A multivariate analysis of the UK house price volatility

A multivariate analysis of the UK house price volatility Kyriaki Begiazi 1 and Paraskevi Katsiampa 2 Abstract: Since the recent financial crisis there has been heightened interest in studying the volatility of the housing market. Using a multivariate GARCH approach we examine volatility linkages among house prices in the UK regions and different property types (flats, terraced, detached and semi-detached and houses). There is evidence that three of the regions have a constant variance, namely North, North West and Scotland. However, the rest of the regions show time-varying variance. According to the results, we find evidence of volatility spillover among the different UK regions with time-varying conditional variances, as well as, among flats, terraced and detached houses. These results have significant implications for appropriate economic policy selection and investment management. Keywords: volatility spillover, GARCH, VECH, contagion effect, causality 1 Department of Accounting, Finance and Economics, Oxford Brookes University, Oxford, OX33 1HX, UK 2 International Business and Economics Research Group (IBERG), Sheffield Business School, Sheffield Hallam University, Sheffield, S1 1WB, UK Kyriaki Begiazi kbegiazi@brookes.ac.uk Paraskevi Katsiampa p.katsiampa@shu.ac.uk

1. Introduction The analysis of real estate markets has long been the subject of interest in different countries. This can be explained by the fact that housing markets play an important role in the economy. Interestingly, house prices share some properties with financial time series (see, e.g., Dolde and Tirtiroglu, 1997; Miles, 2011a; Karoglou et al., 2013; Lin and Fuerst, 2014), and hence appropriate modelling of house prices is of high importance. Recently there has been an increased academic interest in the modelling of house prices by using the class of GARCH models. Previous studies have employed different univariate GARCH-type models for house price volatility, such as GARCH (e.g., Dolde and Tirtiroglu, 1997; Stevenson et al., 2007), the Exponential GARCH (EGARCH) (e.g., Lee 2009; Lin and Fuerst, 2014; Morley and Thomas, 2016), the Component GARCH (CGARCH) (Miles, 2011a; Lee and Reed, 2013; Karoglou et al., 2013), and the Threshold GARCH (TGARCH) model (Katsiampa and Begiazi, 2017). Nevertheless, there is a rather limited literature on volatility linkages between housing markets, or between different property types within specific markets. Information and cross-market hedging could lead to volatility linkages between different markets that affect return expectations and asset demand (Miao et al., 2011). Earlier studies using multivariate models for house prices include the Vector Autoregressive (VAR) (Miller and Peng, 2006; Hossain and Latif, 2009) and the Vector Error Correction model (Damianov and Escobari, 2016), while multivariate GARCH (MGARCH)-type models used in previous studies include the BEKK- MGARCH (Willcocks, 2010; Miao et al., 2011) and the Dynamic Conditional Correlation (DCC) model (Antonakakis et al., 2015). Furthermore, Begiazi et al. (2016) used DCC and BEKK GARCH model to test volatility spillover among global Real Estate Investment Trusts (REITs) and found that the REIT market is becoming increasingly globalised. There is a wide range of literature on the dynamics of the UK housing prices over the last decade (see, e.g., Stevenson et al., 2007; Tsai et al., 2010; Willcocks, 2010; Miles, 2011b; Morley and Thomas, 2011, 2016; Katsiampa and Begiazi, 2017). As it has been previously shown that UK regions exhibit conditional volatilities (see, e.g., Willcocks, 2010; Morley and Thomas, 2016), this paper aims to extend the literature by employing a multivariate GARCH - diagonal VECH model to examine whether housing markets within the UK regions are linked, and, if so, what kind of return and volatility relationships are observed across them. We also

examine whether different property types, namely flats, terraced houses, semi-detached and detached houses, are linked within the whole UK. The remainder of the paper is organised as follows. Section 2 presents the data and methodology used in this study. Section 3 discusses the empirical findings. Finally, the conclusions drawn and the implications for policy making are presented in section 4. 2. Data and methodology This paper uses UK regional quarterly house price data from Nationwide s House Price Index from the fourth quarter of 1973 to the first of quarter 2017, giving a total of 174 observations. Furthermore, we use aggregate UK quarterly series by property type, first quarter 1991 to first quarter 2017. The data are publicly available online at http://www.nationwide.co.uk/about/house-price-index/download-data#tab:downloaddata. The data are converted to natural logarithms, and then we define the housing returns for each property type as R (ln I t, i ln I t 1, ), t, i i where R, is the logarithmic house price index return in quarter t for UK region i or house t i property type i, and I, is the house price index in quarter t for UK region i or house t i property type i. First of all, using the ADF test, we check whether our series have a unit root or not. As will be seen in the next section, all the series are found to be stationary. Next, following Miller and Peng (2006) and Willcocks (2010) each region is modelled by an ARMA (p,q) model. We select he appropriate regional lag structure, by using information criteria, on the basis that there is heterogeneity across different areas. The general ARMA (p,q) is shown below R t,i = μ + φ 1 R t 1 + + φ p R t p,i + ε t,i + θ 1 u t 1,i + θ q u t q,i

The residuals from these models are then tested for the existence of ARCH effects. For those regions with non-constant variances, MGARCH modelling is employed in order to model the conditional variances and examine the linkages among the different areas and property types. MGARCH models enable us to estimate the conditional volatilities of the variables simultaneously. The analysis is based on the fact that simultaneous shocks to variables can be correlated to each other. Furthermore, volatility spillovers might affect the volatility of other related variables. The main reason why we selected a multivariate model is the fact that shocks that increase uncertainty in one market could also increase the uncertainty in another market, and, hence, employing univariate models to study the conditional variance of each market separately is not appropriate for examining interlinkages of different markets. For simplicity we assume only two variables i and j and we consider the following two error processes ε it = v it (h iit ) 0.5 ε jt = v jt (h jjt ) 0.5. As in the univariate case, if we assume that var(v it ) = var(v jt ) = 1, we can think of h iit and h jjt as the conditional covariances of the error. Accordingly, h ijt is the conditional covariance between the two shocks. In order to examine the evolution of co-movements between housing market returns of different UK countries and different property types, we employ the multivariate GARCH - diagonal VECH model of Bollerslev et al. (1988). The idea is to diagonilise the system so that h ijt contains only lags of itself and the cross productsε it ε jt. The model takes the following form H t = C + Α ε t 1 ε t 1 + Β Η t 1. Where the coefficient matrices A, B and C are symmetric and the operator is the element by element (Hadamard) product. Each matrix contains N(N+1)/2 parameters. By parameterising the coefficient matrices as indefinite matrices we have the following model specification (H t ) ij = (C) ij + (Α) ij ε t 1 ε t 1 + (B) ij (H t 1 ) ij

h ijt = c ij + a ij ε it 1 ε jt 1 + β ij h ijt 1. Engle and Kroner (1995) created the Diagonal BEKK model that ensures positive conditional variances H t = CC + Αε t 1 ε t 1 A + ΒΗ t 1 B. The Diagonal BEKK model is identical to the Diagonal VECH model where the coefficient matrices are rank one matrices. The elements of the covariance matrix, Ht, depend only on past values of itself and past values of ε t ε t, which is innovation. The elements of matrix A measure the effects of shocks or news on the conditional variances (ARCH effects). The elements of matrix B explain how past conditional variances affect the current levels of conditional variances, in other words, the degree of volatility persistence in conditional volatility among i and j (GARCH effects). Specifically, the diagonal parameters in matrices A and B measure the effects of own past shocks and volatility on its conditional variance. The volatility spillover measures the cross-market effects of shocks and volatility using the off-diagonal parameters in matrices A and B, respectively. The benefit of this analysis is that we can examine if shocks to the variance of one of the variables could spill over the others. Housing markets play an important role in the economy, so, their behaviour across time is important in order to understand them and improve our investment and risk strategies. 3. Empirical Findings 3.1. UK house prices by region Basic descriptive statistics are presented in Table 1. The results of the Jarque-Bera test (Jarque & Bera, 1980) show that London, North, Outer South East and Scotland do not reject the normal distribution hypothesis at 5% significance. The quarterly mean returns range from 1.58% (Yorkshire & Humberside) to 2.09% (London) and standard deviation from 2.74% (Scotland) to 4.11% (Northern Ireland). The UK as a whole reports 1.76% mean return and 2.59% standard deviation.

Table 1. Descriptive statistics of regional log returns Region Mean Std. Dev. Jarque-Bera East Anglia 0.017781 0.035133 15.38482*** East Midlands 0.017226 0.030589 66.89342*** London 0.020914 0.032357 1.325693 North 0.016026 0.033726 4.629535* Northern Ireland 0.016058 0.041081 19.07336*** North West 0.017011 0.028480 10.87776*** Outer Metropolitan 0.019207 0.029821 7.309514** Outer South East 0.018583 0.031750 2.995706 Scotland 0.016043 0.027434 4.845494* South West 0.018449 0.031353 54.28482*** Wales 0.016026 0.033664 23.86913*** West Midlands 0.016948 0.031269 120.9235*** Yorkshire & Humberside 0.015862 0.033964 12.12583*** UK 0.017642 0.025892 8.462891** We use various diagnostics tests to ensure the validity of the proposed modelling approach. Table 2 summarises the test statistics and lags of the ADF test on the first differenced return series. All of the regions are stationary, so, we can continue our time series modelling. Table 2. ADF tests on first differenced series Region Lag Test Statistic East Anglia 2-12.06032*** East Midlands 1-13.14912*** London 2-11.73600*** North 1-16.49777*** Northern Ireland 6-8.129662*** North West 1-15.07821*** Outer Metropolitan 8-6.051793*** Outer South East 5-7.591686*** Scotland 6-9.067569*** South West 3-7.477850*** Wales 0-22.90741*** West Midlands 2-13.29641*** Yorkshire & Humberside 5-8.215429*** UK 6-7.921928*** The first part of the econometric analysis tests for ARCH effects among the different UK regions. We use ARMA models for each region. Information criteria are used to select the most appropriate model. Different lags are present and this is in accordance with Miller and Peng (2006) and Willcocks (2010), who highlights that the housing market is heterogeneous and buyers form their expectations based on the local experience. Residuals are then tested for heteroscedasticity and ARCH effects. The results are summarised in Table 3. All of the regions have non-constant variances, apart form, North, North West and Scotland. Thus, we use the regions with time-varying variances to create a multivariate GARCH model and examine their inter-linkages.

Table 3. ARCH tests by region Region ARMA (p,q) F statistic nr 2 East Anglia 3,4 4.933375** 4.850649** East Midlands 1,1 15.45985*** 14.33785*** London 4,2 4.641978** 4.571754** North 3,3 0.811591 0.817237 Northern Ireland 3,5 4.786993** 4.710664** North West 6,2 0.033400 0.033786 Outer Metropolitan 3,6 7.933714*** 7.670620*** Outer South East 3,5 5.741229** 5.619008** Scotland 3,3 0.044366 0.044876 South West 5,4 25.98242*** 22.80294*** Wales 3,4 26.09166*** 22.88606*** West Midlands 3,3 35.72150*** 29.86609*** Yorkshire & Humberside 5,2 21.74048*** 19.50221*** UK 4,2 4.519836** 4.454576** Table 4 reports the cross-region volatility spillovers. The conditional variance equation findings for the diagonal VECH model show volatility linkages among the different UK regions. Table 4. Conditional variance and covariance GARCH coefficients by region Region EA EM L NI OM OSE SW W WM YH East Anglia EA 0.6923*** 0.6776*** 0.6867*** 0.7041*** 0.7064*** 0.6731*** 0.7772*** 0.6151*** 0.6151*** 0.7299*** East Midlands EM 0.6632*** 0.6722*** 0.6892*** 0.6914*** 0.6588*** 0.7607*** 0.6021*** 0.7144*** 0.6132*** London L 0.6813*** 0.6985*** 0.7007*** 0.6677*** 0.7710*** 0.6102*** 0.7240*** 0.6215*** Northern Ireland NI 0.7161*** 0.7184*** 0.6846*** 0.7905*** 0.6256*** 0.7424*** 0.6372*** Outer Metropolitan OM 0.7207*** 0.6868*** 0.7930*** 0.6276*** 0.7447*** 0.6392*** Outer South East OSE 0.6545*** 0.7557*** 0.5981*** 0.7097*** 0.6091*** South West SW 0.8725*** 0.6906*** 0.8194*** 0.7033*** Wales W 0.5466*** 0.6485*** 0.5566*** West Midlands WM 0.7695*** 0.6605*** Yorkshire & 0.5670*** Humberside YH 3.2. UK house prices by property type This study also investigates volatility spillovers among different property types, namely flats, terraced houses, semi-detached and detached houses. The following analysis focus on the four property types in the UK as a whole. According to Table 5 the mean returns vary from 1.12% (detached) to 1.41% (terraced) and the standard deviation from 2.23% (detached) to 3.63% (flats). Flats do not reject the null hypothesis of normal distribution. Table 5. Descriptive statistics of UK log returns by property type Property type Mean Std. Dev. Jarque-Bera detached 0.011669 0.022978 1.757053* flats 0.014096 0.036331 1.772687 semidetached 0.012995 0.025056 1.967258* terraced 0.014145 0.025999 1.165501*

Table 6 summarises the results from the ADF stationarity tests. All of the property types are stationary, so, we can use ARMA models to observe their movements over time. The results, as in the regional analysis, present different lags for different property types. That is an indication of heterogeneity not only among regions but also among different property types. Table 6. ADF tests on first differenced series by property type Region Lag Test Statistic detached 7-4.851301*** flats 2-10.57378*** semidetached 6-6.895867*** terraced 1-11.36569*** The best fitted ARMA models for the four property types are presented in Table 7. ARCH effects are evident for detached houses and flats (5% significance), but also for terraced houses (10% significance). The three property types indicating that there is evidence of ARCH effects are therefore modelled using a multivariate GARCH model. Table 7. ARCH tests - UK by property type (5 lags) Property type ARMA (p,q) F statistic nr 2 lag detached 4,3 3.709842** 10.39570** 3 flats 3,3 6.060158** 5.830332** 1 semidetached 4,3 0.714884 0.723916 1 terraced 4,3 2.869240* 2.845228* 1 The values of the transformed variance GARCH coefficients are shown in Table 8 as the transformed value and they suggest that there is considerable persistence in the inter-property conditional volatility. Detached houses and flats report the highest conditional variance coefficient estimates. Table 8. Conditional variance and covariance GARCH coefficients - UK by property type Property type detached flats terraced detached 0.589469*** 0.637274*** 0.310424*** flats 0.624044*** 0.412547*** terraced 0.145158 4. Conclusions The motivating factor for this study was to extend Willcoks' (2010) analysis by using more recent data, but also by considering different property types in the UK (flats, terraced, semidetached and detached houses)as well. Our results indicate that the UK housing market shows

evidence of ARCH effects in all regions apart from North, North West and Scotland. It is also found that regions with time-varying variance report important volatility linkages.. With regards to the different property types, there is evidence of volatility spillover across flats, terraced and detached houses. Consequently, we can conclude that they are not only crossregion effects in the UK, but also cross-property type volatility spillover. Since the recent financial crisis there has been increased interest in studying the volatility of the housing market. As we have seen, the property market can have considerably effects on the whole economy. Our results could have important theoretical and practical implications for investors, regulators and risk managers. As better analytical tools could lead to better decisions, the proposed quantitative analysis of returns and variance of the UK house prices indices could help investors understand not only the cross-regional, but also the cross-property UK housing market.

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