Testing for Causality-In-Mean and Variance between the UK Housing and Stock Markets

This paper employs the two-step procedure to analyze the causality-in-mean and causality-in-variance between the housing and stock markets of the UK. The empirical findings make two key contributions. First, although previous studies have indicated a one-way causal relation from the housing market to the stock market in the UK, this paper discovered a two-way causal relation between them. Second, a causality-in-variance as well as a causality-in-mean was detected from the housing market to the stock market.


Introduction
Although major financial institutions experienced the subprime mortgage crisis and Lehman Brothers went out of business, the market for real estate has grown steadily in the last decade.As indicated in Figure 1, the UK is one of the largest markets in the world, followed by the US, Japan, Australia, and France.In addition, since the UK decided to withdraw from the European Union ("Brexit"), based on a referendum conducted on 23 June 2016, market participants and macroeconomic policymakers have focused more on its impact on the UK real estate market.Therefore, examining the relation between the UK real estate and other financial markets is useful for both practitioners and academic researchers.Many previous empirical studies have explored the relation between the real estate and stock markets.Regarding this relation, we need to understand the following two effects.First, researchers who support the "wealth effect" claim that households benefiting from unanticipated gains in stock prices tend to increase housing demand.Second, researchers who support the "credit price effect" claim that an increase in real estate prices can stimulate economic activity and the future profitability of companies by raising the value of collateral and reducing the cost of borrowing for both companies and households.Thus, identifying the direction of causality between the real estate and stock markets as well as the number of lags is essential.
As mentioned above, many previous empirical studies have analyzed the relation between the real estate and stock markets (e.g., Gyourko and Keim (1992); Ibbotson and Siegel (1984); Ibrahim (2010); Kapopoulos and Siokis (2005); Lin and Fuerst (2014); Liow (2006); Liow (2012); Liow and Yang (2005); Louis and Sun (2013); Okunev and Wilson (1997); Okunev et al. (2000); Quan and Titman (1999); Su (2011);and Tsai et al. (2012)).To the best of our knowledge, no studies have analyzed the causality-in-variance between the real estate and stock markets.As indicated by Ross (1989), volatility provides useful data on the flow of information.For institutional investors such as banks, life insurance companies, hedge funds, and pension funds, deeper knowledge of spillover mechanisms for volatility can be useful for diversifying investments and hedge risk.Table 1 summarizes the previous studies.Academic research on the relation between the real estate and stock markets has been undertaken since the 1980s.In this research, almost all studies have focused on the cointegration relation between the two markets.In recent years, not only a linear cointegration method but also a nonlinear cointegration method has been undertaken (e.g., Liow and Yang (2005); Okunev et al. (2000); Su (2011);and Tsai et al. (2012)).Using data from four major Asian countries (Japan, Hong Kong, Singapore, and Malaysia), Liow and Yang (2005) analyzed the relation between the securitized real estate and stock markets.Moreover, they conducted a fractional cointegration analysis of two asset markets.Furthermore, they revealed that fractional cointegration exists between the securitized real estate and stock markets of Hong Kong and Singapore.Okunev et al. (2000) examined the dynamic relation between the US real estate and S&P 500 stock index from 1972 to 1998 by conducting both linear and nonlinear causality tests.While the linear test results generally indicate a unidirectional relation from the real estate market to the stock market, nonlinear causality tests indicate a strong unidirectional relation from the stock market to the real estate market.Su (2011) used a nonparametric rank test to empirically investigate the long-run nonlinear equilibrium relation within Western European countries.Nonlinear causality test results demonstrated that unidirectional causality from the real estate market to the stock market exists in the Germany, the Netherlands, and the UK.Unidirectional causality from the stock market to the real estate market was observed in Belgium and Italy, and feedback effects were discovered in France, Spain, and Switzerland.Tsai et al. (2012) used nonlinear models to analyze the long-term relation between the US housing and stock markets.Empirical results demonstrated that the wealth effect between the stock and housing markets is more significant when the stock price outperforms the housing price by an estimated threshold level.
This paper uses the cross-correlation function (CCF) approach developed by Cheung and Ng (1996) to examine the causal relation between the housing and stock markets in the UK.This empirical technique has been widely applied in the examination of stock, fixed income, and commodities markets, business cycles, and derivatives. 1While the test of Granger causality techniques examines the causality-in-mean, the CCF approach detects both the causality-in-mean and causality-in-variance. 2 1 Some examples include studies by Hamori (2003), Alaganar and Bhar (2003), Bhar andHamori (2005, 2008), Hoshikawa (2008), Nakajima and Hamori (2012), Miyazaki and Hamori (2013), Tamakoshi and Hamori (2014), and Toyoshima and Hamori (2012).
2 See Hafner and Herwartz (2008) and Chang and McAleer (2017a) for the causality-in-variance analysis using multivariate GARCH models.
The CCF approach can detect the direction of causality as well as the number of leads/lags involved.3Furthermore, it permits flexible specification of the innovation process and nondependence on normality.4The remainder of this paper is organized as follows.The next section presents the CCF approach.In the following sections, we discuss the data, descriptive statistics, and results of the unit root tests and provide a description of the autoregressive-exponential generalized autoregressive conditional heteroskedasticity (AR-EGARCH) specification.Thereafter, we present the empirical results and discuss the findings.Finally, a summary and conclusion are presented in the closing section.

Empirical Techniques
Following Cheung and Ng (1996), suppose there are two stationary and ergodic time series, X t and Y t .When I 1,t , I 2,t , and I t are three information sets defined by I 1,t = (X t , X t−1 , . . .)I 2,t = (Y t , Y t−1 , . . .) and Similarly, X is said to cause Y in the mean if Feedback effect in the mean occurs if Y causes X in the mean and X causes Y in the mean.On the other hand, Y is said to cause X in the variance if where µ X,t denotes the mean of X t conditioned on I 1,t−1 .Similarly, X is said to cause Y in the variance if where µ Y,t denotes the mean of Y t conditioned on I 2,t−1 .Feedback effect in the variance occurs if X causes Y in the variance and Y causes X in the variance.
We impose the following structure in Equation (1) through Equation ( 4) to detect causality-in-mean and causality-in-variance.Suppose X t and Y t are written as where { t } and {ζ t } are two independent white noise processes with zero mean and unit variance.
For the causality-in-mean test, we have the standardized innovation as follows: with ε t and ζ t being the standardized residuals.Since these residuals are unobservable, we use their estimates.Next, using their estimates, we calculate the sample cross-correlation of the squared standardized residual series, r uν (k), and the sample cross-correlation caluculated using the standardized residual series, r εζ (k), at time lag k.
The quantities r εζ (k) and r uν (k) are used to detect causality-in-mean and causality-in-variance, respectively, using the CCF approach.
First, we can detect the null hypothesis that there is no causality-in-mean using the following CCF statistic: If the CCF test statistic is below the critical value calculated using the standard normal distribution, then we cannot reject the null hypothesis.
Second, we can detect the null hypothesis that there is no causality-in-variance using the test statistic, which is given by CCF = If the CCF test statistic is below the critical value calculated using the standard normal distribution, then we cannot reject the null hypothesis.
The CCF approach is divided into two steps.First, we estimate univariate time-series models that consider the time variant conditional means and conditional variances.In this paper, we adopt the AR-EGARCH formulation. 5Second, from the estimated AR-EGARCH model, we calculate the standardized residuals of estimated model and calculate the series of standardized squared residuals by conditional variances.As mentioned above, we use the CCF of these standardized residuals to test the null hypotheses of no causality-in-mean and no causality-in-variance.

Data, Descriptive Statistics, and Results of an Augmented Dickey-Fuller Test
We employ monthly data on the UK housing and stock markets from January 1991 to August 2016.This sample period was chosen based on the availability of data obtained from The Nationwide Building Society. 6able 2 presents the descriptive statistics for the monthly change rate in stock and housing prices.As indicated in Figure 2, the volatility of the stock market is higher than that of the housing market.The measure for skewness and kurtosis, Jarque-Bera statistics, are used to detect whether the housing and the stock monthly change rates are normally distributed. 7The Jarque-Bera statistics reject normality at a 10% significance level in both variables.Table 3 indicates the results of the Augmented Dickey-Fuller test.The results reveal that, while the null hypothesis that the variables have a unit root is accepted in both variables in the level, the null hypothesis is rejected at the first difference.

Estimation of an AR-EGARCH Model
The first step of the CCF approach is to model the monthly change rates in the housing and stock prices.We estimate the AR(k)-EGARCH(p,q) model as follows: Crisis t = 0 (t = Jan 91, . . ., May 07) 1 (t = Jun 07, . . ., Aug16) , ( 12) where z t = ε t /σ t .Note that the left-hand side of Equation ( 13) is the log of the conditional variance.
Using the log form of the EGARCH(p,q) model, it is possible to guarantee the non-negativity constraints without imposing the constraints of the coefficients. 8By including the term z t−i , the EGARCH(p,q) model reflects the asymmetric effect of positive and negative shocks.
The persistence of shocks to the conditional variance is given by ∑ p i=1 β i .Equation (11), which is the conditional mean, is formulated as an autoregressive model of order k.To determine the optimal lag length k for each variables, we use the Schwartz-Bayesian Information Criterion (SBIC). 9.The SBIC is also applied in Equation ( 13) to determine the optimal lag length p and q. 10 Table 4 presents the estimates for the AR(k)-EGARCH(p,q) model.Regarding the standard error, this paper accepts the robust standard error developed by Bollerslev and Wooldridge (1992).First, the EGARCH(1,1) model is chosen for both variables.While all parameters of the EGARCH model in the monthly change rate in the stock price are significant, all parameters excluding γ 1 in the monthly change rate in the housing price are significant at the conventional significance levels.
Furthermore, Table 4 reports the estimates of the coefficient β 1 , which measures the degree of volatility persistence.We find that β 1 is significant at conventional significance levels, and the value of β 1 is close to 1.These estimates lead to the conclusion that the persistence in shocks to volatility is relatively large.Table 2 also indicates the diagnostics of the empirical results of the AR-EGARCH model.While Q( 24) is a test statistic for the null hypothesis that there is no autocorrelation up to order 24 for standardized residuals, Q 2 (24) is a test statistic for the null hypothesis that there is no autocorrelation up to order 24 for standardized residuals squared. 11These tables show that both statistics are statistically significant at 5% level for all cases.Thus, the null hypothesis that there is no autocorrelation up to order 24 for standardized residuals and the standardized residuals squared is accepted.These results empirically support the formulation of the AR-EGARCH model.

Testing for Causality-In-Variance
The second step is to detect causality-in-mean and causality-in-variance, using the calculated sample cross-correlations.Table 4 indicates the empirical results.Lags are measured in months, which range from 1 to 24.For example, in the case of "housing and stock (−k)," the significance of positive lags implies that the causal direction is from the stock market to the housing market.
Table 5 presents significance lags causality for both cases.First, in the case of "housing and stock (−k)," the causality-in-mean exists in lag 6 and the causality-in-variance exists in lags 5 and 9. Second, in the case of "housing and stock (+k)," the causality-in-mean exists in lags 21 and the causality-in-variance exists in lags 4 and 12.The above results provide two interesting findings.First, although Su (2011) indicated a one-way causal relation from the housing market to the stock market in the UK, this paper discovered a two-way causal relation between them.This supports the idea that both a wealth effect and a credit price effect exist between the housing and stock markets.Second, both causality-in-mean and causality-in-variance are detected from the housing market to the stock market.This finding has never been referred to in previous studies and it is useful for both practitioners and academic researchers.

Conclusions
This paper analyzes the causality-in-mean and causality-in-variance between the UK stock and housing markets using monthly data from January 1991 to August 2016.A CCF approach developed by Cheung and Ng (1996) and a causality-in-variance test applied to financial market prices are used as tests (Cheung and Ng 1996).The empirical findings make two key contributions.First, although Su (2011) showed a one-way causal relation from the housing market to the stock market in the UK, this paper discovered a two-way causal relation between them.Thus, both a wealth effect and a credit price effect exist between the housing and stock markets.This paper also detected a causality-in-mean and causality-in-variance from the housing market to the stock market.This point has never been referred to in previous studies and is useful for both practitioners and academic researchers.

Figure 1 .
Figure 1.Market capitalization of the S&P Global REIT (Real Estate Investment Trust) Index in August 2016.Data Source: S&P Capital IQ.

Figure 2 .
Figure 2. Rates of change in the stock and housing indexes.Data Source: Nationwide Building Society, Yahoo Finance.

Table 1 .
Summaries of previous studies.

Table 2 .
Descriptive Statistics: Rates of change in the stock and housing indexes.

Table 3 .
Results of an augmented Dickey-Fuller test.

Table 5 .
Cross-correlation for levels and squares of the standardized residuals.