 + 1) x 1 vector of excess returns, more precisely lo20,
hi20, tbond and div_y, and
= [
1S
t
...
4S
t
]
is the 4 x 1 vector of intercept terms
of
in state
S
t
while
,
is the (3 + 1) x (3 + 1) matrix of autoregressive and
regression coefficients associated with lag 1 in state
S
t
and
=
[
1,t
2,t
3,t
4,t
]~(,
)
.
Table 9
restricted MSVAR(2,1) parameter estimates
84
multivariate restricted MSVAR(2,1)
lo20
hi20
tbond
div_y
1. Intercept term
Regime 1
-0.0032
0.0081
0.0026
0.0004
Regime 2
0.0085
0.0043
0.0018
-0.0009
2. VAR(1) Matrix
Regime 1
lo20
0.3542
-0.0245
-0.0335
-0.0002
hi20
-0.1371
-0.1015
0.0047
-0.0138
tbond
0.3931
0.3656
0.3021
-0.0058
div_y
0.2568
0.0071
-0.0476
0.9875
Regime 2
lo20
0.0664
0.0116
-0.0813
0.0005
hi20
0.2824
0.1214
-0.0255
0.0109
tbond
-0.1335
0.1470
0.3673
0.0164
div_y
-0.0936
-0.1509
0.0040
1.0234
3. Covariance Matrix
Regime 1
lo20
0.001764
hi20
0.000000 0.000955
tbond
0.000000 0.000000 0.000215
div_y
0.000000 0.000000 0.000000 0.000001
Regime 2
lo20
0.007056
hi20
0.000000 0.003310
tbond
0.000000 0.000000 0.001026
div_y
0.000000 0.000000 0.000000 0.000016
4. Transition Probabilities
Regime 1 Regime 2
Regime 1
0.8988****
(0.0000)
0.4873****
(0.0000)
Regime 2
0.1012****
(0.0000)
0.5127****
(0.0000)
**** denotes significance at 0.01%
As it can be seen from Table 9 the model has 64 parameters; 8 intercept terms, 32
regression and autoregressive coefficients, 20 covariance and variance terms
(even though the covariance terms are set to 0 and have not been estimated) and
4 regimes transition probabilities. The 4 regimes transition probabilities are the
only free parameters that have been estimated in the multivariate restricted model
MSVAR(2,1) estimation (their pvalues are reported in parentheses), the totality of
85
the other parameters in the estimation procedure have been set to be equal to the
values estimated from the corresponding univariate MSARH(2,1) models. Table 10
reports the parameter estimates for each one of the four univariate models.
Table 10
single univariate MSARH(2,1) parameter estimates
single univariate MSARH(2,1) models
lo20
hi20
tbond
div_y
1. Intercept term
Regime 1
-0.0032
(0.6758)
0.0081***
(0.0003)
0.0026
(0.2427)
0.0004****
(0.0000)
Regime 2
0.0085
(0.5602)
0.0043
(0.7199)
0.0018
(0.5738)
-0.0009
(0.8432)
2. VAR(1) Matrix
Regime 1
lo20
0.3542****
(0.0000)
-0.0245
(0.6726)
-0.0335
(0.1747)
-0.0002
(0.6582)
hi20
-0.1371
(0.1917)
-0.1015
(0.3295)
0.0047
(0.9060)
-0.0138****
(0.0000)
tbond
0.3931**
(0.0025)
0.3656**
(0.0048)
0.3021****
(0.0000)
-0.0058**
(0.0048)
div_y
0.2568
(0.2551)
0.0071
(0.9415)
-0.0476
(0.5279)
0.9875****
(0.0000)
Regime 2
lo20
0.0664
(0.5200)
0.0116
(0.8995)
-0.0813
(0.6747)
0.0005
(0.9464)
hi20
0.2824
(0.0882)
0.1214
(0.3805)
-0.0255
(0.9199)
0.0109*
(0.0326)
tbond
-0.1335
(0.6511)
0.147
(0.5035)
0.3673***
(0.0005)
0.0164
(0.5724)
div_y
-0.0936
(0.8356)
-0.1509
(0.6978)
0.004
(0.9223)
1.0234****
(0.0000)
3. Variance
Regime 1
0.001764****
(0.0000)
0.000955****
(0.0000)
0.000215****
(0.0000)
0.000001**
(0.0024)
Regime 2
0.007056****
(0.0000)
0.003310****
(0.0000)
0.001026****
(0.0000)
0.000016
(0.1865)
* denotes significance at 5%, ** at 1%, *** at 0.1%, **** at 0.01%
I voluntary omitted from the previous table the transition probabilities estimates
and their relative pvalues because, conversely to the other parameter estimates,
they have not been used to set any parameter values in the estimation of the
multivariate restricted model. It can be noticed that 20 parameters out of 48
86
estimated from the single univariate models are statistically significant at the 10%
level, the latter conclusion implies that the total number of statistically significant
parameters in the multivariate restricted model are 24 considering the 4
statistically significant at the 10% level regimes transition probabilities estimated in
this latter multivariate model. Thus it can be concluded that the number of
statistically significant parameters at the 10% level of the multivariate restricted
model is 24 out of 52; while if we consider the total number of parameters of the
model, which includes also not estimated parameters such as the covariance
terms set to 0, the number of statistically significant parameters is 28 out of 64.
The parameter estimates are summed up in the following model expressions
which have been previously explained in the subparagraph 3.4.7.:
(28)
=
+
,
-
+
Regime 1
[
20
20
_
] = [
-0.0032
0.0081
0.0026
0.0004
] + [
0.3542 -0.1371 0.3931 0.2568
-0.0245 -0.1015 0.3656 0.0071
-0.0335 0.0047 0.3021 -0.0476
-0.0002 -0.0138 -0.0058 0.9875
] [
20
-1
20
-1
-1
_
-1
] + [
020,t
20,t
,t
_,t
]
[
020,
20,
,
_,
] ~ ([
0
0
0
0
] , [
0.001764
0
0
0
0
0.000955
0
0
0
0
0.000215
0
0
0
0
0.000001
])
Regime 2
[
20
20
_
] = [
0.0085
0.0043
0.0018
-0.0009
] + [
0.0664 0.2824 -0.1335 -0.0936
0.0116 0.1214 0.1470 -0.1509
-0.0813 -0.0255 0.3673 0.0040
0.0005 0.0109 0.0164 1.0234
] [
20
-1
20
-1
-1
_
-1
] + [
020,t
20,t
,t
_,t
]
[
020,
20,
,
_,
] ~ ([
0
0
0
0
] , [
0.007056
0
0
0
0
0.003310
0
0
0
0
0.001026
0
0
0
0
0.000016
])
87
Transition probabilities
[
p
11
p
12
p
21
p
22
] = [
0.8988
0.4873
0.1012
0.5127
]
Where the term
p
12
indicates the probability of a regime switch from state 2 at time
t-1 to state 1 at time t. The process described by the model can be interpreted
using the framework in Samuelson (1991) as a momentum process, i.e., a process
that is more likely to continue in the same state rather than transition to the other
state.
3.7 Model restriction test
At this point of the study once the estimation has been performed I decided to
formally test the restriction implied by the model. The restriction, as described
previously, imposes
S
t
= S
,t
for
= 1, ... ,
, in words a unique Markov chain is
assumed to drive simultaneously all the regime switching dynamics of all the asset
classes time series. The restriction
S
t
= S
,t
for
= 1, ... ,
implies that exists a
unique transition matrix and therefore the transition probabilities from one regime
to another are identical for all the asset returns processes. In the two states model
I have estimated the restriction imposes that
11
=
,11
and
22
=
,22
for
=
1, ... , . As suggested by Guidolin in his work "
Modelling, Estimating and
Forecasting Financial Data under Regime (Markov) Switching
", I formally tests the
restriction performing a Likelihood Ratio test. The test compares the maximized
log-likelihood values of the restricted case and the unrestricted case. The
restricted case is the multivariate restricted MSVAR(2,1) model while the
unrestricted case is the general case with potentially separate regime process.
The maximized log-likelihood of the former model is exactly the maximized log-
likelihood value of the multivariate restricted MSVAR(2,1) model while the
maximized log-likelihood of the latter model is equivalent to the sum of the
individually (one for each asset class) maximized log-likelihood function. The log-
likelihood ratio statistics is computed in the following way:
(29)
= 2 (
-
)
where
88
=
(2,1)
= 6066.0830
=
20;(2,1)
+
20;(2,1)
+
;(2,1)
+
_;(2,1)
= 760.9213 +
953.2881 + 1411.5042 + 2951.9365 = 6077.6501
Thus the LRT statistics assumes value 23.1343. The LRT statistics is distributed
as a Chi-squared distribution with g degrees of freedom where g is equal to the
number of equality restrictions imposed by the restricted model. In this study 8
parameters are set to be identical:
11
=
20,11
=
20,11
=
,11
=
_,11
22
=
20,22
=
20,22
=
,22
=
_,22
thus the LRT statistics is distributed as a Chi-squared distribution with 8 degrees
of freedom.
The null hypothesis of the presence of unique Markov chain that is assumed to
drive simultaneously all the regime switching dynamics of all the asset classes
time series cannot be rejected based on the available data; the LRT statistics of
23.1343 is smaller than the critical value at alpha 0.1% (one chance over a
thousand that the test statistics is wrong) which is equal to 26.1245. The pvalue of
the LRT statistics is lower than 0.999 and is equal to 0.9968. To sum up it can be
said that there is a statistical evidence, one chance out of a thousand to be wrong,
that there is the presence of a unique switching dynamics shared by all the asset
classes time series and that the multivariate restricted model MSVARH(2,1)
specification cannot be rejected based on statistical evidence.
Figure 36
scatter of the state 1 smoothed probabilities estimated from the
single univariate markow switching models
89
Figure 36 consists of six scatter diagrams each of one represents the relation
between the state 1 smoothed probabilities of a pair of returns or dividend yield
estimated from single univariate markow switching models. For example the first
scatter diagram in the upper-left corner represents the relation between the state 1
smoothed probability of a single univariate markow switching model estimated for
the large stocks and the state 1 smoothed probability of a single univariate
markow switching model estimated for the small stocks. Overall the predominant
relation that emerges from the scatter diagrams seems to be moderately positive
except for the pair bonds-small stocks, thus it can be said that, although not so
graphically convincing, there is evidence of a positive relation between the state 1
smoothed probabilities of a pair of asset classes or dividend yield estimate from
single univariate markow switching models, in other words the greater the
smoothed probability of an asset class returns or dividend yield to belong to state
1 the greater the probability of the other asset class returns or dividend yield of the
pair to belong to state 1.
At this point I also assessed whether the process is stationary or not. A not regime
switching model is considered stationary if the absence of roots outside the unit
circle is verified for the matrices of autoregressive coefficients. A generalization of
the rule that works for testing stationarity in a not regime switching model, for
example a VAR(1), is applicable in the regime switching framework, in fact, Ang
90
and Bekaert (2002) have showed that formally, it is just sufficient for such a
condition
to
be
verified
in
at
least
one
of
the
k
regimes.
,
=
is the first order matrix of autoregressive and regression coefficients in
state 1 while
,
=
is the state 2 equivalent object. Consider the previously
estimated model, MSVAR(2,1), let
,20
,
,20
,
,
,
,_
be the eigenvalues
of the state i first order matrix of autoregressive and regression coefficients,
,
=
,
these eigenvalues solve the characteristic equation
|
,
=
-
N
| = 0
then, if the
eigenvalues, for just any i, are not all equal and smaller than one in modulus thus
the MSVAR(2,1) process is stable and given that a stable MSVAR(2,1) implies
stationarity then the MSVAR(2,1) process is also stationary. I have calculated and
assessed the modulus of the 8 eigenvalues, the results are shown below:
Regime1
[
1,20
1,20
1,
1,_
]
= [
0.3287 + 0.0933
0.3287 + 0.0933
-0.1035 + 0.0000
0.9884 + 0.0000
]
[
||
1,20
1,20
1,
1,_
||
]
= [
0.3417
0.3417
0.1035
0.9884
]
Regime2
[
2,20
2,20
2,
2,_
]
= [
-0.0312
0.2752
0.3125
1.0221
]
[
||
2,20
2,20
2,
2,_
||
]
= [
0.0312
0.2752
0.3125
1.0221
]
As it can be seen the stability and stationarity of the process in state 1, according
to Ang e Bekaert (2002), guarantees the whole process to be stationary.
3.7 Model description
Table 11
unconditional and state conditional asset class and dividend yield
means
lo20
hi20
tbond
div_y
Regime 1
0.004024 0.008032 0.001947 0.023979
Regime 2
0.004980 -0.000730 0.002509 0.035556
unconditional
0.004188 0.006526 0.002044 0.025969
91
Table 12
unconditional and state conditional covariance matrices adjusted for
the regime structure
lo20
hi20
tbond
div_y
Regime 1
lo20
0.002300
hi20
-0.000001 0.001200
tbond
0.000000 0.000000 0.000297
div_y
0.000001 -0.000009 0.000001 0.000014
Regime 2
lo20
0.004478
hi20
-0.000002 0.002182
tbond
0.000000 -0.000001 0.000631
div_y
0.000003 -0.000025 0.000002 0.000042
Unconditional
lo20
0.002674
hi20
-0.000001 0.001371
tbond
0.000000 -0.000001 0.000355
div_y
0.000002 -0.000014 0.000001 0.000022
Timmerman (2000) derives the moments of a general regime switching process
with constant probabilities; the formulas were originally derived for a univariate
case, I have adapted them to the multivariate case.
Table 11 shows unconditional and state conditional asset class and dividend yield
means calculated as follows:
(30)
E(
+
|S
t
= 1; ) =
= (
-
,
)
-1
(31)
E(
+
|S
t
= 2; ) =
= (
-
,
)
-1
(32)
E(
+
|) =
1
+ (1 -
1
)
The first two formulas represent the means of a VAR(1) model, in fact conditionally
to a current state an MSVAR(1) is equivalent to a VAR(1).
,
and
,
are (4 x 4)
matrices of autoregressive and regression coefficients at lag 1, respectively in
92
state 1 and state 2,
is a (4 x 4) identity matrix while
1
and
2
are respectively
state 1 and state 2 unconditional or ergodic probabilities.
Table 12 shows unconditional and state conditional covariance matrices adjusted
for the regime structure calculated as follows:
(33)
(
+
|S
t
= 1; ) = p
11
1
+ p
21
2
+ p
11
p
21
(
-
)(
-
)
(34)
(
+
|S
t
= 2; ) = p
12
1
+ p
22
2
+ p
12
p
22
(
-
)(
-
)
(35)
(
+
|) =
1
1
+ (1 -
1
)
2
+
1
(1 -
1
)(
-
)(
-
)
The first two formulas
11
represents the state conditional covariance matrices, as it
can be seen the actual covariance matrices consist of two components. The first
component in these formulas are simply weighted averages of
1
and
2
while the
second component takes into account the regime structure, in fact the covariance
matrix at time t+1 depends on the realization of the current regime. The regime
structure adds also a jump component to the conditional covariance matrices due
to the presence of conditional means that change from one regime to the other.
As evidenced by Ang and Timmermann (2012) differences in means across
regimes,
-
, enter the higher moments such as variance. In particular, the
covariance matrix is not simply the average of the covariance matrices across the
two regimes indeed the difference in means also imparts an effect because the
switch to a new regime contributes to volatility. Intuitively, the possibility of
changing to a new regime with different mean introduces an extra source of risk.
Differences in means in addition to differences in covariance matrices can
generate persistence in levels and squared values
12
, causing volatility persistence
observed in many return series. The first formula represents the auto covariance
of levels (mean persistence) while the second formula represents auto covariance
of squared levels (volatility persistence).
(36)
Cov(
,
-
) =
1
(1 -
2
)(
-
)(
-
)
(p
11
+ p
22
- 1)
11
Andrew Ang and Geert Bekaert "How Do Regimes Affect Asset Allocation" (2002) and Massimo
Guidolin and Federica Ria "Regime Shifts in Mean
-Variance Efficient Frontiers: Some International
Evidence" (2010)
12
Andrew Ang and Allan
Timmermann "Regime Changes and Financial Markets" (2012)
93
(37)
Cov(
,
-
) =
1
(1 -
2
)(
-
+
1
-
2
)(
-
+
1
-
2
)
(p
11
+
p
22
- 1)
Again differences in means play an important role in generating autocorrelation in
first moments, without such differences the autocorrelation will be equal to zero.
In contrast, volatility persistence can be induced either by differences in means or
by differences in covariance matrices across regimes. In both cases, the
persistence tends to be greater the stronger the combined persistence of the
regimes, as measured by
(p
11
+ p
22
- 1)
.
The following subparagraph of the paragraph represents an attempt to give an
economic interpretation to the model estimated.
3.8.1 Economic interpretation of regimes
Regime 1 is an extremely persistent bull regime characterized by low realized
volatility and positive average realized excess returns on all assets. Small stocks
and large stocks grow rapidly on average, the average realized excess return on
bonds, likewise the average realized dividend yield level, are more modest and
relatively lower than in regime 2. The average duration is equal to 9.8855 months
and as a result this regime characterizes approximately 85% of the data in the
long run. This statistic is the simple empirical probability of being in state 1
computed by dividing the number of draws associated with state 1 by the total
number of sample draws (540); a draw is considered to be associated to a state i
when the corresponding smoothed probability of state 1 is greater than or equal to
0.5%. In this state the intercept term of lo20 and tbond are equal respectively to -
0.0032 and 0.0026 but not significantly different from zero while the intercept term
of hi20 and the dividend yield are instead statistically different from 0 and are
equal respectively to 0.0081 and 0.0004. The volatility of all the asset classes
returns and the dividend yield are small and lower than in state 2; small stock
returns are the most volatile asset followed by large stocks and bonds. Figure 39
shows that regime 1 capture episodes of the bull market since the mid-sixties, long
periods with growing stock prices during the mid-1990s and the late 1990s and the
94
recent bull market in the mid-2000s. Regime 2 is a not-highly persistent bear
regime characterized by high volatility and large and negative average realized
excess returns on small and large stocks while the average realized excess return
on bonds are significantly positive and relatively larger than in regime 1. The
average duration is equal to 2.0523 months and as a result this regime is
characterized by approximately 15% of the data in the long run. In this state the
intercept terms are all not significantly different from zero. The volatility of all the
asset classes returns and the dividend yield are large and higher than in state 1;
similarly to state 1 small stock returns are the most volatile asset followed by large
stocks and bonds. Figure 39 shows that regime 2 includes two oil shocks in the
1970s, the recession of the early 1980s, the October 1987 stock market crash, the
Kuwait invasion in the early 1990s and the `Asian flu' (
1996-1998), the dot-com
market crash of 2000-2001 (the recent bear market of 2002-2002) and the 2008-
2009 financial crisis.
The cross-regime differences between the estimated intercept terms are all
smaller than 1% except for lo20 which is equal to 1.1740%, the second larger
difference, 0.3768%, belongs to hi20 followed by tbond and div_y ones, equal to
0.0767% and 0.1291% respectively. The cross-regime differences between the
estimated volatilities again tend to be larger for lo20 and hi20, respectively equal
to 0.5290% and 0.2355% while tbond and div_y have smaller ones equal to
0.0811% and 0.0016%.
Table 13
state conditional time series average levels and volatilities
STATE STATISTICS
lo20
hi20
tbond
div_y
div_y
%change
1
average values 0.010920 0.006673 0.000543 0.030128 0.003674
volatilites
0.002330 0.001102 0.000268 0.000121 0.000808
2
average values -0.016294 -0.013784 0.007353 0.036819 -0.021052
volatilites
0.014520 0.006211 0.001168 0.000241 0.004827
According to Table 13 the larger average realized returns go along with the lower
realized volatilities in regime 1 which is considered the bull market state; from the
95
same table it can be clearly stated that regime 2 is a recession or bear state with
high realized volatility and mostly negative average realized returns.
Table 13 shows the state conditional average values and volatilities for small
stocks, large stocks, bonds, dividend yield and changes in dividend yield. The
values are calculated ex post classifying each period as either state 1 or state 2
based on the estimated smoothed state probabilities; a state 1 (state 2) smoothed
probability value greater than 0.5 implicates the classification of that data point as
belonging to state 1 (state 2). It follows that the statistics relative to state 2 are
calculated from a time series with length 89 whereas the data relative to state 1
are calculated from a time series with length 459. At first glance it can be seen that
both the stock returns show a positive average value in state 1 and a negative
value in state 1 whereas the average bonds returns are positive in both state
similarly to the dividend yield ones, however both the bonds returns and the
dividend yield are characterized by higher average values in state 2 than in state
1.
The lo20 and hi20 average levels in state 1 are relative high and above global
average level (1.0920% and 0.6673%) while the volatilities are below global
average (0.233% and 0.1102%); on the contrary in state 2 the lo20 and hi20
average levels are below global average level (-1.6294% and -1.3784%) and their
volatilities are above global average (1.452% and 0.6211%). The tbond time series
has a behavior antithetical to that of lo20 and hi20 regarding the average levels in
state 1 and 2, in fact in state 1 tbond average level is relatively low and below
global average (0.0543%) while in state 2 relative high and above global average
(0.7353%); on the contrary the tbond volatility behaves similarly to those of lo20
and hi20 showing lower volatility below global value in state 1 (0.0268%) and
higher volatility above global value in state 2 (0.1168%). The dividend yield value
is relatively low and below the global mean in state 1 (3.0128%) and the volatility
(0.0121%) is also below average, on the contrary in state 2 the posterior average
dividend yield value is higher than average (3.6819%) as well as the volatility
(0.241%).
I also computed the dividend monthly yield percentage variations and made some
considerations on their state conditional average levels and volatilities; I find out
96
that in state 1 the average dividend yield monthly percentage variation is slightly
positive and above the global average (0.3674%) while in state 2 it assumes a
dramatic negative value (-2.1052%), similarly the volatility of the dividend yield
monthly percentage variation assumes lower value in state 1 than in state 2
(respectively 0.0808% and 0.1481%). The changes in dividend yield present a
positive average value in state 1 and a negative average value in state 2, this
feature can be explained by the transient nature of the dividend yield (from
frequent and long period of low values in state 1 to rare and short high values in
state 2) that impose to its changes to be moderately positive in state 1 and
significantly negative in state 2. As it can be clearly seen from Table 13, all the
asset classes returns, the dividend yield and the changes in dividend yield show
larger volatility values in state 2 than in state 1.
Table 14
regimes conditional risk premia
STATE
lo20-hi20
lo20-tbond
hi20-tbond
1
0.004247
0.010377
0.006130
2
-0.002510
-0.023647
-0.021137
Table 14 shows the regimes conditional risk premia calculated as the state
conditional differences between the average realized returns on two different asset
classes.
As it can be easily seen all the asset classes present big difference in the risk
premia among regimes, in fact all the risk premia are positive in state 1 and
negative in state 2. State 1 and state 2 identify a size effect in stock returns. In
state 1 the average realized return of small stocks exceeds that of large stocks of
about 0.4247% per month , while this get reversed in state 2 where the average
realized return of large stocks exceeds that of small stocks of about 0.251% per
month. It can be seen that the average realized of small stocks and large stocks
exceed that of bonds of about 1.0377% and 0.613% respectively in state 1, while
in state 2 the situation get reversed with bond returns now exceeding small stocks
and large stocks average realized returns of about -2.364% and -2.1137%
respectively. The entire risk premia structure is reversed in state 2, this means that
97
for each pair of asset classes and for each regime, there is a different asset class
that beats the other in term of higher average realized return, i.e. in state 1 small
stocks beat large stocks and bonds and similarly large stocks beat bonds whereas
in state 2 large stock beats small stocks and similarly bonds beat small stocks and
large stocks. The considerations above brings to the conclusion that in regime 1 it
seems reasonable to invest in equity, whose average realized return outperforms
that of bonds, while in regime 2 holding bonds seems to be more profitable. The
presence of a size effect brings to the further conclusion that, regarding the stock
market, a strategy that invests in small stocks in regime 1 and large stocks in
regime 2 seems to be appropriate and profitable.
In this section of the subparagraph some considerations about the liner correlation
between returns are provided.
Table 15
regimes conditional linear correlation
STATE
lo20-hi20
lo20-tbond
hi20-tbond
1
0.607525
0.027858
0.177233
2
0.811323
0.268023
0.296864
unconditional
0.718089
0.121093
0.207156
As shown in Table 15 correlations between returns appear to vary substantially
across regimes. The realized correlation between large (hi20) and small (lo20)
stocks'
returns varies from a high of 0.811323 in state 2 to a low of 0.607525 in
state1. The correlation between returns on small stocks and bonds varies from a
high of 0.268023 in state 2 to a low of 0.027858 in state 1 while the correlation
between returns on large stocks and bonds goes from 0.177233 in state 1 to
0.296864 in state 2. The large cross-regime differences in the realized linear
correlation coefficients occurs between small stocks and bonds, equal to
0.240164, while the smaller one occurs between large stocks and bonds, equal to
0.119631. Small stock returns have a larger cross-regime difference in realized
correlation with bonds returns (0.240164) than large stock returns (0.203798)
while large stock returns have it larger with small stock returns (0.203798) than
with bonds returns (0.119631). As it can be seen from Table 15 the unconditional
98
correlations assume values between the two states conditional correlation values,
this is consistent with the evidence of regime time-varying correlations found in
monthly equity returns by Ang and Chen (2002). The ability of this model to
identify a correlation close to 0 between small stocks and bonds returns in state 1 -
which the linear model is unable to do
is a sign of the potential value of adopting
a regime switching model in the portfolio construction context. As evidenced by
recent works by Andersen, Bollerslev, Diebold, and Vega (2004) stock and bond
returns move together more than how measured by a single state linear
correlation; the reason is that the correlation switches sign across different
regimes and may appear spuriously small when averaged across states. The
results on realized linear correlation coefficients between returns, conditional to my
model states estimates, are consistent with the existing regime switching literature,
in fact many studies (e.g., Hong, Tu and Zhou 2007) on asymmetric co-
movements between asset returns and market indices suggest that stocks are
more likely to move with the market when the market goes down than when it goes
up, in fact conditional to the states estimates of my model, all pair-wise realized
linear correlation coefficients between returns are structurally higher in state 2
(bear state) than in state 1 (bull state), for instance the average correlation in state
1 is 0.270872 vs. an average of 0.458737 in state 2. As suggested by Jun Tu
(2010) Regime-dependent co-movements could be important for examining the
economic value of regime switching for portfolio decisions since, thought standard
investment theory advises portfolio diversification under pricing model uncertainty,
the value of this advice might be questionable if all stocks tend to fall a lot as the
market falls in a bear regime.
3.8.2 Regimes and predictability from the dividend yield
Table 16
regime conditional liner correlations between stocks and
bonds returns, and dividend yield
STATE
lo20-div_y
hi20-div_y
tbond-div_y
1
0.030009
-0.033811
-0.013665
2
0.002636
0.110058
-0.083438
unconditional
-0.011503
-0.015227
-0.011708
99
As suggested by Vo and Maurer (2013) the estimated regime switching model can
also model the asymmetry in the dividend yield's ability to hedge shocks in the
investment opportunity set. As shown in Table 16 each state is characterized by a
marked realized correlation structure. During expansion periods both large stock
returns and bonds returns tend to move in the opposite direction of the dividend
yield, whereas small stock returns offer an hedge against adverse movements of
the investment opportunity set since it is characterized to be, although very
weakly, negatively correlated with dividend yield. During recession periods both
small stock returns and large stock returns co-move with the dividend yield with
the latter one showing a considerably higher linear realized correlation coefficient
than the former one, conversely the bonds returns offer an hedge against stock
returns since it is characterized by a negative, although very weak, realized linear
correlation coefficient. Vo and Maurer (2013) also observed that an high
correlation between stock returns and the dividend yield implicates a weaker
hedge from stocks holdings against adverse movements of the investment
opportunity set due to the fact that an unexpected decline in the dividend yield is
accompanied by unexpected higher realized stock returns. At the same time
however, mean reversion in the dividend yield process lower the investor's
expectation about future stock returns. Thus as investment opportunity turn sour,
holding a greater amount of stocks represents a more valuable hedge the lower
the correlation between stocks and the dividend yield.
Table 17
regime conditional correlations between stocks and
bonds returns, and changes in dividend yield
STATE
lo20-
%
div_y
hi20-
%div_y
tbond-
%div_y
1
0.591008
0.507782
0.156789
2
0.693650
0.764479
0.241957
unconditional
0.654009
0.649582
0.161971
I also studied the state conditional realized linear correlation coefficients between
stock and bond returns, and changes in dividend yield. The results are shown in
Table 17. As it can be seen all the asset classes returns show in both regimes
100
positive realized linear correlation coefficients between them and the changes in
dividend yield. A possible explanation for this evidence may be provided by the
fact that, as shown in Table 13, in state 1 both the changes in dividend yield and
stock returns have positive average values whereas in state 2 they are
characterized by negative average values, this reasonably lead to positive realized
linear correlation coefficients. The correlation coefficient between changes in
dividend yield and bonds returns in state 2 is more challenging to explain and
interpret, at first glance, based on Table 13 a negative realized correlation
coefficient might be guessed, however the correlation coefficient between changes
in dividend yield and bonds returns is weak and its magnitude is not comparable
with the correlation coefficients between changes in dividend yield and stock
returns. All the considerations regarding the realized linear correlation coefficients
provided so far are also supported by the following scatter diagrams.
Figure 37
scatter of the dividend yield changes vs. small, large and bonds
returns
Figure 38
scatter of the dividend yield vs. small, large and bonds returns
101
My results are also similar to what discovered by Henkel et al. (2011), they find
predictability to be stronger in recessions; I found out that 9 out of 16 state 2
VAR(1) coefficients are larger than the state 1 counterparts in absolute terms, and
that 7 out of 16 state 2 VAR(1) coefficients, against only 4 out of 16 in state 1, are
significantly different from zero. The estimated state dependent VAR(1) matrix
shows significant time variations in the ability of the dividend yield to predict future
stock returns and in the prediction itself, it can be seen that higher dividend yields
forecast higher stock returns in state 1, but negative ones in state 2. The
autoregressive coefficients suggest significant predictive power of bonds and
dividend yield in both states 1 and 2, while the autoregressive coefficients of large
stock returns fail to be significant in both states and the autoregressive coefficients
of small stock returns indicate significant predictability only in state 1. Lagged
bonds returns have the strongest predictive power in state 1, while in state 2,
characterized by less predictability of returns, there is statistical evidence that only
large stocks and bonds lagged returns affect respectively small stocks and bonds
returns.
The different properties of the dividend yield across the two state affect the
conditional distribution of the other asset classes returns, even though in both
states the estimates suggests a low predictability on the dividend yield, in fact the
regression coefficients of any expected return time series, except for the div_y on
102
its own lag, are not statistically significant. The estimate of the regime dependent
VAR(1) matrix suggests that a higher dividend yield in state 1 forecasts higher
lo20 and hi20 returns, and lower tbond returns while in state 2 it forecasts lower
lo20 and hi20 returns, and higher tbond returns. The dividend yield is highly
persistent in fact its autoregressive coefficient estimate is 0.9875 in state 1 and
1.0234 in state 2. Given the high persistence in the dividend yield time series, a
single lag is required for the model.
3.8.3 A comparative analysis between the estimated regimes and the
NBER USA recession indicator
In the following subparagraph a comparative analysis between the occurrence of
recession periods estimated by my model and the NBER recession periods is
illustrated. The analysis has been conducted using data provided by the National
Bureau of Economic Research in addition to the data provided by the estimation of
my model. The NBER based Recession Indicators for the United States from the
Peak through the Period preceding the Trough is an interpretation of US Business
Cycle Expansions and Contractions data provided by The National Bureau of
Economic
Research (NBER) and realized by the Federal Reserve Bank of St.
Louis, the indicator essentially classifies each month either as a recession (value
equal to 1) or an expansion period (value equal to 0). I also used another time
series provided by the Federal Reserve Bank of St. Louis namely The Smoothed
U.S. Recession Probabilities that is obtained from a dynamic-factor Markov
switching model applied to four monthly coincident variables: non-farm payroll
employment, the index of industrial production, real personal income excluding
transfer payments, and real manufacturing and trade sales. This model is
developed by Piger, Jeremy Max, Chauvet and Marcelle
13
. This time series
essentially indicates for each month the probability that the United States economy
is experiencing a recession. Starting from these two time series provide by the
Federal Reserve Bank of St. Louis I constructed two time series that represent the
NBER expansion indicator and a NBER smoothed U.S expansion probabilities; the
13
This model was originally developed in Chauvet, M., An Economic Characterization of Business
Cycle Dynamics with Factor Structure and Regime Switching, International Economic Review,
1998, 39, 969-996.
103
former one is the complement to 1 of the NBER recession indicator assuming
value 1 during expansion period and 0 during recession periods, the latter one is
the complement to 1 of the NBER smoother U.S. recession probabilities.
Figure 39
estimated states from the multivariate markow switching model
Figure 40
NBER based Recession Indicators for the United States from the
Peak through the Period preceding the Trough (USRECP)
Figure 41 estimated smoothed states probabilities
104
Figure 42
NBER Smoothed U.S. Recession Probabilities
Figure 43 asset classes returns, dividend yield, dividend yield changes and
state 2 probabilities
105
Figure 44
states frequency occurrence by year
The figure above represents how often the estimated model stays in a regime on
average in a year. Each annual value relative to a certain state is the ratio
106
between the number of months in which the model has been in a that state and
the number of months in a year, i.e. 12.
Table 18
frequencies of common NBER and MODEL's recession and
expansion periods
Model recession periods
Model expansion periods
NBER recession
periods
0.518519
0.089325
NBER expansion
periods
0.481481
0.910675
As evidenced in Table 18, I found that approximately 52% of the periods classified
as recession (state 2) by my model occur in an NBER recession period,
conversely approximately 91% of the periods classified as expansion (state 1)
occur in an NBER expansion period.
Table 19
regression of estimated state 1 smoothed probabilities on NBER
recession and expansion indicator
NBER recession indicator
NBER expansion indicator
smoothed state 1
probabilities
-0.484702
0.484702
smoothed state 2
probabilities
0.484702
-0.484702
As shown in Table 19 correlation between estimated smoothed state probabilities
and NBER recession dates are -0.484702 for state 1 and 0.484702 for state 2; as
observed by Guidolin and Timmermann (2005) since the state probabilities sum to
one, by construction, if some correlations are positive, others must be negative.
This evidences bring other support to the fundamental idea that state 1 occurs
around official recession periods indicated by the NBER recession indicator.
Figure 45
regression of estimated state 1 smoothed probabilities on NBER
recession indicator
107
Figure 46
regression of estimated state 2 smoothed probabilities on NBER
recession indicator
The regression analysis of estimated state probabilities on the NBER recession
indicator, illustrated in Figure 45 and Figure 46, leads to the conclusion that in the
NBER recession periods the estimated state 1 smooth probabilities assumes large
values while state 2 smooth probabilities assumes small values; the opposite
situation occurs in the NBER expansion periods. In fact the regression analysis of
108
states probabilities on the NBER indicator shows a negative coefficient for state 1
and a positive coefficient for state 2.
Figure 47
regression of estimated 1 month lagged state 1 smoothed
probabilities on NBER recession indicator
Figure 48
regression of estimated 1 month lagged state 2 smoothed
probabilities on NBER recession indicator
109
Figure 49
regression of estimated 3 months lagged state 1 smoothed
probabilities on NBER recession indicator
Figure 50
regression of estimated 3 months lagged state 2 smoothed
probabilities on NBER recession indicator
Figure 51
regression of estimated 6 months lagged state 1 smoothed
probabilities on NBER recession indicator
110
Figure 52
regression of estimated 6 months lagged state 2 smoothed
probabilities on NBER recession indicator
As suggested by Guidolin and Timmermann (2005) it may be argued that the state
probabilities estimated from financial returns should lead economic recession
months, for this reason I also conducted a regression analysis of state 1 and state
2 probabilities lagged 1, 3 and 6 months on the NBER recession indicator. The
results are shown in Figure 47, 48, 49, 50, 51 and 52. It can be clearly concluded
that both the state 1 and state 2 regression coefficients show a decrease in
111
absolute terms when the lag become larger, therefore the larger the lag in the
state probabilities time series the weaker the positive relation between the lagged
recession probabilities and the presence of a recession period in the NBER
recession indicator, this means that there is no evidence of an improved capacity
of the estimated lagged recession probabilities, compared to the not lagged one,
to forecast the presence of a recession in the NBER recession indicator.
Figure 53
regression of estimated state 1 (expansion) smoothed probabilities
on NBER expansion probabilities
Figure 54
regression of estimated state 1 (expansion) smoothed probabilities
on NBER recession probabilities
112
Figure 55
regression of estimated state 2 (recession) smoothed probabilities on
NBER expansion probabilities
Figure 56
regression of estimated state 2 (recession) smoothed probabilities on
NBER recession probabilities
113
I conducted an additional regression analysis of both the state 1 and state 2
probabilities on the NBER recession and NBER expansion probabilities, the
results are shown in Figure 53, 54, 55 and 56. The NBER recession probabilities is
represented by the Smoothed U.S. Recession Probabilities supplied by the
American National Bureau of Economic Research with monthly frequency from
June 1967. This analysis further supports the fundamental idea that the estimated
state 2 represents recession periods while state 1 represents expansion period.
The regression analysis of the estimated state 1 probabilities on the NBER
expansion probabilities and NBER recession probabilities respectively shows a
large and positive and a large and negative coefficient. The antithetical situation
occurs with the regression analysis of the estimated state 2 probabilities.
Table 20
correlations between estimated smoothed state probabilities and
NBER recession and expansion probabilities
NBER recession probabilities
NBER expansion probabilities
state 1 smoothed
probabilities
-0.533960
0.533960
state 2 smoothed
probabilities
0.533960
-0.533960
114
The same conclusion is also supported by the correlations between the estimated
smoothed state probabilities and the NBER recession and expansion probabilities
shown in Table 20, it can be seen that there is evidence of a positive relation
between the state 2 (bear market, recession) estimated smoothed probabilities
and the NBER recession probabilities and a negative relation between the state 1
(bull market, expansion) estimated smoothed probabilities and the NBER
recession probabilities. All the evidences presented so far suggest that the regime
switching estimate of my model appears to be related to the underlying economic
fundamentals and business cycle to some extent. As suggested by Jun Tu (2010)
a possible explanation of the divergence between the regime switching and the
underlying business cycle may come from the fact that stock markets also react to
sectoral or shorter-lived contractions in the economy that are not designated as
recessions by NBER.
3.8.4 A dynamic correlation analysis of asset classes returns
In this subparagraph I am going to illustrate the findings relative to the dynamic
correlation analysis I have realized. I compute and plot the dynamics of the
conditional correlations implied by the model using only real time information (i.e.,
using filtered and not smoothed probabilities). In computing dynamic correlations it
has been necessary to adopt some adjustment as suggested by Guidolin in
"
Modelling, Estimating and Forecasting Financial Data under Regime (Markov)
Switching
", in order
to take into account the effects of both variances and
covariances of the joint presence of switches in expected excess returns.
The covariance matrices of the regime switching restricted MSVAR(2,1) model I
have estimated are characterized by the fact that the covariances in both regimes
are restricted to be zero; as a consequence the only source of correlation between
the asset classes time series in the system is due to the presence of a unique
Markov switching dynamic, in common for all the asset returns, which drives the
moments of three asset classes time series. The additional source of correlation
due to the presence of a common Markov switching dynamic comes from the fact
that the Markov state moves the means in the same direction, except or the large
115
stocks, and at the same time, which makes the standard correlation an imperfect
measure of comovements.
The dynamic correlation formula is here illustrated:
(38)
,
=
,1
1t
+
,2
2t
+
1t
2t
(E(
+1|
,
t
;
)-E())(E(
+1|
,
t
;
)-E())
((
,1
2
1t
+
,2
2
2t
+
1t
2t
(E(
+1|
,
t
;
)-E())
2
)(
,1
2
1t
+
,2
2
2t
+
1t
2t
(E(
+1|
,
t
;
)-E())
2
))
0.5
where
and
are two asset classes time series,
it
is the filter probability of
regime i at time t,
,
is covariance between
and
in state i,
,
2
is the variance
of
in state i,
E() = E(
t+1
|) =
1
,
=1
+ (1 -
1
)
,
=2
is the unconditional
mean of
(formula (32)), while
E(
+1
|
,
t
; )
is the conditional mean of
based
on the information available at time t. The information available at time t consists
of
,
a vector that contains the values of
,
and
at time t, and the filtered
probabilities
1t
and
2t
. The conditional mean formula is here illustrated:
(39)
E(
+1
|
,
t
; ) = (
1t
p
11
+
2t
p
12
)E(
+1
|S
t
= 1,
,
t
; ) + (
1t
p
21
+
2t
p
22
)E(
+1
|S
t
= 2,
,
t
; )
where (assuming that
is modeled by a MSVAR(2,1) and that there are another
asset,
, and a predictor
)
(40)
E(
+1
|S
t
= ,
,
t
; ) =
S
t
=
+
S
t
=
,
t
+
S
t
=
,
t
+
S
t
=
,
t
The conditional mean term also represents, as illustrated at the beginning of the
paragraph, the mean of the normal distribution
(41)
(
|
t-1
; ) =
p
2
=1
2
=1
it-1
which represents the conditional density of the t-th observation. The above
correlation coefficient is calculated for each period, as a result a time series of
length 540 is obtained for the dynamic correlation between any two asset classes
time series.
116
The numerator in the ratio calculation is the filtered covariance at time t adjusted to
take into account the effects of regime switches while the terms in the denominator
are the filtered standard deviation at time t adjusted to take into account the effects
of regime switches. Similarly to the conditional mean, the conditional standard
deviation is here illustrated:
(42)
(
+
|
t
; ) =
,+1|
t
;
= (
1t
p
11
+
2t
p
12
)
,S
t
=1
+ (
1t
p
21
+
2t
p
22
)
,S
t
=2
Figure 57
model implied dynamic correlation between small stocks and bonds
Figure 58
model implied dynamic correlation between small stocks and large
stocks
117
Figure 59
model implied dynamic correlation between large stocks and bonds
Figure 60
dynamic correlation between asset classes returns and dividend
yield
118
Table 21
unconditional and state conditional average dynamic correlation
coefficients
STATE
lo20 vs.
tbond
lo20 vs.
hi20
hi20 vs.
tbond
lo20 vs.
div_y
hi20 vs.
div_y
tbond vs.
div_y
state 1
-0.0007
0.0004
0.0024
0.0070
-0.0013
-0.0062
state 2
-0.0040
0.0006
0.0035
0.0109
-0.0109
-0.0234
unconditional -0.0012
0.0005
0.0026
0.0076
-0.0027
-0.0088
Table 21 shows the unconditional and state conditional average dynamic
correlation coefficients. The first two records of the table contain the state
conditional average dynamic correlation coefficients, each of them calculated as
the average values of the corresponding dynamic correlation time series,
conditionally on the state value contained in the first column ; the third record of
the table shows the unconditional average dynamic correlation coefficient
calculated as the average value of the corresponding whole dynamic correlation
time series. Overall, it can be seen from Table 21, and more clearly from Figure
93, that the average dynamic correlation coefficients are, in absolute term,
substantially larger in state 2 than in state 1, this finding confirms the already
discussed tendency of the return to comove greatly in the recession regime (state
119
2). For each pair of time series the unconditional average dynamic correlation
coefficient assumes a value between the two corresponding state conditional
average values. From a comparison between Table 21 and Table 16 it emerges
that 5 out of 6 correlation coefficients between the dividend yield and an asset
class share the same arithmetic sign, thus the two different methodologies used to
produce the results in the two tables bring to the same conclusion; the exception is
represented by the correlation coefficient between the large stocks and the
dividend yield in state 2. As it can be seen from Table 11, the dividend yield and
the small stocks state conditional means are larger in state 2 than in state 1, while
small stocks and bonds state conditional means are larger in state 2 than in state
1. In this context, during a recession period (state 2) the dividend yield is above its
unconditional mean as well as the small stock returns, thus a positive
comovement, as illustrated in Table 21, occurs; at the same time the large stocks
perform worse in state 2 than in state 1, thus the dividend yield and the large stock
returns move to different direction; lastly the comovement in the opposite direction
of bonds and dividend yield can be explained by the significant positive
comovement between large stocks and bonds that make bonds returns grow,
together with large stock returns, when the economy is experiencing an expansion
period and the dividend yield is low. The dynamic just explained is partly originated
by the time-variations in the dividend yield, which may induce a large hedging
demand.
3.8.5 Regime shifts in the asset classes returns means and volatilities
The following charts show the asset classes returns and their relative time varying
conditional mean and time varying conditional standard deviations estimated by
the model.
Figure 61
asset classes returns and estimated time varying conditional means
120
Figure 62
estimated time varying conditional standard deviations
Figure 63
estimated smoothed state probabilities
121
The two plots above show that the three asset classes expected excess returns,
their conditional means and their conditional standard deviations are to some
extent synchronized across time series, reflecting the shapes of the evolution of
smoothed probabilities.
3.9 A single regime VAR(1) model estimates
In order to evaluate the potential benefit of a multiple regimes model against a
more simple single regime model, a first order vector autoregressive model
VAR(1) has been estimated. A comparison of the two model performances, both in
the in-sample and out-of-sample framework, has been conducted.
The Markov Switching Vector Autoregressive model MSVAR is a general class of
models which nests the standard VAR model but additionally accounts for
nonlinear regime shifts. As a consequence the VAR(1) model is a special case of
the MSVAR(2,1) in which the intercepts terms are, the VAR(1) autoregressive
coefficients and the covariance matrix are not regime-dependent, and then the
model is homoskedastic. This model configuration shows a static and
simultaneous contagious effect through the off-diagonal elements of the variance
and covariance matrix, a static and linear one through the VAR coefficients. The
VAR(1) model is based on the following expression:
(43)
= +
-
+
where
= (
)
is a (3 + 1) x 1 vector that contains lo20, hi20 and tbond
returns and an external predictor
which is represented by the dividend yield
122
div_y;
= [
1
...
4
]
is the intercept vector for
,
is a (3 + 1) x (3 + 1) matrix
of autoregressive and regression coefficients while
= [
1t
...
4t
]~(, )
.
The model has been estimated using the in-sample data from January 1965 to
December 2010; Table 22 contains the parameter estimates.
Table 22
single regime VAR(1) parameter estimates
VAR(1)
lo20
hi20
tbond
div_y
1. Intercept term
-0.0040
(0.0075)
-0.0026
(0.0052)
0.0005
(0.0023)
0.0007
(0.0002)
2. VAR(1) Matrix
lo20
0.1665
(0.0600)
0.1234
(0.0905)
0.1428
(0.1376)
0.2910
(0.2244)
hi20
0.0434
(0.0412)
-0.0225
(0.0622)
0.2631
(0.0945)
0.1780
(0.1542)
tbond
-0.0509
(0.0181)
0.0096
(0.0272)
0.3235
(0.0414)
0.0280
(0.0675)
div_y
-0.0007
(0.0016)
-0.0148
(0.0024)
-0.0098
(0.0036)
0.9823
(0.0058)
3. Covariance Matrix
lo20
0.003959
hi20
0.001974
0.001869
tbond
0.000167
0.000155
0.000358
div_y
-0.000037
-0.000030
-0.000005
0.000003
The stability and invertibility of the model have been tested; given that all
eigenvalues of the associated lag operators have modulus less than 1 it can be
said that the model is stable and invertible.
123
CHAPTER 4 - ASSET ALLOCATION
4.1 Unconditional and state conditional asset classes returns
distributions and efficient frontiers based on a MSVAR(2,1)
model
The following charts show the asset classes returns and their relative time varying
conditional means and time varying conditional standard deviations estimated by
the model. The unconditional distributions are mixtures of the two state conditional
normal distributions in which the weighting factors are equal to the corresponding
state ergodic probabilities. As it can be seen the mixture of the two normals
generates a distribution characterized by skewness and kurtosis. Given that the
investor is not interested in investing in the dividend yield but only in the small
stocks, large stocks and bonds, the expected unconditional mean for each t
consists of the first three elements of the following vector:
(44)
14
E(
+
|) =
1
+ (1 -
1
)
and the unconditional covariance matrix adjusted for the regime structure is equal
to the (3 x 3) left-upper part of the following matrix:
(45)
15
Var(
+
|) =
1
+ (1 -
1
)
+
1
(1 -
1
)(
-
)(
-
)
T
where
(46)
= E(
|S
t
= 1; ) = (
-
,
)
-1
and
(47)
= E(
|S
t
= 2; ) = (
-
,
)
-
represent the state conditional asset classes means while
1
and
2
= (1 -
1
)
are the state ergodic probabilities. The following figures plot for each asset class
the probability density function corresponding to the mixture of two normals that
14,15
Massimo Guidolin "Modelling, Estimating and Forecasting Financial Data under Regime
(Markov) Switching"
download at
"http://didattica.unibocconi.it/mypage/dwload.php?nomefile=Lecture_7_
-
_Markov_Switching_Models20130520235704.pdf"
124
draws from state 1 density function with probability
1
and state 2 density function
with probability
2
.
Figure 64
small stocks state conditional and unconditional estimated density
function
Figure 65
large stocks state conditional and unconditional estimated density
function
125
Figure 66
bonds state conditional and unconditional estimated density function
As pointed out by Ang and Timmermann (2012) a natural question might be which
portfolios should be optimally held in each regime, and whether there is an optimal
portfolio to hedge against the risk of regime changes. The first paper to examine
asset allocation with regime changes was Ang and Bekaert (2002a), the paper
126
examines portfolio choice for a small number of countries. They exploited the
ability of the regime switching model to capture higher correlations during market
downturns and examine the question of whether such higher correlations during
bear markets negate the benefits of international diversification. They find there
are still large benefits of international diversification. The costs of ignoring the
regimes is very large when a risk-free asset can be held; investors need to be
compensated approximately 2 to 3 cents per dollar of initial wealth to not take into
account regime changes. The model I have estimated is characterized by two
regimes: the high volatility regime has the lowest Sharpe ratio and its mean-
standard deviation frontier is the red one while the low volatility regime has the
highest Sharpe ratio and its mean-standard deviation frontier is the green one. The
unconditional mean-standard deviation frontier averages across the two mean-
standard deviation frontiers and is the blue one and it has been constructed using
the asset classes unconditional moments. The state 1 conditional efficient frontier
is shown as the green line while the state 2 conditional efficient frontier is shown
as the red line. As pointed out by Ang and Timmermann (2012) an investor who
ignores regimes sits on the unconditional frontier, thus an investor can do better by
holding a higher Sharpe ratio portfolio when the low volatility regime prevails.
Conversely, when the bad regime occurs, the investor who ignores regimes holds
too high a risky asset weight. She would have been better off shifting into the risk-
free asset when the bear regime hits. Clarke and de Silva 1998 stated that the
presence of two regimes and two frontiers means that the regime switching
investment opportunity set dominates the investment opportunity set offered by
one frontier. In other words as affirmed by Ang e Bekaert (2002) portfolio
allocations based on regime switching estimates have the potential to out-perform
because they set up a defensive portfolio in the bear regime that hedges against
high correlations and low returns.
Figure 67
regime-dependent mean-standard deviation efficient frontiers
127
Figure 67 shows the unconditional and the regime-dependent mean-standard
deviation efficient frontiers of portfolios that consist of Small-Cap Equities, Large-
Cap Equities and bonds. Then means and variances have been annualized. The
intercepts of the tangency lines are equal to zero because the vertical axis unit of
the Figure 67 is expressed as excess returns over risk-free rate, alike the asset
classes returns, consequently the excess return of the risk-free rate is equal to
zero. Again the efficient frontiers have been estimated from the state conditional
and unconditional asset classes density functions which are characterizes by an
expected unconditional mean estimates vector equal to the first three elements of
the following vector:
(48)
16
E(
+
|) =
1
+ (1 -
1
)
and state conditional mean estimates vectors equal to:
(49)
17
= E(
|S
t
= 1; ) = (I
4
-
,
)
-1
and
(50)
18
= E(
|S
t
= 2; ) = (I
4
-
,
)
-1
16, 17, 18
Massimo Guidolin
"Modelling, Estimating and Forecasting Financial Data under Regime
(Markov) Switching"
download at
"http://didattica.unibocconi.it/mypage/dwload.php?nomefile=Lecture_7_
-
_Markov_Switching_Models20130520235704.pdf"
128
while the unconditional covariance matrix adjusted for the regime structure is
equal to the (3 x 3) left-upper part of the following matrix:
(51)
19
Var(
+
|) =
1
+ (1 -
1
)
+
1
(1 -
1
)(
-
)(
-
)
T
and state conditional covariance matrices equal to
and
.
4.2 In-sample asset allocation exercise
In this paragraph I am going to illustrate the in-sample portfolio construction
processes that have been adopted and both the realized and the expected asset
allocation and portfolio statistics, using firstly the MSVAR(2,1) and secondly the
VAR(1) to model the first two moments of the asset classes returns distributions.
4.2.1 In-sample asset allocation exercise based on a MSVAR(2,1)
model
The regime switching, state conditional, efficient frontiers illustrated in Figure 67
are the actual efficient frontiers in which the investor sits only if she knows which
regime applies at each time. Given that the model I have estimated is
characterized as having a state variable driven by a hidden Markov process, the
regimes cannot be identified in real time. Then, the underlying regime is treated as
a latent variable that is unobserved and an agent can learn about regimes
employing a filtering algorithm. Since the return distribution is very different in the
bull and bear state, the state probability perceived by investors is a key
determinant of their asset holdings. The filtering algorithm uses a Bayesian rule to
update beliefs according to how likely new observations are drawn from different
regimes, which are weighted by prior beliefs concerning the previous regimes. The
higher the persistence of the regimes, the greater the weight on past data.
In Figure 67, the risk-return trade-offs of each single regime are known because
the first two moments of the asset classes returns distributions have been
19
Massimo Guidolin
"Modelling, Estimating and Forecasting Financial Data under Regime (Markov)
Switching"
download at "http://didattica.unibocconi.it/mypage/dwload.php?nomefile=Lecture_7_
-
_Markov_Switching_Models20130520235704.pdf"
129
estimated by the model. However, as pointed out by Ang and Timmermann
(2012), an investor who knows the parameters can infer which regime prevails at
each time. Then the updating of the probability of the current regime, given all
information up to time t, can be computed using methods similar to a learning
problem. The investor infers the regime probability from the current information, in
particular she computes the probability that a certain regime prevails at the current
time, which can be easily computed as a by-product of the estimation of the
regimes switching model. This regime probabilities are called filtered probabilities.
A filtered probability is the best inference of the current state, based on real time
information. Given these probabilities an investor is able to build the filtered
dynamic variance-covariance matrix and the filtered dynamic conditional mean
vector.
Given that the investor is not interested in invests in the dividend yield but only in
the small stocks, large stocks and bonds, the filtered dynamic conditional mean for
each t is a (3 x 1) vector equal to the first three elements of the following vector:
(52)
E(
|
-
; ;
t
) =
t|t-1
=
1t
E(
|
-
; S
t
= 1; ;
t
) +
2t
E(
|
-
; S
t
= 2; ;
t
)
and
(53)
E(
|
-
; S
t
= ; ;
t-1
) =
t|t-1;
=
=
=
+
,
=
-
for each t is a (4 x 1) vector that represents the lo20, hi20, tbond and div_y
expected returns at time t conditional on time t-1 returns, time t state (
S
t
= )
and
all the past information up on time t-1. Here
=
is the vector of intercept terms of
in state
S
t
=
,
,
=
is the matrix of autoregressive and regression coefficients
associated with lag 1 in state
S
t
=
and
t
is all the past information up to time t
the except the returns at time t. In computing the vector of the expected returns at
time t=1, conditional on time t-1=0, I assumed that
is a vector of zeros
20
and
that
10
=
20
= 0.5
.
20
Massimo Guidolin
"Modelling, Estimating
and Forecasting Financial Data under Regime (Markov)
Switching"
download at "http://didattica.unibocconi.it/mypage/dwload.php?nomefile=Lecture_7_
-
_Markov_Switching_Models20130520235704.pdf"
130
The filtered dynamic variance-covariance matrix for each t is equal to the (3 x 3)
left-upper part of the following matrix::
(54)
(
|
-
; ;
t
) =
1t
1
+
2t
2
+
1t
2t
(
t|t-1;
=
-
t|t-1;
=
)(
t|t-1;
=
-
t|t-1;
=
)
again here is assumed that
10
=
20
= 0.5
.
Given
that
1
1t
and
2
2t
then
E(
|
-
; ;
t
) E(
+
|)
.
I then used the filtered dynamic variance-covariance matrix and the filtered dynamic
conditional mean vector to build five in-sample recursive optimal portfolios.
Figure 68
lo20 and its filtered dynamic conditional mean
Figure 69
hi20 and its filtered dynamic conditional mean
131
Figure 70
tbond and its filtered dynamic conditional mean
I estimated two in-sample dynamic recursive efficient portfolios that maximize the
Sharpe among portfolios on the efficient frontier; in the first in-sample dynamic
recursive portfolio the budget constraint is opened up to permit between 0% and
100% in the riskless asset while the second one requires fully-invested portfolios
whose weights must sum to 1; in addition short selling, thus negative asset class
weights are not allowed. Specifically, a portfolio that maximizes the Sharpe ratio is
132
also the tangency portfolio on the efficient frontier from the mutual fund theorem;
such portfolios are called tangency portfolios since the tangent line from the
riskless asset rate to the efficient frontier touches the efficient frontier at portfolios
that maximize the Sharpe ratio. The other three in-sample dynamic recursive
portfolios have been chosen as those who maximize the investor utility function
with three different risk aversion coefficient subject to non negative weights and
opened upper budget constraint. One of the factors to consider when selecting the
optimal portfolio
for a particular investor is degree of risk aversion. This level of
aversion to risk can be characterized by defining the investor's indifference
curve.
This curve consists of the family of risk/return pairs defining the trade-off
between the expected return and the risk. It establishes the increment in return
that a particular investor requires to make an increment in risk worthwhile. Typical
risk aversion coefficients range from 2.0 through 4.0, with the higher number
representing lesser tolerance to risk. I choose values of risk aversion coefficients
equal t o 1, 3 and 5. I then computed the optimal risky portfolios by generating the
efficient frontier from the asset data and then finding the optimal risky portfolio and
compute the optimal allocation of funds between the risky portfolio and the riskless
asset based on the risk-free rate, the borrowing rate, and the investor's degree of
risk aversion. The actual proportion assigned to each of these two investments
(the risky portfolio and the riskless asset) is determined by the degree of risk
aversion characterizing the investor. If the sum of the computed asset classes
weights exceeds 100%, implying that the risk tolerance specified allows borrowing
money to invest in the risky portfolio, and that no money is invested in the riskless
asset, as a result borrowed capital is added to the original capital available for
investment. Tobin's mutual fund theorem (Tobin 1958) says that the portfolio
allocation problem can be viewed as a decision to allocate between a riskless
asset and a risky portfolio. In the mean-variance framework an efficient portfolio on
the efficient frontier serves as the risky portfolio such that any allocation between
the riskless asset and this portfolio dominates all other portfolios on the efficient
frontier. This portfolio is called a tangency portfolio because it is located at the
point on the efficient frontier where a tangent line that originates at the riskless
asset touches the efficient frontier. The optimal choice for an investor is the point
133
of tangency of the highest indifference curve to the Capital Allocation Line CAL, it
follows that the slope of the indifference curve is equal to the slope of the CAL.
= - 0.005
The risk aversion coefficient is a number proportionate to the amount of risk
aversion of the investor and is usually set to integer values less than 6, and 0.005
is a normalizing factor to reduce the size of the variance, while variance is the
square of the standard deviation, a measure of the volatility of the investment and
therefore its risk. This equation is normalized so that the result is a yield
percentage that can be compared to investment returns, which allows the utility
score to be directly compared to other investment returns. Here is assumed that
the riskless asset has variance of
0
and is completely uncorrelated with all other
assets. The set of all portfolios with the same utility score plots as a risk-
indifference curve. An investor will accept any portfolio with a utility score on her
risk-indifference curve as being equally acceptable. Where one of the curves
intersects the efficient frontier or the CAL at a single point is the portfolio that will
yield the best risk-return trade-off for the risk that the investor is willing to accept.
Again, here short selling and negative asset class weights are not allowed. I wrote
all the Matlab scripts that carry out the entire computational process.
There are a number of ways to implement a portfolio mean-variance optimization.
As already stated, I choose to implement an excess return framework, thus each
asset classes returns are expressed as excess returns over the risk-free rate; the
risk-free rate is known at each period and is equal to the 1-month US T-Bill
monthly returns. Hence, the risk free-rate varies over time as I implement the
allocation program. The tangency portfolio, the 100% risky portfolio (the one that
invests only in risky assets and not in cash), will hence move over time as well. I
also use a borrowing rate represented by the monthly Bank Prime Loan Rate
21
made available by the Board of Governors of the Federal Reserve System (US). It
21
Board of Governors of the Federal Reserve System (US), Bank Prime Loan Rate [MPRIME],
retrieved from FRED, Federal Reserve Bank of St. Louis
https://research.stlouisfed.org/fred2/series/MPRIME, February 12, 2016.
134
is the rate posted by a majority of top 25 (by assets in domestic offices) insured
U.S.-chartered commercial banks and it is one of the several base rates used by
banks to price short-term business loans. Given that I adopted an excess returns
framework also the borrowing rate has been reduced by the amount of the risk
free rate, in practice when some money are borrowed to be invested in the risky
portfolio then the portfolio excess returns over the risk free (lending) rate are
reduced by an amount equal to the excess of the borrowing rate over the risk free
(lending) rate; in the time slot analyzed by this study the borrowing rate is always
higher than the lending rate (risk free rate), as a consequence borrowing money is
always costly in terms of portfolio expected returns.
Figure 71
lending rate (risk free rate), borrowing rate (Prime loan rate) and
excess borrowing rate (excess Prime loan rate)
It is noticed that if the risk free lending and borrowing rates are equal, the optimum
risky portfolio is obtained by drawing a tangent to the portfolio frontier from the
level of risk free lending and borrowing rate, however when the two rates are
different the CAL is not unique, one starts form the lending rate and the one that
starts from the borrowing rate, in this study, is slightly kinked as the borrowing rate
is higher than the risk-free lending rate. If the risky fraction exceeds 1 (100%),
implying that the risk tolerance specified allows borrowing money to invest in the
risky portfolio, and that no money is invested in the risk-free asset. This borrowed
135
capital is added to the original capital available for investment. As a consequence
the efficient frontier of portfolios with borrowing and lending rate consists of a line
segment equal to the CAL that starts from the lending rate for portfolios
characterized by a risky fraction smaller than 1, a curve portion of the efficient
frontier with neither borrowing nor lending for those portfolios characterized by a
risky fraction equal to 1 and finally another line segment equal to the CAL that
starts from the borrowing rate for portfolios characterized by a risky fraction
greater than 1.
In the following section of the subparagraph the portfolio construction process
results are exposed.
136
Figure 72
maximum Sharpe ratio portfolios weights (1 = 100%) with opened
lower budget constraint (permit to invest in the riskless asset)
137
Figure 73
maximum Sharpe ratio portfolios weights (1 = 100%) with budget
constraint (not permit to invest in the riskless asset)
138
Figure 74
overall optimal portfolio weights (1 = 100%) with capital allocation
and risk aversion coefficient = 1
139
Figure 75
overall optimal portfolio weights (1 = 100%) with capital allocation
and risk aversion coefficient = 3
140
Figure 76
overall optimal portfolio weights (1 = 100%) with capital allocation
and risk aversion coefficient = 5
141
Figures 72 to 76 show the overall recursive dynamic weights of the five different
built portfolios. As it can be seen the average borrowing factor is very high for the
three portfolio built optimizing the expected utility using the capital allocation line
with lending and borrowing option, as it might have been largely expected the
greater the risk aversion coefficient, the smaller the leverage factor. Conversely,
the portfolios built with the first two methods, which is portfolios chosen among
those of the efficient frontiers in order to maximize the expected Sharpe ratio
(possibly with a share allocated to the riskless asset for the second portfolio
construction method), did not resort to any leverage.
Figure 77
risky optimal portfolio weights (1 = 100%) with capital allocation and
risk aversion coefficient = 1
Figure 78
risky optimal portfolio weights (1 = 100%) with capital allocation and
risk aversion coefficient = 3
142
Figure 79
risky optimal portfolio weights (1 = 100%) with capital allocation and
risk aversion coefficient = 5
Figures 77 to 79 show the optimal weights of the tangency portfolios between the
capital allocation line (either the lending capital allocation line, or the borrowing
capital allocation line) and the efficient frontier.
143
Figure 80
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset)
Figure 81
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with budget constraint (not permit to invest in the
riskless asset)
144
Figure 82
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1
145
Figure 83
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3
Figure 84
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5
146
It is interesting to notice from Figure 80 that is present a sort of substitution effect
between lo20 and hi20 optimal weights, when more than half of the portfolio is
allocated to the former one, and to a small extent between tbond and hi20 optimal
weights when tbond weights are larger than 50%. This relation between lo20 and
hi20 optimal weights might be explained by the fact that during bull markets the
growth in the lo20 allocation, as long as lo20 weights are low, it is coupled with a
reduction of the less profitable asset class, namely the tbond, however when lo20
weights represent a conspicuous share of the portfolios, a further growth of its
share in the portfolio is possible only at expense of the most profitable asset class,
namely the hi20. Similarly, the substitution effect between tbond and lo20 optimal
weights might be originated by volatility reasons since the tbond asset class is the
less volatile and the lo20 the most volatile. Volatility can also explain the marked
positive relation between hi20 and tbond optimal weights in Figures 82, 83 and 84;
weights on those two asset classes jointly grow as the investor flies from the most
volatile asset class, namely lo20, to the less volatile asset classes, indeed hi20
and tbond.
147
Figure 85
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset) against state 1 smoothed probability
Figure 86
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with budget constraint (not permit to invest in the
riskless asset) against state 1 smoothed probability
148
Figure 87
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 against state 1 smoothed
probability
Figure 88
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 against state 1 smoothed
probability
149
Figure 89
scatter of the optimal weights(1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 against state 1 smoothed
probability
Figures 85 and 86 shows how lo20 and hi20 optimal weights grow as a function of
the perceived probability of being in state 1 since their unconditional means in this
state are higher than the tbond ones, which, indeed, decrease as a function of
state 1 filtered probability. The same dynamic is less obvious in Figures 87, 88 and
89 because of the leverage factor. Nevertheless, the three figures highlight how
more often the investor resorts to borrowing money when he perceives a high
probability of being in state 1 rather than in state 2 (low state 1 filtered probability
corresponds to high state 2 filtered probability).
Figure 90
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset) against the dividend yield
150
Figure 91
scatter of the optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with budget constraint (not permit to invest in the
riskless asset) against the dividend yield
Figure 92
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 against the dividend yield
151
Figure 93
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 against the dividend yield
Figure 94
scatter of the optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 against the dividend yield
152
From Figures 90 to 94 it seems that lo20 optimal weights, on the average,
increase as a function of the dividend yield while hi20 and tbond optimal weights
decrease as a function of that; the relations just discussed however are not that
strong.
Figure 95
small stocks optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset) as a function of the filtered probability of being
in state 1, and of the dividend yield at the time the portfolio is chosen
153
Figure 96
large stocks optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset) as a function of the filtered probability of being
in state 1, and of the dividend yield at the time the portfolio is chosen
Figure 97
bonds optimal weights (1 = 100%) of the maximum Sharpe ratio
portfolios with opened lower budget constraint (permit to invest in the
154
riskless asset) as a function of the filtered probability of being in state
1, and of the dividend yield at the time the portfolio is chosen
Figure 98
riskless asset optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with opened lower budget constraint (permit to invest
in the riskless asset) as a function of the filtered probability of being
in state 1, and of the dividend yield at the time the portfolio is chosen
155
Figure 99
small stocks optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with budget constraint (not permit to invest in the
riskless asset) as a function of the filtered probability of being in state
1, and of the dividend yield at the time the portfolio is chosen
Figure 100 large stocks optimal weights (1 = 100%) of the maximum Sharpe
ratio portfolios with budget constraint (not permit to invest in the
riskless asset) as a function of the filtered probability of being in state
1, and of the dividend yield at the time the portfolio is chosen
156
Figure 101 bonds optimal weights (1 = 100%) of the maximum Sharpe ratio
portfolios with budget constraint (not permit to invest in the riskless
asset) as a function of the filtered probability of being in state 1, and
of the dividend yield at the time the portfolio is chosen
Figure 102 small stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
157
Figure 103 large stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 104 bonds optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 as a function of the filtered
158
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 105 riskless asset optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 1 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
159
Figure 106 small stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 107 large stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
160
Figure 108 bonds optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 109 riskless asset optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 3 as a function of the filtered
161
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 110 small stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
162
Figure 111 large stocks optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figure 112 bonds optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
163
Figure 113 riskless asset optimal weights (1 = 100%) of the portfolios with capital
allocation and risk aversion coefficient = 5 as a function of the filtered
probability of being in state 1, and of the dividend yield at the time the
portfolio is chosen
Figures 95 to 113 represent the optimal weights (1 = 100%) of the five different
portfolios as a function of the filtered probability of being in state 1, and of the
dividend yield at the time the portfolio is chosen. The graphs have been obtained
164
fitting 4th degrees bisquare robust polynomial surfaces (4 degrees for each input
dimensions) using the Matlab Curve Fitting Tool. As it can be seen for the first
portfolio, the one that maximizes the expected Sharpe ratio with opened lower
budget constraint, the lo20 optimal weights on average grow as a function of state
1 perceived probability especially for high dividend yield values; the same overall
trend can be spotted in the hi20 optimal weights surface even though its slope is
steeper and less sensitive to the dividend yields; conversely, because of its
hedging properties due to low volatility and a high state 2 mean return, the tbond is
characterized on average by decreasing optimal weights as a function of state 1
filtered probability. The conclusions that can be drawn from the observation of the
surfaces of the portfolio that maximizes the expected Sharpe ratio with budget
constraint are pretty similar to the ones just discussed. From the analysis of the
last three portfolios, those chosen to maximize the expected utility function given
three different risk aversion levels, emerges that on average the hi20 optimal
weights always increase as a function of the state 1 filtered probability due to its
high mean return in this state, in addition it can be seen that conditionally on the
dividend yield the hi20 optimal weights are higher for average dividend yield
values while are lower for dividend yield values smaller than its mean. The lo20
optimal weights seem to be always pretty constant, while the tbond optimal
weights decrease as function of the perceived state 1 probability because of its
attractiveness due to its higher mean and low volatility in state 2.
Table 22
in-sample average portfolio overall weights (1 = 100%)
avg. weights maximum Sharpe ratio portfolio with opened lower budget
constraint
unconditional
state 1
state 2
lo20
0.1944 (0.2522)
0.2011 (0.2499) 0.1566 (0.2633)
hi20
0.2045 (0.2209)
0.2294 (0.2276) 0.0634 (0.0919)
tbond
0.1979 (0.2284)
0.1706 (0.1935) 0.3528 (0.3296)
risk-free asset
0.4032 (0.2603)
0.3989 (0.2547) 0.4272 (0.2908)
avg. weights maximum Sharpe ratio portfolio with budget constraint
unconditional
state 1
state 2
lo20
0.3153 (0.3566)
0.3243 (0.3514) 0.2640 (0.3834)
hi20
0.3509 (0.3028)
0.3909 (0.3011) 0.1241 (0.1937)
tbond
0.3338 (0.3320)
0.2848 (0.2836) 0.6119 (0.4362)
risk-free asset
0.0000 (0.0000)
0.0000 (0.0000) 0.0000 (0.0000)
165
avg. optimal weights with capital allocation and risk aversion coefficient = 1
unconditional
state 1
state 2
lo20
3.8171 (5.9090)
4.3355 (6.2342) 0.8796 (1.5717)
hi20
4.7067 (5.3590)
5.3730 (5.4988) 0.9307 (1.8449)
tbond
6.5800 (11.7046)
7.0450 (12.2000) 3.9450 (7.9150)
risk-free asset -14.1038 (18.7596) -15.7535 (19.5087) -4.7552 (9.3859)
avg. optimal weights with capital allocation and risk aversion coefficient = 3
unconditional
state 1
state 2
lo20
1.3565 (1.9411)
1.5402 (2.0394) 0.3160 (0.5346)
hi20
1.6341 (1.7505)
1.8653 (1.7839) 0.3239 (0.6183)
tbond
2.2351 (3.8847)
2.3525 (4.0647) 1.5695 (2.5608)
risk-free asset
-4.2257 (6.1148)
-4.7580 (6.3676) -1.2094 (2.9932)
avg. optimal weights with capital allocation and risk aversion coefficient = 5
unconditional
state 1
state 2
lo20
0.8576 (1.1625)
0.9745 (1.2168) 0.1953 (0.3271)
hi20
1.0164 (1.0405)
1.1593 (1.0548) 0.2069 (0.3789)
tbond
1.3668 (2.3238)
1.4162 (2.4370) 1.0870 (1.5134)
risk-free asset
-2.2408 (3.5994)
-2.5499 (3.7525) -0.4892 (1.7382)
Table 22 shows unconditional and state conditional average optimal weights for
each one of the five different portfolios, their standard deviations are reported in
parentheses.
As it can be seen, for the first two portfolios, the hi20 represents the largest share
of the portfolio, disregarding the riskless asset, conditionally on state 1 while tbond
does it conditionally on state 2; hi20 and tbond have the highest mean returns in
state 1 and state 2 respectively, in addition tbond is also the less volatile asset
class conditionally on state 2. As a consequence the investor flies from the high
profitable stock markets to the low volatile bond market during state 2, in fact both
the lo20 and hi20 shares are higher in state 1 than state 2 while the opposite is
true for the tbond. Taking a look at the unconditional weights it can be seen that
hi20 again represents the largest share for the second portfolio while the riskless
asset does it for the first portfolio, this can be explained by the fact that hi20 has
the highest unconditional mean return, however it must be that when it is permit to
invest in the riskless asset, the investor ends up to be bettered off by holding a
large share of his or her portfolio in the riskless asset in order to rise the expected
Sharpe ratio. The last three portfolios, those realized by maximizing the expected
utility with lending and borrowing capital allocation line and with three different risk
166
aversion coefficients, are characterized by average optimal weights far larger than
100%. The optimal weights of these three portfolios are characterized by the same
rank regardless of whether the optimal weights are considered unconditionally or
conditionally on the state. The tbond always represents the largest portfolio share
followed by the hi20 and lo20 respectively; the riskless asset constantly represents
important short weights given the wide use of borrowing. The rank just discussed
might be explained by the fact that even though the tbond has the lowest mean
return it is the less volatile asset class followed indeed by the hi20 and at the end
the lo20, in addition hi20 has a mean returns higher than lo20 in state 1 which
occurs most of the time, contributing to show off investor the hi20 returns more
desirable than the lo20. Taking a look at Table 22 it can be also noticed that the
average borrowing is a decreasing function of the risk aversion and that it is on
average higher during state 1 due to the low volatility nature of this regime that
favors higher market exposures.
Table 23
in-sample average portfolio risky weights (1 = 100%)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 1
unconditional
state 1
state 2
lo20
0.3519 (0.3886)
0.3654 (0.3856)
0.2753 (0.3985)
hi20
0.3424 (0.3253)
0.3859 (0.3273)
0.0959 (0.1649)
tbond
0.2612 (0.3264)
0.2073 (0.2631)
0.5670 (0.4583)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 3
unconditional
state 1
state 2
lo20
0.3495 (0.3856)
0.3634 (0.3832)
0.2703 (0.3920)
hi20
0.3435 (0.3235)
0.3866 (0.3256)
0.0989 (0.1636)
tbond
0.2626 (0.3273)
0.2086 (0.2646)
0.5691 (0.4576)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 5
unconditional
state 1
state 2
lo20
0.3454 (0.3821)
0.3602 (0.3802)
0.2618 (0.3844)
hi20
0.3452 (0.3217)
0.3880 (0.3236)
0.1027 (0.1664)
tbond
0.2649 (0.3270)
0.2104 (0.2644)
0.5738 (0.4546)
167
Table 23 again shows that the investor during state 1 prefers to hold a risky
portfolio that mainly consists of hi20 while during state 2 a portfolio that mainly
consists of tbond; these risky portfolio are then leveraged by a borrowing factor
which in turn is influenced by the lending and borrowing rates, the risk aversion
coefficient and the market returns. From a comparison of Table 22 and 23
emerges some conflicting information; although the overall tbond optimal weights
are the largest, from Table 22 emerges that the share of this asset class in then
average risky portfolios is not the largest, this might be explained by the fact that
the investor might have largely resort to leverage when he or she has invested a
prevailing part of the portfolio in the tbond asset class, this is the case during state
2, in fact in this state tbond weights dominate the other by a great extent, in
addition the same large resort to borrowing might have been occurred also in
certain periods classified as state 1 when tbond represented a large portfolio
share, as a result the contribution of the leverage at certain periods caused the
relative weight of the tbond asset class to grow, compared with the other asset
classes weights, making weights on average the largest in each one of the three
maximum expected utility portfolio.
Table 24
in-sample portfolios expected moments
maximum Sharpe ratio portfolio with opened lower budget constraint
unconditional
state 1
state 2
avg. expected
return
0.0073 (0.0071) 0.0072 (0.0071) 0.0075 (0.0068)
avg. expected
standard deviaton
0.0167 (0.0123) 0.0155 (0.0106) 0.0234 (0.0179)
avg. Sharpe ratio
0.4668 (0.3309) 0.4877 (0.3374) 0.3484 (0.2631)
maximum Sharpe ratio portfolio with budget constraint
unconditional
state 1
state 2
avg. expected
return
0.0113 (0.0093) 0.0112 (0.0094) 0.0117 (0.0088)
avg. expected
standard deviaton
0.0280 (0.0138) 0.0258 (0.0112) 0.0407 (0.0195)
avg. Sharpe ratio
0.4634 (0.3363) 0.4846 (0.3426) 0.3434 (0.2706)
optimal weights with capital allocation and risk aversion coefficient = 1
unconditional
state 1
state 2
avg. expected
return
0.2056 (0.3995) 0.2254 (0.4224) 0.0933 (0.1965)
168
avg. expected
standard deviaton
0.3144 (0.3199) 0.3381 (0.3269) 0.1799 (0.2370)
avg. Sharpe ratio
0.4005 (0.2982) 0.4153 (0.3050) 0.3167 (0.2409)
optimal weights with capital allocation and risk aversion coefficient = 3
unconditional
state 1
state 2
avg. expected
return
0.0719 (0.1325) 0.0784 (0.1401) 0.0351 (0.0649)
avg. expected
standard deviaton
0.1121 (0.1009) 0.1195 (0.1033) 0.0702 (0.0736)
avg. Sharpe ratio
0.4085 (0.2970) 0.4236 (0.3034) 0.3229 (0.2416)
optimal weights with capital allocation and risk aversion coefficient = 5
unconditional
state 1
state 2
avg. expected
return
0.0450 (0.0792) 0.0489 (0.0838) 0.0233 (0.0388)
avg. expected
standard deviaton
0.0710 (0.0581) 0.0752 (0.0596) 0.0474 (0.0421)
avg. Sharpe ratio
0.4154 (0.2969) 0.4310 (0.3030) 0.3275 (0.2434)
Table 24 shows, unconditional and state conditional, average expected returns,
average expected standard deviations and average Sharpe ratios of the five
different portfolios. The average expected returns are the single period portfolios
expected returns, function of the underlying asset classes expected returns and
portfolio weights, averaged over the whole in-sample period. Coherently with the
average expected returns, the average expected standard deviations are the
single period standard deviations of the underlying asset class in which the
portfolios invest, weighted by the portfolios weights and corrected for the regime
switching dynamic, and averaged over the whole in-sample period; the Sharpe
ratios again are the single period ratios, function of the portfolio expected returns
and expected standard deviations, averaged over the whole period.
It can be
seen, as largely expected, that for the last three portfolios the average expected
return during state 1 is higher than during state 2, surprisingly the opposite is true
for the first two portfolios even thought the differences are very small. Surprisingly,
the average expected portfolio standard deviations of the last three portfolios are
higher in state 1 than in state 2 and not the opposite, as widely expected and
indeed happened for the first two portfolios where the state 2 portfolio average
expected deviations are almost twice as large as in state 1. This might be
explained by the wide optimal weights oscillations due to the large resort to
169
borrowing, this conclusion can be confirmed by the fact that, both the
unconditional and the state conditional, average portfolios expected standard
deviations decrease as a function of the increase of the risk aversion coefficient
which in turn means that the higher the risk aversion coefficient, the lower the
average resort to borrowing, the lower the average portfolio expected standard
deviations. Taking a look at the unconditional average portfolio expected returns
and standard deviations it can be noticed that, as largely expected, the higher the
risk aversion coefficient the less profitable and less risky in terms of standard
deviations the resulting portfolio. Interestingly giving the investor the chance to
hold some riskless asset when it maximize the Sharpe ratio leads to a lower
average portfolio expected returns and average portfolio expected standard
deviations that contributes to the realization of a higher expected Sharpe ratio. The
last three portfolios are characterized by an increasing unconditional average
expected Sharpe ratio as a function of the amount borrowed to invest in the risky
assets. Overall it is clearly noticed that the average state 1 expected Sharpe ratios
greatly exceed the correspondent state 2 measures which means that the
estimated regimes have a clear economic interpretation observable also from the
optimal weights estimated for the five different portfolios.
Figure 114 maximum Sharpe ratio portfolios (with and without budget constraint)
expected returns
170
Figure 115 expected returns of the maximum expected utility portfolios (capped
at 1 = 100%)
In this section of the subparagraph the in-sample portfolio realized results are
shown.
Table 25
in-sample realized portfolios statistics
maximum Sharpe ratio portfolio with opened lower budget constraint
unconditional
state 1
state 2
average realized return
0.0070
0.0073
0.0052
realized returns standard deviaton
0.0264
0.0210
0.0463
realized returns Sharpe ratio
0.2650
0.3469
0.1126
maximum Sharpe ratio portfolio with budget constraint
unconditional
state 1
state 2
average realized return
0.0102
0.0112
0.0041
realized returns standard deviaton
0.0407
0.0310
0.0751
realized returns Sharpe ratio
0.2492
0.3620
0.0544
optimal weights with capital allocation and risk aversion coefficient = 1
unconditional
state 1
state 2
average realized return
0.1979
0.2193
0.0769
realized returns standard deviaton
0.5934
0.6232
0.3628
realized returns Sharpe ratio
0.3336
0.3519
0.2119
optimal weights with capital allocation and risk aversion coefficient = 3
unconditional
state 1
state 2
171
average realized return
0.0690
0.0763
0.0278
realized returns standard deviaton
0.1985
0.2082
0.1239
realized returns Sharpe ratio
0.3477
0.3665
0.2245
optimal weights with capital allocation and risk aversion coefficient = 5
unconditional
state 1
state 2
average realized return
0.0432
0.0476
0.0186
realized returns standard deviaton
0.1200
0.1257
0.0767
realized returns Sharpe ratio
0.3600
0.3784
0.2421
As it can be seen, on average, the portfolios realized returns during state 1
overperform those realized during state 2, as largely expected. Similarly to what
has already been discussed regarding Table 24, the higher realized portfolio
standard deviations during state 1 might be explained by the wide optimal weights
oscillations due to the large resort to borrowing, this conclusion can be confirmed
by the fact that, both the unconditional that the state conditional, average portfolio
expected standard deviations decrease as a function the increase of the risk
aversion coefficient which in turn means that the higher the risk aversion
coefficient, the lower the average resort to borrowing, thus the lower the average
portfolio expected standard deviations. From Table 24 it can be seen that the
realized Sharpe ratio increases as a function of the risk aversion coefficient
reaching a peak of 0.36 when the former is equal to 5. Again, interestingly, giving
the investor the chance to hold some riskless asset when it maximize the Sharpe
ratio leads to a lower average portfolio expected returns and average portfolio
expected standard deviations that contributes to the realization of a higher
expected Sharpe ratio. Overall it can be deduced that the conclusions obtained
from the observation of the expected portfolio returns are here substantially
reproduced.
Figure 116 in-sample maximum Sharpe ratio portfolio with opened lower budget
constraint realized returns
172
Figure 117 in-sample maximum Sharpe ratio portfolio with opened lower budget
constraint cumulative realized returns
Figure 118 in-sample maximum Sharpe ratio portfolio with budget constraint
realized returns
173
Figure 119 in-sample maximum Sharpe ratio portfolio with budget constraint
realized cumulative returns
Figure 120 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 1 realized returns
174
Figure 121 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 1 cumulative realized returns
Figure 122 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 3 realized returns
175
Figure 123 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 3 cumulative realized returns
Figure 124 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 5 realized returns
176
Figure 125 in-sample optimal portfolio with capital allocation and risk aversion
coefficient = 5 cumulative realized returns
4.2.2 In-sample asset allocation exercise based on a VAR(1) model
In this subparagraph the in-sample portfolio construction results using the VAR(1)
are shown. The same five in-sample recursive optimal portfolios as in the
MSVAR(2,1) case have been built.
177
Figure 126 lo20 and its VAR(1) predicted conditional mean
Figure 127 hi20 and its VAR(1) predicted conditional mean
Figure 128 tbond and its VAR(1) predicted conditional mean
178
Figure 129 VAR(1) maximum Sharpe ratio portfolios weights (1 = 100%) with
opened lower budget constraint (permit to invest in the riskless asset)
Figure 130 VAR(1) maximum Sharpe ratio portfolios weights (1 = 100%) with
budget constraint (not permit to invest in the riskless asset)
179
Figure 131 VAR(1) overall optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 1
Figure 132 VAR(1) overall optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 3
180
Figure 133 VAR(1) overall optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 5
Figure 134 VAR(1) risky optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 1
181
Figure 135 VAR(1) risky optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 3
Figure 136 VAR(1) risky optimal portfolio weights (1 = 100%) with capital
allocation and risk aversion coefficient = 5
182
Table 26
VAR(1) in-sample average portfolio overall weights (1 = 100%)
avg. weights maximum Sharpe ratio portfolio with opened lower budget
constraint
lo20
hi20
tbond
risk-free asset
0.2384 (0.3162) 0.0361 (0.1220) 0.2876 (0.3118)
0.4380 (0.2953)
avg. weights maximum Sharpe ratio portfolio with budget constraint
lo20
hi20
tbond
risk-free asset
0.4204 (0.4514) 0.0712 (0.2042) 0.5084 (0.4404)
0.0000 (0.0000)
avg. optimal weights with capital allocation and risk aversion coefficient =
1
lo20
hi20
tbond
risk-free asset
0.9902 (1.5068) 0.3331 (0.9524) 4.4857 (9.7123) -4.8090 (10.8207)
avg. optimal weights with capital allocation and risk aversion coefficient =
3
lo20
hi20
tbond
risk-free asset
0.5321 (0.5839) 0.1268 (0.3394) 1.6242 (3.2006) -1.2831 (3.4641)
avg. optimal weights with capital allocation and risk aversion coefficient =
5
lo20
hi20
tbond
risk-free asset
0.3991 (0.4368) 0.0794 (0.2149) 1.0621 (1.9066) -0.5406 (2.0231)
Table 27
VAR(1) in-sample average portfolio risky weights (1 = 100%)
avg. optimal risky weights with capital allocation and risk aversion
183
coefficient = 1
lo20
hi20
tbond
risk-free asset
0.4470 (0.4645) 0.0662 (0.2108) 0.4164 (0.4492) 0.0000 (0.0000)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 3
lo20
hi20
tbond
risk-free asset
0.4415 (0.4621) 0.0646 (0.2016) 0.4235 (0.4470) 0.0000 (0.0000)
avg.optimal risky weights with capital allocation and risk aversion
coefficient = 5
lo20
hi20
tbond
risk-free asset
0.4343 (0.4592) 0.0595 (0.1866) 0.4358 (0.4454) 0.0000 (0.0000)
Table 28
VAR(1) in-sample portfolios expected moments
maximum Sharpe ratio portfolio with opened lower budget constraint
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0060 (0.0065)
0.0208 (0.0173)
0.2985 (0.2491)
maximum Sharpe ratio portfolio with budget constraint
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0097 (0.0097)
0.0372 (0.0202)
0.2859 (0.2701)
optimal weights with capital allocation and risk aversion coefficient = 1
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0677 (0.2105)
0.1396 (0.2049)
0.0808 (4.1512)
optimal weights with capital allocation and risk aversion coefficient = 3
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0276 (0.0699)
0.0623 (0.0639)
-0.2686 (12.4132)
optimal weights with capital allocation and risk aversion coefficient = 5
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0190 (0.0423)
0.0440 (0.0385)
-0.6195 (20.6788)
Figure 137 VAR(1) maximum Sharpe ratio portfolios (with and without budget
constraint) expected returns
184
Figure 138 VAR(1) expected returns of the maximum expected utility portfolios
(capped at 1 = 100%)
Table 29
VAR(1) in-sample realized portfolios statistics
maximum Sharpe ratio portfolio with opened lower budget constraint
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0153
0.0271
0.5649
maximum Sharpe ratio portfolio with budget constraint
185
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0236
0.0425
0.5546
optimal weights with capital allocation and risk aversion coefficient = 1
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.1824
0.6388
0.2855
optimal weights with capital allocation and risk aversion coefficient = 3
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0740
0.2135
0.3465
optimal weights with capital allocation and risk aversion coefficient = 5
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0510
0.1304
0.3909
Figure 139 VAR(1) in-sample maximum Sharpe ratio portfolio with opened lower
budget constraint realized returns
Figure 140 VAR(1) in-sample maximum Sharpe ratio portfolio with opened lower
budget constraint cumulative realized returns
186
Figure 141 VAR(1) in-sample maximum Sharpe ratio portfolio with budget
constraint realized returns
Figure 142 VAR(1) in-sample maximum Sharpe ratio portfolio with budget
constraint realized cumulative returns
187
Figure 143 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 1 realized returns
Figure 144 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 1 cumulative realized returns
188
Figure 145 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 3 realized returns
Figure 146 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 3 cumulative realized returns
189
Figure 147 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 5 realized returns
Figure 148 VAR(1) in-sample optimal portfolio with capital allocation and risk
aversion coefficient = 5 cumulative realized returns
190
A comparison of the in-sample realized portfolio returns obtained using both the
MSVAR(2,1) and the VAR(1) can be conducted from the observation of Table 25
and Table 29. Overall it emerges that none of the two models used to estimate the
asset classes first two moments leads to a substantial portfolio overperformance
over the other model. In fact, among the portfolios built using the MSVAR(2,1)
estimates only the one that maximizes the expected utility function with risk
aversion coefficient equal to 1 overperfoms the portfolios built using the more
simple VAR(1) model; however if the comparison is conducted in terms of realized
portfolio Sharpe ratio it can be seen that the portfolios built using the two states
model overperform those built using the single state model two out of three times.
4.3 Out-of-sample asset allocation exercise
In this paragraph I am going to illustrate the out-of-sample portfolio construction
processes that have been adopted and both the realized and the expected asset
allocation and portfolio statistics, using firstly the MSVAR(2,1) and secondly the
VAR(1) to model the first two moments of the asset classes returns distributions.
The out-of-sample period starts at January 2010, lasts 5 years and end at
December 2014. For the portfolios built using the asset classes first two expected
moments estimated from a MSVAR(2,1) model and from a VAR(1) model in the
first paragraph the portfolios weights results are shown as well as some expected
191
portfolio statistics while in the second subparagraph the portfolio realized returns
are exhibited and commented.
4.3.1 Out-of-sample asset allocation exercise based on a
MSVAR(2,1) model
The out-of-sample exercise required me to estimate the asset classes first two
expected moments using the MSVAR(2,1) already estimated. Although the two
state conditional VAR matrices regressive and autoregressive coefficients have
been estimated using the in-sample returns, from 1965 to 2010, the time t filtered
probabilities in this out-of-sample exercise are no more a by-product of the
likelihood function maximization, indeed they must be calculated with the
Hamilton's filter updating prior beliefs
regard them, formulated at time t-1, with the
arrival of new information at time t according to the following formula:
(55)
|
=
-|-
(
|
t-1
;
)
where
|
is a (1 x 2) vector and P is (2 x 2) matrix that governs the transition
dynamic (its coefficients have been estimated using the in-sample returns),
is a
(2 x 1) vector whose j-th element is the density in regime j of the observation
:
(56)
= (
|S
t
= ,
t-1
; )
while the following formula is the unconditional density of the observation
:
(57)
22
(
|
t-1
; ) =
(
-|-
)
In the out-of-sample exercise that I perform, the last available estimated and not
updated filtered probability is
relative to December 2009. As a consequence
any next probability in the out-of-sample exercise has been estimated with the
Hamilton
's filter according to formula (55).
Figure 149 out-of-sample predicted probabilities
22
James D. Hamilton "Regime
-Switching Models. Palgrave Dictionary of Economics
" (2005) and
Birger Nilsson and Andreas Graflund
"Dynamic Portfolio Selection: The Relevance of Switching
Regimes and Investment Horizon" (2001)
192
Given that the investor is not interested in investing in the dividend yield but only in
the small stocks, large stocks and bonds, the predicted dynamic conditional mean
for each t+T from January 2010 to December 2014 is equal to the first three
elements of the following vector:
(58)
E(
+
|
+-
; ;
t+T-1
) =
t+T|t+T-1
=
1,t+T|t+T-1
E(
+
|
+-
; S
t
= 1; ;
t+T-1
) +
2,t+T|t+T-1
E(
+
|
+-
; S
t
= 2; ;
t+T-1
)
(59)
E(
+
|
+-
; S
t
= ; ;
t+T-1
) =
t+T|t+T-1;
=
=
=
+
,
=
+-
As it can be seen, similarly to what have been done for the expected in-sample
vector of dynamic conditional mean, the one step ahead prediction (t+1) uses the
current returns (t); it follows that the time t+T prediction, which for example might
be the prediction for December 2011 (t=540 that corresponds to December 2009
and T=12) uses the November 2011 (t=540 that corresponds to December 2009
and T=11) realized returns. Here
=
is the vector of intercept terms of
+
in
state
S
t
=
,
,
=
is the matrix of autoregressive and regression coefficients
associated with lag 1 in state
S
t
=
; their values have been estimated using the in-
sample returns.
The predicted dynamic variance-covariance matrix for each t+T from January 2010
to December 2014 is equal to the (3 x 3) left-upper part of the following matrix:
193
(60)
(
+
|
+-
; ;
t+T-1
) =
1,t+T|t+T-1
+
2,t+T|t+T-1
+
1,t+T|t+T-1
2,t+T|t+T-1
(
t+T|t+T-1;
=
-
t+T|t+T-1;
=
)(
t+T|t+T-1;
=
-
t+T|t+T-1;
=
)
where
t+T|t+T-1;
=
and
t+T|t+T-1;
=
are the predicted dynamic conditional means
calculated above.
Figure 150 out-of-sample lo20 and its predicted dynamic conditional mean
Figure 151 out-of-sample hi20 and its predicted dynamic conditional mean
Figure 152 out-of-sample tbond and its predicted dynamic conditional mean
194
I then used the predicted dynamic variance-covariance matrix and the predicted
dynamic conditional mean vector to build the same five in-sample recursive
optimal portfolios. The results are here illustrated.
Figure 153 out-of-sample maximum Sharpe ratio portfolios weights (1 = 100%)
with opened lower budget constraint (permit to invest in the riskless
asset)
Figure 154 out-of-sample maximum Sharpe ratio portfolios weights (1 = 100%)
with budget constraint (not permit to invest in the riskless asset)
195
Figure 155 out-of-sample overall optimal portfolio weights (1 = 100%) with
capital allocation and risk aversion coefficient = 1
Figure 156 out-of-sample overall optimal portfolio weights (1 = 100%) with
capital allocation and risk aversion coefficient = 3
196
Figure 157 out-of-sample overall optimal portfolio weights (1 = 100%) with
capital allocation and risk aversion coefficient = 5
Table 30
out-of-sample average portfolio overall weights (1 = 100%)
avg. weights maximum Sharpe ratio portfolio with opened lower budget
lo20
hi20
tbond
risk-free asset
0.2677 (0.2783) 0.1535 (0.1346) 0.2619 (0.3331)
0.3169 (0.2565)
avg. weights maximum Sharpe ratio portfolio with budget constraint
197
lo20
hi20
tbond
risk-free asset
0.3852 (0.3536) 0.2459 (0.2062) 0.3689 (0.3858)
0.0000 (0.0000)
avg. optimal weights with capital allocation and risk aversion coefficient = 1
lo20
hi20
tbond
risk-free asset
1.8252 (2.3385) 1.8493 (3.5482) 3.2800 (7.4712) -5.9545 (10.4384)
avg. optimal weights with capital allocation and risk aversion coefficient = 3
lo20
hi20
tbond
risk-free asset
0.7042 (0.7785) 0.6822 (1.1746) 1.1844 (2.4722)
-1.5707 (3.3469)
avg. optimal weights with capital allocation and risk aversion coefficient = 5
lo20
hi20
tbond
risk-free asset
0.4541 (0.4891) 0.4496 (0.7038) 0.7851 (1.4767)
-0.6888 (1.9459)
Table 31
out-of-sample average portfolio risky weights (1 = 100%)
avg. optimal risky weights with capital allocation and risk aversion
coefficient=1
lo20
hi20
tbond
risk-free asset
0.4721 (0.4136) 0.2242 (0.2708) 0.3036 (0.4114) 0.0000 (0.0000)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 3
lo20
hi20
tbond
risk-free asset
0.4520 (0.3960) 0.2422 (0.2665) 0.3058 (0.4099) 0.0000 (0.0000)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 5
lo20
hi20
tbond
risk-free asset
0.4169 (0.3738) 0.2550 (0.2509) 0.3281 (0.4023) 0.0000 (0.0000)
Figure 158 out-of-sample maximum Sharpe ratio portfolios (with and without
budget constraint) expected returns
198
Figure 159 out-of-sample expected returns of the maximum expected utility
portfolios (capped at 1 = 100%)
Table 32
out-of-sample portfolios expected moments
maximum Sharpe ratio portfolio with opened lower budget constraint
avg. expected return avg. expected standard deviaton
avg. Sharpe ratio
0.0083 (0.0055)
0.0266 (0.0164)
0.3559 (0.2220)
maximum Sharpe ratio portfolio with budget constraint
avg. expected return avg. expected standard deviaton
avg. Sharpe ratio
199
0.0122 (0.0070)
0.0386 (0.0169)
0.3559 (0.2220)
optimal weights with capital allocation and risk aversion coefficient = 1
avg. expected return avg. expected standard deviaton
avg. Sharpe ratio
0.0884 (0.1756)
0.2100 (0.1994)
0.3075 (0.2013)
optimal weights with capital allocation and risk aversion coefficient = 3
avg. expected return avg. expected standard deviaton
avg. Sharpe ratio
0.0335 (0.0578)
0.0806 (0.0595)
0.3208 (0.1982)
optimal weights with capital allocation and risk aversion coefficient = 5
avg. expected return avg. expected standard deviaton
avg. Sharpe ratio
0.0222 (0.0346)
0.0527 (0.0337)
0.3305 (0.1973)
In this section of the subparagraph the out-of-sample portfolio realized results are
shown.
Table 33
out-of-sample realized portfolios statistics
maximum Sharpe ratio portfolio with opened lower budget constraint
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0084
0.0246
0.3406
maximum Sharpe ratio portfolio with budget constraint
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0115
0.0305
0.3788
optimal weights with capital allocation and risk aversion coefficient = 1
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0729
0.2437
0.2990
optimal weights with capital allocation and risk aversion coefficient = 3
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0266
0.0832
0.3203
optimal weights with capital allocation and risk aversion coefficient = 5
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0181
0.0509
0.3555
Figure 160 out-of-sample maximum Sharpe ratio portfolio with opened lower
budget constraint realized returns
200
Figure 161 out-of -sample maximum Sharpe ratio portfolio with opened lower
budget constraint cumulative realized returns
Figure 162 out-of -sample maximum Sharpe ratio portfolio with budget constraint
realized returns
201
Figure 163 out-of -sample maximum Sharpe ratio portfolio with budget constraint
realized cumulative returns
Figure 164 out-of -sample optimal portfolio with capital allocation and risk
aversion coefficient = 1 realized returns
202
Figure 165 out-sample optimal portfolio with capital allocation and risk aversion
coefficient = 1 cumulative realized returns
Figure 166 out-of -sample optimal portfolio with capital allocation and risk
aversion coefficient = 3 realized returns
203
Figure 167 out-of -sample optimal portfolio with capital allocation and risk
aversion coefficient = 3 cumulative realized returns
Figure 168 out-of -sample optimal portfolio with capital allocation and risk
aversion coefficient = 5 realized returns
204
Figure 169 out-of -sample optimal portfolio with capital allocation and risk
aversion coefficient = 5 cumulative realized returns
4.3.2 Out-of-sample asset allocation exercise based on a VAR(1)
model
In this subparagraph the out-of-sample portfolio construction results using the
VAR(1) are shown. The VAR(1) model parameters have been estimated using the
in-sample returns. The same five out-of-sample recursive optimal portfolios as in
the MSVAR(2,1) case have been built. Similarly to what have been done for the
MSVAR(2,1) out-of-sample vector of dynamic conditional mean, the one step
ahead prediction (t+1) uses the current returns (t); it follows that the time t+T
205
prediction, which for example might be the prediction for December 2011 (t=540
that corresponds to December 2009 and T=12) uses the November 2011 (t=540
that corresponds to December 2009 and T=11) realized returns while the predicted
variance-covariance matrix for each t+T from January 2010 to December 2014 is
constant since the VAR(1) variance-covariance is not time varying.
Figure 170 out-of-sample lo20 and its VAR(1) dynamic conditional mean
Figure 171 out-of-sample hi20 and its VAR(1) predicted conditional mean
206
Figure 172 out-of-sample tbond and its VAR(1) predicted conditional mean
Figure 173 VAR(1) out-of-sample maximum Sharpe ratio portfolios weights (1 =
100%) with opened lower budget constraint (permit to invest in the
riskless asset)
Figure 174 VAR(1) out-of-sample maximum Sharpe ratio portfolios weights (1 =
100%) with budget constraint (not permit to invest in the riskless
asset)
207
Figure 175 VAR(1) out-of-sample overall optimal portfolio weights (1 = 100%)
with capital allocation and risk aversion coefficient = 1
Figure 176 VAR(1) out-of-sample overall optimal portfolio weights (1 = 100%)
with capital allocation and risk aversion coefficient = 3
208
Figure 177 VAR(1) out-of-sample overall optimal portfolio weights (1 = 100%)
with capital allocation and risk aversion coefficient = 5
Table 34
VAR(1) out-of-sample average portfolio overall weights (1 = 100%)
avg. weights maximum Sharpe ratio portfolio with opened lower budget
constraint
lo20
hi20
tbond
risk-free asset
0.2709 (0.3148) 0.0051 (0.0128) 0.3107 (0.3411)
0.4132 (0.2747)
209
avg. weights maximum Sharpe ratio portfolio with budget constraint
lo20
hi20
tbond
risk-free asset
0.4788 (0.4640) 0.0085 (0.0210) 0.5126 (0.4557)
0.0000 (0.0000)
avg. optimal weights with capital allocation and risk aversion coefficient =
1
lo20
hi20
tbond
risk-free asset
0.8531 (1.0363) 0.0382 (0.1527) 3.6249 (10.5283) -3.5162 (10.5595)
avg. optimal weights with capital allocation and risk aversion coefficient =
3
lo20
hi20
tbond
risk-free asset
0.5457 (0.4914) 0.0127 (0.0509) 1.3633 (3.4813) -0.9218 (3.3899)
avg. optimal weights with capital allocation and risk aversion coefficient =
5
lo20
hi20
tbond
risk-free asset
0.3943 (0.3845) 0.0076 (0.0305) 0.9320 (2.0789) -0.3339 (1.9918)
Table 35
VAR(1) out-of-sample average portfolio risky weights (1 = 100%)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 1
lo20
hi20
tbond
risk-free asset
0.5355 (0.4712) 0.0023 (0.0124) 0.4123 (0.4621) 0.0000 (0.0000)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 3
lo20
hi20
tbond
risk-free asset
0.5340 (0.4725) 0.0023 (0.0124) 0.4137 (0.4636) 0.0000 (0.0000)
avg. optimal risky weights with capital allocation and risk aversion
coefficient = 5
lo20
hi20
tbond
risk-free asset
0.5098 (0.4663) 0.0023 (0.0124) 0.4380 (0.4602) 0.0000 (0.0000)
Figure 178 VAR(1) out-of-sample maximum Sharpe ratio portfolios (with and
without budget constraint) expected returns
210
Figure 179 VAR(1) out-of-sample expected returns of the maximum expected
utility portfolios (capped at 1 = 100%)
Table 36
VAR(1) out-of-sample portfolios expected moments
maximum Sharpe ratio portfolio with opened lower budget constraint
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0061 (0.0055)
0.0222 (0.0165)
0.3034 (0.2459)
maximum Sharpe ratio portfolio with budget constraint
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
211
0.0101 (0.0077)
0.0386 (0.0212)
0.3007 (0.2496)
optimal weights with capital allocation and risk aversion coefficient = 1
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0588 (0.2214)
0.1163 (0.1976)
0.2747 (0.2288)
optimal weights with capital allocation and risk aversion coefficient = 3
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0251 (0.0733)
0.0581 (0.0622)
0.2800 (0.2285)
optimal weights with capital allocation and risk aversion coefficient = 5
avg. expected return
avg. expected standard deviaton
avg. Sharpe ratio
0.0175 (0.0441)
0.0410 (0.0376)
0.2860 (0.2282)
Table 37
VAR(1) out-of-sample realized portfolios statistics
maximum Sharpe ratio portfolio with opened lower budget constraint
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0054
0.0210
0.2560
maximum Sharpe ratio portfolio with budget constraint
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0101
0.0325
0.3093
optimal weights with capital allocation and risk aversion coefficient = 1
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0438
0.2651
0.1651
optimal weights with capital allocation and risk aversion coefficient = 3
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0203
0.0922
0.2204
optimal weights with capital allocation and risk aversion coefficient = 5
average realized return
realized returns standard
deviaton
realized returns Sharpe
ratio
0.0146
0.0581
0.2518
Figure 180 VAR(1) out-of-sample maximum Sharpe ratio portfolio with opened
lower budget constraint realized returns
212
Figure 181 VAR(1) out-of -sample maximum Sharpe ratio portfolio with opened
lower budget constraint cumulative realized returns
Figure 182 VAR(1) out-of -sample maximum Sharpe ratio portfolio with budget
constraint realized returns
213
Figure 183 VAR(1) out-of -sample maximum Sharpe ratio portfolio with budget
constraint realized cumulative returns
Figure 184 VAR(1) out-of -sample optimal portfolio with capital allocation and
risk aversion coefficient = 1 realized returns
214
Figure 185 VAR(1) out-sample optimal portfolio with capital allocation and risk
aversion coefficient = 1 cumulative realized returns
Figure 186 VAR(1) out-of -sample optimal portfolio with capital allocation and
risk aversion coefficient = 3 realized returns
215
Figure 187 VAR(1) out-of -sample optimal portfolio with capital allocation and
risk aversion coefficient = 3 cumulative realized returns
Figure 188 VAR(1) out-of -sample optimal portfolio with capital allocation and
risk aversion coefficient = 5 realized returns
216
Figure 189 VAR(1) out-of -sample optimal portfolio with capital allocation and
risk aversion = 5 cumulative realized returns
Conversely to the in the in-sample, in the out-of-sample exercise the portfolios
based on the MSVAR(2,1) asset classes first two moments estimates did largely
overperform those built on the VAR(1) model. What emerges from a comparison of
Table 33 and Table 37 is that in all the five types of portfolios the MSVAR(2,1)
217
leads to greater average portfolio realized returns and to greater realized Sharpe
ratios compared to the VAR(1).
4.4 Forecasting ability comparison
In this paragraph an assessment of the in-sample and the out-of-sample one-step-
ahead forecasting ability of both the single regime VAR(1) and the regime
switching MSVAR(2,1) model is illustrated and discussed. Three different criteria
have been adopted in this comparison analysis.
4.4.1 RMSE
The one-step-ahead average root-mean squared error is given by the formula:
(61)
=
1
(
+
- E(
+
))
2
=1
where
+
are the actual small stock, large stocks and bond realized returns while
E(
+
)
represents the forecasted returns; both vectors are constructed as a
sequence of the single asset classes returns, thus if the length of the sample is 60
periods and the model has three asset classes (as in the out-of-sample exercise)
the length of either
+
or
E(
+
)
is 180. The RMSE is a frequently used
measure of the differences between values predicted by a model and the values
actually observed; it represents the sample standard deviation of the differences
between predicted values and actual realized returns. The RMSE is a good
measure of accuracy to compare forecasting errors of different models, thus
smaller RMSE are preferred.
Table 38
in-sample and out-of-sample RMSE comparison
VAR(1)
In-sample RMSE
Out-of-sample RMSE
0.0387
0.0416
MSVAR(2,1)
In-sample RMSE
Out-of-sample RMSE
0.0451
0.0411
218
Table 38 shows both the in-sample and the out-of-sample VAR(1) and
MSVAR(2,1) RMSE values in order to assess their forecasting ability. As it can be
seen the simple VAR(1) model performs better than the more complicated
MSVAR(2,1) model in the in-sample period but there is no evidence of this relative
advantage in the out-of-sample period where the two models substantially have
the same forecasting ability.
4.4.2 Pearson correlation coefficient
The second measure I adopted is the Pearson correlation coefficient between the
forecasted return and the actual realized ones. The closer the slope coefficient to
1, the better the forecasting ability of the model.
Table 39
in-sample and out-of-sample Pearson correlation coefficient between
realized and forecasted returns by the VAR(1) and the MSVAR(2,1)
VAR(1)
In-sample correlation
Out-of-sample correlation
0.7816
-0.0751
MSVAR(2,1)
In-sample correlation
Out-of-sample correlation
0.2590
-0.0048
In the in-sample period both models seem to forecast returns in the right direction,
however the correlation coefficient is smaller in absolute value for the regime
switching model. In the out-of-sample period both models seem to have very weak
but negative predictive power, which means that both models forecast returns in
the wrong direction. This finding substantially confirms the conclusion based on
the RMSE comparison.
4.4.3 Regression analysis
As a third measure I regress the realized returns for each risk factor on the
forecasted returns and an intercept.
(62)
+
= + E(
+
)
If the forecasts are accurate, I expect
= 0
and
= 1
.
219
Table 40
results of the in-sample and out-of-sample regression of the realized
returns on the returns forecasted by the VAR(1) and the MSVAR(2,1)
VAR(1)
Intercept
Slope
In-sample
-0.0093
3.3181
Out-of-sample
0.0110
-0.3402
MSVAR(2,1)
Intercept
Slope
In-sample
-0.0005
0.9363
Out-of-sample
0.0100
-0.0182
A comparison of the intercept for the regime switching model and the single state
model reveals that in the in-sample period both models underestimate future
returns, while in the out-of-sample period they overestimate them. The slope
coefficients indicate that the relationships between forecasted returns and realized
returns in the in-sample period is stronger in the MSVAR(2,1) model than in the
VAR(1) model while in the out-of-sample period there is no substantial difference
between the two models.
4.5 Asset allocation exercise conclusions
In conclusion it can be said that while the construction of portfolios based on the
two regimes MSVAR(2,1) model estimates do not lead to results that clearly
overperform those based on the single regime VAR(1) model estimates in the in-
sample framework, it significantly does it in the out-of-sample framework.
Therefore the adoption of a sophisticated multi regimes model greatly improves
the portfolios performance in the out-of-sample exercise for all types of portfolios
constructed.
220
CHAPTER 5 - ASSET ALLOCATION BASED ON THE
COPULA-OPINION POOLING APPROACH
The purpose of this paragraph is to investigate and make a comparison of the
behavior of the MSVAR and the VAR in the COP framework in terms of both
expected and realized portfolio returns and Sharpe ratio in the context of mean-
variance and conditional value-at-risk (CVaR) portfolio optimization. In the first part
of the each subparagraph the results of an out-of-sample portfolio optimization
exercise based on the copula opinion pooling approach applied both to the simple
single regime VAR(1) model and to the two regimes MSVAR(2,1) model are
shown; in the second part of each subparagraph some conclusions regarding the
performances of the portfolios based on the two models are drawn.
5.1 The copula-opinion pooling approach
The recent financial crisis has pointed out the "Risk
-on/Risk-
off" notion,
characterized by periods when the market is not driven any more by the
fundamental analysis. Consequently, a question about the effectiveness of the
classic portfolio optimization models has risen: the strong rise of cross asset
correlations suggests integrating new market anomalies into models, such as the
non-normality of returns, the non-linear dependence, the sharp rise of correlations
between assets historically uncorrelated that occurred during the recent recession
or the fact that the linear codependence modeled by a variance-covariance matrix
in an asset returns normality framework are not well adapted for the stressed
scenarios that have been observed over the out-of-sample years, namely from
January 2010 to December 2014.
The Markov regime switching models characteristically fit these returns features
given their non-linear nature; but what is further needed is a more complex
portfolio optimization approach that can be able to take into account and to
manage these recently observed asset returns features.
As a consequence it seems necessary to widen the possibilities of the Black-
Littermann model to take these anomalies into account. As Meucci (2005)
reminded, the pathbreaking technique by Black and Litterman (1990) (BL) allows
221
portfolio managers to compute a posterior market distribution that smoothly blends
their subjective views on the market with a prior market distribution, under the
assumption that all the distributions involved be normal. It is based on both
Bayesian markets forecasts and mean-variance optimization, it solves the issue of
stability on expected returns, and proposes a way to rely both on fundamental and
quantitative analysis. In reality, as already established, the markets are in general
highly non-normal. Furthermore, practitioners might wish to input their views in
less informative ways than the alpha + normal noise prescription of BL, for
instance by means of uniform distributions on ranges.
In addition, the COP approach allows the investor to set views on the market
instead of the market parameters.
In theory, these issues are fully addressed by the copula-opinion pooling (COP)
approach, see Meucci (2006).
It turns out that the implementation in practice of the COP approach under any
distributional assumption is straightforward: by representing the prior market
distribution in terms of a very large number of Monte Carlo scenarios, an equal
number of scenarios from the COP posterior market distribution can be computed
in a fraction of a second. Also, Monte Carlo methods are not cursed by
dimensionality: therefore the COP approach can handle markets with any number
of risk factors.
The intuition behind the COP approach is to determine the marginal distribution of
each view separately, whereas the joint co-dependence, i.e. the copula, among
the views is directly inherited from the prior market structure. Subsequently the
joint distribution of the views is translated into a joint posterior distribution for the
market. Thus, the first objective of the model is to simulate the expected future
evolution of the market, the so called posterior market distribution, mixing the
results of a quantitative model based on historical data, the so called prior market
distribution, and the views of the investor. These two cumulative distribution
functions are then averaged according to weights depending on the confidence
levels the investor has on his views.
222
The representation of the prior distribution depends on two steps, namely
modeling and simulation. The former one has been performed by the estimation of
two alternative models, i.e. the VAR(1) and the MSVAR(2,1). In general the
investor can estimate the prior market distribution by means of any techniques and
models.
It can be easily said that the in-sample 540 data points (from January 1965 to
December 2009) are not enough to motivate the use of empirical distribution.
Meucci (2006) suggested parametric bootstrap where the data matrix is first fitted
into a multivariate or univariate model, which is later used to simulate another
market matrix large enough to motivate the use of empirical distribution. As a
consequence the prior market distribution has been obtained by J = 500000 Monte
Carlo simulations generating the market prior distribution M of size (J x N) where N
is equal to the number of time series of the model the investor is interested to
invest in, i.e. small stocks, large stocks and bonds. The prior distributions for both
VAR(1) and MSVAR(2,1) models have been only created once and stored for
further calculations. Thus M represents the returns on the three asset classes. As
explained by Meucci (2006) I represent these returns in terms of their multivariate
probability distribution function (pdf):
(63)
~
M
I subsequently depart from these distributions according to the subjective views
produced by the either the MSVAR(2,1) or the VAR(1) model.
As already explained I empirically represent the prior market distribution by means
of a large number of Monte Carlo simulations. More precisely, the prior market
distribution is represented by a (J × N) panel M:
(64)
~
M
Each N-dimensional row represents one of the J simulated joint scenarios for the
market variable M.
The second input of the COP approach are the practitioner's views. Any
parametric or non-parametric representation of the cdf of the views is a viable
223
option. Meucci in his works remarks that in the COP approach this distribution can
take on any shape: from the alpha + normal noise specification as in BL to an
uninformative uniform specification. Also, in order to complete the specification of
the views it is needed to set the confidence levels in each view. It is also clear that
the views have a huge impact on the final allocation, more or less depending on
the confidence level.
I defined uniformly distributed views over the given ranges which can be modeled
by the following uniform distribution:
(65)
F
k
(v) = {
0 a
k
v-a
k
b
k
-a
k
[a
k
, b
k
], k = 1,2,3,4
1 v b
k
This choice probably is not the best one but it is the most general and
uninformative one. In my case, for any time over the out-of-sample period, the
boundaries of the ranges read:
(66)
(
lo20
- 0.5
lo20
hi20
- 0.5
hi20
tbond
- 0.5
tbond
)
(67)
(
lo20
+ 0.5
lo20
hi20
+ 0.5
hi20
tbond
+ 0.5
tbond
)
where
and
respectively represent at any data point the asset class expected
return and standard deviation produced by either the MSVAR(2,1) model or the
VAR(1) model . Thus, each time over the out-of-sample period is characterized by
a different time varying views since both the MSVAR(2,1) model and the VAR(1)
model produce time varying asset class expected returns and standard deviations
over time. As a consequence, given the views based on any model, during the out-
of-sample period a different returns posterior distribution is obtained for any model
in any period, as well as different portfolio optimization results
. The view's uniform
ranges are centered on the asset class expected returns and have width equal to
a standard deviation.
224
A practitioner can have K = N subjective views on linear combinations of the
market M. These linear combinations are represented by a (K × N) dimensional
pick matrix P: the generic k-th row of the pick matrix determines the weights of the
k-th view. As I express one absolute view for each asset class at any time over the
out-of-sample period, the pick matrix P reads:
(68)
[
1 0
0 1
0 0
0 0
0 0
0 0
1 0
0 1
]
The resulting invertible matrix:
(69)
(
)
defines the view-adjusted market coordinates.
can be any (N - K) × N matrix
such that P is invertible. However, for computational efficiency and accuracy
Meucci suggests to impose that the rows of
be orthogonal and that their norm
equal one. In other words, the N-dimensional random vector:
(70)
is fully equivalent to the market M. Then I rotate the market into the views
coordinates as in (70). In terms of the empirical panel representation of the market
prior (64) this operation corresponds to a simple matrix multiplication:
(71)
Each row of the (J × N) panel
is an independent joint realization of the market in
the views coordinates. The practitioner's views correspond to statements on the
first K entries of
. Each of these statements, i.e. the generic k-th subjective view,
is expressed in terms of a cumulative distribution function (cdf):
(72)
()
{
} = 1, ... ,
The prior distribution (63) also implies a distribution for each view, which is
represented by its cdf:
225
(73)
()
{
} = 1, ... ,
In gen
eral the practitioner's
views (72) are different than the respective market
implied distributions (73). I thus sort the first K columns of the panel (71) and
obtain two (J × K) panels W and C. The prior cdf (73) of the k-th view evaluated at
the sorted values is approximated with extreme accuracy by its empirical
counterpart:
(74)
(
,
)
+1
The generic element
,
of the panel C represents the ranking of the respective
entry of the panel
within its column, normalized by the total number of
simulations. Each row of the panel C represents a joint simulation of the formula
(77). In other words, the ranking panel C represents the copula of the views:
(75)
copula ranking
The COP approach resolves this dichotomy by means of opinion pooling
techniques. The posterior cdf is defined as a weighted average:
(76)
+ (1 -
)
= 1, ... ,
where the weights
represent the confidence in the respective view. Formula
(76) defines the posterior marginal distribution of each view. In the COP approach
the copula of the posterior distribution of the views is inherited from the prior
copula:
(77)
(
1
) =
d
(
1
(
1
)
(
)
)
The posterior marginal cdf's are approximated with extreme accuracy by their
empirical counterparts. From (69) it follows that the (J × K) panel
defined as:
(78)
,
(
,
) + (1 -
)
+1
226
represents
the posterior marginal cdf's (76
) evaluated in correspondence to the
respective entries in the sorted panel W. Therefore the posterior joint distribution
of the views is defined as follows:
(79)
(
V
1
V
K
) =
d
(
F
1
-1
(C
1
)
F
K
-1
(C
K
)
)
I can produce a (J × K) panel
of joint scenarios from the posterior distribution of
the views (79) by means of the linear interpolator. Finally, to determine the
posterior distribution of the market
~
M
I apply (70) backwards and rotate the
views back into the market coordinates:
(80)
=
d
-
where the first K entries of
follow the posterior distribution (79) and the
remaining entries are left unaltered. Empirically it is equivalent to rotate these
simulations back into the market coordinates as in (80) by means of a simple
matrix multiplication:
(81)
-
Each N-dimensional row represents an independent joint scenario from the COP
posterior market distribution.
As Meucci (2006) has already summarized, the COP approach starts from a fully
generic prior distribution (63) and yields a posterior distribution (81) that reflects
the pr
actitioner's views (
72). As in BL, the COP posterior distribution represents a
gentle, consistent twist of the original prior distribution, see Meucci (2006):
therefore, once fed into an optimization algorithm, it gives rise to sensible
allocations. In order to obtain the posterior it is needed to perform the
marginalizations (73), the copula factorization (77) and the quantile inversions
(79). None of these operations is feasible in closed analytical form, except in trivial
cases. Nevertheless, it is straightforward to perform these operations numerically,
as I proceed to show.
227
The COP posterior market distribution obtained for each period during the out-of-
sample is then used into a mean-variance and a conditional value-at-risk (CVaR)
portfolio optimization.
5.2 Out-of-sample mean-variance asset allocation exercise
based on COP and MSVAR(2,1) and VAR(1)
In this subparagraph the results of the mean-variance portfolio optimization based
on the COP posterior market distributions are shown.
The MSVAR(2,1) and VAR(1)
COP posterior multivariate market distributions obtained are then used to compute, for
each period, the asset classes expected returns and standard deviations that are fed into
a mean-variance portfolio optimization.
The same five recursive optimal portfolios as
in the in-sample period case have been built over the out-of-sample period.
Table 41
out-of-sample average optimal mean-variance portfolios expected
returns based on empirical copula opinion pooling asset classes
returns distribution
maximum Sharpe Ratio portfolio with opened lower budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0030
0.0024
30%
0.0044
0.0041
50%
0.0156
0.0074
70%
0.0070
0.0071
90%
0.0058
0.0054
maximum Sharpe Ratio portfolio with budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0036
0.0038
30%
0.0057
0.0059
50%
0.0338
0.0115
70%
0.0133
0.0082
90%
0.0093
0.0066
optimal weights with capital allocation and risk aversion coefficient = 1
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0049
0.0044
30%
0.0073
0.0070
50%
0.0706
0.0270
70%
0.0261
0.0136
228
90%
0.0172
0.0107
optimal weights with capital allocation and risk aversion coefficient = 3
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0063
0.0057
30%
0.0091
0.0082
50%
0.1798
0.0766
70%
0.0631
0.0303
90%
0.0397
0.0210
optimal weights with capital allocation and risk aversion coefficient = 5
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0068
0.0056
30%
0.0111
0.0093
50%
0.7413
0.3647
70%
0.2509
0.1266
90%
0.1529
0.0790
Table 42
out-of-sample average optimal mean-variance portfolios expected
Sharpe ratio based on empirical copula opinion pooling asset classes
returns distribution
maximum Sharpe Ratio portfolio with opened lower budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.2262
0.1432
30%
0.2262
0.1432
50%
0.1368
0.1209
70%
0.1847
0.1263
90%
0.2084
0.1300
maximum Sharpe Ratio portfolio with budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.2911
0.1923
30%
0.2911
0.1923
50%
0.1895
0.1526
70%
0.2226
0.1645
90%
0.2484
0.1710
optimal weights with capital allocation and risk aversion coefficient = 1
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4008
0.2699
30%
0.4008
0.2699
229
50%
0.2915
0.2178
70%
0.3128
0.2277
90%
0.3310
0.2372
optimal weights with capital allocation and risk aversion coefficient = 3
view confidence
MSVAR(2,1)
VAR(1)
10%
0.5982
0.4052
30%
0.5982
0.4043
50%
0.4734
0.3292
70%
0.4829
0.3354
90%
0.4924
0.3416
optimal weights with capital allocation and risk aversion coefficient = 5
view confidence
MSVAR(2,1)
VAR(1)
10%
1.1810
0.8065
30%
1.1810
0.8006
50%
0.9804
0.6624
70%
0.9847
0.6666
90%
0.9889
0.6709
A comparison of the out-of-sample expected portfolio returns and the expected
Sharpe ratio obtained using the COP posterior market distribution based on both
the MSVAR(2,1) and the VAR(1) can be conducted from the observation of Table
41 and Table 42. What emerges from an analysis of Table 41 is that in almost all
the cases, given any type of portfolios and any level of confidence in the views, the
portfolios based on the MSVAR(2,1) model greatly overperform those based on
the VAR(1) model in terms of greater average portfolio expected returns.
If the comparison is conducted in terms of realized Sharpe ratio, as depicted in
figure 42, again it can be seen that the portfolios built using the two states model
greatly overperform those built using the single state model. It can be seen that
the realized Sharpe ratio increases as a function of the risk aversion coefficient
and decreases as a function of the confidence level in the views, perhaps as a
consequence of the higher variability of the portfolio returns due to the posterior
market distribution which greatly depends on the time varying views.
230
Table 43
out-of-sample average optimal mean-variance portfolios realized
returns based on empirical copula opinion pooling asset classes
returns distribution
maximum Sharpe Ratio portfolio with opened lower budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0067
0.0044
30%
0.0091
0.0078
50%
0.0384
0.0141
70%
0.0148
0.0118
90%
0.0119
0.0072
maximum Sharpe Ratio portfolio with budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0068
0.0053
30%
0.0097
0.0098
50%
0.0591
0.0235
70%
0.0228
0.0111
90%
0.0164
0.0077
optimal weights with capital allocation and risk aversion coefficient = 1
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0079
0.0065
30%
0.0102
0.0100
50%
0.0979
0.0250
70%
0.0359
0.0136
90%
0.0238
0.0104
optimal weights with capital allocation and risk aversion coefficient = 3
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0066
0.0069
30%
0.0110
0.0095
50%
0.1606
0.0275
70%
0.0594
0.0167
90%
0.0390
0.0142
optimal weights with capital allocation and risk aversion coefficient = 5
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0073
0.0039
30%
0.0113
0.0096
50%
0.5634
0.1311
70%
0.1931
0.0504
90%
0.1190
0.0339
231
Table 44
out-of-sample average optimal mean-variance portfolios realized
Sharpe ratio based on empirical copula opinion pooling asset classes
returns distribution
maximum Sharpe Ratio portfolio with opened lower budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4904
0.3667
30%
0.4871
0.4429
50%
0.3097
0.2878
70%
0.3560
0.2588
90%
0.3925
0.1978
maximum Sharpe Ratio portfolio with budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4779
0.2729
30%
0.3969
0.3395
50%
0.3075
0.3484
70%
0.3546
0.2582
90%
0.4110
0.2138
optimal weights with capital allocation and risk aversion coefficient = 1
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4181
0.2760
30%
0.3874
0.3295
50%
0.2943
0.1789
70%
0.3229
0.2379
90%
0.3531
0.2443
optimal weights with capital allocation and risk aversion coefficient = 3
view confidence
MSVAR(2,1)
VAR(1)
10%
0.3453
0.3076
30%
0.3831
0.3083
50%
0.2407
0.0961
70%
0.2675
0.1656
90%
0.2922
0.2156
optimal weights with capital allocation and risk aversion coefficient = 5
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4058
0.1903
30%
0.3794
0.3089
50%
0.2433
0.1324
232
70%
0.2503
0.1521
90%
0.2573
0.1693
Overall it can be deduced from Table 43 and Table 44 that the conclusions drawn
from the observation of the expected portfolio returns and expected Sharpe ratio
are here substantially reproduced even though the benefits derived from the two
states model are decreased to some extent.
Table 45
out-of
sample optimal mean-variance portfolio cumulative realized
returns based on empirical copula opinion pooling asset classes
returns distribution (initial wealth = 1)
maximum Sharpe Ratio portfolio with opened lower budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
1.4873
1.2940
30%
1.4893
1.3613
50%
1.5845
1.4492
70%
1.4653
1.4911
90%
1.5343
1.2459
maximum Sharpe Ratio portfolio with budget constraint
view confidence
MSVAR(2,1)
VAR(1)
10%
1.7063
1.5822
30%
1.7534
1.7495
50%
1.8033
1.7699
70%
1.8772
1.7189
90%
1.9139
1.7265
optimal weights with capital allocation and risk aversion coefficient = 1
view confidence
MSVAR(2,1)
VAR(1)
10%
6.0868
2.1617
30%
9.9021
3.5597
50%
-13.0966
2.4393
70%
24.1332
0.1452
90%
-1.52E+09
125.5950
optimal weights with capital allocation and risk aversion coefficient = 3
view confidence
MSVAR(2,1)
VAR(1)
10%
2.2983
1.9021
30%
3.4282
1.8403
50%
5.7637
2.0457
233
70%
5.5426
2.0094
90%
-144.8532
0.2081
optimal weights with capital allocation and risk aversion coefficient = 5
view confidence
MSVAR(2,1)
VAR(1)
10%
1.9821
1.4811
30%
2.5359
1.5258
50%
3.5870
1.7607
70%
5.8193
2.0566
90%
-66.8947
2.2802
5.3 Out-of-sample conditional value-at-risk (CVaR) asset
allocation exercise based on COP and MSVAR(2,1) and
VAR(1)
In this subparagraph the results of the conditional value-at-risk portfolio
optimization based on the COP posterior market distributions derived from both
the MSVAR(2,1) model and the VAR(1) model are shown. In this exercise, for
each out-of-sample period the simulated multivariate scenarios are fed into a
CVaR portfolio optimization to obtain five different portfolios equally distributed
along the time dependent CVaR - expected return efficient frontier; with I being the
least risky and V the most risky portfolios. The exercise has been repeated for
values of the confidence in the views that span from 0.1 to 0.9. The CVaR
probability level has been set to 0.95 for each portfolios.
Table 46
out-of-sample average optimal CVaR portfolios expected returns
based on empirical copula opinion pooling asset classes returns
distribution
Risk level I
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0000
0.0000
30%
0.0000
0.0000
50%
0.0000
0.0000
70%
0.0000
0.0000
90%
0.0000
0.0000
Risk level II
234
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0016
0.0018
30%
0.0020
0.0019
50%
0.0025
0.0021
70%
0.0031
0.0024
90%
0.0036
0.0027
Risk level III
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0032
0.0037
30%
0.0039
0.0038
50%
0.0050
0.0043
70%
0.0061
0.0048
90%
0.0073
0.0054
Risk level IV
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0048
0.0055
30%
0.0059
0.0057
50%
0.0075
0.0064
70%
0.0092
0.0072
90%
0.0109
0.0081
Risk level V
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0063
0.0073
30%
0.0078
0.0076
50%
0.0100
0.0085
70%
0.0123
0.0096
90%
0.0146
0.0108
Table 47
out-of-sample average optimal CVaR portfolios expected Sharpe
ratio based on empirical copula opinion pooling asset classes returns
distribution
Risk level I
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0068
0.0103
30%
0.0146
0.0068
50%
0.0059
0.0013
70%
0.0285
-0.0080
90%
0.0863
0.0476
235
Risk level II
view confidence
MSVAR(2,1)
VAR(1)
10%
0.1105
0.0726
30%
0.1412
0.0933
50%
0.1928
0.1245
70%
0.2874
0.1773
90%
0.7093
0.4133
Risk level III
view confidence
MSVAR(2,1)
VAR(1)
10%
0.1105
0.0725
30%
0.1412
0.0932
50%
0.1928
0.1241
70%
0.2873
0.1766
90%
0.7086
0.4132
Risk level IV
view confidence
MSVAR(2,1)
VAR(1)
10%
0.1084
0.0665
30%
0.1385
0.0872
50%
0.1869
0.1182
70%
0.2746
0.1685
90%
0.6789
0.4034
Risk level V
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0811
0.0592
30%
0.0978
0.0775
50%
0.1475
0.1060
70%
0.2236
0.1522
90%
0.5637
0.3553
Similarly to what have been shown in the last paragraph, a comparison of the out-
of-sample expected portfolio returns and the expected Sharpe ratio obtained using
the COP posterior market distribution based on both the MSVAR(2,1) and the
VAR(1) can be conducted from the observation of Table 46 and Table 47. What
emerges from an analysis of Table 46 is that in almost all the cases, given any
portfolio risk level and any level of confidence in the views, the portfolios based on
the MSVAR(2,1) model greatly overperform those based on the VAR(1) model in
terms of greater average portfolio expected returns.
236
If the comparison is conducted in terms of realized Sharpe ratio, as depicted in
figure 49, again it can be seen that the portfolios built using the two states model
greatly overperform those built using the single state model.
Table 48
out-of-sample average optimal CVaR portfolios realized returns
based on empirical copula opinion pooling asset classes returns
distribution
Risk level I
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0000
0.0000
30%
0.0000
0.0000
50%
0.0000
0.0000
70%
0.0000
0.0000
90%
0.0000
0.0000
Risk level II
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0034
0.0038
30%
0.0033
0.0033
50%
0.0035
0.0030
70%
0.0036
0.0024
90%
0.0035
0.0025
Risk level III
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0068
0.0076
30%
0.0067
0.0065
50%
0.0070
0.0059
70%
0.0071
0.0048
90%
0.0069
0.0049
Risk level IV
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0100
0.0110
30%
0.0097
0.0098
50%
0.0102
0.0095
70%
0.0103
0.0074
90%
0.0097
0.0071
Risk level V
237
view confidence
MSVAR(2,1)
VAR(1)
10%
0.0114
0.0129
30%
0.0111
0.0111
50%
0.0124
0.0124
70%
0.0121
0.0127
90%
0.0119
0.0112
Table 49
out-of-sample average optimal CVaR portfolios realized Sharpe ratio
based on empirical copula opinion pooling asset classes returns
distribution
Risk level I
view confidence
MSVAR(2,1)
VAR(1)
10%
0.1255
0.1372
30%
0.2258
-0.1101
50%
0.1185
0.0852
70%
0.2216
0.1135
90%
0.0453
0.2176
Risk level II
view confidence
MSVAR(2,1)
VAR(1)
10%
0.5084
0.5011
30%
0.4098
0.3938
50%
0.3705
0.3672
70%
0.3615
0.3115
90%
0.3669
0.3070
Risk level III
view confidence
MSVAR(2,1)
VAR(1)
10%
0.5084
0.2464
30%
0.4098
0.2287
50%
0.3705
0.2163
70%
0.3587
0.1815
90%
0.3590
0.1844
Risk level IV
view confidence
MSVAR(2,1)
VAR(1)
10%
0.4630
0.3565
30%
0.3825
0.3465
50%
0.3562
0.3441
70%
0.3433
0.2781
238
90%
0.3231
0.2684
Risk level V
view confidence
MSVAR(2,1)
VAR(1)
10%
0.3080
0.2342
30%
0.2693
0.2403
50%
0.3084
0.2945
70%
0.3027
0.3190
90%
0.2959
0.2846
Overall it can be deduced from Table 49 and Table 50 that the conclusions drawn
from the analysis of the expected portfolio returns and expected Sharpe ratio are
here substantially confirmed even though the benefits derived from the adoption of
the two states model are decreased to some extent, especially for the riskiest
portfolio.
Table 50
out-of
sample optimal CVaR portfolio cumulative realized returns
based on empirical copula opinion pooling asset classes returns
distribution (initial wealth = 1)
Risk level I
view confidence
MSVAR(2,1)
VAR(1)
10%
1.0000
1.0000
30%
1.0000
1.0000
50%
1.0000
1.0000
70%
1.0000
1.0000
90%
1.0000
1.0000
Risk level II
view confidence
MSVAR(2,1)
VAR(1)
10%
1.2237
1.2568
30%
1.2181
1.2127
50%
1.2292
1.1947
70%
1.2353
1.1547
90%
1.2286
1.1583
Risk level III
view confidence
MSVAR(2,1)
VAR(1)
10%
1.4925
1.5625
30%
1.4770
1.4621
239
50%
1.5021
1.4158
70%
1.5111
1.3235
90%
1.4907
1.3306
Risk level IV
view confidence
MSVAR(2,1)
VAR(1)
10%
1.7901
1.8725
30%
1.7484
1.7570
50%
1.7954
1.7212
70%
1.7967
1.5218
90%
1.7370
1.5009
Risk level V
view confidence
MSVAR(2,1)
VAR(1)
10%
1.8974
1.9781
30%
1.8427
1.8204
50%
1.9953
1.9867
70%
1.9665
2.0340
90%
1.9364
1.8605
5.4 COP asset allocation exercise conclusions
In conclusion it can be said that the adoption of a regime switching model, rather
than a single regime model, in the attempt to model the market returns and to
provide out-of-sample market forecasts, which the construction of portfolios was
based on, in the copula-opinion pooling approach proved to be an overall
profitable choice at any level of confidence in the views and with any portfolio
optimization method implemented.
CHAPTER 6
CONCLUSIONS
From the point of view of this dissertation, it is important to note that all the time
series displayed significant deviation from the normal distribution benchmark, as
evidenced by the statistically significant normality tests results, and then they can
be considered not normally distributed. This is a clear indication that the returns on
the three asset classes cannot be captured by linear models to reinforce the idea
of a regime switching model, which is more flexible in accommodating the mixing
240
of several empirical distributions. The regime switching model is also supported by
the results of normality tests, as all null hypotheses are strongly rejected. All the
evidences presented so far suggest that the regime switching estimate of my
model appears to be related to the underlying economic fundamentals and
business cycle to some extent. From the analysis of the parameter estimates,
during regime 1 it seems reasonable to invest in equity, whose average realized
return outperforms that of bonds, while in regime 2 holding bonds seems to be
more profitable. The presence of a size effect brings to the further conclusion that,
regarding the stock market, a strategy that invests in small stocks in regime 1 and
large stocks in regime 2 seems to be appropriate and profitable.
From the analysis of the first part of the asset allocation exercise it can be said
that while the construction of portfolios based on the two regimes MSVAR(2,1)
model estimates does not lead to results that clearly overperform those based on
the single regime VAR(1) model estimates in the in-sample framework, it
significantly does it in the out-of-sample framework. Therefore the adoption of a
sophisticated multi regimes model greatly improves the portfolios performance in
the out-of-sample exercise for all types of portfolios constructed.
If the views based on the asset pricing models are taken into account, as in the
second part of the asset allocation exercise, it can be said that the adoption of a
regime switching model, rather than a single regime model, in the attempt to
model the market returns and to provide out-of-sample market forecasts, which the
construction of portfolios was based on, in the copula-opinion pooling approach
proved to be a profitable choice at any level of confidence in the views and almost
at any level of portfolio risk.
The findings reveal that when regime switching is taken into account, the optimal
portfolio weights deviate substantially from those that obtain under the single-state
under various portfolio selection frameworks. Thus the investment policies are
very steep with respect to the state variables, which results in volatile portfolio
241
policies. As a consequence, portfolio managers should be able to exploit these
regime changes to increase profitability.
This paper has illustrated the potential benefit of a portfolio asset allocation
exercise based on market returns forecasts produced by a markov regime
switching model, which implies its expected returns, their volatility, and their
correlations to be time varying and state dependent. On the one hand, I
exaggerate the performance of the model due to the fact that I do not take
transaction costs into account. However, I underestimate the potential of regime
switching models since I did not to estimate the best possible model, indeed I did
an extensive model search amongst only vector autoregressive regime-switching
models.
To sum up, the regime switching model has proven to be superior than the single
regime model due to its performance evaluated by Sharpe ratio and both average
expected and average realized portfolio returns. This evidences witness that
incorporating the regime switching features into an asset pricing model, which is
then used to feed the portfolio optimization process, is an effective and profitable
strategy under both favorable and adverse market conditions.
242
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Frequently asked questions
What is the main focus of this document?
This document is a language preview which comprehensively covers various aspects of asset allocation, particularly focusing on regime switching models and their effectiveness compared to single-regime models.
What key themes are explored?
The key themes include dynamic asset allocation, Markov regime switching models, multivariate time series analysis, portfolio optimization (mean-variance and CVaR), the Copula-Opinion Pooling approach, and analyzing the influence of economic factors like dividend yield and NBER recession indicators on asset returns.
What is the structure of the document?
The document is structured as follows:
- Introduction
- Literature Overview on regime-switching models
- Data description, including sources and descriptive statistics
- Model Estimation, including a theoretical background for Markov regime switching models and interpretation of results
- Asset Allocation, examining in-sample and out-of-sample performances based on various asset pricing models
- Asset Allocation based on the Copula-Opinion Pooling approach
- Conclusions
What kind of data was employed?
The data includes US small stocks (lo20), US large stocks (hi20), US Treasury bonds (tbond), the US 1-month T-Bill rate (risk_free), and the S&P 500 Dividend Yield (div_y). The time period under consideration is January 1965 – December 2010, followed by an out-of-sample period of January 2010 to December 2014.
What are the core objectives of the analysis?
The core objectives are to compare the performance of multivariate regime switching models against single regime models in the context of dynamic asset allocation and to investigate and make a comparison of the behavior of the regime switching model and the single state model in the COP framework.
What models are estimated?
Various Markov Switching Vector Autoregressive (MSVAR) models are estimated, including MMSIAH(k,p), MMSIA(k,0), MMSIAH(k,0), MMSIA(k,p), MMSIH(k,p), MMSI(k,p), and a multivariate restricted MSVAR(K,1) model. A single regime VAR(1) model is also estimated for comparison purposes.
What model selection criteria are used?
Akaike Information Criterion (AIC), Schwartz Information Criterion (SIC), Bayesian Information Criterion (BIC), and Hannan-Quinn criterion (HQC) are utilized to choose the best model specification.
What performance measures are examined?
The study examines Root Mean Squared Error (RMSE), Pearson correlation coefficient, regression analysis, expected portfolio returns, realized portfolio returns, and Sharpe ratios for portfolio performance comparison.
What is the Copula-Opinion Pooling (COP) approach?
The Copula-Opinion Pooling approach allows the investor to input views on the market, which are then combined with a prior market distribution obtained from the asset pricing model (MSVAR or VAR) to generate a posterior market distribution. This posterior distribution is used for portfolio optimization.
What are the key findings?
Key findings suggest the superiority of regime switching models against single-regime models due to their ability to capture economic regime changes. The investment portfolio performance heavily relies on the accuracy of asset returns forecasts, so a model that can account for relevant market features is crucial. However, there may be conflicting evidence of performance superiority from the two pricing models when analyzing Sharpe ratio or weights.
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen. -
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Laden Sie Ihre eigenen Arbeiten hoch! Geld verdienen und iPhone X gewinnen.
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