A multivariate analysis of United States and global real estate investment trusts

Using daily data for the period February 2006 to July 2013 we examine the return and volatility linkages between the two main United States REIT sub-sectors and global linkages between the Americas, Europe and the Asia Pacific regions using the BEKK-GARCH and the DCC-GARCH models. We find that there is no evidence of any volatility spillovers between the US sub-sectors. By contrast, we find evidence of volatility spillovers between the Asia Pacific and the Americas, the Asia Pacific and Europe but no spillovers between the United States and Europe. Our results suggest that the REIT market is becoming increasingly globalized and that investors need to consider time varying volatility and correlations across different regions of the world when forming their optimal portfolio-allocations.


Introduction
Volatility and correlations of asset returns are significant inputs in the calculation of risk in modern portfolio theories and in the analysis of risk management, strategic financial planning and asset allocation. This paper is one of the first to look at the issue of the globalization of the real estate investment trusts (REITs) market by comparing the volatility interactions in the form of co-movements and spillovers between USA subsectors with that between the Americas, Europe and Asia Pacific regions using and their conditional variance were negatively related to REITs excess returns. In terms of returns, He (1998) and Lee and Chiang (2004), find evidence to support the existence of commonality between US equity and mortgage REITs. These results are disputed by Cotter and Stevenson (2006) using monthly returns and the multivariate VAR-GARCH technique on REIT sub-sectors who find that the influence of various US equity series and the correlations are weak. 1 There is a growing literature examining the relationship between REITs sector and broader equity market indices such as Subrahmanyam (2007) who finds that stock market returns are negatively related to REIT order flows and that the real estate market is a substitute for investments in the stock market.
When it comes to global REITs there has been a limited but growing literature focusing on correlations in returns but next to nothing on correlations between volatilities of returns. Bond et al. (2003) examine the risk and return characteristics of publicly traded real estate companies for 14 countries for the period 1990-2001 using monthly data and find substantial variation in mean returns and standard deviations. They also detect evidence of a strong global market risk component using the MSCI world index. Yunus and Swanson (2007) examine the short run and long run relationships between the Asia Pacific region (Australia, Hong Kong, Japan and Singapore) and the US for the period January 2000 to March 2006. Their short run causality tests show no evidence of significant lead-lag relationships suggesting the potential for significant benefits from international portfolio diversification. They find that from a long run perspective Hong Kong and Japan provide the best diversification benefits. Liow et al. (2009) examine correlation and volatility dynamics of publicly traded real estate securities using monthly data for the period 1986-2006. They find that correlations between real estate security returns are lower than those between stock markets. They also detect significant positive connections between real estate securities market correlations and their conditional volatilities and that the international correlation structure of real estate securities and stock markets are linked to each other. In a recent paper Chang and Chen (2014) look for evidence of contagion using daily REITs for 16 countries covering the period 2006-2010. To do this they look to see if correlation coefficients increase significantly during the crisis period 2007-2010. Their results show significant evidence of contagion in global REITs during the crisis. However their evidence looks at transmission from the US market to the other countries and not vice-versa.
To effectively diversify their portfolios investors are interested in spreading their investments across international markets. As such, the interdependence between domestic and global financial markets is very important for them. Volatility spillover is present when a market shows significant signs of co-movement with other global markets and this is important for policy makers because it affects the financial system and the economic performance. Consider for example, Mishkin (2005) and Singh et al. (2010) who find that interdependence is accompanied by speedy transmission of volatility shocks linking the domestic and global stockmarkets. To date research that examines volatility spillovers and return co-movements between the national REIT markets in different regions of the globe has been very limited. 2 This is the gap that the current research is trying to address.

Data and methodology
The empirical tests conducted in this paper utilize the FTSE EPRA/NAREIT daily indices. The dataset used comprises of daily data for the period February 2006 to July 2013. The first dataset consists of the two main US REIT sub-sectors, the FTSE NAREIT All Equity REITs Index (FNRE)  Some descriptive statistics of the respective series are outlined in Table 2 detailing the first four moments of each series, and the correlation matrix between the series. The values of the coefficients of skewness, kurtosis together with the large Jarque-Bera statistics lead to the rejection of the null hypothesis of a normal distribution. From the correlation matrix we can see that the correlation among our variables are positive and high (>0.7). Therefore, some degree of multicollinearity is unavoidably present. Heteroskedasticity is also present as indicated by the high value of the LM-statistic. It is clear that all series exhibit cases of volatility clustering requiring that the estimation should include ARCH-type processes.
The empirical analysis is undertaken using an MGARCH framework. To examine the relationship between the Equity and Mortgage US REITs we use a bivariate (restricted and unrestricted) GARCH model and a trivariate (restricted) GARCH model for the Global indices.
First we have to determine the suitability of the BEKK model. This requires the existence of heteroskedastic effects in the return series. Using the Engle (1982) LM test for ARCH (p) effects, we find strong evidence of ARCH effects for all cases. The following mean equations were estimated for each index.
where r t is an 2 × 1 vector of daily returns at time t for each index, and ε t |ε t − 1~N (0, H t ) is an 2 × 1 vector of random errors for each index at time t. This model helps us in the examination of any volatility transmission. The main advantage of the BEKK model is that it has few parameters and ensures positive definiteness of the conditional covariance matrix to ensure non-negative estimated variances. The bivariate version of the BEKK GARCH specification (Engle and Kroner 1995) is defined as: where y t is a 2 × 1 vector of random variables incorporating the returns and ε t is a normally distributed error term. H t , denotes the conditional variance-covariance matrix at t and matrices B and A as well as the diagonal elements of C have to be positive. The elements of the covariance matrix H t , depends only on past values of itself and past values of ε t /ε t , which is innovation. Each matrix C, A and B dimension is 2 × 2 and C is restricted to be upper triangular. The elements of  matrix A measure the effects of shocks or "news" on the conditional variances (ARCH effects). The 2 × 2 square matrix B shows how past conditional variances affect the current levels of conditional variances, in other words, the degree of volatility persistence in conditional volatility among the markets (GARCH effects). 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. This model is suitable for cross dynamics of conditional covariances because A and B do not need to be diagonal. We assume that a 11 > 0 and b 11 > 0 due to the uniqueness of the BEKK representation. Then, if K = 1 there exists no other C, B, A in the model that will give an equivalent representation. The purpose of the restrictions is to eliminate all other observationally equivalent structures. The amount of parameters to be estimated is N (5 N + 1)/2, thus in a bivariate model (N = 2, with p = q = 1) 11 parameters should be estimated. We can differentiate between two alternative specifications presented analytically below: a) Bivariate Unrestricted Specification GARCH (1,1) -BEKK, N = 2: The matrix multiplication is presented as: b) Trivariate Restricted Specification Consider the BEKK GARCH (1,1) with N = 3. In the restricted trivariate model the matrices A and B are diagonal and the amount of parameters to be estimated is 24. The matrix multiplication is presented below.
The last restricted specification of the BEKK model restrict the off diagonal elements in A and B, that measure the volatility spillover, to zero. Consequently, each conditional variance depends only on past values of itself and the lagged cross-product of residuals.
For our volatility analysis, we use the DCC-GARCH model proposed by Engle (2002). This is a generalized Bollerslev's (1990) constant conditional correlation model by making the conditional correlation matrix time-dependent. This method takes as input the standardized residuals, which are simply the data series residuals divided by the GARCH conditional standard deviation to estimate DCC conditional correlations. 4 The model for two assets is defined as: ρ 12;t ¼ q 12;t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 11;t q 22;t p ð12Þ q 12;t ¼ R 12 1−α−β ð Þþαs 1;t−1 s 2;t−1 þ βq 12;t−1 ð13Þ In this model ρ 12,t is the DCC-model conditional correlation, R 12 is the average realized correlation, s 1,t − 1 and s 2,t − 1 are the lagged GARCH standardized residuals and the "quasi-correlations" are represented by q values. The first term R 12 1−α−β ð Þ≡ω is restricted to be constant. This is known as correlation targeting and reduces the number of unknown parameters to only α and β. To estimate α and β parameters, we use maximum likelihood estimation. As noted by Engle (2009), the log-likelihood function in this case applies to a pair of assets, which is given by: 4 Empirical results  * indicates significance at the 5% level or higher subscripts 11 for asset 1, and 22 for asset 2. Similarly, cross-market effects are given by subscripts 21 and 12 for asset 1 and asset 2 respectively. All the past shocks and past volatility are significant. The result that |α ii | < |b ii |, suggests that the behavior of current variance and covariance is not so much affected by the magnitude of past innovations as by the magnitude of lagged variances and covariances. In the conditional variance equation the α ii coefficients represent ARCH effects, while the b ii coefficients represent GARCH effects. As would be expected in the volatility equation, current returns and volatility are affected from their own past series returns. The coefficients are significant revealing that autocorrelation and volatility clustering is present in the returns with an autocorrelated relationship in the second moment of the distribution. Autoregressive and time dependent volatility effects incur for each series as shown by the α 11 , α 22 , b 11 , b 22 parameters. The off-diagonal elements of matrices A and B capture the cross-market effects such as shock and volatility spillover. In documenting the shock transmission between the main US REIT subsectors, we find evidence of unidirectional linkage between Equity and Mortgage REITs running from equity to mortgage (i.e., only the off-diagonal parameter α 12 is statistically significant). In other words, equity shocks affected mortgage mean returns. Second, we did not identify any volatility spillover between them (the offdiagonal parameters of matrix B are statistically insignificant). The use of different data frequency can lead to very contrasting empirical findings as outlined in Cotter and Stevenson (2006). It is possible that the use of the higher frequency data masks more of the fundamental relationships, with general market sentiment coming more to the force.  Table 4 reports the results of the unrestricted bivariate BEKK GARCH (1,1) estimations that examine the global linkage of REIT markets. First, we estimate three pairwise models using a bivariate GARCH framework and adopting a BEKK The off-diagonal elements of matrices A and B capture the cross-market effects such as shock and volatility spillover. In documenting the shock transmission globally, we find a bidirectional correlation between America and Asia Pacific since the off-diagonal elements α 12 and α 21 are statistically significant. This indicates a strong connection between them. Further, we find evidence of unidirectional linkage between Europe and Asia Pacific running from Asia Pacific to Europe since only the α 21 coefficient is statistically significant. In other words, Asia Pacific shocks affected Europe mean returns. No mean effects were found between Americas and Europe. Second, we identify bidirectional volatility linkages between Americas-Asia Pacific and Europe-Asia Pacific; the pairs of off-diagonal parameters, b 12 and b 21 , are both statically significant. These results provide strong evidence of the global REIT market's integration. Next, we proceed by estimating a restricted trivariate BEKK GARCH (1,1) model. Table 5 presents the estimated coefficients of the variance-covariance matrix of a trivariate M-GARCH BEKK model employed for analyzing volatility relationship between global REITs. The results show that all the conditional variance coefficients are significantly positive. As seen above, all the past shocks and past volatility are significant. Since we observe that |α ii | < |b ii |, we conclude that the behavior of current variance and covariance is not so much affected by the magnitude of past innovations as by the magnitude of lagged variances and covariances. Moreover, the statistical significance of the GARCH b ii parameters reveals the large extent of volatility clustering. Finally, from the dynamic correlations series presented in Fig. 3, we observe that the correlation stays within 0.87 and −0.15 (mean: 0.35) for Americas and Europe, 0.61 and

DCC-GARCH specification
The final test involves estimating the DCC-GARCH model. The results of this analysis are presented in Table 6. We obtain the model parameters (α, β) for any given pair of assets. The coefficients α and β refer to the DCC (1,1) estimates. The estimated GARCH-DCC model appears to provide a good representation of the conditional variance of the data. The persistence of the conditional correlations, measured by α and β, is close to unity that is between 0.92 and 0.99. The β coefficient is always significant and above 0.90 and α is below 0.04 revealing slight response to innovations and major persistency.
Only the Americas and Europe report an insignificant parameter α. All the other parameters α and β are positive and statistically significant suggesting evidence of a strong interaction between the returns of the indices. It is worth noting that all significant coefficients highlight the time varying nature of conditional variances and covariances. Figure 4 presents the graphs of the conditional correlation coefficients, as estimated using the GARCH-DCC (1,1) procedure, for each pairing of the REIT time-series. A number of issues are of interest; however, there is no evident consistency across the different pairs. When the majority of the correlations are relatively low, this implies diversification potential across REIT sectors and when they display a relatively high level of spread across the correlations, this confirms the findings for the unconditional coefficients (0.0-0.2: very weak, 0.2-0.4 weak, 0.4-0.7 moderate, 0.7-0.9 strong, 1 perfect correlation).
As expected, the strongest correlations are reported in the two main US subsectors. US Equity and US Mortgage REITs report high conditional correlations in general, ranging from 0.5 to 0.8 (moderate) and the lowest conditional correlation (0.27) is reported around the end of 2012. In Americas-Asia Pacific the conditional correlations follow a downward trend from 2006 to 2009 (very weak) and a strong upward trend from 2009 to 2013 (weak) that clearly indicate that the sector has undergone a distinct shift over the last few years. On the other hand Europe-Asia Pacific and Americas-Europe (weak) correlations tend to be far more tightly banded. Indeed, only positive conditional correlations are reported in all the tested pairs.

Conclusions
This paper examined the linkage of REITs. The starting point we examine the linkage between the two main US REIT subsectors (Equity and Mortgage) and then we extended our analysis globally for Americas, Europe and Asia Pacific REITs. We employ GARCH-BEKK and DCC models based on daily return indices from 2006 to 2013. As would be expected both ARCH and GARCH effects are present. In the volatility equation, past own series returns affect current returns and volatility. The appropriate coefficients are significant, supporting the findings of autocorrelation in the returns. Regarding the two main US subsectors we find no evidence of any volatility spillover between them. This result leads to the assumption that investors can benefit from risk diversification. Another factor which may play an important role in this interdependence is the fact that fewer than 10 % of REITs are mortgage sector in the United States. Therefore the equity sector is clearly larger than the mortgage and this may also explain why equity shocks affect mortgage mean returns. Depending on their variance analysis the Mortgage index variation shoots through the rough during 2007-2009, after that through 2009-2010 the Equity index had a greater variation and from 2010 to 2013 they are moving together.
As far as the global linkage of REIT market and the shock transmissions are concerned, we find a bidirectional correlation of Americas and Asia Pacific that indicates a strong connection between them. Further, we find that Asia Pacific shocks affected Europe mean returns; while there are no mean effects between Americas and Europe. Second, we identify bidirectional volatility linkages between Americas-Asia Pacific and Europe-Asia Pacific. These results provide convincing evidence of the global REITs markets integration. The global REIT market is a financial market with particular characteristics and each REIT system has its own legislation. The Asia Pacific REIT index that consists of most constituents according to Table 1 seems to be the more influential for both the Americas and Europe. The absence of any cross market effects between Americas and Europe implies that investors can significantly benefit from a reduction of diversifiable risk.
The most immediate implication of the DCC model is that there is a strong interaction between the returns of the indices that highlights the time varying nature of conditional variances and covariances. DCC coefficients also reveal a slight response to innovations and a major persistency. Conditional correlations show the way that the returns of one REIT index correlate with the returns of another REIT index over time. While Americas-Europe and Europe-Asia Pacific weak correlations follows a more stable trend, Americas-Asia Pacific correlations have undergone a distinct upward shift over the last few years. This is an indication that Americas and Asia Pacific have become more integrated post 2009 and that they have lost some of their diversification properties. On a local base in the US REIT market strangely we discover that the diversification potential within the two main sectors has slightly risen in the last years with the subsectors behaving not as homogeneous as in the period 2006-2012. . It is commonly argued that REITs should adopt a focused investment strategy in order that investors can make their own diversification decisions. However, this is based on an underlying assumption that performance does differ and that the share prices of REITs reflect the fundamentals of the underlying property sectors. Recent research by Philippas et al. (2013) and Chong et al. (2012) have concluded that REITs are behaving in a more homogenous manner than the past and this calls into question the investment based argument for REITs to be focused.
The results of this study will help investors in their portfolio selection to incorporate time varying volatility and correlations and can be extended in several directions. Empirical research on the matter can possibly apply other multivariate techniques such as constant correlation (CC) or time-varying correlation (VC). Also, it would be interesting to apply the current methodology in more secondary US REIT subsectors and in a more analytical global based analysis; however, these issues are left for further research.