Für neue Autoren:
kostenlos, einfach und schnell
Für bereits registrierte Autoren
Masterarbeit, 2017
124 Seiten
1 THEORETICAL BACKGROUND AND MOTIVATION
1.1 Motivation
1.2 Relevant Literature on Forecasting
1.3 Comparative Statics in the Light of the Theoretical Framework
2 ARIMA MODELLING
2.1 ARMA / ARIMA
2.2 ARMAX / ARIMAX
2.3 ARIMA Error Model
2.4 The Augmented Dickey-Fuller Test
2.5 Information Criteria
2.6 Granger Causality
3 DATA
4 METHODS AND RESULTS
4.1 Ordinary Least Squares (OLS) Method
4.2 ARIMA Models
4.2.1 Building Steps and Unit Root Testing
4.2.2 Modelling Implications and Forecasting Horizon
4.2.3 Technical Data Analysis
4.2.4 Model Specification
4.2.5 Model-Based Outlier Adjustment and Forecasting Results
4.2.6 Residual Diagnostics and Unit Circle
4.3 ARIMA Error Models
4.3.1 Building Steps and Technical Analysis
4.3.2 Multicollinearity Testing
4.3.3 Causality Detection
4.3.4 Modelling Benchmarks
4.3.5 ARIMA Error (1,1,0) with a Drift
4.3.6 ARIMA Error (1,1,0) without a Drift
4.3.7 Experimental Model
5 CONCLUSIONS
Figure 1. Four Quadrant Model
Figure 2. Decrease in Capitalization Rate / Required Rate of Interest
Figure 3. Increase in Demand of Space
Figure 4. Increase in Supply of Space
Figure 5. Raw Data Levels of the Research Variables over the Training Set 1995-2015 (N=80)
Figure 6. The Real Estate Price Index in Finland 1995-2015
Figure 7. Logarithmic Levels of the Research Variables over the Training Set 1995- (N=80)
Figure 8. Forecasting Results for an Equivalent I(1) OLS Regression (1995.25-2016.75)
Figure 9. Building Mechanism of ARIMA Models
Figure 10. Seasonal ARIMA(1,1,0) with a Drift Forecasted 50 Periods Ahead (Raw Data) .
Figure 11. Seasonal ARIMA(1,1,0) Forecasted 50 Periods Ahead (Raw Data)
Figure 12. ARIMA(1,1,0)(0,1,0) Forecasted 50 Periods Ahead (Raw Data)
Figure 13. Non-Cumulative Density of Non-Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram
Figure 14. Logarithmic Differences of Real Estate Index in Finland (1995-2015)
Figure 15. Non-Cumulative Density of Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram
Figure 16. Logarithmic Seasonal Differences of Real Estate Index in Finland (1995-2015)
Figure 17. Non-Cumulative Density of the Differences of Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram
Figure 18. Differences of Logarithmic Seasonal Differences of Real Estate Index in Finland (1995-2015)
Figure 19. Trend Component of the House Price Series 1995-2015 (Logarithms)
Figure 20. Seasonal Component of the House Price Series 1995-2015 (Logarithms)
Figure 21. Random Variation Component of the House Price Series 1995-2015 (Logarithms)
Figure 22. Autocorrelation Function (ACF) for House Price Index in Finland 1995-2015 ..
Figure 23. Simulated MA(1) Process with a Coefficient Value of 0.7 for Randomly Generated Data Based on the Normal Distribution
Figure 24. Simulated MA(1) Process with a Coefficient Value of -0.7 for Randomly Generated Data Based on the Normal Distribution
Figure 25. Partial Autocorrelation Function (PACF) for House Price Index in Finland 1995- 2015
Figure 26. Simulated AR(1) Process with a Coefficient Value of 0.7 for Randomly Generated Data Based on the Normal Distribution
Figure 27. Simulated AR(2) Process with Coefficient Values of 0.35 for both AR Parameters and Randomly Generated Data Based on the Normal Distribution
Figure 28. ARIMA 1 Outliers in Logarithmic Form
Figure 29. ARIMA 2 Outliers in Logarithmic Form
Figure 30. AR(1) Outliers in Logarithmic Form
Figure 31. Final Forecasting Results for the ARIMA 1 and Holt & Winters Specifications with Long-Term Trend Dynamics Against the Control Model (2015.25-2016.75)
Figure 32. Final Forecasting Results for the ARIMA 2 and AR(1) Specifications without Long-Term Trend Dynamics Compared Against Realized Price Development (2015.25-2016.75)
Figure 33. Residuals Generated by ARIMA 1 Adjusted for Outliers
Figure 34. Residuals Generated by ARIMA 2 Adjusted for Outliers
Figure 35. Residuals Generated by AR(1) Adjusted for Outliers
Figure 36. Unit Circle for ARIMA 1
Figure 37. Unit Circle for ARIMA 2
Figure 38. Unit Circle for AR(1)
Figure 39. Building Mechanism of ARIMA Error Models
Figure 40. Seasonal Moving Averages of Logarithmic Research Variables (1995-2015)
Figure 41. Outliers for ARIMA Error (1,1,0) with a Drift
Figure 42. Forecasting Results Generated by ARIMA Error (1,1,0) with a Drift Against the Realized Price Evolution in Index Points (1995.25-2016.75)
Figure 43. Residuals Generated by ARIMA Error (1,1,0) with a Drift
Figure 44. Unit Circle for ARIMA Error (1,1,0) with a Drift
Figure 45. ARIMA Error (1,1,0) without a Drift Outliers
Figure 46. Forecasting Results Generated by ARIMA Error (1,1,0) without a Drift Against the Realized Price Evolution in Index Points (1995.25-2016.75)
Figure 47. Residuals Generated by ARIMA Error (1,1,0) without a Drift
Figure 48. Unit Circle for ARIMA Error (1,1,0) without a Drift
Figure 49. Ad Hoc Mortgage Rate Variable (Y-Axis) Employed in Experimental Model and Mortgage Rate in Terms of Raw Data (X-Axis)
Figure 50. Logarithmic Mortgage Rate Variable, Simple Logarithmic Mortgage Rate and Mortgage Rate in Terms of Raw Data (1995-2015)
Figure 51. Experimental Model (2013) Outliers in Logarithmic Form
Figure 52. Forecasting Results for Experimental Model Against the Actual Price Evolution in Terms of Index Points (1995.25-2016.75)
Figure 53. Residuals Generated by Experimental Model
Figure 54. Unit Circle for the Experimental Model
Table 1. OLS Regression Statistics Based on the Logarithmic Variables (1995-2015)
Table 2. ADF Test Results for Construction, Wages, Mortgage Rate and House Prices for AR(l4) Lag Length (Logarithms)
Table 3. ADF Test Results for Construction, Wages, Mortgage Rate and House Prices for AR(l12) Lag Length (Logarithms)
Table 4. Parameter Values for ARIMA Models Before Outlier Adjustment
Table 5. ARIMA Model Comparison Before Outlier Adjustment (1995-2015)
Table 6. Forecasting Errors for Endogenous Models Expressed in Index Points Before Outlier Adjustment (2015.25-2016.75)
Table 7. ARIMA Model Comparison After Outlier Adjustment (1995-2015)
Table 8. Parameter Values for Outlier-Adjusted ARIMA models
Table 9. Forecasting Errors for Endogenous Models in Terms of Real Index Points After Outlier Adjustment (2015.25-2016.75)
Table 10. Ljung-Box Test for Candidate Models
Table 11. Direct Granger Causality Test for the Independent Variables
Table 12. ARIMA Error (1,1,0) with a Drift
Table 13. Ljung-Box Test for ARIMA Error (1,1,0) with a Drift
Table 14. ARIMA Error (1,1,0) without a Drift
Table 15. Ljung-Box Test for ARIMA Error (1,1,0) without a Drift
Table 16. Experimental Model for ARIMA-Based Modelling
Table 17. Ljung-Box Test for the Experimental Model
There is an abundance of existing literature regarding time series forecasting and the housing market. This thesis evaluates the performance and statistical adequacy of several time series models in the context of real estate price forecasting in Finland. Each statistical model is applied so that forecasts are generated over the seven quarters following the training sample. The resulting forecasts are compared against realized price development. Model evaluation is carried out from the viewpoint of forecasting errors in the validation period, statistical fit, modelling constraints and success in the light of the theoretical framework.
It was concluded that scarce ARIMA-based models are suitable for short-term real estate price forecasting in the concerned setting. The models were built on logarithmic I(1) nominal data and augmented with seasonal dummy variables. The Chen & Liu (1993) structural anomaly detection method enhanced statistical fit in the training period and forecasting accuracy in the validation period. The inclusion of a drift parameter generally led to inflated forecasting results in the validation period. The RMSE and MAE error statistics produced by the best ARIMA-based models remained well below 0.5 % in the validation period. The Holt & Winters and I(1) OLS models were also outperformed by the most adept ARIMA-based models.
The independent variables were chosen along with the four quadrant framework introduced by DiPasquale & Wheaton (1992). Introduction of exogenous factors generally improved the forecasting performance exhibited by ARIMA(p,1,q)-based models, which realized in terms of forecasting error statistics. Theoretical equivalence to the four quadrant framework was achieved to a large extent. Mortgage rate displayed negative correlation with real estate prices whereas disposable income was positively correlated with the price level. However, the number of new construction permits displayed a positive relationship with the FIN real estate price index.
Keywords
Forecasting, Real Estate, ARIMA
Additional information
This thesis aspires to forecast real estate prices in the short-term with univariate and multivariate statistical models. Most of the models introduced in this thesis belong to the category of autoregressive integrated moving average (ARIMA) models. ARIMA-based models are adaptive and they can be augmented with exogenous factors.
There are many historical examples of economic upswings and crises having resulted from the housing market, subprime mortgage crisis being one of the most notable recent examples of such occurrences. House prices are a significant factor from the viewpoint of overall economy. The aggregate of mortgages also supports the entire economy through financial multiplier effects.
Investors in the real estate sector include pension funds, real estate funds and households. Obtaining an accurate outlook on the developments of rents and prices is essential from the viewpoint of institutional investors and their portfolio decisions. The issue of real estate prices is also significant from the perspective of banks holding loans and mortgages secured with real estate assets. Real estate prices display persistent trends over time and are susceptible to sizeable up and downswings. (Al-Marwani 2014: 1, 3, 37)
Real estate prices also influence the consumption tendencies of households, which has a significant impact on macroeconomic circumstances. The financial behavior of households is affected by lifetime consumption smoothing. The position was introduced by Milton Friedman (1957) in his permanent income hypothesis. The permanent income hypothesis posits that people smooth their consumption over lifetime so that it matches the expectations regarding their income in the future. (Friedman 1957: 25-30.)
Consumer’s income and consumption are constituted of transitory and permanent components. The relationship between permanent income and consumption is regulated by the rate of interest at which the consumer can borrow and lend, wealth and consumer utility gained by consuming with respect to utility from wealth additions. (Friedman 1957: 25-30.)
According to the permanent income hypothesis, households will generally consume more if their perceived wealth stored in property values increases. This pattern of consumer behavior is referred to as “wealth effect” by Miller et al. (2011). The consumption of households is also dependent on the collateral effect of real estate prices, which means that the aggregate borrowing constraint faced by households is loosened when real estate prices increase ceteris paribus. Empirical evidence is mostly consistent with these two theories. (Miller et al. 2011: 2-4, 24-25)
According to Miller et al. (2011), real estate prices are a significant factor in gross metropolitan product (GMP) growth. The growth effects of predictable changes in house prices are analogous to collateral effect, whereas wealth effect is manifested in unpredictable price changes. The effects on GMP growth are approximately three times stronger for collateral effect compared to wealth effect. Moreover, the wealth effect decreases and the collateral effect is emphasized when households face tough financing conditions. The growth effects of house prices last up to eight quarters and peak four quarters after the concerned hike or decline. (Miller et al. 2011: 2-4, 24-25)
Miller et al. (2011: 2-4, 24-25) infer that if an economic recession was caused by liquidity issues of households (collateral effect) instead of a decline in their perceived wealth (wealth effect), outcomes of financial stimulus policies would yield more favorable results. However, if a recession is caused by the fact that households have a lesser view of their wealth, financial stimulus may not shore up economic recovery in a sufficient way. It ought to be noted that property values should be supported by fundamentals and not just perceived wealth in order to orchestrate successful central bank policies.
Moreover, Leamer (2007) states that a downturn in residential investment predicts a following macroeconomic decline. The housing market has been involved in the onset every US recession since WW2 except for the one that was related to Korean war in the early 1950’s. Long-run growth patterns cannot be attributed to housing, but shifts in real estate investment do anticipate short-term macroeconomic fluctuation. (Leamer 2007: 10-11, 17)
T-statistics and temporal regression analysis based on the components of GDP point out that residential investment, consumer durables, consumer non-durables and consumer services have a significant contribution to GDP growth. Because all the variables are measured in terms of money, it can be stated that an abnormal contribution to GDP from the residential investment sector predicts twice as much contribution to GDP in the next quarter. The beta coefficient for the variable was two which was higher than for any other sectoral variable. (Leamer 2007: 24.)
In addition to residential investment, consumer goods variable had a multiplier effect on GDP growth with a beta coefficient of 1.7. The rest of the beta coefficients remained below one and the coefficient for consumer durables was actually negative and significant, which is probably due to the temporal and discreet nature of the regression. Some of the variables, such as business spending on equipment and software, did not contribute to GDP growth in a significant way. However, the simple regression is of transitory nature and residential investment is not a large factor when it comes to long-run economic growth. (Leamer 2007: 24, 8)
Leamer (2007) further deduces that housing cycles should be in the least taken into account as monetary policy is being implemented. From the perspective of monetary policy, the output gap of the Taylor Rule could be replaced with the number of housing starts and change in them reflecting the level of new building in a forward-looking way. Taking residential investment into account in the decision-making process faced by the incumbent parties would assist the officials as they decide on the key interest rates and monetary policy. (Leamer 2007: 3.)
It would be beneficial to use residential investment as a proxy for monetary policy instead of Taylor’s output gap because this would allow more proactive central bank conduct earlier on in the business cycle. The housing market is not as sensitive to interest rates in the middle of monetary expansion as it is towards the end of it. Hence, anti-inflation policies should be executed early enough from the viewpoint of the housing market, so that monetary contraction would not severely damage the overall economy through excessively inflated housing market. This type of conduct would decrease the frequency of recessions and make them less serious. (Leamer 2007: 3.)
On the aggregate macroeconomic level, the housing market is dependent on sales volumes and not on house prices. The housing market is extremely sticky to adjust downward. Hence, the adjustment realizes through a change in sales volumes and not via a decline in real estate prices. The aggregate volume of sales in monetary terms is important for the overall macroeconomic situation. Price variation along with time is higher for older homes in comparison with newer ones, which indicates that the volatile component of real estate is the land. (Leamer 2007: 25-28.)
Vishwakarma (2013) compared out-of-sample forecasting capabilities of ARIMA, ARIMAX and ARIMAX-GARCH models. ARIMAX and ARIMAX-GARCH can be viewed as desirable models because they both consider exogenous variables. Volatility is taken into account in an advantageous way by the ARIMAX-GARCH model fitted for the ARIMAX residuals.
The variables employed by Vishwakarma (2013) were used in the log-difference form and they included Canadian real estate index (RREI), being the dependent variable, gross domestic product (GDP), the difference between the long-term and short-term interest rates, consumer price index (RCPI) and the RCAD/USD exchange rate. The use of macroeconomic covariates was justified by previous research relating to equity forecasting. (Vishwakarma 2013: 4-11.)
Vishwakarma (2013: 4) states that the use of gross domestic product as an explanatory variable was appropriate in the light of previous literature. Karolyi and Sanders (1998: 248-260) determined that real estate investment trust (REIT) returns are highly susceptible to economic variables while the exposure to bond and stock market risk premiums is lower. Furthermore, Chen, Roll and Ross (1986: 387) also implemented a consumption variable in the context asset price forecasting.
Vishwakarma (2013: 4) concluded that the difference between long-term and short-term interest rates is predictive of house prices in the sense that investors holding long-term equities, such as real estate, are rewarded with comparatively better yields when the gap between these two rates is higher. The position was justified by Fama & French (1992: 449-464) and Stock & Watson (1989: 150-160).
Vishwakarma (2013: 6-8) performed unit root testing by utilizing the Dickey-Fuller and Augmented Dickey-Fuller methods in order to verify variable stationarity. Furthermore, the Box-Jenkins (1976) methodology for ARIMA modelling was implemented in the research. The research concluded that ARIMA-based models performed well in short-term forecasting, but lacked sufficient accuracy for long-term forecasting.
Vishawakarma (2013: 7-12) used a modified ARMA order for ARIMAX modelling and fitted a GARCH model for the resulting ARIMAX error terms. The research concluded that the ARIMAX model produced the most accurate projections regarding short-term price variation in the Canadian real estate market as ARIMAX(1,1,1) was tested against the ARIMA(5,1,1) and ARIMAX-GARCH(1,1) models. The ARIMAX model performed consistently well in terms of predicting trends and turning points of real estate prices while the other two ARIMA family models failed to generate reliable short-term forecasts in alternating circumstances. The difference between the forecasts produced by ARIMA and ARIMAX-GARCH methodologies was broad as the forecasted values remained far apart from each other. (Vishwakarma 2013: 12.)
A flaw relating to the research of Vishwakarma (2013) is that the way seasonality was addressed in the research was not specified nor was a reason pointed out not to address seasonality. Multicollinearity testing was not delineated by Vishwakarma (2013) either. Variance inflation factor (VIF) or correlation matrix could have been used for the purpose. Possible correlations between the independent variables pose a threat of inaccuracy when it comes to the relations between individual coefficient values. Structural change parameterization was not elaborated in the research either, which could have been carried out by employing the Chen & Liu (1993) method.
Karakozova (2004: 51-73) found that ARIMAX models were more suitable for return forecasting in the Helsinki office market in comparison with regression and error correction models (ECM). The employed ECM models only performed adequately over some periods while the ARIMAX methodology generally succeeded better in the detection of market irregularities. Hence, the ARIMAX model delivered a more consistent forecasting performance. ARIMAX models involve independent variables in addition to moving average and autocorrelation terms. In the research, the concerned covariates were the previous values of capital growth, GDP growth and increase in service sector employment.
Al-Marwani (2014) conducted real estate price forecasting in the United Kingdom and employed the Box-Jenkins methodology for ARMA modelling instead of merely relying upon black box information criterion values, such as the AIC and BIC. OLS based models were also employed in the analysis. Real estate price data used in the research was very extensive spanning from the first quarter of 1952 until the first quarter of 2011. However, the data reflecting other variables, such as property type, neighborhood and economic statistics, was of shorter length with a time span from the 1990’s to the early 2010’s depending on the dataset. The data used in the research was of quarterly kind. (Al-Marwani 2014: 49-65.)
The employed forecasting methodology did not apply the standard seasonal ARIMA, or the so-called “SARIMA” modelling, but instead utilized ARIMA(p,1,q) and OLS models augmented with seasonal dummy variables. ARIMA(p,1,q) is equivalent to ARMA built on the first differences. In the methodology, ARMA residuals were fitted in a multiple regression with seasonal dummy variables to inspect if the model had captured seasonality without further considerations. Models were created so that the balance between model scarcity and sufficient explanatory power was maintained. (Al-Marwani 2014: 47-68.)
Model residuals were expected to follow the normal distribution. The implemented methodology was advantageous enough to consider residual autocorrelations and independence. When it comes to model evaluation, the R^{2} statistic was used for the simple OLS regression model whereas the adjusted R^{2} was utilized in the context of ARIMA models. The F-test was used to determine the significance of the adjusted R^{2} statistic. In the out-of-sample forecasting stage, models were compared against the naïve forecasting model based on the root-mean-squared error (RMSE) statistics. (Al-Marwani 2014: 49-59)
Al-Marwani (2014: 61-122) investigated the time series properties associated with the UK house price index and different real estate property types, such as detached houses, terraced houses and flats. The price development of each property type was compared against the overall real estate price evolution in the United Kingdom. Various ARMA models were built and specified for different property types and the concerned house price index.
In the last section of the research, Al-Marwani (2014: 123-156) synthesizes geographic information systems (GIS) with time series forecasting in a novel way. In this section, it was investigated how real estate price formation is connected to social and economic factors. The associated factors included, for example, environment, health and economic welfare variables, such as income.
The research conducted by Al-Marwani (2014: 157-162) pointed out that local price forecasting is more accurate compared to forecasting a larger aggregate, such as a country-wide index which is a common forecasting setting. Furthermore, the research pointed out that higher local real estate prices predict more green spaces and a more favorable socio-economic situation in a given area. The research could have benefitted from further consideration when it comes to structural changes in the data. It is usually recommended to test multicollinearity and cointegration relationships as well when multivariate time series forecasting is carried out. However, the research was useful and it really addressed the setting in which real estate time series forecasting is carried out.
Crawford and Fratantoni (2003: 223-243) compared out-of-sample real estate price forecasting capabilities of GARGH, ARIMA and regime switching univariate models in several parts of the United States. According to their findings, ARIMA models are appropriate for out-of-sample forecasting, but their performance is worse in point forecasting. The researchers also found that ARIMA models are more suitable for time series forecasting than overly complex models.
Tse (1997) applied ARIMA models in the forecasting of Hong Kong office and industrial building prices. This type of a forecasting setting was chosen in order to prevent interference caused by government interventions. In the research, quarterly data was employed spanning from the first quarter of 1980 until the second quarter 1995. The Dickey-Fuller and augmented Dickey-Fuller tests were utilized to address data stationarity. (Tse 1997: 153-160.)
Tse (1997: 153) pointed out that forecasts built on disaggregated data are inclined to be more accurate compared to the use of aggregate data, which was in concord with the work of Al-Marwani (2014). For example, property types can be forecasted separately. Nominal data was deflated by Tse (1997: 154) in order to ease stationarity attainment. The chosen method was considered favorable from the viewpoint of over-differencing avoidance. Excessive differencing is a source of information loss.
The price level of real estate is susceptible to trends over time (Al-Marwani 2014: 50). Tse (1997: 154) states that mere differencing is the most popular approach for de-trending and stationarity achievement when it comes to time series forecasting. Tse (1997: 153-160) determined that the data could not be adjusted for structural changes due to data limitations. Furthermore, it was denoted that ARIMA models are adaptive enough to sustain structural changes. This position is highly controversial considering the work of Chen & Liu (1993) and Junttila (2001).
Tse (1997) concluded that the ARIMA (2,1,1) model was suitable for real estate price forecasting in the Hong Kong setting under scrutiny. This is a remarkable finding considering the simplicity of the model. Forecasts were compared against the naïve forecast. Root-mean-square error (RMSE) was used for forecasting accuracy evaluation. (Tse 1997: 156-160.)
Van den Bossche et. al (2004) investigated how weather, policy and economic conditions pertain to traffic accidents in Belgium from the perspective of severity and frequency. The model was used in the production of 12 months ahead out-of-sample forecasts. The researchers implemented ARIMA error modelling for the purpose and monthly data was utilized in the research paper. The ARIMA error model provided a valid statistical fit within the 95 % confidence interval. The most important finding of the research was that weather conditions and political regulations are major explanatory factors when it comes to severity and frequency of traffic-related accidents. On the contrary, economic conditions are not a major contributor in the accident formation process. (Van den Bossche et. al 2004: 5, 13-19)
Oikarinen (135-149) studied short-run and long-run real estate price dynamics in the Helsinki metropolitan area. He examined the long-run relationship between real house prices, real aggregate income, real after-tax mortgage rate and loan-to-GDP ratio. The trace type Johansen test was availed in order to investigate the existence of long-run relationship between the research variables. The long-run relation was positive for real income and loan stock, but negative for interest rate, which is intuitive and expected. The long-run relationship implied a slight overvaluation of the housing market in the Q2 of 2006.
Furthermore, Oikarinen (2007: 139-145) employed error correction models (ECM) in order to explain the short-run dynamics of the concerned housing market. Over 60 % of the variation in house prices was explained by lagged variables only. Furthermore, movements in house prices were considered highly predictable in the short-term. However, the parsimonious ECM model employed by Oikarinen (2007: 143-145) could not anticipate the stern price increase that took place over the Q3 and Q4 of the same year.
Oikarinen (2007: 147-149) concluded that less than 10 % of the deflection from the long-run relation vanishes over a quarter as a result of the adjustment in house prices. The research also pointed out that there is a two-way interaction between borrowing and house prices. Because the adjustment of house prices is sluggish and statistical models can be primed with backward-looking capabilities, short-term movements in house prices can be predicted by market fundamentals.
This thesis addresses house price evolution in the short-term. The housing market dynamics are relevant from the viewpoint of understanding the causalities behind the issue of real estate prices. Wheaton and DiPasquale (1992: 181-197) approached real estate price formation by introducing the so-called “four quadrant framework”. (DiPasquale & Wheaton 1992: 188-189.)
The model is presented in Figure 1. The two quadrants on the right represent the market of space, while the left-hand quadrants depict the asset market for real estate. The level of rent is defined in the property market and the asset market turns this level into real estate prices. Demand for space is a function of rent and macroeconomic conditions. The supply side is given by the housing stock which adjusts through construction as a reaction to prevailing asset prices. The asset and property markets are in equilibrium when the starting stock equals the finishing stock. If the finishing stock is lower (higher) than the starting stock, real estate prices, rents and construction must rise (decrease) in order to reach a new equilibrium. (DiPasquale & Wheaton 1992: 181- 197)
illustration not visible in this excerpt
Figure 1. Four Quadrant Model
Rent level (R) is determined in the NE quadrant and obtained from the equilibrium in the property market. In this equilibrium, the demand and supply for space coincide so that (D = S). The demand of space depends on economic factors, such as income, population dynamics and rent. The supply of space is given by the housing stock. (DiPasquale & Wheaton 1992: 187-188.)
The price level (P) of real estate is determined in the NW (north-west) quadrant. The quadrant has two dimensions: rent and price. The formula of real estate price formation is (P = R / i) in which R represents rent defined in the NE quadrant whereas i is for capitalization rate. (DiPasquale & Wheaton 1992: 187-188.)
From the viewpoint of the model, capitalization rate is an important concept. It is depicted by the ray passing from the origin to the NW quadrant in a 45-degree angle with respect to Y axis. Capitalization rate represents the relevant rent-to-price ratio and the yield required by investors in order to hold real estate assets in their portfolios. Lower capitalization rate implies higher real estate values since investors demand less return. (DiPasquale & Wheaton: 187-188.)
The SW quadrant depicts new construction dependent on real estate prices. The ray drawn through the SW quadrant represents the cost of construction that is a function of construction f(C). Cost of construction is expected to increase along with building rate, which explains the direction of the ray with respect to price and construction. The line intersects with the price axis at the price required to build e.g. one square meter of dwelling space. The line does not start from the origin since there is always a positive price associated with construction. The level of new construction is determined by the asset market equilibrium in the NW quadrant. In the equilibrium, construction costs, depicted in the SW quadrant, equal asset prices. Construction is regulated by real estate prices. (DiPasquale & Wheaton 1992: 188-189.)
The long-term housing stock adjustment through construction and depreciation is determined in the SE quadrant and defined by housing stock (S), stock depreciation (d) and construction level (C) as Δ S = C - dS . In the steady state of construction, new construction equals stock depreciation and Δ S = 0. Hence, the level of construction in the steady state can be defined as C = dS and the stock equals C/d = S. (DiPasquale & Wheaton 1992: 188-189.)
The line that emanates from the origin through the SE quadrant represents the steady state of stock at which building equals depreciation. The SE quadrant uses construction as an input and gives the value of the housing stock which would be achieved if that building rate was continued to the infinity. If the starting stock exceeds the finishing stock, construction, real estate prices and rents have to rise in order for the market to reach a new equilibrium. On the contrary, the opposite is true for a situation in which the original stock is smaller than the finishing stock. (DiPasquale & Wheaton 1992: 188-189.)
The first comparative static is displayed in Figure 2. A general decrease in interest rates, improved perceived risk characteristics of real estate or more favorable tax treatment lowers the required return on real estate assets, which generates a counterclockwise rotation in the ray drawn through the NW quadrant. Lesser required return on real estate assets increases the price level of real estate. Given the transition in the capitalization rate, the resulting situation is followed by new construction and stock adjustment in the SW and SE quadrants, respectively, which will eventually lower the rent level. (DiPasquale & Wheaton 1992: 192-193.)
illustration not visible in this excerpt
Figure 2. Decrease in Capitalization Rate / Required Rate of Interest
In this thesis, the previous transition is considered analogous to a decrease in mortgage rate. Real estate for personal use is often considered “the largest investment of one’s life” and a great deal of national wealth has been invested in real estate assets. Households, real estate funds and pension funds also own real estate assets which have been acquired purely for rental income and appreciation purposes. The real estate assets purchased by households and investment funds are often financed with debt. Mortgage rate is expected to be negatively correlated with the price level.
illustration not visible in this excerpt
Figure 3 demonstrates a situation in which the demand curve for space (D) shifts out from the original location due to improvements in macroeconomic conditions or local population growth. The transition leads to higher rents, which in turn increases the price level (P) of real estate determined by the asset market. The housing stock in the SE quadrant will eventually adjust to the positive shock through construction. (DiPasquale & Wheaton 1992: 191.)
illustration not visible in this excerpt
Figure 3. Increase in Demand of Space
In this thesis, the previous transition is interpreted analogous to increase in disposable income. Aggregate disposable income also considers population dynamics. Disposable income is expected to be positively correlated with real estate prices, which is logical and in accordance with the four quadrant framework.
A positive shock in the supply of space may occur due to eased short-term credit market conditions faced by the construction industry, favorable zoning decisions, improved construction technology or enhanced supply of labor. This type of changes cut construction costs. The housing stock also increases in size as a result of new construction. Rents and real estate prices eventually decrease as a consequence of a positive supply shock. Figure 4 portrays these dynamic transitions. (DiPasquale & Wheaton: 194-196.)
illustration not visible in this excerpt
Figure 4. Increase in Supply of Space
Wheaton and DiPasquale (1992: 194) point out that the real estate sector in the United States experienced a credit shortage in the early 1990’s despite generally lower short-term rates. This is an example of a negative shock in the supply of space through new construction ceteris paribus. (DiPasquale & Wheaton 1992: 194-196.)
The housing stock is modelled through construction in this thesis because of better data availability in comparison with the existing housing stock. Construction is expected to display a negative correlation with house prices in the light of the theoretical framework. The adjustment of the housing stock is still gradual, which may not be fully captured by the construction variable.
The four quadrant framework describes comparative statics that take place in the real estate market. The theoretical assumptions set by the framework are intuitive and largely falsifiable. DiPasquale & Wheaton (1992) posit that house price formation is based on macroeconomic circumstances, interest rate and space conditions.
The presentation of AR(p) process displayed in Equation 1. In the equation, x t reflects the dependent variable at moment t and ϕ p stands for an autocorrelation parameter at moment t-p. Furthermore, µ = mean and ε t ∼ i.i.d(0, σ^{2} ). Autocorrelation refers to a situation in which present time series values correlate with past observations. (Hamilton 1994: 53-55, Hyndman & Athanasopoulos 2014: 223-224.)
illustration not visible in this excerpt
The presentation of MA(q) process is displayed in Equation 2. In the equation, θ q stands for a moving average parameter at moment t-q. Furthermore, µ = mean and ε t ~ i.i.d(0, σ^{2} ). The moving average model assumes that present values can be forecasted based on previous error terms. (Hamilton 1994: 48-51, Hyndman & Athanasopoulos 2014: 224-225.)
illustration not visible in this excerpt
ARMA(p,q) order denotes the number of autocorrelation and moving average parameters with letters p and q. In the presentation of ARMA(p,q) displayed in Equation 3, moving average and autocorrelation parameters are defined as earlier. Furthermore, µ is a mean parameter and ε t ~ i.i.d(0, σ^{2} ). The model can be viewed as a combination of AR and MA processes. (Greene 2012: 983, Shmueli & Lichtendahl 2016: 152, Athanasopoulos 2014: 214-234.)
illustration not visible in this excerpt
The distinction between ARMA and ARIMA models is that ARIMA models include the degree of differencing required to make the underlying data stationary . Therefore, ARIMA order is hence defined as (p,d,q), in which d denotes the degree of differencing. Differencing procedure can be perceived through the use of backshift operators. Backshift operator, B, is defined so that Bhxt = xt-h. Differencing procedure is hence defined so that (1-Bh)xt, in which h is the length of differencing. These two notations are important from the perspective of ARIMA-based models introduced in this thesis. (Mellin 2007: 27-34, Hyndman & Athanasopoulos 2014: 215-223.)
ARIMA models are commonly presented with backshift operators. Standard ARIMA(p,d,q) model is displayed in Equation 4 by using backshift operators. Equation 5 is an abridged version of the previous equation. In these equations, µ = mean, B = backshift operator and d = order of differencing. The rest of the model parameters are defined as earlier in this section. (Mellin 2007: 27-34, Athanasopoulos 2014: 220-242.)
illustration not visible in this excerpt
The complete order is specified as (p,d,q)(P,D,Q)s for an ARIMA model which includes seasonal and non-seasonal elements. The non-seasonal ARIMA order is defined as earlier, but the concerned capital letters stand for the equivalent seasonal autocorrelation, seasonal order of integration and seasonal moving average parameters while s displays the length of season. It is to be noted that the seasonal and non-seasonal components are actually multiplicative. (Hyndman & Athanasopoulos 2014: 242-258, Mellin 2007: 35-40, 135.)
Double differences, the non-seasonal and seasonal schemes, can be considered as well, which translates into (1-B)(1-Bs) differences. It is also relatively common to consider (1-Bs)(1-B) instead when ARIMA modelling is performed. The seasonal ARIMA presentation (SARIMA) has been written by using backshift operators in Equation 6 for the sake of completeness. The non-seasonal ARMA parameters, order of differencing, mean and error term are defined as earlier in this section, but the equivalent seasonal parameters are specified as follows: = seasonal moving average parameter, Φ = seasonal autocorrelation parameter, s = length of season and D = order of seasonal differencing. (Hyndman & Athanasopoulos 2014: 242-258.)
illustration not visible in this excerpt
The Box-Jenkins methodology can be utilized when ARIMA modelling is carried out. The first stage of the methodology is model identification. Sufficient stationarity has to be reached in this stage. The number of consecutive spikes in the autocorrelation function (ACF) indicates the MA(q) order whereas the number of sequential spikes in the partial autocorrelation function (PACF) indicates the AR(p) order. Significant spikes in these graphs at seasonal intervals indicate SMA(Q) and SAR(P) processes. (Reagan 1984: 21-24.)
The second part of the Box & Jenkins methodology is the model estimation stage. Non-linear functions are estimated often based on the maximum likelihood model, which is commonly carried out with statistical software or programming language. Model parameters should be significant, uncorrelated, stationary and invertible. (Reagan 1984: 25-29.) Candidate models are ranked according to relevant information criteria and modelling benchmarks. Outliers and structural anomalies can be left out of the sample or parameterized so that outlier effects are properly considered. (Shmueli & Lichtendahl 2016: 154-155, Chen & Liu 1993: 285.)
The third part of the Box & Jenkins methodology is model evaluation. The estimated model is sufficient if the concerned residuals display white noise. Residual autocorrelation must be tested by implementing, for example, the Ljung-Box (1978) test. If the model is concluded insufficient, the reviewer returns to the identification stage to discover a more suitable statistical model. (Reagan 1984: 27-29.) Time series can be divided into training, validation and forecasting parts to compare the forecasting performance of different models (Shmueli & Lichtendahl 2016: 150).
ARIMAX(p,d,q) model is a covariate-augmented version of the standard ARIMA presentation. In Equation 7, general ARIMAX model is presented in order to render an intuitive and equivalently relevant mathematical presentation of the model along with the example of Vishwakarma (2013: 5). In the model, µ = mean, zt-i stands for a vector of explanatory variables and β i provides the concerned parameter coefficient conditional upon the model structure. Lag length for the vector of exogenous variables is denoted with k. The rest of the parameters are defined as earlier in this section. The variables are treated as differences in the presentation.
illustration not visible in this excerpt
ARIMAX(p,d,q)(P,D,Q)s can be rendered by applying seasonal differencing, but two is generally treated as the maximum degree of differencing. Another approach to account for seasonal variation in the context of ARIMAX modelling is the use of seasonal dummy variables when the first differences are considered.
The ARMAX model for stationary data can be written with zero mean by using backshift operators as well as the endogenous ARMA presentation. The backshift modification has been carried out in Equation 8 and Equation 9. Backshift operator, B, and model parameters are defined as earlier in the section 2.1.
illustration not visible in this excerpt
Van den Bossche et. al (2004: 11-13) implemented the ARIMA error model in the examination of severity and frequency of traffic-related deaths in Belgium. The model with ARMA errors can be considered a linear multiple regression in which the behavior of error terms is explained by an ARMA process, which allows more intuitive mathematical presentation in comparison with the ARMAX model. (Hyndman & Athanasopoulos 2014: 259-271.)
The ARMA error model displayed in Equation 10 and Equation 11. Equation 10 includes a vector of explanatory variables defined as zt while β is the associated factor coefficient . Moreover, forecasting errors produced by the baseline model are included and depicted in Equation 11. As presented, the error term defined as Nt follows an ARMA process with autocorrelation and moving average parameters relating to past forecasting errors produced by the linear baseline model. Moving average and autocorrelation parameters are defined as earlier in the section 2.1.
illustration not visible in this excerpt
The model with ARMA errors can be re-written in an intuitive way by using backshift operators and zero mean. The ARMA error model specification with backshift operators is displayed by Equation 12. An abridged version of the same specification is shown in Equation 13. (Hyndman & Athanasopoulos 2014: 259-271.)
illustration not visible in this excerpt
The transition to the model with ARIMA errors requires that the dependent and the explanatory series are differenced before fitting the model with ARMA errors. Differencing operator is defined as (1-B). The effect of this procedure is identical to the mere inclusion of a differencing operator before the autocorrelative series so that (1-B) ϕ (B). In Equation 14, the matrix of independent variables is denoted with zt-i. The number of included lags is indicated with letter k. It is implied in the presentation that lag t-0 is the first lag to be considered in the model. The rest of the parameters are defined as earlier. (Hyndman & Athanasopoulos 2014: 259-271.)
illustration not visible in this excerpt
1] When the XREG argument is specified for exogenous variables in the R environment, the Arima() [forecast], auto.arima() [forecast] and arima() [stats] functions produce an ARIMA error model. Seasonality can be parameterized and even forced as these functions are called. The forecasting results are essentially analogous to the ARIMAX methodology.
The possibility of unit roots has to be taken into account in order to verify data stationarity when ARIMA-based models are created i.e. variance and mean cannot change over time when time series forecasting is carried out. Stationarity can be examined by performing the Augmented Dickey-Fuller test on the dependent and explanatory variables.
The ADF test is presented mathematically in Equation 15. In the specification,
µ = constant, β = trend coefficient, p = autoregressive process lag order,
ε t i.i.d.(0, σ^{2} ). The process under scrutiny follows a random walk with a drift if
β = 0 and equivalently a random walk if µ = 0 is imposed too. Trend stationary specification leaves both parameters free. (Greene 2012: 988-997, Hyndman & Athanasopoulos 2014: 220-221.)
illustration not visible in this excerpt
Conventional t ratio is a common way to calculate the ADF test statistic and it is depicted in Equation 16. Under the null hypothesis γ = 1. The concerned test statistic is compared against a suitable table of critical values as unit root existence is evaluated. If it can be concluded that γ < 1, the null hypothesis of unit root presence is rejected. Failure to rejected the null is taken as evidence of a unit root. Another approach for ADF test statistic calculation is displayed in Equation 17. (Greene 2012: 994.)
illustration not visible in this excerpt
An alternative formulation of the test can be obtained by subtracting yt-1 from both sides of the equation. The formulation after the transition has been displayed in
+ +
Equation 18, in which ϕ j = GHjR' \G and γ * = ( FH' \F) - 1. In this form, the ADF
test can be performed by testing the null hypothesis of γ*= 0 against γ*< 0.^{10}. The DFτ test statistic can be used for the purpose. (Greene 2012: 994.)
+
illustration not visible in this excerpt
The augmented Dickey-Fuller test differs from the original Dickey-Fuller test in the sense that it is augmented with lagged autocorrelations. Choosing the correct lag length (p) is an important question when unit roots are tested with the ADF test. Lag length can be adjusted according to information criteria, such the AIC or BIC, or a fixed number of lags can be used.
For example, one might use four lags for quarterly data. However, the choice of lag length is always ad hoc. Excessive lag length makes the ADF test inefficient whereas the inclusion of too few lags does not properly factor in all the autocorrelations. (Brooks 2008: 327-329.) Non-stationarity can be dealt with by obtaining logarithms of the variables. If the data remains non-stationary despite the logarithmic quality, it can be further differenced to resolve the problem (Pfaff 2011: 57-59).
Joint density of the observation set for x 1 , ... ,xt, is specified so that f(x1, x2 .. xt)| θ, in which x is the observed outcome which is conditioned on the parameter set denoted with θ . Furthermore, ∫ ... ∫ f(x1, x2 ... xn; θ )dx1 … dxn =.1 . Joint density function is written as a function of the data conditioned on the parameters. The likelihood function is written L(θ|xt), depicting likelihood parameters, θ, given the outcome of x instead. Hence, it is reverse presentation of joint density function: joint density is treated as a function of likelihood parameter vector values conditioned on the underlying data.
(Greene 2012: 549-551.)
An important mathematical implication is that ∫...∫ f( θ ; x1, x2 .. xn)d θ 1 … d θ n ≠ 1. However, the likelihood function is always positive. The model can be extended to multivariate case. Maximized likelihood is used in many common economic applications. The maximum likelihood technique finds the values of parameters to maximize the probability of reaching the observed data. (Greene 2012: 549-551; Hyndman & Athanasopoulos 2014: 225.)
The logarithmic likelihood function is used when information criterion values are calculated. Information criteria are black box values that reflect statistical fit provided by a given statistical model. The Akaike information criterion (AIC) is a commonly used measure of statistical fit utilized as a benchmark in this thesis along with other modelling aspects. An ideal statistical model minimizes the AIC and maximizes the log likelihood function with respect to competing candidate models. The AIC value is calculated based on Equation 19. In the equation, L represents the maximized value of likelihood function with respect to the observed data and k stands for the number of free parameters to be estimated. (Akaike 1981: 3-14; Greene 2012: 573.)
2m - 2op(q) (19)
The Bayesian Information Criterion (BIC) is a different way to measure statistical fit. The most significant difference between the AIC and BIC is that the latter one penalizes parameter inclusion more severely. The BIC is calculated as shown by Equation 20 in which n stands for the number of observations, k is the number of model parameters and L represents the maximized likelihood function. A suitable statistical model minimizes the BIC value and maximizes the log likelihood function. (Greene 2012: 573.)
op(p)m - 2op(q) (20)
A common approach for statistical fit evaluation is to consult both of these benchmarks instead of solely relying on either one of them. However, it can be erroneous to rely only on the black box criteria when time series modelling is carried out because there are other modelling benchmarks of which endorsements can be in contradiction with information criteria.
The granger test can be utilized to find out if the lagged values of a time series can be used to forecast another time series when autocorrelations are included. The method is based on linear regression. The correlations of lagged values can be tested in order to address the issue. The test is specified as in Equation 21, in which x and y represent two separate time series, t represents moment and ε t ∼ i.i.d(0, σ^{2} ) stands for the error term. Furthermore, µ is a constant parameter. The test can be carried out at different lag lengths. Excessive lag length may cause spurious regression issues. Including too few lags may cause bias due to the occurrence of residual autocorrelation. (Granger 1969: 431, Brooks 2008: 298.)
illustration not visible in this excerpt
Engle-Granger representation theorem exists for more advanced long-run dynamics. Error correction model (ECM) can be formulated in the I(1) form if a stationary residual series can be obtained from the level regression yt ~ µ + xt + ε t . The I(0) series of residuals is lagged and then used as an error correction term (ECT) in the I(1) regression for the lagged first differences of x and y. Negative coefficient sign of ECT indicates a cointegration relationship whereas coefficient volume is for the adjustment speed towards the long-run equilibrium. The method can be augmented to multivariate case. (Pfaff 2011: 77-78.)
A fairly simple model that displays similar supply and demand dynamics present in the theoretical framework was used by Drake (1993: 1225-1228). Furthermore, Berglund (2007: 9-10) employed a similar forecasting model. The set of exogenous factors employed in this thesis was inspired by these models. The forecasting setting with respect to independent variables is demonstrated through a simple OLS model. In Equation 22, the price level of real estate (HPI) is the dependent variable, β 0 stands for a constant. Granted construction permits (C), disposable income (I) and mortgage rate (M) are independent variables.
illustration not visible in this excerpt
It certainly is attractive to deem that these three explanatory factors are pivotal from the viewpoint of real estate price formation. The exogenous variables were chosen along with the work of Drake (1993: 1225-1228), Oikarinen (2007: 135-148), Berglund (2007: 9-10) and Viswakarma (2013: 7). The research variables used in this thesis are also present in the theoretical framework introduced by Wheaton and DiPasquale (1992). The employed variables include 1. Construction, 2. Income,
The total length formed by the training and validation samples is 87 observations. In this thesis, future forecasting was not carried out due to the limited sample. The training period includes 80 observations spanning from the first quarter of 1995 until the third quarter of 2016. Forecasts were produced seven periods ahead and compared against the realized price development.
Tse (1997: 160) states that ARIMA forecasting requires 50 observations. Hyndman & Kostenko (2007: 12-15) pointed out that the significance of sample size is highly case-specific for seasonal models. From a technical viewpoint, the required sample size is dependent on the degree of random variation associated with the data. The more random variation there is, the larger sample is necessitated to carry out ARIMA-based time series forecasting.
An analysis that bears a certain degree of resemblance to this thesis was carried out by Berglund (2007: 3) with less than 50 annual observations. The approach can be considered statistically inappropriate, but possibly useful from a practical viewpoint. Real estate market forecasting is essentially about finding relevant market factors and considering suitable modelling approaches. (Miller & Sklarz 1986: 99-108.)
The research variables are displayed in Figure 5 as raw data. Quarterly form was used in the graph. New construction is the number of granted construction permits, disposable income is measured in millions of euros, mortgage rate is presented in percentage points and house price index (HPI) is measured in index points. The number of building permits was aggregated to the quarterly frequency for the graphical presentation, but it was originally obtained as monthly data. The house price index was modified so that it starts from the Q1 of 1995. The way research variables were implemented in this thesis will be delved into hereafter in this chapter.
illustration not visible in this excerpt
Figure 5. Raw Data Levels of the Research Variables over the Training Set 1995-2015 (N=80)
Log transformation was the minimum data adjustment used in the statistical models of this thesis excluding one control model produced on raw data. Logarithmic data was utilized by Leamer (2007: 7), Berglund (2007: 10-15) and Vishwakarma (2013: 5-7). Logarithms were used in order to reduce heteroscedasticity, smooth variation, turn multiplicative seasonality into additive seasonality and to make each series well-behaved. Economic processes are often modelled as logarithms in the field of time series forecasting.
Number 10 was used as a base for the logarithms of income, construction and house prices in every graphical presentation and calculation of this thesis because of better intuition in absolute terms compared to the Napier’s constant. No mathematical grounds exist to prefer either one of these base numbers nor does the base number influence parameter values reached when maximum likelihood estimates are calculated. Napier’s constant is often used in the field of econometrics because interest rate compounding over time can be carried out based on the Napier’s constant.
Vishwakarma (2013: 4) considered inflation in the building process of ARIMAX. Furthermore, Tse (1997: 154) deflated the data with consumer price index to achieve stationarity. In this thesis, all the variables are used in the nominal form because nominal prices are the point of interest. Real estate prices are measured in terms of money and not in terms of consumer price index (CPI). Berglund (2007: 3) used nominal variables too. Logarithmic transformation also imposes similar effects with deflating to a certain extent.
Consumer price index also includes confounding factors due to the inflation measurement technique employed in Finland. For example, mortgage rate is an input variable of the CPI basket. This is confounding because mortgage rate is also one of the employed independent variables. A decline in average mortgage rate would decrease the observed CPI, which would further inflate real house prices ceteris paribus in an arbitrary way. This would increase the coefficient sign of mortgage rate in absolute terms. On the contrary, if CPI was considered as a separate variable, multicollinearity problem would be evident. Inflation adjustment was not needed to strengthen stationarity either because log transformation and differencing sufficiently verified stationarity.
The dependent variable used in this thesis is the house price index in Finland for all the housing types. The data was obtained from Statistics Finland (2017). Year 2000 was treated as a base year in the original index series so that 2000=100. The data was converted into an index that starts from 100 index points at the start of 1995 for research purposes. The dependent variable is especially meaningful in the context of ARIMA-based modelling and charged with a great deal of explanatory significance. Real estate price index values over the training period are displayed in Figure 6 as raw data. The variable was still implemented as a logarithm in the modelling stage. One limitation associated with the house price variable is that the housing market is sticky downward (Leamer 2007: 25). Oikarinen (2007: 139) states that sticky downward adjustment applies in the least to nominal prices which are the point of interest in this research.
illustration not visible in this excerpt
Figure 6. The Real Estate Price Index in Finland 1995-2015
In this thesis, the reserve of dwellings proposed by Drake (1993: 1225-1228) was not considered because of poor data availability. Depreciation could not have been considered in a satisfactory way for the complete housing reserve either. Furthermore, the number of dwellings on the market is not a constant share of the housing stock.
Hence, the housing stock was not used as an explanatory variable and construction was considered instead. The variable represents new construction in the theoretical framework and not the total housing stock in Finland. The use of a construction variable was considered a logical and executable approach in the light of the four quadrant framework. Changes in supply are expected to be negatively correlated with the price level of real estate based on the theoretical foundation.
However, Leamer (2007: 42) states that housing starts are probably negatively correlated with unemployment, which in indicates that the short-term relation may be positive after all. Furthermore, the use of a construction variable was analogous to Berglund (2007: 10), but the correlation sign of “new dwellings” obtained in the research was in contradiction with the theoretical assumptions set by the four quadrant framework.
The description of the construction variable is the number of admitted building permits for residential dwellings on the monthly basis in Finland (Statistics Finland 2017). This is the form in which the data was obtained. The variable was employed as the average number of units per month on the quarterly basis, which provided better equivalence to the rest of the variables in terms of coefficient value. The approach also allowed more intuitive graphical presentation for logarithms. Construction was used as a logarithm along with the rest of the variables. The number of new building permits is clearly a seasonal process, so logarithmic transformation is expected to be beneficial from the viewpoint of statistical modelling.
Vishwakarma (2013: 7) used gross domestic product (GDP) as an explanatory variable to reflect demand in the context of ARIMA-based forecasting with exogenous factors. The use of income was appropriate based on the work of DiPasquale and Wheaton (1992), Drake (1993: 1225-1228), Case et al. (2006: 17-29), Oikarinen (2007: 135-145), Berglund (2007: 10-15) and Al-Marwani (2014: 41-44, 156-161) too. Disposable income was treated as a quarterly accrual variable and it was hypothesized to be positively correlated with real estate prices. The data was obtained from Statistics Finland (2017).
The relationship between house prices and mortgage rate is expected to be negative based on the four quadrant framework. Oikarinen (2007: 136-141) observed a negative impulse response on house prices from mortgage rate and sluggish adjustment. Changes in the loan stock were also positively correlated with house prices, which was indicated by the short-run ECM models. Vishwakarma (2013: 4) pointed out that the difference between short-term and long-term interest rates was negatively correlated with house prices. The lifecycle model used by Berglund (2007: 15) also predicted a negative relationship between the rate of interest and the price level of real estate.
Hence, mortgage rate on new contracts was used as one of the three independent variables in this thesis. Mortgage rate data was attained from the Bank of Finland (BOF). The variable reflects the interest rate on new mortgage contracts. The variable considers the current borrowing position faced by consumers that aspire to purchase real estate at a given moment (t). The mortgage rate variable does not depict the aggregate average rate in the portfolio held by the banking sector. Mortgage rate has remained above 1.1 % across the entire sample (N=87) including the validation period. Many reference rates, such as Euribor 12M, have recently displayed negative values. The direct utilization of mortgage rate is accurate in comparison with a reference rate because the method includes the profit margin imposed on borrowers by the banking sector.
Berglund (2007: 14) implemented real interest as a logarithmic series and stated that this was possible because the concerned rate has not been negative in the research period. The research posits that a common way to operate with the logarithm of interest rate is to consider a semi logarithmic relationship instead. Another typical approach is to consider logarithm so that Log10(1+r), in which r = 0.01 for 1 %. The approach was adopted in this thesis. Number 10 was employed as a base number in order to maintain equivalence to the rest of the variables and sufficient intuition in absolute and graphical terms. Interest rate compounding is often carried out by using Napier’s constant as a base number instead, but no mathematical grounds exists to prefer natural logarithm over the chosen approach.
Disposable income is measured in millions of euros on the quarterly basis, so the scale of variation vastly differs from the equivalent fluctuation interval of mortgage rate, which explains the difference in the coefficient signs associated with these two variables in the statistical models introduced in this thesis. Simple logarithms of the research variables were considered in Figure 7 in order to render an intuitive graphical presentation. Number 10 was used as a base number. The real estate price index starts from 100 index points, which translates to two (2) in logarithmic terms.
illustration not visible in this excerpt
Figure 7. Logarithmic Levels of the Research Variables over the Training Set 1995-2015 (N=80)
2] Figures 5-7 were produced by first turning the data into time series objects with the ts() [stats] function and then specifying the plot() [graphics] and legend() [graphics] functions in the R environment.
The first model of this thesis was very similar to the simple model introduced by Drake (1993) with 80 observations (N=80). Berglund (2007: 10) also employed an OLS method in the same context. The concerned research variables were chosen to be used in the level OLS model because these variables are expected to exhibit similar comparative statics present in the four quadrant framework. The real estate index is treated as the dependent variable whereas construction, income and mortgage rate are the independent variables. The utilization of seasonal dummy variables was considered too risky in the level OLS context. Implementation could have also led to erroneous inflection points in the I(1) forecasting stage because of the absence of ARMA parameters and lagged variable values.
The regression results are displayed in Table 1 which has been computed for logarithmic level variables. The results were statistically significant considering the high R-squared value and p-values within the 99 % significance bounds for every research variable excluding construction of which statistical significance was merely within the 95 % significance level.
Based on the logarithmic level OLS regression results, it is apparent that income has the most significant effect on the house price index. The coefficient value for income was remarkably high (1.007). The coefficient is extreme in the light of the variation scale that is 4.232 - 4.645. The concerned T-statistic was also very significant and it was recorded at 19.981. The regression sign of income was according to the hypothesis set by the theoretical framework and previous literature.
Construction does not have a very notable impact on real estate prices according to the OLS regression results. Moreover, the correlation sign of construction was positive and not negative as it was hypothesized in the theory section. The intuition is that whenever the monthly average of new construction permits adds one, the logarithm of real estate price index increases by 0.042. This type of a transition would require extreme fluctuation considering that adding one is equivalent to being multiplied by number 10 when it comes to logarithms.
Berglund (2007: 10) made similar observations regarding the sign of construction that suffers from the lack of logic in the light of the theoretical foundation. The house price index is calculated for used dwellings. However, there are plausible explanations for the positive correlation sign. Construction companies may simply anticipate demand in a correct way. Residential investment and building predict economic upswings and downturns as proposed by Leamer (2007: 24, 8). In addition to macroeconomic situation, building is dependent on interest rate environment. When new real estate is purchased old dwellings are often renovated and new consumer goods are acquired, which stimulates economic activity and also directly increases property values associated with the housing stock through improvement in technical and aesthetic condition of dwellings.
Moreover, consumers who buy new real estate are often wealthy and end up selling their expensive dwellings right before buying a freshly constructed home, which can distort the statistics in favor of a positive correlation between the real estate index and construction. New dwellings of freshly built housing complexes may also end up sold forward as used ones shortly after building has taken place, which could also explain the slightly positive relationship.
Furthermore, imperfect information regarding the price of newish dwellings is mitigated when new housing complexes are completed close to them and sold to consumers, which can facilitate trade of newish and more expensive used dwellings. Metropolitan dynamics also improve as a result of new construction and zoning decisions, which increases the valuation characteristics of the existent stock.
3] The OLS regression was produced in the R environment by parameterizing the lm() [stats] function.
The relationship between mortgage rate and real estate prices was negative according to the OLS level regression as it was hypothesized. The relation was also significant in terms of p-value and statistical impact. The coefficient value displayed by the variable was also remarkable (-3.910). The rate of interest was considered in the orthodox Log10(1+r) form, in which r = 0.01 = 1 %. The variation scale of the mortgage rate variable has been minuscule in the training period, which explains the extreme coefficient. T-Statistic value was still relatively high (-5.546) for mortgage rate.
An analogous OLS model for the first differences was produced based on the same regression dynamics present in the level OLS model. Time series forecasting is commonly carried out in the differenced context. The I(1) OLS for the research variables was forecasted seven periods ahead. The model performed moderately well in terms of forecasting when it was used in the I(1) form. The forecasting performance has been displayed in Figure 8.
Table 1. OLS Regression Statistics Based on the Logarithmic Variables (1995-2015)
illustration not visible in this excerpt
N is the number of observations for each variable. R^{2} represents the coefficient of determination. Statistical significance for each variable of the OLS regression is presented in the following way: *** Statistically significant at the 1 % significance level,
** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level. (P-values)
illustration not visible in this excerpt
Figure 8. Forecasting Results for an Equivalent I(1) OLS Regression (1995.25-2016.75)
The use of endogenous ARIMA models is a common way to model time series data. The ARIMA modelling of this thesis takes part in the broader tradition of ARIMA-based time series forecasting. Building scheme of ARIMA specifications in the univariate setting is presented in Figure 9.
illustration not visible in this excerpt
Figure 9. Building Mechanism of ARIMA Models
Since outlier and structural change detection is model-based in this thesis, stationarity of research variables was evaluated before the introduction of ARIMA-based models. Research variables were first reviewed visually based on graphs and autocorrelation functions. Logarithmic levels appeared non-stationary based on this inspection.
The possibility of unit root presence was further addressed by employing unit root tests. The ADF test was carried out for the dependent and independent variables in the logarithmic, log-difference, logarithmic seasonal difference and difference of logarithmic seasonal difference forms. Separate results were produced under three different assumptions regarding each time series. These assumptions were:
1. No Constant or Trend 2. Constant 3. Trend. Different critical values apply to different ADF test types classified according to these assumptions.
Unit root test results were produced for the AR(l4) and AR(l12) lag length benchmarks. These lag benchmarks are equivalent to four (4) and twelve (12) lags. The employed benchmarking concept was introduced by Schwert (1989: 147-155). Tse (1997: 155) used one and four lags in the ADF testing stage. The choice of lag length is ad hoc. Lag length could have been alternatively specified according to the AIC or BIC.
The ADF test results are displayed in Tables 2 and Table 3. Lag specification is adjustable according to information criteria or a fixed number of lags can be used. The null hypothesis in the ADF test is unit root presence. The more negative the test statistic is, the less likely unit root presence is considered.
4] ADF testing was conducted in the R environment by implementing the ur.df() [urca] function.
Table 2. ADF Test Results for Construction, Wages, Mortgage Rate and House Prices for AR(l4) Lag Length (Logarithms)
illustration not visible in this excerpt
N is the number of observations for each variable under scrutiny. The sequence of value starts from the first quarter of 1995. Lag length was selected based on the AR(l4) benchmark. Statistical significance of the ADF test for each variable is presented in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level,
* Statistically significant at the 10 % significance level. (P-values)
Table 3. ADF Test Results for Construction, Wages, Mortgage Rate and House Prices for AR(l12) Lag Length (Logarithms)
illustration not visible in this excerpt
N is the number of observations for each variable under scrutiny. The sequence of value starts from the first quarter of 1995. Lag length was selected based on the AR(l12) benchmark. Statistical significance of the ADF test for each variable is presented in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level,
* Statistically significant at the 10 % significance level. (P-values)
In the ADF testing stage, it was discovered that none of the level variables were stationary without the inclusion of a constant or a trend. However, trend and drift inclusion generally improved the results in terms of stationarity, but did not render the set of variables stationary. All the first differences and seasonal differences of logarithmic level variables were stationary within the 95 % confidence interval without the inclusion of a trend or a constant in the AR(l4) lag specification. By using the (1-B^{4} )(1-B) differencing scheme, 99 % certainty of stationarity was achieved for all the variables in both lag selection schemes. The Chen & Liu (1993) outlier parameterization method was expected to further improve stationarity characteristics of the time series. Seasonality inclusion with dummy variables will further harmonize variation.
Unit root testing evoked implications that relate to constant inclusion for ARIMA models. ARIMA-based models can be adjusted so that a constant parameter is allowed. However, constant inclusion has to be justified by the underlying data. Adding a constant to differenced logarithmic data implies a polynomial trend which is called “drift” in the context of ARIMA(p,1,q) models.
The following equations demonstrate drift effects. ARIMA(p,d,q) model can be written: (1 - ϕ 1 B - … - ϕ p Bp)(1 − B)d yt = µ + (1 + θ 1 B + … + θ q Bq) ε t . In the model d = order of differencing, B = backshift operator, ϕ p = autoregressive parameter, θ q = moving average parameter, µ = constant and ε t ~ (i.i.d). It is to be noted that yt is differenced by using a differencing operator (1 − B)d. An alternative way to write the same model is: (1 - ϕ 1 B - … - ϕ p Bp)(1 − B)d (yt - α td/d!) = (1 + θ 1 B + … + θ q Bq) ε t. Constant, µ , is defined as follows: µ = α (1 - ϕ 1 - … - ϕ p ), in which α is the sample mean for the (1−B)d yt differences. As demonstrated by these ARIMA dynamics, the inclusion of a constant parameter in the I(1) context renders a polynomial trend in terms of I(0) data. The polynomial trend is of order determined by the order of differencing. The slope of this trend is analogous to the sample mean associated with the first differences in the context of ARIMA(p,1,q) models. (Hyndman & Athanasopoulos 2014: 238-239, 260.)
Drift imposes trend development when it comes to the long-term forecasting horizon whereas long-term forecasts built on I(1) data without a drift will exhibit an implicit non-zero constant and flat development in the long-run without further trend dynamics. The short-term forecasting horizon is still affected by recent impulses. Short-term forecasts are, hence, dependent on these impulses with respect to ARMA parametrization. (Hyndman & Athanasopoulos 2014: 238-239, 260.)
The level form of an ARIMA error model is based on the following regression presentation: Yt = β 0 + β 1 t + Nt, in which t = time and Nt = ARIMA(p,1,q). ARIMA order posits that the data has to be differenced once in order to reach stationarity and the I(1) form. The resulting equation is now Yt = Yt-1 + β 1 + N't. Thereby, the I(0) constant drops out when the first differences are considered so it does not influence the forecasts or statistical fit exhibited by the model. However, the trend parameter remains in the I(1) equation as a drift parameter. (Hyndman & Athanasopoulos 2014: 265-266.)
It is appropriate to demonstrate the dynamics imposed by drift inclusion graphically. An univariate ARIMA(1,1,0) model with a drift and seasonal dummy variables was built on raw real estate price data. The model was forecasted 50 periods ahead from the end of the training set (N=80). Mean forecast is reflected along with the 80 % and 95 % confidence intervals in Figure 10. The application of a drift parameter increased likelihood considering the underlying training set.
5] When the Arima() [forecast] function is implemented in the R environment, mean inclusion is set by default “TRUE” for any undifferenced time series. On the contrary, mean inclusion is set to “FALSE” when a differenced time series is considered, which does not influence the forecasts or statistical fit displayed by any ARIMA model in differences, which was demonstrated in this section. The auto.arima() [forecast] function considers a drift parameter if drift inclusion improves information criterion values for the model. Figures 10-12 were produced with the forecast() [forecast] and plot() [graphics] functions in the R environment. (Hyndman & Yasmeen 2017: 10, 13.)
illustration not visible in this excerpt
Figure 10. Seasonal ARIMA(1,1,0) with a Drift Forecasted 50 Periods Ahead (Raw Data)
Likelihood-based AIC and BIC information criteria supported constant inclusion for I(1) ARIMA models built on logarithmic data, but there are other modelling benchmarks to consider as well. The data will not necessarily exhibit a trend based on the mean displayed by the first differences in the training set, which can be attributed to recent developments of the time series. The drift setting was employed for the ARIMA 1 model of which long-term forecasting horizon prospect is analogous to the one displayed in the previous figure.
Valipour et al. (2012: 332-333) specified ARIMA models with and without a constant in order to investigate if the data necessitates a constant. In the research, optimal model structure was determined by RMSE statistics. When it comes to this thesis, error statistics could deteriorate in the validation period as a result of drift inclusion. The ARIMA 2 and AR(1) models were built under an expectation of price saturation. The inclusion of a drift was deemed inappropriate for these models built on logarithmic data because constant inclusion would inflate the forecasting results and lead to erroneous outcomes considering the flat recent development of house prices. Considering that the dependent series has displayed a negative growth pattern in the most recent quarters of the training period (N=80) and flat slope, drift exclusion was justified.
The effects of a drift parameter could not be properly compensated by employing structural change parameterization nor would such a parameterization process be justified by the structural anomaly detection method of Chen & Liu (1993). The ARIMA 2 and AR(1) model for the first differences will render forecasts that are flat in the long-term because they are dependent on the latest observations.
The models are still very relevant from the perspective of short-term forecasting. The forecasting horizon is very short as it only consists seven quarters. The exclusion of a drift parameter from such a model imposes an expectation of stable development because there implicitly exists a non-zero constant for level data when the first differences are considered without a drift. (Hyndman & Athanasopoulos 2014: 238- 239, 260, 265.)
ARIMA(1,1,0) built on raw data with seasonal dummy variables was forecasted 50 periods ahead in order to demonstrate the effects rendered by drift exclusion in the long-term. The forecasting results are displayed in Figure 11. The 80 % and 95 % confidence bounds are reflected along with the mean forecast. This type of an “impulse forecasting method” works well when it comes to short-term forecasting despite flat price expectation in the long-term.
illustration not visible in this excerpt
Figure 11. Seasonal ARIMA(1,1,0) Forecasted 50 Periods Ahead (Raw Data)
For the sake of completeness, modelling based on the second differences was also addressed in this thesis. A drift parameter in the I(2) context would produce a quadratic trend in levels. The AIC and BIC values did not support drift inclusion for an I(2) univariate ARIMA model. It is remarkable that the development of real estate prices has been negative when forecasting is carried out based on the second differences i.e. acceleration of house prices. (Hyndman & Athanasopoulos 2014: 238-239, 260.)
The ARIMA(1,1,0)(0,1,0) model displayed in Figure 12 was built on the (1-B^{4} )(1-B) differences with an illustrative purpose and the model was forecasted 50 periods ahead. The mean forecast produced by the model is displayed alongside with merely 20 % confidence intervals. Greater confidence intervals could not be depicted because the forecasts produced with the model displayed such low likelihood. The forecasting results propose negative development because they were calculated based on the acceleration of the series. The results were also utterly illogical considering that the model implied a large probability of negative index values in a relatively short-term. Logarithmic transformation would have omitted the possibility of negative values and improved likelihood, but the forecasting setting would have still remained inappropriate. Excessive differencing is a source of information loss and an undesirable characteristic of data (Tse 1997: 154).
illustration not visible in this excerpt
Figure 12. ARIMA(1,1,0)(0,1,0) Forecasted 50 Periods Ahead (Raw Data)
It was concluded that it is appropriate to produce ARIMA-based models only for the first degree of differencing i.e. only ARIMA(p,1,q) models were considered in this thesis. The stationarity of I(1) house price index can be questioned, but structural change parameterization will further strengthen stationarity in addition to logarithmic transformation and differencing.
The (1-B) differences were deemed sufficiently stationary based on the section 4.2.1 of this thesis. Vishwakarma (2013: 5) also considered a time series of house prices as the first differences of logarithms. The use of (1-B^{4} )(1-B) differences is a generic way to operate in the field of seasonal ARIMA models, but too much information would have been lost as a result of such transition. (Hyndman & Athanasopoulos 2014: 215-217, 242),
When it comes to independent variables and the implications of unit root testing, the deterministic component could be separated by de-trending the independent variables for a constant or a trend in order to foster stationarity. Trend and constant components can be excluded from a time series. The total series can be modelled through the following equation: yt = µ + δ T + ε t in which µ = constant, δ = trend, T = time and ε t = error term.
It can be necessary to de-trend independent variables in order to avoid spurious regression. Such phenomenon may occur, for example, when an independent variable keeps increasing, but this pattern is not directly connected to the dependent variable. For example, the magnitude of globally employed aggregate cloud storage space is highly likely to increase and this pattern is somewhat independent from other phenomena. Thus, deviations from the postulated trajectory can be significant instead of the combined motion of the trajectory and deviations. However, independent variables were not de-trended in this thesis because it was maintained that the independent variables sufficiently explain the dependent in the light of the theoretical framework.
6] Trend and constant removal can be carried out in the R environment by employing the detrend() [pracma] function.
Figure 13 and Figure 14 illustrate the distribution for the first differences of house prices. Figure 15 and Figure 16 illustrate the distribution for the first seasonal differences of house prices. Figures 17 and Figure 18 display the distribution for the (1-B^{4} )(1-B) differences of real estate prices. Each differencing scheme is displayed as a histogram fitted in the normal distribution and as a time series plot. Al-Marwani (2014: 74-86) portrayed differences as histograms too.
It is notable that first order single differences, (1-B), are positive at the start of the series, but exhibited fluctuation around zero in the most recent quarters of the training period. The equivalent seasonal differences, (1-B^{4} ), displayed a slight recovery at the end of the training set, but still remained negative. The (1-B^{4} )(1-B) differences have recently turned slightly positive despite a relatively long negative period that preceded the current development in the training set. The second degree of differencing is analogous to acceleration.
7] Figures 13-18 were produced by specifying the fitdist() [fitdistrplus] and plotdist() [fitdistrplus] functions in the R environment.
illustration not visible in this excerpt
Figure 13. Non-Cumulative Density of Non-Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram.
illustration not visible in this excerpt
Figure 14. Logarithmic Differences of Real Estate Index in Finland (1995-2015)
illustration not visible in this excerpt
Figure 15. Non-Cumulative Density of Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram.
illustration not visible in this excerpt
Figure 16. Logarithmic Seasonal Differences of Real Estate Index in Finland (1995-2015)
illustration not visible in this excerpt
Figure 17. Non-Cumulative Density of the Differences of Seasonal Differences of Real Estate Prices in Finland 1995-2015 Fitted in Normal Distribution as a Histogram.
illustration not visible in this excerpt
Figure 18. Differences of Logarithmic Seasonal Differences of Real Estate Index in Finland (1995-2015)
The dependent series can be divided into the trend, seasonal and random variation components in order to achieve a better comprehension of the data. From the viewpoint of endogenous modelling, it is useful to point out that the data contains these three components. Each component was extracted and plotted separately and they have been displayed in Figures 19-21.
Logarithmic series was employed for the purpose because this is the form in which the data was dealt with. The trend component was counted with a simple moving average. The implemented time window was four for quarterly data. The definition of trend is now different from the previous section and only dependent on the latest four observations.
Relatively weak seasonality was extracted by averaging after trend removal, which is the reason for the symmetric shape of the graph. Seasonality exhibited by the series was considered additive because the degree of seasonal variation was not affected by increase in quantity when the logarithmic series was considered. A moderate degree of random variation remained in the data after the trend and seasonal components had been removed.
The implications evoked by the decomposition process were that a suitable short-term statistical model has to consider seasonality and the recent developments of the time series. A moderate degree of random variation implies that the sample size is sufficient for ARIMA modelling (Hyndman & Kostenko 2007: 12-15). Seasonality of real estate prices was considered by extending the I(1) ARIMA models with seasonal dummy variables.
8] The three patterns associated with house prices were extracted in the R environment by employing the decompose() [stats] function. The data was handled in logarithmic form. The function first determines the trend component by counting a simple moving average with a quarterly time window. The seasonal component is counted after this procedure by averaging. Seasonality was parameterized as “additive” in the function specification. Random variation remains after these two components have been removed.
illustration not visible in this excerpt
Figure 19. Trend Component of the House Price Series 1995-2015 (Logarithms)
illustration not visible in this excerpt
Figure 20. Seasonal Component of the House Price Series 1995-2015 (Logarithms)
illustration not visible in this excerpt
Figure 21. Random Variation Component of the House Price Series 1995-2015 (Logarithms)
The first model is used as a competing exponential smoothing specification against more advantageous ARIMA models. Holt (1957) and Winters (1960) exponential smoothing model was chosen for the purpose. The model is referred to as “Holt & Winters model” hereinafter. The Holt & Winters model can and should be implemented for non-stationary data. The model was specified for logarithmic levels of real estate prices.
The Holt & Winters method can be understood through the presentation of simple exponential smoothing technique shown in Equation 23. Exponential smoothing is specified so that α is a weight coefficient for the most recent value and st is defined as a weighted average of the most current observation (xt) and the previously smoothed statistic (st-1). Furthermore, 0 ≤ α ≤ 1. (Shmueli & Lichtendahl 2016: 87-91, Hyndman & Athanasopoulos 2014: 172-175.)
illustration not visible in this excerpt
The reason why the method is called exponential smoothing can be understood through recursive substitution presentation. The series is decomposed backwards in Equation 24. The importance of past observations geometrically declines in the presentation.
illustration not visible in this excerpt
Holt (1957) and Winters (1960) augmented the exponential smoothing model so that seasonality and trend are included. Seasonality was relatively weak for the dependent variable, but it is still relevant to consider the seasonal component. Seasonality was not of multiplicative kind so it was classified as “additive”. One of the advantages associated with the logarithmic transformation is that seasonality is smoothed so that it is statistically includable. The model has been delved into by Hyndman & Athanasopoulos (2014: 183-194).
The additive Holt and Winters method is presented in equations 25-28 as it was implemented in this thesis. The method employs a trend component and the resulting forecasts will not be flat. In Equation 25, h+m verifies that the seasonal index estimates are from the final year of the sample (N). The most recent observation is denoted with yt and st-m stands for seasonally adjusted observation. Furthermore, ℓ t is the level component that is an estimate of the level at time t and bt stands for the trend component. (Hyndman & Athanasopoulos 2014: 183-194.)
illustration not visible in this excerpt
Equation 26 is the level equation that displays a weighted moving average of seasonally adjusted observation (yt - st-m) and non-seasonal forecasting result ( ℓ t-1 + bt-1) for moment t that is based on the estimates of the level component and trend from the previous period. Moreover, α operates as a smoothing parameter in the weighted moving average equation. It has to be noted that 0 ≤ α ≤ 1.
Furthermore, bt is the estimated slope (trend) at time t and β * is a smoothing parameter in Equation 27 for which applies that 0 ≤ β * ≤ 1. The trend component displays that bt is, in fact, a weighted average of the trend at time t-1, indicated with bt-1, and change in the level component notated with ℓt − ℓt-1.
Seasonality is captured in the model by Equation 28 for st. The seasonal component can be understood as a weighted moving average of the seasonal index and the seasonal index in the same period (t-m) last year. The seasonal index consists of the most recent observation, the level component of the previous period and the previous trend estimate so that it is stated as yt - ℓ t-1 - bt-1. Seasonal index for the same period last year is st-m. The concerned smoothing parameter used in the weighted moving average equation is γ for which applies 0 ≤ γ ≤ 1 . For quarterly data used in this thesis it also applies that the length of season is four so that m = 4.
Since there is a trend and seasonal variation associated with the data, gamma and beta were parameterized when the Holt & Winters forecasting method was performed. The values of α, β* and γ were estimated to be 1, 0.84 and 0.04, respectively. The equivalent coefficient values for the level, trend and four seasonal coefficients were 2.421, -0.002, 0.000, 0.002, 0.000 and -0.002, respectively.
The level component is clearly the most important determinant of forecasting in terms of coefficient value. The value of alpha for the model is extremely high (1), which points out that the estimated level is completely based on the most recent seasonally adjusted observation. When it comes to the level equation, the time series is adjusted by subtracting the seasonal component. Gamma value was low, which indicates that the seasonal component is mostly based on seasonal indices beyond the current seasonal index. Beta value is high, so the trend component receives a very large impulse from the difference between the two latest level components. The associated trend coefficient was negative and minor. The forecasting result produced by the model was compared against the results produced by the three endogenous ARIMA models specified hereafter in this chapter.
ARIMA processes are flexible enough to mimic different exponential smoothing methods (Shmueli & Lichtendahl 2016: 87-91). Simple exponential smoothing is actually equivalent to fitting an ARIMA(0,1,1) model. Furthermore, double exponential smoothing is analogous to fitting an ARIMA(0,1,2). (Mellin 2007: 153.)
According to Reagan (1984: 21-24), the inspection of autocorrelation and partial autocorrelation functions is a crucial part of the Box-Jenkins methodology. Al-Marwani (2014: 74-86) also portrayed autocorrelation plots for the dependent series. Defining ARIMA order is of pivotal importance in the process of ARIMA forecasting. The autocorrelation (ACF) function and 95 % confidence interval bounds depicted in Figure 22. Autocorrelation for variable y at lag p is defined as Corr(yt,yt-p) . Because the ACF decays slowly, the data has to be differenced in order to achieve stationarity. (Jain & Mallick 2016: 59, Hyndman & Athanasopoulos 2014: 229-230.)
9] The Holt & Winters model was produced by using the HoltWinters() [stats] function in the R environment.
illustration not visible in this excerpt
Figure 22. Autocorrelation Function (ACF) for House Price Index in Finland 1995-2015
When an equivalent ACF was created for the first differences of house prices, a significant spike at lag one was observed, which implied that the data could be concerned with an MA(1) process. Spikes beyond the first lag were not discerned. However, the ACF graph for the first differences was sinusoidal, which would enable the use of an ARIMA(p,1,0) process too. There was no fluctuation outside the 95 % confidence bounds in the ACF for the first differences of house prices. (Hyndman & Athanasopoulos 2014: 229-230.)
The presentation of ARIMA(0,1,1) is displayed in Equation 29 with zero mean expectation. Error term, ε t, is defined as the difference between the value of x t and the expected value of xt so that ε t = xt - xet. If the moving average parameter, θ 1, is expected to be positive and the observed outcome is higher than prediction, the deviation from the expected value has an impact of positive direction in the following period. When θ 1 is negative, the situation is reversed and the concerned influence is of opposite direction from the error term.
illustration not visible in this excerpt
Positive MA(1) coefficient implies a model aligned with fluctuation of the latest error term whereas a negative coefficient implies movement to reverse direction from the error term. In the latter case, the concerned process revolves around the zero mean.
The MA(1) process has been simulated with coefficient values 0.7 and -0.7 presented in Figure 23 and Figure 24, respectively. The data generation method was based on the normal distribution and standard deviation was set to be 0.1. The simulations were generated over 100 periods. Memory capabilities of MA processes are very limited because error term correlation exists for the most recent forecasting errors.
illustration not visible in this excerpt
Figure 23. Simulated MA(1) Process with a Coefficient Value of 0.7 for Randomly Generated Data Based on the Normal Distribution
illustration not visible in this excerpt
Figure 24. Simulated MA(1) Process with a Coefficient Value of -0.7 for Randomly Generated Data Based on the Normal Distribution
The partial autocorrelation function (PACF) graph displays autocorrelation uniquely explained by each particular lag under consideration. The PACF and 95 % confidence bounds are displayed in Figure 25. The graph implies AR(1) characteristics considering the unique explanatory power associated with the autocorrelation at lag one.
illustration not visible in this excerpt
Figure 25. Partial Autocorrelation Function (PACF) for House Price Index in Finland 1995-2015
When an equivalent PACF was considered for the first differences, ARIMA(1,1,0) signature seemed probable judging by the significant spike at lag one in the I(1) and little random variation beyond the first lag. The Chen & Liu (1993) method will streamline the I(1) data so that it fully enables the fitting of ARIMA(1,1,0). The PACF for I(1) data also indicated relatively weak seasonal autocorrelation which can be overcome by implementing seasonal dummy variables. (Jain & Mallick 2016: 59, Hyndman & Athanasopoulos 2014: 229-230.)
The AR(1) process receives a large impulse from the previous observation. In the AR(2) process, frequency of the noise decreases when both coefficients are positive. The memory of AR processes is longer than it is for MA processes, which is true because autocorrelation is considered for all the previous values. The AR effects deteriorate as less recent observations are considered because correlation coefficients decrease. The AR(1) and AR(2) processes were simulated to describe AR effects.
The following simulations were produced 100 periods ahead. Data generation was based on the normal distribution and the standard deviation was set to be 0.1. The AR(1) coefficient was specified as 0.7 and the process is depicted in Figure 26. The AR(2) process was also simulated in an equivalent setting and the process is displayed in Figure 27. The coefficients of both AR processes were specified as 0.35.
illustration not visible in this excerpt
Figure 26. Simulated AR(1) Process with a Coefficient Value of 0.7 for Randomly Generated Data Based on the Normal Distribution
illustration not visible in this excerpt
Figure 27. Simulated AR(2) Process with Coefficient Values of 0.35 for both AR Parameters and Randomly Generated Data Based on the Normal Distribution
Overlapping ARMA processes are hard to detect by purely relying on visual inspection of autocorrelation diagrams. Thus, different models have to be specified and evaluated in terms of information criteria, coefficient values, parameter standard errors, residual diagnostics and forecasting errors. It is to be noted that the ACF and PACF both involved somewhat unclear fluctuation on the verge tail off pattern and significance bounds, which could enable the use of an ARIMA process instead of solely relying on AR(p) or MA(q). (Hamilton 1994: 43-72, Jain & Mallick 2016: 59.)
The most suitable endogenous ARIMA models were specified based on the modelling benchmarks implemented in this thesis. The concerned benchmarks were considered as follows: 1. Parameter coefficient values in terms of size 2. Statistical significance of parameters in terms of standard errors 3. Absence of redundant parameters 4. Low root-mean-square-error (RMSE) and mean average error (MAE) for the entire model in the validation period 5. Low AIC and BIC 6. For MA(1) model: −1 < θ1 < 1 and MA(2) model: −1 < θ 2 < 1, θ2 + θ1 > −1, θ1 - θ2 < 1 7. For AR(1) model: -1 < ϕ1 < 1 and AR(2) model: -1 < ϕ2 < 1, ϕ1 + ϕ2 < 1, ϕ2 - ϕ1 < 1 8. Parsimonious models and over-fitting avoidance 9. Residual independence. (Hyndman & Athanasopoulos 2014: 72-73, 224-225; Andrews et. al 2013: 24-25, 35-36.)
The candidate ARIMA models were built on logarithmic I(1) data, which was analogous Vishwakarma (2013: 7). Seasonal dummies were employed in the models to extend them with seasonality. Festa (2016: 15-30) concluded that this type of seasonality consideration method for ARIMA-based models is favorable from the viewpoint of residual variance. Seasonality was clearly of additive kind after logarithmic adjustment performed on the data.
10] Figures 22-27 were created with acf() [forecast], pacf() [forecast], arima.sim() [stats] and plot() [graphics] R functions. Several ARIMA models were tested in the R environment by using the Arima() [forecast] and forecast() [forecast] functions. The anterior one of these functions calculates parameter values based on maximum likelihood estimates and the latter one was used for forecasting.
ARIMA 1 model was specified as ARIMA(1,1,0) with a drift and seasonal dummy variables. This type of a simple model produced the lowest AIC and BIC values and, hence, it was used as one of the candidate models. These two likelihood-based information criteria endorsed the inclusion of a drift parameter because the data exhibits a long-term trend if the entire training set is considered.
The ARIMA 2 model was delineated as ARIMA(1,1,3) with seasonal dummy variables. All the parameters of ARIMA 2 displayed high statistical significance excluding the MA(2) process. An extreme coefficient value (0.991) was recorded for the AR(1) process, but the model did not display characteristics of non-stationarity from the viewpoint of unit circle dynamics. The inclusion of AR(1) process reversed the coefficients of the three MA processes. The MA characteristics charged the model with frequent fluctuation in the opposite direction from the error terms. The model considers the long-run upward trend ceased.
The third ARIMA model is referred to as “AR(1)” herein and it was specified as ARIMA(1,1,0) without a drift, so the structure differed from ARIMA 1. The model was extended with seasonal dummy variables along with the rest of the models, which improved the likelihood-based information criterion values. The model was also very scarce and had the best performance out of the three ARIMA models in terms of error statistics in the validation period.
Descriptive statistics for each ARIMA model are presented in Table 4. Statistical fit and error figures in the validation period are displayed in Table 5 and Table 6, respectively. The AR(1) model delivered the best forecasting performance whereas ARIMA 1 endorsed by information criteria performed the worst. The error statistics of the Holt & Winters model are presented alongside with the three ARIMA models in order to render a comparative performance. The Holt & Winters was the only model that predicted downward development in the price level of real estate, which can be attributed to the model structure that emphasizes the most recent observations. The model was outperformed by the ARIMA 2 and AR(1) specifications built without drift dynamics under the expectation of long-term price saturation.
Table 4. Parameter Values for ARIMA Models Before Outlier Adjustment
illustration not visible in this excerpt
N is the number of observations in the data set. Statistical significance of each parameter is presented in the following way:
*** Standard error ≤ 10 %, ** Standard error ≤ 30 %, * Standard error ≤ 50 %. Absolute standard errors are displayed in brackets below the coefficient value. Three decimal places were used in the calculations. Quarters are displayed as Q1, Q2 and Q3.
Table 5. ARIMA Model Comparison Before Outlier Adjustment (1995-2015)
illustration not visible in this excerpt
N is the number of observations for each model. AIC displays the Akaike’s information criterion value for each model. The BIC value displays the Bayesian information criterion value for each model. P-Values are presented for the models as well.
Table 6. Forecasting Errors for Endogenous Models Expressed in Index Points Before Outlier Adjustment (2015.25-2016.75)
illustration not visible in this excerpt
Mean absolute error points out MAE for each model. MAE was encoded in the R environment so that MAE = mean(abs(error)). Root-mean-squared error displays RMSE for each model. RMSE was encoded in the R environment so that RMSE = sqrt(mean(error^{2} )).
Generic strategies for outlier detection in the context of linear models include observation omission based on deviation in residuals with respect to residual standard deviation, deviation in the first differences with respect to the standard deviation of the series and deviation in the first differences compared against a statistical distribution. Outliers can be considered in the light of one of these yardsticks so that, for example, observations displaying deflection above 95th percentile are labelled as outliers. Structural breaks can be assessed through simple intervention analysis too. Intervention analysis could consider, for example, minor recession (1996), burst of the dotcom bubble (2001-2002), financial crisis (2007-2008) and economic hardship (2013-2015) potential structural breaks.
Many scholars consider anomaly detection useful from the viewpoint of time series modelling. According to Oikarinen (2007: 78-79), structural breaks, such as liberalization of the financial markets and capital flows, can deteriorate long-run relationships between research variables. Furthermore, Junttila (2001) investigated structural anomalies in the context of ARIMA forecasting and observed improvement in inflation forecasts for the final intervention models compared against simple rolling regression technique. His findings applied to some of the cases, but not for all of them.
Tse (1997: 160) stated that ARIMA models can accommodate structural changes. Despite the directional benefits provided by common outlier detection strategies, structural anomalies are more of a complex issue in the context of adaptive ARIMA models. Outlier adjustment is ad hoc. In general, outlier and anomaly parameterization makes data more manageable.
In this thesis, structural anomalies associated with ARIMA-based forecasting models were estimated and parameterized by using the iterative three-stage method for structural anomalies introduced by Chen & Liu (1993). The approach is iterative in the sense that it requires model parameter re-estimation based on the outlier detection results. Parameter estimates are produced by using the maximum likelihood method. When the Chen & Liu (1993) procedure is implemented, the test statistic values are calculated based on outlier impact, dynamic outlier pattern and residual standard deviation. The underlying ARIMA orders were separately estimated in the section
A time series subject to a non-repetitive event has been displayed by Equation 30. Y* is the target series subject to outlier effects. In the equation, Yt is a time series that follows a general ARIMA process. Standard assumptions apply to unit circle dynamics of the general ARIMA process. It (t 1) is the outlier impact indicator function in which t 1 is the possible location of an outlier. A(B)/[G(B)H(B)] notation stands for the dynamic pattern associated with each outlier type that varies from type to another. Hence, parameters A, G and H change according to outlier type and they are polynomials of B. Finally, w represents outlier magnitude. (Chen & Liu 1993: 285.)
illustration not visible in this excerpt
Outliers are classified according to their effects. For an additive outlier (AO), the dynamic pattern can be defined so that A(B)/[G(B)H(B)] = 1. Hence, it is essentially a transitory spike in the data. For a level shift (LS) outlier, the dynamic pattern can be defined as A(B)/[G(B)H(B)] = 1/(1-B), which implies that the change is of permanent kind: LS effects render a permanent change in the model structure.
Dynamic pattern of a temporary change (TC) outlier can be illustrated through the concept of delta (δ), which relates to the statistical parameterization technique as well. TC outlier impact is influenced by delta value in a way that can be written as A(B)/[G(B)H(B)] = 1/(1-δB). For a temporary (TC) outlier, outlier effects decay gradually in a logarithmic manner. In the rare case of an innovational outlier (IO), the dynamic pattern can be written as follows by using backshift notations and ARMA parameters: A(B)/[G(B)H(B)] = θ(B)/ϕ(B)α(B). (Chen & Liu 1993: 285.)
Outlier coefficients can be either negative or positive, which is important to understand. When it comes to outlier parameterization, it was carried out by including suitable variable values from zero (0) to one (1) so that outlier effects and the associated dynamic patterns are reflected by the parameterization. Additive outlier (AO) is a single deviation in a series of dummies from zero (0) to one (1). Level shift (LS) is a permanent change from zero (0) to one (1) in an equivalent series. Temporary change (TC) takes full effect at a point in time and then deteriorates logarithmically from one (1) to zero (0) at a logarithmic rate defined by the delta value. The deceleration coefficient of TC outliers was specified as 0.7 in this thesis. (Chen & Liu 1993: 285.)
When it comes to the Chen & Liu (1993) method structure, in the first iteration, the model parameter values are calculated based on maximum likelihood estimates and the original series is used for the purpose. Adjusted series is used afterwards in the procedure. Outliers are assessed with respect to residual standard deviation and initial outlier effects that are utilized as inputs when the test statistics are calculated for possible outliers. The standardized statistics approximately follow the normal distribution. If the test statistic exceeds a chosen critical threshold value, possibility of an outlier exists. Different test statistics exists for each outlier type. If outliers are found, parameter estimates are updated accordingly and re-iteration(s) are carried out. (Chen & Liu 1993: 286-288.)
In the second stage of the procedure, outlier effects are considered jointly in the light of the multiple regression model used by Chen & Liu (1993: 287). Adjusted series is obtained by removing the outlier effects. After the removal of outliers that are considered significant compared against critical values, new maximum likelihood estimates for the model parameters are estimated for the adjusted series. If the relative change of residual standard error is greater than the chosen control value, one should return to previous step and perform further iterations. (Chen & Liu 1993: 287-288.)
In the final stage, the outlier effects and parameter estimates are evaluated jointly. Series residuals are obtained by filtering the series based on the most recent parameter estimates obtained in the second stage. These residuals are used to iterate through the first and the second stage, again. The series is no longer adjusted and outliers based on the final parameter estimates are achieved. The outlier effects obtained in the second stage of the process are the final estimates of the concerned effects. In this thesis, maximum number of iterations was specified as four (4) for each loop. The iteration process relies on the adjusted residuals of previous iterations. (Chen & Liu 1993: 287-288.)
Outlier parameterizations for each ARIMA model were achieved by implementing the Chen & Liu (1993) method. These parameterizations were used as exogenous variables when outlier-adjusted ARIMA models were specified. Outlier effects for each model are graphically depicted in Figures 28-30. The associated graphical presentation is logarithmic because this is the form in which the data was employed. Hence, 2.0 is equivalent to 10^{2} or 100 in terms of real index points.
11] The Chen & Liu (1993) three-stage method was performed in the R environment by specifying the tso() [tsoutliers] function. The ARIMA specifications were used as baseline models in the implementation of the technique. Outlier effects were first assessed with the tso() [tsoutliers] function and then parameterized by using the outliers.effects() [tsoutliers] function. Tso() takes in the dependent variable, external factors, ARIMA structure, delta value and the number of iterations as inputs. The function requires that the data is first converted into time series objects by employing the ts() [stats] function. The resulting structural anomaly parameterizations were further used as independent variables in the re-estimation process of the ARIMA models which was performed by specifying the XREG argument in the Arima() [forecast] function. The parameterizations had to be declassified with the unclass() [base] and data.frame() [base] functions in order to carry out re-estimation. The Chen & Liu (1993) parameterization method can be alternatively employed without the use of the tso() [tsoutliers] function by executing the following sequence of R functions: residuals() [stats], coefs2poly() [tsoutliers], locate.outliers() [tsoutliers] and outliers.effects() [tsoutliers].
Outlier-adjusted fit statistics are displayed in Table 7 whereas the descriptive statistics for each model are reflected in Table 8. The ARIMA 1 and AR(1) models built on logarithmic data yielded one TC(9) outlier when the Chen & Liu (1993) method was implemented. Temporary change effects were preferred because the outlier effects of AR(1) built on the first differences are sustained over a relatively long period. The ARIMA 2 model yielded two level shift outliers: LS(9) and LS(56). These effects are perpetual.
ARIMA models recalculate regression based on the entire series including the anomalous observations which can actually be significant from the viewpoint of modelling. Tse (1997: 160) states that ARIMA processes are adaptive enough to cope with structural changes and that they are not susceptible to structural anomalies because the models can be commenced at any point of a series. Based on this position relating to structural anomalies in the ARIMA context, implemented modelling benchmarks and the ad hoc nature of structural changes, only the LS(9) outlier was used in the ARIMA 2 model. The decision was made in order maintain sufficient statistical significance of the MA process coefficient values. The inclusion of a level shift outlier at point 56 would have rendered the MA(1) and MA(2) processes redundant, which was not appropriate from the viewpoint of the modelling benchmarks. The adaptiveness provided by MA processes charge the model with crucial forecasting capabilities from the viewpoint of overall forecasting performance.
Outlier inclusion improved in-sample estimation accuracy in terms of likelihood for the ARIMA 1, ARIMA 2 and AR(1) models, which realized as improved information criterion values and parameter coefficients. Forecasting results were also re-calculated for outlier-adjusted models and the results improved for each ARIMA model, which is remarkable. The concerned error statistics are displayed in Table 9.
A relatively minor flaw associated with outlier inclusion was that the statistical significance of MA(1) process deteriorated in the ARIMA 2 model. However, the total statistical significance improved for the model. Furthermore, neither of the three models exhibited evidence of having become over-primed in terms of coefficient values as a result of the Chen & Liu (1993) procedure. It sometimes happens that models become susceptible to parameter non-stationarity from the viewpoint of unit circle dynamics when parameter coefficients are inflated.
The forecasting performances of the ARIMA 1 and Holt & Winters models with trend dynamics were graphically compared against a control model, which is presented in Figure 31. The control model was specified as ARIMA(1,1,0) with a drift built on raw data. Information criteria did not support seasonality inclusion for the control model. Outlier detection was still carried out for the control model and it was parameterized with LS(9), LS(56), AO(57) and TC(68) structural anomalies. The stationarity of I(1) raw data was also verified at four lags within the 95 % confidence interval by employing the ADF test.
The ARIMA 1 and Holt & Winters models were outperformed by the control model built on raw data. Inflated forecasting performance exhibited by the ARIMA 1 model can be attributed to drift inclusion. The model was built on logarithmic data, which further heightened the drift effects in terms of raw data. However, logarithmic transformation is one of the most standard data adjustments in the field of time series forecasting and it generally makes data more manageable. Linearization is often warranted as phenomena related to economic growth are modelled.
The forecasting performance of ARIMA 1 was very weak despite extremely solid statistical fit within the training sample judging by information criterion values. The estimated slope associated with the trend expectation was based on the first differences of the entire training set, which deteriorated the forecasting results with respect to the realized fluctuation in the price level of real estate. Forecasting ability still improved as a result of structural anomaly adjustment for the model.
The only short-run endogenous model which forecasted a decline in real estate prices was the Holt & Winters model built on non-stationary logarithmic data and exponential smoothing capabilities. The Holt & Winters model included trend dynamics and predicted a decline in prices because the model puts a great deal of emphasis on the most recent observations, which is evident in the light of parameter coefficients present in the model.
The ARIMA 2 and AR(1) models were built without drift dynamics. These two models maintained an expectation of price saturation and flat long-term development of real estate prices. The models were compared against actualized real estate price evolution over the seven following quarters. Graphical comparison is displayed in Figure 32.
The ARIMA 2 and AR(1) models equipped with seasonal features were able to forecast the direction and inflection points of price evolution seven periods ahead despite minor inaccuracy based on autoregressive and moving average impulses. The AR(1) model displayed the best forecasting performance out of the univariate models introduced in this thesis, which realized in terms of forecasting errors and the final index point value. However, ARIMA 2 captured the shape of price evolution in the validation period more accurately. The overall forecasting results improved by the Chen & Liu (1993) method for both of these models.
12] Visualizations displayed in Figures 28-30 were produced with the tso() [tsoutliers] and plot() [graphics] functions.
Figure 28. ARIMA 1 Outliers in Logarithmic Form
Figure 29. ARIMA 2 Outliers in Logarithmic Form
Figure 30. AR(1) Outliers in Logarithmic Form
Table 7. ARIMA Model Comparison After Outlier Adjustment (1995-2015)
illustration not visible in this excerpt
N is the number of observations for each model. AIC displays the Akaike’s information criterion value for each model. BIC displays the Bayesian information criterion for each model. P-Values are presented for each model as well.
Table 8. Parameter Values for Outlier-Adjusted ARIMA models
illustration not visible in this excerpt
N is the number of observations in the data set. Statistical significance of each parameter is presented in the following way:
*** Standard error ≤ 10 %, ** Standard error ≤ 30 %, * Standard error ≤ 50 %. Absolute standard errors are displayed in brackets below the coefficient value. Three decimal places were used in the calculations. Quarters are displayed as Q1, Q2 and Q3.
Table 9. Forecasting Errors for Endogenous Models in Terms of Real Index Points After Outlier Adjustment (2015.25-2016.75)
illustration not visible in this excerpt
Mean absolute error points out MAE for each model. MAE was encoded in the R environment so that MAE = mean(abs(error)). Root-mean-squared error points displays RMSE for each model. RMSE was encoded in the R environment so that RMSE = sqrt(mean(error^{2} )).
illustration not visible in this excerpt
Figure 31. Final Forecasting Results for the ARIMA 1 and Holt & Winters Specifications with Long-Term Trend Dynamics Against the Control Model (2015.25-2016.75)
illustration not visible in this excerpt
Figure 32. Final Forecasting Results for the ARIMA 2 and AR(1) Specifications without Long-Term Trend Dynamics Compared Against Realized Price Development (2015.25- 2016.75)lo
13] Measurement intervals used in the graphs associated with the forecasts generated in this thesis may differ from the ones employed by e.g. Statistics Finland in the sense that the furthest time point associated with the measurement result is considered. Hence, the nth quarter of year X is equivalent to n x 0.25 + X. Nominal prices were considered in this thesis. The house price index is a re-indexed version of the base year 2000 HPI in Finland.
Shmueli & Lichtendahl (2016: 151) state that if autocorrelation remains in the residuals produced by a given statistical model, the model has not captured all the autocorrelation information correctly. Residual autocorrelation was tested by implementing the Ljung-Box (1978) test. The statistical procedure is described by Greene (2012: 962-963) and Hyndman & Athanasopoulos (2014: 56-57).
Degree of freedom for the serial correlation test was adjusted according to the standard principle, h-p-q, in which h represents the number of lags, p represents the number of associated autocorrelative parameters and q stands for the associated moving average parameters. (Hyndman & Athanasopoulos 2014: 56-57.) Lag length for the Ljung-Box test was specified as 10 because the house price data was only concerned with weak seasonality. In turn, income and construction have clear and strong seasonal patterns. Vishwakarma (2013) did not consider seasonality for real estate prices in any way despite the same form of data. Lag length decision is ad hoc. Considering 10 lags instead of using the 2M principle and eight lags did not have a remarkable impact on the serial correlation results.
Evidence of residual autocorrelation was very feeble for all the ARIMA models. However, it was weakest for the ARIMA 2 model. The second strongest evidence of serial correlation was observed for the ARIMA 1 model with a drift. The degree of freedom used in the Ljung-Box test was still significantly lower for ARIMA 2 because of the larger number of ARMA parameters, so the setting was clearly more favorable for the ARIMA 1 model.
All the ARIMA models still did well when it comes to serial correlation, but the ARIMA 2 model outperformed the AR(1) and ARIMA 1 models, which realized despite the largest number of ARMA parameters. Drift parameter improved the result of ARIMA 1 with respect to AR(1), which was expected based on higher likelihood statistics. The output of the Ljung-Box test has been displayed in Table 10 and the residuals were also individually plotted in Figures 33-35.
14] The Ljung-Box test was carried out in the R environment by specifying the Boxtest() [stats] function.
Table 10. Ljung-Box Test for Candidate Models
illustration not visible in this excerpt
N is the number of observations for each variable under scrutiny. The sequence of values starts from the first quarter of 1995. The null hypothesis of no residual autocorrelation is rejected in the Ljung-Box test if the displayed p-value is less than 0.1. The statistical significance associated with each chi-squared result is expressed in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level.
illustration not visible in this excerpt
Figure 33. Residuals Generated by ARIMA 1 Adjusted for Outliers
illustration not visible in this excerpt
Figure 34. Residuals Generated by ARIMA 2 Adjusted for Outliers
illustration not visible in this excerpt
Figure 35. Residuals Generated by AR(1) Adjusted for Outliers
Model-based Chen & Liu (1993) outlier parameterization sometimes overly inflates parameter coefficient values, which weakens ARIMA models from the viewpoint of root modulus stationarity. Parameter coefficient values may approach the limitations set by the modelling benchmarks when the procedure is carried out.
An invertible causal ARIMA model should have every AR and MA root lying outside the unit circle. On the contrary, all the equivalent inverse roots should lie inside the unit circle for such a model. When the stationarity conditions of ARIMA modelling are met, modulus < 1. (Hyndman & Athanasopoulos 2014: 223-225.)
Unit circle was plotted for all the endogenous ARIMA models, which has been displayed in Figures 36-38. The concerned inverse AR and MA roots laid inside the unit circle for all the ARIMA models indicating model stationarity from the viewpoint of parameter roots. However, the ARIMA 2 model was relatively close to nonstationarity judging by the unit circle and parameter coefficient values.
illustration not visible in this excerpt
Figure 36. Unit Circle for ARIMA 1
illustration not visible in this excerpt
Figure 37. Unit Circle for ARIMA 2
illustration not visible in this excerpt
Figure 38. Unit Circle for AR(1)
Data Analysis
Carry out tecnhical data analysis. VerifyCausality.
Evaluate multicollinearity. Justify the inclusion of lagged covariates.
Model Estimation
Use (1-B) differences to avoid
over-differencing. Consider seasonality with dummy variables.
Re-estimate ARIMA order. Include lags for the independent variables.
Evaluation
Evaluate models in the light of the Use outlier detection (Chen & Liu 1993) and modelling benchmarks (ad hoc) and the employ serial correlationcontrol theoretical framework. (Box-Ljung 1978).
Figure 39. Building Mechanism of ARIMA Error Models.
Van den Bossche et. al (2004: 9-18) exploited the model with ARIMA errors in order to investigate severity and frequency of road traffic accidents in Belgium. Models with ARIMA errors are used in this thesis instead of the ARIMAX methodology due to more apparent mathematical intuition. Forecasting results produced by models with ARIMA errors and exogenous variables are essentially analogous to the results generated by an equivalent ARIMAX model. The mathematical intuition ARIMA error model is described by Hyndman and Athanasopoulos (2014: 260-261).
The data is concerned with several seasonal patterns associated with the research variables that do not directly follow one another. In order to illustrate the seasonally adjusted levels of the four variables, all the series were smoothed by using a simple moving average with a time window of four quarters. The implemented simple smoothing method is described by e.g. Shmueli & Lichtendal (2016: 80-87).
Each training series has been plotted by employing moving average smoothing in Figure 40 to demonstrate the levels of the research variables without seasonality. Number 10 was used as a base of the logarithms in order to render an intuitive presentation in absolute terms. Mortgage rate was used as a simple logarithm along with the rest of the variables so that instantaneous graphical intuition and equivalence to the rest of the variables were achieved.
illustration not visible in this excerpt
Figure 40. Seasonal Moving Averages of Logarithmic Research Variables (1995-2015)
15] Figure 40 was produced by employing the ma() [forecast], ts() [stats] and autoplot() [ggplot2] functions in the R environment.
There is a negative correlation relationship between mortgage rate and real estate prices. The positive relation between income and house prices is also evident. These two correlation signs are in concord with the theoretical framework of this thesis.
Possible reasons behind the positive correlation of house prices and construction were analyzed in the section 4.1, but the long-run relation may be negative. Cointegration was tested with the Johansen procedure to inspect the long-run relation. The OLS regressions of the restricted VECM did not endorse a cointegration model due to low statistical significance. Impulse response analytics for the underlying level VAR still supported a slightly negative relationship in the long-run. (Pfaff 2011: 80-87.)
The underlying VAR model pointed out that differences have significant explanatory power over the dependent variable. Hence, ARIMA Error (p,d,q) and VAR(p) methodologies are suitable forecasting methods. VAR(p) methodology would have been afflicted by the lack of economic intuition in terms of coefficient values. VAR modelling is not concerned with a structure that imposes theoretical assumptions on variables either. (Pfaff: 2011: 36-43, Hyndman & Athanasopoulos 2014: 267-268.)
The ARIMA Error (p,d,q) models were built on the first differences of the research variables so that the differencing scheme can be specified as (1-B) with a backshift operator. Vishwakarma (2013: 5) used the logarithmic (1-B) differences as well. The implementation of seasonal differencing was considered sub-optimal because of excessive information loss. The ARIMA error setting imposes a great threat of over-differencing in comparison with the univariate stage because some of the independent datasets appeared nearly stationary after a mere logarithmic transformation.
16] The Johansen method was produced with the ca.jo() [urca] function. The OLS regressions of the restricted VECM were returned by the ca.jorls() [urca] function. The underlying VAR model was extracted with the vec2var [urca] function.
The stationarity of the (1-B) differences was verified in the least within the 95 % confidence level for all the differenced variables in the unit root testing section of this thesis (4.2.1) when the ADF test was utilized at AR(l4) lag length. The probability of unit roots was considered sufficiently low.
The ARIMA Error (p,1,q) models introduced in this thesis were augmented with seasonal dummy variables in order to account for seasonality, which was endorsed by the AIC and BIC values. The employed seasonality consideration method was similar to Festa (2016: 15-30).
Hyndman & Athanasopoulos (2014: 265) fitted an ARIMA error model built on exogenous dynamics without a constant parameter. Implicit mean expectation exists in the I(1) context, so the use of three seasonal dummy variables is statistically appropriate. The implementation of four dummies is not technically possible when the first differences are considered because there is an implicit mean expectation. ARIMA error modelling is carried out with ARMA parameters based on the error process of the baseline regression, which adds certain flexibility to the procedure.
A drift parameter was only included in one of the ARIMA error models in order to get comparative forecasting results from a model with a drift. The inclusion of a drift also slightly improved information criteria for the model. Drift model includes an upward deterministic trend expectation when it comes to I(0) data. The expectation does not depend on the independent variables, which is not intuitive from the viewpoint of the theoretical framework.
17] Implementation of ARIMA Error (p,d,q) models is feasible in terms of software and programming environment availability in comparison with the ARIMAX methodology. The Arima() [forecast] function holds several useful modelling capabilities. The function is also compatible with other R functions which provide practical forecasting and visualization features. (Hyndman & Athanasopoulos 2014: 259-271.)
The rest of the models were specified without a drift in order to maintain that the independent variables and ARMA elements bear sufficient explanatory power over the dependent variable. The reasons not to include a drift for I(1) univariate models are mentioned in the section 4.2.2 of this thesis and they apply to ARIMA Error (p,1,q) models with exogenous variables as well.
Because ARIMA error models are concerned with exogenous factors, multicollinearity between the independent variables had to be researched. Relevant multicollinearity would render the coefficient values unreliable. Multicollinearity was tested with variance inflation factor (VIF).
Furthermore, it had to be verified that lagged values of the independent variables forecast the dependent variable in order to have sufficient grounds to include any lagged values in ARIMA error models. Lagged values are expected to be significant based on the work of Vishwakarma (2013: 7) and Oikarinen (2007: 139-145).
Multicollinearity among the independent variables was assessed by utilizing the variance inflation factor (VIF) test for the exogenous variables which was specified to address possible correlations between the independent variables. OLS regression cannot be used for this purpose because, in the case of high multicollinearity, standard errors of the estimates become too large and cause exaggerated p-values. (Andrews et al. 2013: 36, 47.)
The concept of VIF is specified in the following way: VIF = 1/(1-R^{2} ). Values below 10 are considered acceptable level of multicollinearity (Andrews et al. 2013: 36). VIF values were tested for the logarithmic levels of construction, income and mortgage rate.
The associated VIF values for the independent variables were 1.129, 3.348 and 3.577, respectively. The VIF values were much smaller for the first differences. It was concluded that the independent variables do not display VIF values above the acceptable threshold. The implementation of each research variable was considered appropriate.
18] VIF testing for the exogenous variables was carried out in the R environment by performing the vif() [car] function in order to investigate multicollinearity. The function takes in an OLS regression as a parameter. The function was called with the following OLS regression for logarithmic level variables: House Prices ~ µ + Construction + Income + Mortgage rate + ε .
Direct Granger causality between the dependent and independent variables was tested for the first differences of logarithmic levels. The method was implemented to verify relevant lagged causality between the dependent and independent variables when lagged autocorrelation is taken into account. Andrews et al. (2013: 41) tested also reverse causality which is an undesirable property of independent variables that were excluded based on the results accordingly.
None of the variables could be omitted in this thesis since the main research objective was to test if the four quadrant framework is valid in reality. Because none of the independent variables could be excluded, testing was restricted to direct Granger causality. Andrews et al. (2013: 35-36) considered direct causality a desirable property of independent variables. Granger causality was tested at lags 1-4. The rate of interest was considered in the orthodox gross logarithmic form. Lag length was sequentially increased in the procedure and individual lags were not tested. The Granger causality results are presented in Table 11.
Table 11. Direct Granger Causality Test for the Independent Variables
illustration not visible in this excerpt
N is the number of observations for each variable under scrutiny. The sequence of value starts from the first quarter of 1995. F-values are presented for each variable in Table 11. Statistical significance of the Granger test results for each variable is presented in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level. (P-values)
19] Granger causality testing was carried out in the R environment by employing the grangertest() [lmtest] function.
The dependent variable received lagged impulses from all the independent variables based on the large and statistically significant F-statistic values, which is attractive from viewpoint of ARIMA modelling (Andrews et al. 2013: 35-36). Mortgage rate was especially significant at the first lag, which alone is enough of an argument to include lagged values in ARIMA error models. Symmetric and sequential lag inclusion was preferred in the ARIMA error models introduced in this thesis in order to avoid spurious regression.
The use of income and construction at further lags would overly smooth predictions and lead to over-fitting. The method would probably improve theoretical equivalence in the light of coefficient values judging by the Granger test results. The Granger causality results are merely directional because the setting does not consider seasonality or outlier adjustment. It is also important to note that the test does not address causality at lag t-0, so the setting is not equivalent to the modelling stage.
Vishwakarma (2013: 7) considered the independent variables at lags t-0 and t-1 in a similar context. The approach was adopted in this thesis as well. Sluggish adjustment of house prices with respect to the independent variables was expected in the light of Oikarinen (2007: 140).
Hyndman & Athanasopoulos (2014: 269-271) describe circumstances under which lagged predictors can be used in statistical models i.e. when lagged values have explanatory power over the dependent variable. Sometimes it is relevant to consider lagged values merely for the reason that some of the independent variables can have incremental effects.
The three ARIMA error models introduced in this thesis are referred to as “ARIMA Error (1,1,0) with a Drift”, “ARIMA Error (1,1,0) without a Drift” and “Experimental Model”. These models consider the independent variables at lags t and t-1, which is analogous to the ARIMAX(1,1,1) model of Vishwakarma (2013: 6-7). The use of ARIMA Error (1,1,1) order would have led to parameter redundancy for the MA(1) process, so it was left out from the models. The AR modelling signature displayed higher likelihood contrasted against MA-based models. A sufficient statistical model fulfils the modelling benchmarks, but it is highly improbable to find an optimal model that outperforms all the alternative models in every relevant field. However, despite the tradeoff, it is possible to find a good balance and compromise when it comes to the fulfilment of the various modelling benchmarks.
The following modelling benchmarks were considered in the building process of ARIMA error models: 1. Significant coefficient values of ARMA parameters
2. Significant coefficient values of the exogenous factors 3. Sequential and symmetric lag selection for independent variables and spurious regression avoidance 4. Scarce models and over-fitting avoidance 5. Low standard errors with respect to coefficient values 6. Theoretical equivalence 7. Low AIC and BIC 8. Validation period performance in terms of MAE and RMSE error statistics 9. For MA(1) model: −1 < θ1 < 1 and MA(2) model: −1 < θ2 < 1, θ2 + θ1 > −1, θ1 - θ2 < 1 10. For AR(1) model:
illustration not visible in this excerpt
ARIMA Error (1,1,0) with a Drift is concerned with autocorrelation dynamics. Mortgage rate was considered in the Log10(1+r) form, in which r = 0.01 for 1 %. This is the orthodox method to consider logarithmic interest rate. The method inflates the coefficient value of interest rate because the variable and variation are minuscule in absolute terms, but the variable bears significant explanatory power over real estate prices. The model was specified with a drift because drift inclusion slightly improved the likelihood of the model when mortgage rate was considered this way. The forecasting results were expected to be inflated as a result of drift inclusion.
ARIMA Error (1,1,0) with a Drift is presented in Equation 31 and Equation 32. The applied differencing procedure was used for every variable and it can be defined with a backshift operator so that Δ = (1-B). The error term is expected to be identically and independently distributed: ε t ~ i.i.d(0, σ^{2} ) . Autocorrelative parameter, ϕ, is defined as in the theory section (2.1) of this thesis. In Equation 31, µ = drift, HPI = house price index, Ct = construction, It = income and Mt = mortgage rate at moment t. Furthermore, ê, \ and ë are coefficients for the independent variables. St is a 4x1 vector of seasonal dummy variables defined as Q1, Q2, Q3 and 0. Q4 is treated as the default quarter. Finally, β1 is an equivalent 1x4 coefficient vector for the seasonal dummy variables. Coefficient for the fourth quarter is zero.
illustration not visible in this excerpt
The model was adjusted for outliers in order to account for structural changes associated with the data reflected against the model. The model yielded one LS(8) outlier and it was re-parameterized with this structural anomaly in order to further improve information criterion values, which succeeded. Outlier effects have been displayed in Figure 41. It is to be noted that this outlier is actually at time point 9/80 because lagged values imposed series abridgement. The entire training set consists of 80 observations (N=80).
The coefficients of independent variables were remarkable in terms of volume and according to the theoretical framework for income and mortgage rate. Autocorrelation still mitigated the effects of income. Only mortgage rate and construction displayed high statistical significance when it comes to parameter standard errors. The coefficient value of mortgage rate was inflated as a result of the standard logarithmic form in which it was considered. The income variable displayed statistical significance, but not strong statistical significance i.e. standard error below or equal to 50 % of coefficient.
ARIMA Error (1,1,0) with a Drift did well from the viewpoint of information criterion values, but the resulting forecasting performance was relatively weak because the model was specified with a drift parameter. The concerned information criterion values, error statistics in the validation period and parameter coefficients are displayed in Table 12. The forecasting performance exhibited by the model compared against the realized price development is graphically illustrated in Figure 42.
ARIMA Error (1,1,0) with a Drift residuals were sufficiently independent and they are displayed in Figure 43. Negative residuals at the end of the series seem questionable based on visual inspection. However, residual autocorrelation was not observed in the Ljung-Box test, which is shown in Table 13. Lag length for the Ljung-Box test was adjusted to 10 because the associated seasonality was weak. The degree of freedom was set to 8 according to h-p-q principle, in which h = lag length, p = number of MA parameters and q = number of AR parameters. Residual independence was established. The model was also stationary in the light of the unit circle i.e. modulus < 1. The AR parameter inverse root laid inside the unit circle, which is displayed in Figure 44. (Hyndman & Athanasopoulos 2014: 56-57, 223-225.)
illustration not visible in this excerpt
Figure 41. Outliers for ARIMA Error (1,1,0) with a Drift
Table 12. ARIMA Error (1,1,0) with a Drift
illustration not visible in this excerpt
N is the number of observations in the data set. Statistical significance of each parameter is presented in the following way:
*** Standard error ≤ 10 %, ** Standard error ≤ 30 %, * Standard error ≤ 50 %. Three decimal places were used in the calculations. Absolute standard errors are displayed in brackets below the coefficient value. Quarters are displayed as Q1, Q2 and Q3. Lagged values are indicated with lag order in brackets after variable name. AIC and BIC stand for the Akaike and Bayesian information criterion, respectively. RMSE indicates root-mean-square error and MAE indicates mean absolute error.
illustration not visible in this excerpt
Figure 42. Forecasting Results Generated by ARIMA Error (1,1,0) with a Drift Against the Realized Price Evolution in Index Points (1995.25-2016.75)
illustration not visible in this excerpt
Figure 43. Residuals Generated by ARIMA Error (1,1,0) with a Drift
Table 13. Ljung-Box Test for ARIMA Error (1,1,0) with a Drift
illustration not visible in this excerpt
N is the number of observations for the variable under scrutiny. The sequence of values starts from the second quarter of 1995. The null hypothesis of no residual autocorrelation is rejected in the Ljung-Box test if the displayed p-value is less than 0.1. The statistical significance associated with the chi-squared result is expressed in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level.
illustration not visible in this excerpt
Figure 44. Unit Circle for ARIMA Error (1,1,0) with a Drift
The second ARIMA error model was analogous to the first specification without a drift parameter. The logarithm of mortgage rate was again considered as Log10(1+r), in which r = 0.01 for 1 %. ARIMA Error (1,1,0) without a drift is presented in Equation 33 and Equation 34. The applied differencing procedure was employed for every variable and it can be defined with a backshift operator so that Δ = (1-B). Autocorrelation parameter, ϕ t, is defined as in the theory section (2.1) of this thesis and ε t ~ i.i.d(0, σ^{2} ) . In Equation 33, HPIt = house price index in Finland, Ct = construction, It = disposable income and Mt = mortgage rate at moment t. Moreover, ê, \ and ë are coefficients for the independent variables. St is a 4x1 vector of seasonal dummy variables that contains the quarters Q1, Q2, Q3, 0. Q4 is again treated as the default quarter. Finally, β1 is an equivalent 1x4 coefficient vector for the seasonal dummy variables indicating each quarter. (Hyndman & Athanasopoulos 2014: 265.)
illustration not visible in this excerpt
The Chen & Liu (1993) method yielded the same level shift anomaly which was used in the previous model. The effects of this structural anomaly are displayed in Figure 45. Descriptive statistics were produced for the model and they are shown in Table 14. The coefficient value improved for income and it was on the verge of high statistical significance. The rate of interest was again statistically significant and remarkable from the viewpoint of the associated coefficient sign. Theoretical equivalence clearly improved when the drift parameter was left out from the model. The forecasting performance of the model also improved as a result of drift exclusion. Forecasting ability in the validation period was excellent, which has been illustrated in Figure 46.
Residuals of the model were inspected with the Ljung-Box procedure. Lag length for the Ljung-Box test was adjusted to 10 because the underlying data only exhibited very slight seasonal variation. The degree of freedom used in the residual independence test was specified according to the standard principle. The residuals displayed independence, which is shown in Table 15. The model residuals are presented as a time series in Figure 47. The model was stationary in the light of the unit circle i.e. modulus < 1. The inverse root of the AR parameter laid inside the unit circle and model stationarity was confirmed. The unit circle is displayed in Figure 48. (Hyndman & Athanasopoulos 2014: 56-57, 223-225.)
illustration not visible in this excerpt
Figure 45. ARIMA Error (1,1,0) without a Drift Outliers
Table 14. ARIMA Error (1,1,0) without a Drift
illustration not visible in this excerpt
N is the number of observations in the data set. Statistical significance of each parameter is presented in the following way:
*** Standard error ≤ 10 %, ** Standard error ≤ 30 %, * Standard error ≤ 50 %. Three decimal places were used in the calculations. Absolute standard errors are displayed in brackets below coefficient values. Quarters are displayed as Q1, Q2 and Q3. Lagged values are indicated with lag order in brackets after variable name. AIC and BIC stand for the Akaike and Bayesian information criterion, respectively. RMSE indicates root-mean-square error and MAE indicates mean absolute error.
illustration not visible in this excerpt
Figure 46. Forecasting Results Generated by ARIMA Error (1,1,0) without a Drift Against the Realized Price Evolution in Index Points (1995.25-2016.75)
illustration not visible in this excerpt
Figure 47. Residuals Generated by ARIMA Error (1,1,0) without a Drift
Table 15. Ljung-Box Test for ARIMA Error (1,1,0) without a Drift
illustration not visible in this excerpt
N is the number of observations for the variable under scrutiny. The sequence of values starts from the second quarter of 1995. The null hypothesis of no residual autocorrelation is rejected in the Ljung-Box test if the displayed p-value is less than 0.1. The statistical significance associated with the chi-squared result is expressed in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level.
illustration not visible in this excerpt
Figure 48. Unit Circle for ARIMA Error (1,1,0) without a Drift
Miller & Sklarz (1986: 108) indicate that time series forecasting is essentially concerned with data science and fundamental analysis. The fundamental side includes variables that relate to economic, regional and population dynamics when it comes to the housing market. Technical approaches to market trending are still very beneficial from the viewpoint of forecasting and they should be used in conjunction with economic fundaments.
Experimental Model was parameterized as ARIMA Error (1,1,0). The model was built on similar lag dynamics to the ARIMAX model of Vishwakarma (2013: 7) in order to avoid the threats of over-fitting and spurious regression caused by further lag inclusion. Experimental Model does not produce flat forecasts in the long-term because future values are dependent on exogenous variables.
Drift inclusion would have led to the deterioration of likelihood-based information criteria and undesirable deterministic model traits from the perspective of the theoretical framework. The inclusion of a drift parameter did not improve information criterion values because mortgage rate was considered in a different way compared to the previous ARIMA error models.
The effects of simple logarithmic mortgage rate were slightly reduced near zero for Experimental Model. The ad hoc mortgage rate variable was considered as Log(1+r), in which r = 1 = 1 %. In other words, plain number one was added to raw rate before taking logarithm in order to remove some of the heightened variation in the variable as raw values approach zero. The ad hoc mortgage rate variable gets value zero (0) when raw mortgage is zero because Log(1+0) = 0. Napier’s constant was used as a base number because of better coefficient value intuition as a result of wider variation scale in comparison with 10-base logarithm. Base number has no influence on the forecasting results because proportional variation remains unchanged.
The levels of mortgage rate logarithm considered this way (Y-Axis) are reflected against the equivalent levels of raw mortgage rate (X-Axis) in Figure 49. The method considers sensitivity to interest rate displayed by the housing market in a novel way. The ad hoc procedure works relatively well in this time series, but it cannot be generalized.
illustration not visible in this excerpt
Figure 49. Ad Hoc Mortgage Rate Variable (Y-Axis) Employed in Experimental Model and Mortgage Rate in Terms of Raw Data (X-Axis)
The ad hoc approach effects are reflected in Figure 50 which displays the variation in the ad hoc mortgage rate variable, simple logarithm of mortgage rate and mortgage rate in terms of raw data over the training period (N=80). Simple logarithm of mortgage rate is defined as Log(r), in which r = 1 for 1 %. The utilization of the ad hoc method will be grounded on relevant literature and real economic events hereafter in this section.
illustration not visible in this excerpt
Figure 50. Logarithmic Mortgage Rate Variable, Simple Logarithmic Mortgage Rate and Mortgage Rate in Terms of Raw Data (1995-2015)
Leamer (2007: 1-3, 32-34, 39) concludes that the real estate market is more sensitive to interest rates towards the end of monetary expansion. The phenomenon is attributed to risky behavior of the banking sector as a result of which households will eventually start to rely on decreasing interest rates, increasing property values and rising incomes. When it comes to the credit market environment over the span of the research sample, it can be concluded that mortgage rate has decreased largely as a result of monetary expansion, which improves the theoretical relevance of the ad hoc approach for the rate of interest.
Furthermore, Miller et al. (2011: 24-25) posit that the impact of borrowing constraint faced by households is heightened in economic hardship. National economy of Finland experienced a major economic downturn (2008-2009) and further years of economic hardship (2013-2015) in the training period. The ad hoc approach increases the effects of interest rate over these periods, which also adds theoretical relevance to the ad hoc method of interest rate consideration.
On the contrary, the ad hoc approach mitigates the effects of mortgage rate in the period of high economic growth (1995-2000). The statistical intuition is that the increase in house prices which took place at that time is largely explained by fast disposable income growth. Leamer (2007: 3-7, 33, 39) states that the housing market is not sensitive to interest rates at the start of a business cycle, so the ad hoc method yields an intuitive outcome from this viewpoint.
The ad hoc approach for mortgage rate emphasizes the impact of the global interest rate hike that partially led to the financial crisis of 2007-2008 from the short-term perspective, which is relatively logical. Especially generous income policy agreements since 2007 and sticky downward adjustment could not prevent the housing market from declining at that time.
The positive economic effects of resulting central bank stimulus in 2009-2010 incited a swift and temporary economic recovery in Finland, which is highlighted by the ad hoc approach. Impact of the euro crisis rate hike in 2011-2012 on the housing market is also elevated when the ad hoc approach for mortgage rate is considered.
The current state of national economy in Finland is also heavily dependent on low interest rates, which is captured by the logarithmic interest rate variable. Favorable interest rate environment encourages property investment and private ownership of housing, but the current situation can be considered sensitive to interest rates.
If the ad hoc procedure turns out to render the housing market overly sensitive to the rate of interest as raw data values approach zero, the Chen & Liu (1993) structural anomaly adjustment is expected to catch anomalous deviation. The procedure operates as a safe valve in Experimental Model.
Experimental approach is presented in Equations 35 and 36. The applied differencing procedure was used for every variable and it can be defined with a backshift operator so that ∆ = (1-B). The error term is expected to be identically and independently distributed, so ε t ~ i.i.d(0, σ^{2} ). Autocorrelative parameter, ϕ t, is defined as in the theory section (2.1) of this thesis. In Equation 35, HPI = house price index, C = construction, I = income and M = mortgage rate. Furthermore, ê, \ and ë are coefficients of the independent variables. St is a 4x1 vector of seasonal dummy variables that contains the quarters Q1, Q2, Q3, 0 and, hence, Q4 is treated as the default quarter. β1 is an equivalent 1x4 coefficient vector for the seasonal dummy variables. (Hyndman & Athanasopoulos 2014: 265.)
illustration not visible in this excerpt
The model yielded LS(8) and LS(76) structural anomalies in the Chen & Liu outlier detection procedure. It is to be noted that the series was truncated by one observation because lagged predictor values were included, so these variables were actually at points 9/80 and 77/80 when the entire training set (N=80) is considered. Outlier effects are graphically displayed in Figure 51 and these outliers were used in the model re-specification stage. The previous models included less structural anomalies, which is detrimental from the viewpoint of Experimental Model.
Experimental Model did extremely well in terms of parameter coefficients values. The coefficients of the independent variables were aligned with the theoretical framework and logical excluding construction. Income and mortgage rate displayed high statistical significance, which improved from the previous ARIMA error specifications. Experimental Model exhibited better statistical fit than ARIMA Error Model (1,1,0) with a Drift, which realized in terms of the AIC and BIC values. The descriptive statistics yielded by Experimental Model are displayed in Table 16.
The forecasting performance exhibited Experimental Model has been graphically displayed in Figure 52. The forecasting results were very accurate as they were contrasted against the actualized real estate price development. Previously introduced ARIMA Error (1,1,0) without a Drift still surpassed Experimental Model in terms of forecasting ability in the validation period judging by RMSE and MAE error statistics. The final value forecasted by Experimental Model was still closer to the actualized level of index seven quarters ahead.
Residuals diagnostics were produced for the model along with the rest of the descriptive statistics. Model residuals are shown in Figure 53. These residuals were further inspected with the Ljung-Box test, which is one of the most standard residual independence tests for ARIMA-based models. The test results are displayed in Table 17. Evidence of serial correlation was not observed and the model surmounted the previous specification in terms of residual independence. Unit Circle for the model has been portrayed in Figure 54. The associated inverse AR(1) root laid inside the unit circle and modulus remained below one for the model.
illustration not visible in this excerpt
Figure 51. Experimental Model (2013) Outliers in Logarithmic Form
Table 16. Experimental Model for ARIMA-Based Modelling
illustration not visible in this excerpt
N is the number of observations in the data set. Statistical significance of each parameter is presented in the following way:
*** Standard error ≤ 10 %, ** Standard error ≤ 30 %, * Standard error ≤ 50 %. Absolute standard errors are displayed in brackets below the coefficient value. Three decimal places were used in the calculations. Quarters are displayed as Q1, Q2 and Q3. Lagged values are indicated with lag order in brackets after variable name. AIC and BIC stand for the Akaike and Bayesian information criterion, respectively. RMSE indicates root-mean-square error and MAE indicates mean absolute error.
illustration not visible in this excerpt
Figure 52. Forecasting Results for Experimental Model Against the Actual Price Evolution in Terms of Index Points (1995.25-2016.75)
illustration not visible in this excerpt
Figure 53. Residuals Generated by Experimental Model
Table 17. Ljung-Box Test for the Experimental Model
illustration not visible in this excerpt
N is the number of observations for the variable under scrutiny. The sequence of values starts from the second quarter of 1995. The null hypothesis of no residual autocorrelation is rejected in the Ljung-Box test if the displayed p-value is less than 0.1. The statistical significance associated with the chi-squared result is expressed in the following way: *** Statistically significant at the 1 % significance level, ** Statistically significant at the 5 % significance level, * Statistically significant at the 10 % significance level.
illustration not visible in this excerpt
Figure 54. Unit Circle for the Experimental Model
ARIMA-based models are advantageous because they 1. Are adaptive and flexible
2. Consider short-run developments 3. Can be augmented with certain type of trend expectations 4. Are specifiable with external components. Scarce ARIMA-based models generally produced reliable short-term forecasts seven periods ahead as they were employed in this thesis. ARIMA-based models were deemed suitable for real estate price forecasting in the short-run. Vishwakarma (2013) and Karakozova (2004) got similar results. Modelling setting greatly influenced outcomes achieved in the process of ARIMA-based time series forecasting.
The Chen & Liu (1993) structural anomaly detection procedure was implemented for every ARIMA-based model. The method improved forecasting accuracy, parameter stature and likelihood-based information criteria. The application of the procedure led to deterioration of other modelling benchmarks in few cases, which realized as parameter redundancy and non-stationarity approximation. The problem was overcome with ad hoc structural anomaly parameterization when necessary. However, it is concluded that the Chen & Liu (1993) methodology is an extremely powerful tool for structural anomaly detection.
When it comes to short-term predictions, it seems that univariate ARIMA models tailored for a specific time series generally produce reliable forecasts with the least effort. It is still questionable how to determine the persistency of recent developments and long-run trends. Relying upon the AIC and BIC values can lead to erroneous outcomes from the viewpoint of current developments. Specifying structural changes alleviates the issue, but does not resolve it in the ARIMA context.
The AR(1) model was the best univariate model from the perspective of forecasting performance in the validation period. The model was specified as ARIMA(1,1,0) for logarithmic data with seasonal dummy variables and one TC(9) structural anomaly. It is to be noted that the AR(1) model produces flat forecasts in the long-term excluding slight seasonal variation because the model did not include long-term trend dynamics.
This type of “impulse forecasting” generally outperformed univariate modelling approaches concerned with a drift which set up upward long-term trend expectations. In general, ARIMA-based models concerned with a drift parameter failed to forecast the house price index in the short-term. Drift inclusion led to inflated forecasting results in the validation period, but improved information criteria and likelihood in the light of the training sample. Logarithmic data heightened drift effects.
Dynamics provided by independent variables can add explanatory power in the ARIMA context especially if exogenous factors display unexpected variation. The introduction of independent variables generally improved forecasting results. Different variations of ARIMA Error (1,1,0) with independent variables outperformed all the other model types in the forecasting stage. Standard I(1) OLS, Holt & Winters and univariate ARIMA(p,1,q) methodologies produced less precise forecasts and exhibited worse statistical fit in comparison with this model type.
Experimental Model built on the ARIMA Error (1,1,0) signature exhibited the best theoretical relevance out of the introduced ARIMA-based models, which realized in terms of statistically significant coefficients for the income and mortgage variables. The inclusion of a deterministic drift would have decreased the associated likelihood which was high for the specification. Strong residual independence was also established within the training set. Experimental Model considered the rate of interest in an unorthodox way with respect to the rest of the statistical models, which is questionable.
However, ARIMA Error (1,1,0) without a Drift surmounted Experimental Model in terms of forecasting performance. The model produced extreme forecasting performance as MAE and RMSE were recorded at 0.712 and 0.793 index points, respectively. These forecasting results were extremely accurate considering that the index values were recorded above 260 index points in the validation period. ARIMA Error (1,1,0) without a Drift also produced less outliers in comparison with Experimental Model.
The use of exogenous factors for forecasting purposes still requires pre-information concerning the future values of the independent variables. It is proposed that these values can be forecasted separately by implementing the VAR(p) methodology. Another way to operate is to rely on the estimates provided by public institutions, such as central banks, statistics offices and ministries.
The theoretical assumptions set by the four quadrant framework were mostly consistent with the results achieved in this thesis. However, the short-run correlation of construction and house prices was positive, which was not according to the hypothesis proposed by the theoretical foundation. Possible reasons for the positive relationship were analyzed in this thesis. The coefficient signs of income (positive) and mortgage rate (negative) acted according to the hypotheses imposed by the theoretical framework.
The statistical appropriateness of using error correction terms (ECT) in ARIMA-based models should be further investigated in order to consider long-term relationships in the field of ARIMA forecasting. Future research is also required to create dynamic forecasting models that simulate the fact that the housing market is sticky and possibly discontinuous downward. Varying sensitivity to interest rates displayed by the real estate market requires further research as well.
According to HYPO, real estate prices will increase in 2017 (1.3 %) and 2018 (1.5 %) at a steady pace, but there will be a great deal geographical divergence. Real estate prices are expected to rise fast in growth centers and especially in the metropolitan area compared to the rest of Finland due to urbanization effects. (YLE: 1.9.2017.) Shocks in market fundamentals can still reverse the current growth pattern of house prices (Oikarinen 2007: 147-149). One of the possible threats associated with the housing market is that the market has got used to decreasing interest rates. More research is required to inspect the relationship between house prices and urbanization rate.
Akaike, H. (1981). Likelihood of a model and information criteria. Journal of Econometrics, Vol. 16, Issue 1, p. 3-14.
Al-Marwani, H., A. (2014). An Approach to Modelling and Forecasting Real Estate Residential Property Market. Brunei University.
Andrews, B., Dean, M., D., Swain, R. & Cole, C. (2013). Building ARIMA and
ARIMAX Models for Predicting Long-Term Disability Benefit Application Rates in the Public/Private Sectors. University of Southern Maine.
Berglund, J. (2007). Determinants and Forecasting of House Prices. Uppsala University.
Brooks, C. (2008). Introductory Econometrics for Finance (2nd Edition). Cambridge and New York: Cambridge University Press.
Case, K, E. Quigley, J., M. & Shiller, R., J. (2006). Comparing Wealth Effects: Stock Market vs. the Housing Market . Cowles Foundation. Paper No. 1181
Chen, N., Roll, R. & Ross, S. (1986). Economic Forces and the Stock Market. Journal of Business, Vol. 59, Issue 3, p. 383-403.
Chen, C. & Liu, L. (1993). Joint Estimation of Model Parameters and Outlier Effects in Time Series. Journal of the American Statistical Association, Vol. 88, Issue 421, p. 284-297.
Granger, C. W. J. (1969). Investigating Causal Relations by Econometric Models and Cross-spectral Methods. Econometrica, Vol. 37, Issue 3, p. 424-438.
Crawford, G., W. & Fratantoni, M., C. (2003). Assessing the Forecasting Performance of Regime-Switching, ARIMA and GARCH Models of House Prices . Real Estate Economics, Vol. 31, Issue 2, p. 223-243.
Greene, W. H. (2012). Econometric Analysis (7th Edition). Upper Saddle River: Prentice Hall.
DiPasquale, D. & Wheaton, W. (1992). The Markets for Real Estate Assets and Space: A Conceptual Framework. Journal of the American Real Estate and Urban Economics Association, Vol. 20, Issue 1, p. 181-197.
Drake, L., (1993). Modelling UK House Prices Using Cointegration: An Application of the Johansen Technique, Applied Economics, Vol. 25, Issue 9, p. 1225-1228.
Fama, E.F. & K.R. French (1992). The Cross-section of Expected Stock Returns. Journal of Finance, Vol. 47, Issue 2, p. 427-465.
Festa, E. (2016) Seasonal Mortality Forecasting based on Influenza Epidemics in The Netherlands. University of Amsterdam.
Fox, J., Weisberg, S., Adler, D., Bates, D., Baud-Bovy, G., Ellison, S., Firth, D.,
Friendly, M., Gorjanc, G., Graves, S., Heiberger, R., Laboissiere, R., Monette, G., Murdoch, D. (2017). “ Car ” R Package Documentation.
Friedman (1957). A Theory of the Consumption Function. Princeton University Press.
Hamilton, J., D. (1994). Time Series Analysis. Princeton University Press.
Hothorn, T., Zeileis, A., Farebrother, R., W., Cummins, C., Millo, G., Mitchell,
D. (2017). “ Lmtest ” R Package Documentation.
Hyndman, R., J. & Kostenko, A. (2007). Minimum Sample Size Requirements for Seasonal Forecasting Models. Foresight, Spring 2007, Issue 6, p. 14-15.
Hyndman, R., J. & Athanasopoulos, G. (2014). Forecasting: Principles and Practice. OTexts.
Hyndman, R., J. & Yasmeen, F. (2017). “ Forecast ” R Package Documentation.
Jain, G. & Mallick, B. (2017). A Study of Time Series Models ARIMA and ETS. Modern Education and Computer Science, Vol 4., p. 57-63.
Junttila, Juha. (2001). Structural breaks, ARIMA model and Finnish inflation
forecasts. International Journal of Forecasting, Vol. 17., Issue 2, p. 203-230.
Karakozova, O. (2004). Modelling and Forecasting Office Returns in the Helsinki Area. Journal of Property Research, Vol. 21, Issue 1.
Karolyi, G., A. & Sanders, A., B. (1998). The Variation of Economic Risk Premiums in Real Estate Returns. Journal of Real Estate Finance and Economics, Vol. 17, p. 245-262.
Leamer, E. (2007). Housing IS the Business Cycle. National Bureau of Economic Research (NBER), Working Paper No. 13428.
López-de-Lacalle, J. (2017). “ Tsoutliers ” R Package Documentation.
Mellin, I., (2007). Lecture Slides: ARMA-mallit. Helsinki University of Technology.
Miller, N. & Sklarz, M. (1986) A Note on Leading Indicators of Housing Market Trends. The Journal of Real Estate Research, Vol. 1, Issue 1, p. 99-109.
Miller, N., Peng, L. & Sklarz, M. (2011). House Prices and Economic Growth. The Journal of Real Estate Finance and Economics, Vol. 42, Issue 4, p. 522-541 .
Official Statistic: Average Mortgage Rate on New Contracts. Bank of Finland. [reference 17.6.2017].
Official Statistic: Disposable Income (B6N) , Quarterly National Accounts. Helsinki: Statistics Finland. [reference: 5.7.2017].
Official Statistic: Number of Residential Construction Permits. ISSN=1796-3257. Helsinki: Statistics Finland. [reference: 26.4.2017].
Official Statistic: Prices of Privately Occupied Dwellings, 100 = 2000 index. ISSN=2323-878X. Helsinki: Statistics Finland. [reference: 3.5.2017]
Oikarinen, E. (2007). Studies on House Pricing Dynamics. Turku School of Economics.
Pfaff, B. (2011). Analysis of Integrated and Cointegrated Time Series with R. Second Edition.
Pfaff, B., Zivot, E., Stigler, M. (2016). ” Urca ” R Package Documentation.
Reagan, K., M. (1984). An Evaluation of ARIMA (Box-Jenkins) Models for
Forecasting Wastewater Treatment Process Variables. University of California.
Schwert, G., W. (1989). Tests for Unit Roots: A Monte Carlo Investigation. Journal of Business and Economic Statistics, Vol. 7, Issue 2, p. 147-160.
Shmueli, G. & Lichtendahl, K., C. (2016). Practical Time Series Forecasting with R. Second Edition.
Stock, J.H. & Watson M.W. (1989). New Indexes of Coincident and Leading
Indicators. Olivier Blanchard and Stanley Fischer (eds.) NBER Macroeconomics Annual, The MIT Press.
Tse, R., Y., C. (1997). An application of the ARIMA model to real-estate prices in Hong Kong. Journal of Property Finance, Vol. 8, Issue 2, p. 152-163.
Valipour, M., Banihabib, M., E. & Behbahani, S., M., R. (2012). Parameters Estimate of Autoregressive Moving Average and Autoregressive Integrated Moving Average Models and Compare Their Ability for Inflow Forecasting. Journal of Mathematics and Statistics, Vol. 8, Issue 3, p. 330-338.
Van den Bossche, F., Wets, G. & Brijs, T. (2004). A Regression Model with ARIMA Errors to Investigate the Frequency and Severity of Road Traffic Accidents. Steunpunt Verkeersveiligheid.
Vishwakarma, V., K. (2013). Forecasting Real Estate Business: Empirical Evidence from the Canadian Market. Global Journal of Business Research, Vol. 7, Issue 3, p. 1-14.
Wickham, H. (2017). ” Ggplot2 ” R Package Documentation.
YLE 1.9.2017. HYPO: Kaupungistuminen kasvaa voimakkaasti - Asuntojen hinnat nousevat 1,3 prosenttia.
Masterarbeit, 99 Seiten
Bachelorarbeit, 57 Seiten
Diplomarbeit, 179 Seiten
Bachelorarbeit, 70 Seiten
Masterarbeit, 127 Seiten
Masterarbeit, 99 Seiten
Bachelorarbeit, 57 Seiten
Diplomarbeit, 179 Seiten
Bachelorarbeit, 70 Seiten
Masterarbeit, 127 Seiten
Der GRIN Verlag hat sich seit 1998 auf die Veröffentlichung akademischer eBooks und Bücher spezialisiert. Der GRIN Verlag steht damit als erstes Unternehmen für User Generated Quality Content. Die Verlagsseiten GRIN.com, Hausarbeiten.de und Diplomarbeiten24 bieten für Hochschullehrer, Absolventen und Studenten die ideale Plattform, wissenschaftliche Texte wie Hausarbeiten, Referate, Bachelorarbeiten, Masterarbeiten, Diplomarbeiten, Dissertationen und wissenschaftliche Aufsätze einem breiten Publikum zu präsentieren.
Kostenfreie Veröffentlichung: Hausarbeit, Bachelorarbeit, Diplomarbeit, Dissertation, Masterarbeit, Interpretation oder Referat jetzt veröffentlichen!
Kommentare