3.2 Results
3.2.1 The Actual Housing Price Model
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Figure 5: Housing price model with actual interest rates (authors’ own creation).
To further explain the dynamics of the actual housing price model, the creation of all included variables are elaborated on one by one in the following sub-sections, in addition to their individual coefficient estimate value.
Lagged Change in Real Housing Price
The variable lagged change in real housing price consisted of the 2004-indexed price of sold single-family houses relative to the private consumption deflator lagged one quarter. The regressor was given in logarithmic terms. The variable was expected to work explanatory for the depended variable, as historical changes in real housing prices is assumed determinative for future real housing prices.
The ADF test on the differentiated variable revealed no signs of a unit root. However, economic theory would claim that there is a significant trend in the change in housing prices, as outlined in 2.1.2. Observing housing prices on the market would clearly show signs of trend mirroring economic conditions. The coefficient estimate was highly significant even at a 1 % level. This result could be said to be expected according to the argument that next the previous period's price is the best guess for the next period’s price. Increasing the lagged change in real housing prices by 1 % would cause a rise in the depended variable by approximately 0.56 %.
First-year Payment
First-year payment has had an impact on the housing prices as it reflects the household's costs the first year. The effect of short-term interest rates, the introduction of adjustable-rate mortgages and the mortgage instalment relief combined have impacted the housing demand. Property tax was also included in the equation for first-year payment. It is worth repeating that the impact of frozen property taxation has had a larger impact on the first-year payment and user cost in the larger cities than on a national level (Dam et al., 2011, p. 2). Reducing the first-year payment through adjustable-rate
Variable Name Coefficient Std.Error t-value p-value
Lagged change in real housing price Dlog(KP/PCP)- 1 0.55750 0.05280 10.559 <2E-16 ***
First-year payment RENTEMIN+SSATS+AFDR -1.88565 0.38895 -4.848 2.86E-06 ***
Change in interest and property taxation rate D(RENTE30+SSATS)*DB98K4 -1.17290 0.32556 -3.603 0.000416 ***
User cost (RENTE30+SSATS-DPCPE-DRKPE)- 4 -0.19675 0.09376 -2.098 0.037393 *
Disposable income relative to housing stock (log(YDH/PCP)-log(AIH))- 6 0.04542 0.02563 1.772 0.07826 .
Constant 0.04788 0.02706 1.769 0.078723 .
Signif. Codes: 0.001 '***' 0.01 '**' 0.05 '*' 0.1 '.'
Estimation period: 1973Q1-2017Q2 R-squared: 0.5805 Adjusted R-squared: 0.5677 Change in Real Housing Prices - Actual Housing Price Model
63 mortgages and instalment reliefs have made it possible for more people to raise loans and hence, the demand for houses has amplified. According to Danmarks Nationalbank, the effect of reducing the first-year payment helped explain 60 % of housing demand from the beginning of 2000 to the 2008 financial crisis (Dam et al., 2011, p. 14).
The importance and effect of the inclusion of the first-year payment variable in a housing price model is a topic for discussion. The variable was not included in the MONA model from 2003, but the model as of 2011 emphasises first-year payment as an important component in housing price development.
The time period in which the housing price model was constructed can be argued to be characterised with a liberalisation of the financial market. The liberalisation helped to increase the flexibility in relation to financing home purchases, hereunder, an option of reducing the first-year payment (Dam et al., 2011, p. 6). In later times, the central bank of Denmark's prominent focus on this variable has led to a discussion about the importance of first-year payment in relation to housing prices. Still, the variable exhibited a significant and negative effect on housing prices based on the estimation output.
Both economic intuition and statistic tests support an assumption of a stochastic trend. This was justified by the fact that the reduction of first-year payment in the beginning of 2000 clearly led to a subsequently lower variable level from this point in time, among other things. The unit root was therefore adjusted for. As mentioned in the paragraph above, the variable was highly significant, even at a 99 % confidence level. The negative impact on housing prices was expected as an increase in the first-year payment involve a higher expense for the buyers and likely a reduction in demand.
If first-year payment were to rise by 1 %, it would cause a negative effect on real housing prices by 1.89 %.
The Granger causality test of first-year payment was conducted by using only one lag. The number of lags is usually decided through information criteria such as AIC (Grosche, 2014, p. 287). The result of the Granger causality test displayed a F-value of 0.15 and a probability of 0.7. This means according to the test, that first-year payment was not a useful predictor of housing prices. For variables with a short-term effect on housing prices, the result of no Granger causality is a typical result. Granger tests are sensitive to the specification and estimation of the time series and may display wrong conclusions in relation to the null hypothesis in cases where the dependent variable is highly volatile. Additionally, the Granger causality test has proved to fail when the relationship between the regressors in the simple regression is non-linear (Grosche, 2014, p. 293). All of these
64 factors may be argued to be present in the relationship between first-year payment and housing prices, and hence, the result that first-year payment did not Granger-cause housing prices was not surprising.
Change in Interest and Property Tax Rate
The combined interest rate and property tax rate variable was an expression for the total difference in the bond yield after tax and the imputed rate of property tax (Dam et al., 2011, p. 75). This variable had short-term impacts on the housing prices in several periods from 1972 until late 1990’s.
However, the impact of this variable was proved to be insignificant after the 4th quarter of 1998, and the variable's influence was therefore eliminated after that point in time through a dummy variable.
A unit root test showed no signs of stochastic trend. The coefficient was significant at a 1 % level and had a negative impact on real housing prices. If the variable increased by 1 %, the change real housing prices would decrease by -1.17 %. As substantiated in the previous paragraph, the effect of this variable was only visible until the end of 1998. The real housing prices in the following time periods were not affected by the interest and property tax rate variable.
The result of the Granger causality test showed no signs of predictability. Based on the theory section, this conclusion appears to be incorrect. Both literature and observations of historical development show that the housing prices decrease in time periods of high interest rates due to a lower demand caused by a higher cost of debt. The test results were therefore disregarded, which is justified by the weakness argument of this type of test.
User Cost
The user cost variable in the original housing price model includes the interest rate of a mortgage bond with 30 years of maturity, the property tax rate, and the expected inflation. However, constructing the user cost variable in accordance with this did not result in statistical significance.
This was true even though the variable was adjusted with correct differentiation and lags based on statistical evaluations.
65 To make user cost statistically significant, the variable for the expected change in housing prices (DRKPE) was included. This decision was made in accordance with the argumentation that the user cost variable should take future price expectations into account (Dam et al., 2003, p. 15). The expected change in housing prices was implemented in addition to the expected change in inflation, as predictions related to future housing price changes can be perceived as a determinant for households’ user cost. The user cost equation is presented in figure 5.
The variable displayed signs of a unit root and time trend. In order to make the variable a better fit to the model and solve the unit root problem, the time series was lagged and de-trended through taking its first difference. The variable was eventually significant at a 95 % confidence level with a coefficient of approximately -0.20. The negative sign of user cost was expected as an increase in the cost of investing and keeping a house decrease the price people would be willing or able to pay.
The Granger causality test of the relationship between user cost and housing prices showed that user cost had no predictive content for housing prices even when including several periods. Though, in the case of the user cost variable, the argument of how the variable was constructed can be an influencer of why the variable is not Granger-causing housing prices. This is because complex combinations of variables can be a source of insignificant Granger test result.
Disposable Income Relative to Housing Stock
The disposable income in relation to housing stock variable reflected the long-term effect on real housing prices. The variable consisted of two parts, where the first part represented households’
disposable income relative to private consumption, while the second part was the relevant housing stock, expressed as accumulated net housing investment.
Disposable income relative to housing stock comprised the correction term of the error-correction model that made up the applied housing price model. One of the features of the error term is that it involves two or more time series with stochastic trends that move together so closely over the long run that they appear to have a common trend (Stock and Watson, 2012, p. 691). Hence, it was vital to investigate whether or not the variables disposable income relative to private consumption and housing stock were co-integrated. The error-correction model proposes that if the two time series are co-integrated, the trend can be removed by subtracting one of the terms, or parts
66 of the term, from the other. By doing so, the trend in both terms should be eliminated, and the result is a spread exhibiting no trend (Stock and Watson, 2012, p. 691).
To assess whether or not disposable income relative to private consumption and housing stock had a common stochastic trend, it was conducted an Engle-Granger Augmented Dickey-Fuller test (EG-ADF test). The procedure of an EG-(EG-ADF test is a Dickey-Fuller t-test on the residuals with an intercept and no time trend. The residuals were based on a model where disposable income relative to consumption acted as the dependent variable and housing stock as the independent variable.
Even though the test result displayed an absence of a common stochastic trend in the two time series8, economic intuition and visible interpretation revealed a different result. It is reasonable to expect that a similar increasing trend would be observed in housing stock when disposable income relative to consumption increase. This is due to the fact that if the ratio of disposable income relative to consumption rises, it implies that people have a more substantial purchasing power. An improved purchasing power induces higher demand and consequently higher prices. Higher prices make it more profitable for investors to construct new houses, which increase the future housing stock. A graph illustrating the development of disposable income relative to private consumption against housing stock is presented below.
Figure 6: The variable describing the long-term relation in the housing price model (1973Q1-2017Q2).
8This finding is aligned with the results of ADF tests conducted on the individual time series, where it was only detected a unit root in disposable income and not in housing stock.
67 The estimated coefficient for disposable income relative to housing stock was significant at a 10 % level. Comparing the outcome to the original housing price estimates found by Dam et al. (2011), the identical coefficient was only significant at a 10 % as well. Consequently, this indicates that the obtained estimate for the long-term relation coefficient was acceptable. The coefficient had a positive sign, which means that an increase in the variable would lead to a growth in housing prices. In other words, if the ratio of disposable income relative to private consumption is greater than the accumulated net housing investment, it causes an increase in housing prices. This is in line with economic intuition, as an improvement in income level will induce more people to enter the housing market and thereby put pressure on prices. The coefficient of the long-term variable was 0.05, indicating that an increase in this variable of 1 % would cause an increase in the change of real housing prices of 0.05 %. It is sensible to expect a lower percentage change in the long-term variable compared to the short-term variables, as stated in section 2.1.4.
A Granger causality test of the long-term was expected to indicate a rejection of the null hypothesis.
The reasoning for this argument is based on the content of the variable. The variable reflects the long-term effect on real housing prices and is the error-correction term in the model. Clearly, past values of this variable should contain information that is useful for forecasting changes in housing prices. However, the Granger causality test presented a p-value above 50 % when including six periods back in time, indicating that the null hypothesis of no predictability was true.
The Constant
If all independent variables were zero at the same time, the intercept would be the expected mean value of the change in real housing prices. However, it is highly unlikely that all the variables are zero simultaneously, so the intercept in the model is argued to have no real intrinsic power. The constant term of the housing model had a positive sign and was significant at a 10 % level.
Concluding Remarks
ADF tests were applied as indicators for unit roots in variables. Although test results in several cases proved contradictory, the perception of the existence of stochastic trends, and hence decisions to handle such issues, were based mainly on economic sense and visual interpretation due to the low
68 power argument. Results of ADF tests and automatic ARIMA modelling were carefully analysed and adjusted for, leading to justifiable and satisfying results.
Additional tests were also conducted on the final model to ensure that the model did not suffer from multicollinearity between regressors, influential outliers, or that it contained autocorrelation in residuals. As stated in section 3.1, a VIF test-value of five or above indicates multicollinearity. The result of this test showed no signs of multicollinearity as all the estimation results had a value of around one. The test results are attached in appendix G. A Cook’s D plot of the estimated housing model displayed only one significant outlier (see appendix H). This outlier occurred in the 4th quarter of 1998. However, this outlier was caused by the dummy that changed from 1 to 0 in that exact time period (see sub-section 2.1.4). The respectable dummy was implemented to the model to eliminate the short-term effect of more flexible financing. There were also some other influential outliers close to the boundary. These were detected in the 2nd quarter and 3rd quarter of 1983, and 1st quarter of 1987. As mentioned in 2.1.2, in 1982 to 1983 the Danish economy was slowly recovering from the recession caused by the second oil crisis in 1979. Hence, the outliers in 1983 might reflect an improvement in the general economy. The outlier in 1987 can be caused by the impact of implementing new political initiatives, hereunder the Potato cure and the new tax reform introduced respectively in 1986 and 1987. In regards to autocorrelation, Durbin-Watson test results evidenced no such occurrences in residuals. In other words, the independent variables covered all explanatory information in the model. The test results are attached in appendix I.
The authors believe that a general debate on the Granger causality test results is necessary for further discussion. The fact that none of the model variables contained predictive content for the estimation of housing prices seemed dubious (see appendix J). The most likely explanation for failing to reject the null hypothesis of no predictive content can be attributed to the dependent variable's volatility. As elaborated on in the theory section, housing prices have been vastly volatile throughout the estimation period, with fluctuations ranging several percentage points from one period to another during the years of most economic instability.
The estimated housing price model is argued to overall be a reasonable predictor for housing price levels in Denmark from the 1st quarter of 1973 to the 2nd quarter of 2017. The model required some adjustments to variables in comparison to the original model specified by Dam et al. (2011).
However, all coefficients estimates proved significant and had reasonable signs in relation to intuitive
69 expectations. R2 amounted to 0.5805, which indicates a satisfying explanatory power of above 50%.
This seems reasonable in light of the aforementioned volatile phases in the first part of the estimation period, among other things. It is difficult to obtain a model that is capable of explaining the development to a significantly larger extent than the presented model did.
The graphical interpretation of the estimation results plotted against the true housing prices is displayed in figure 7. The results of the housing price development of the actual housing price model are further deliberated on in section 4.1.
Figure 7: Actual model compared with true housing prices from 1974Q4 to 2017Q2 (Statistics Denmark and authors’ own creation).