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During recent subprime crisis period Chiming Wu^{1} Abstract The purpose of this paper is to explore Asia REITs potential diversification value to global equity investors during recent 2007 subprime crisis. This study examines return level and volatility linkage between Asia REITs and global equity using both a multivariate GARCH based model and an univaiate GARCH copula model. The motivation is for global investors to incorporate timevarying volatilities, correlations, and beta of Asia REITs in their portfolio selection. First part of paper, a bivariate GARCH based model is fitted for each of three pairs: Taiwan REITs vs. global equity, Asia REITs vs. global equity, and Asia (ex. Japan) REITs vs. global equity using BEKK estimation method. Then the timevarying conditional correlations, volatilities, and betas are estimated and compared across various time periods and pairs to examine global diversification benefits of Taiwan and Asia REITs especially during recent subprime global financial crisis. The Second part of the paper is to examine “Tail dependence”, which characterizes the cross market linkages during stressful times. Analyzing tail dependence is of primary interest to portfolio managers who systematically monitor the comovements of asset markets. However, the relevant literature on Asia REITs’ tail dependence on global equity is very thin. The Second part of paper extends the literature by using semiparametric empirical copula and parametric copula fitting test to examine whether lower tail dependence exist between Asia REITs and global equity during recent financial crisis period. In implementing the empirical copula, we model the marginal distributions of returns through a semiparametric method, to the best of my knowledge, which has never been applied to Asia REITs returns before The major findings of this paper suggest that Asia REITs low and nontrend correlations make this a possible proposition for investors in their portfolio selection. Besides, Asia REITs’ lower beta, combining lower correlation (and tail dependence) coefficients and relative lower own volatility (comparing with Taiwan REITs and Asia (ex. Japan) REITs), provide better diversification value at recent global financial crisis. Only using the correlation coefficients, portfolio managers might make wrong asset selection decisions. Our findings have important implications for portfolio tail diversifications, portfolio selections, portfolio risk management and hedging strategies. Keywords: Asia REITs, Multivariate GARCH , BEKK, Tail dependence, Empirical copula. 1. Introduction The recent global financial crisis (since 2008) is initiated by US and Europe bubblebroken real estate market. The increased volatility in crisis period puts pressure on global investors to try to diversify some of this risk. The real estate investment trust (REIT) market in the United States offers investors a way to invest in real estate without the problems of illiquidity, intense management and large lot size/high unit cost (Ciochetti, Craft and Shilling, 2002). The combination of factors such as the limitations on REITs in relation to dividend payments and the strong relative performance of the sector in the aftermath of the collapse of the technology bubble have resulted in increased fund flows into the REITs sector (Liang and Naranjo, 2003). Furthermore, the low correlation of REITs with the U.S. stock market in the late 1990s led to claims that REITs offered excellent diversification benefits to a mixedasset portfolio as well as improved return performance (NAREIT, 2002). As Lee and Stevenson (2005) note, REITs to some extent provide a hybrid investment form, standing between equities and the fixedincome sector. In addition, the asset maintains strong links with the direct private real estate market. These interlinkages provide the asset with unique characteristics. Although many Asian blue chips are property developers, REITs are new to Asia—nearly four decades behind the United States and Australia. The trend began with Japan in 2001 and continued with Singapore, South Korea, Hong Kong, Taiwan, Malaysia, and Thailand. China is expected to have its first by the end of 2010. Table 1 shows the global and Asian REITs statistics. Taiwan have the first REIT in 2005 and up to the present, there are eight REITs at the market. (Table 1 shown here) To evaluate the comovement, it has been a common practice to examine linear correlation coefficient (ρ), often based on multivariate GARCHtype models. The advantage of the use of a multivariate framework is that not only does such an approach provide an analysis of volatility and the accompanying interlinkages between the assets concerned, but it also allows an estimation of the timevarying covariance’s and correlations. This allows an investor to incorporate timevarying volatility and correlations in their portfolio selection. Increased (decreased) correlation imply lower (higher) diversification effects impacting investor portfolio selection. The practice of using correlation coefficient as an allpurpose dependence measure has been questioned in recent years (see, e.g. Embrechts et al. 2003; Longin and Solnik 2001; Rachev et al. 2005 and the references therein). There are several reasons. First, at the theoretical level, it has been shown that ρ is an appropriate measure of dependence if and only if the joint distributions of financial returns are Gaussian (or more generally elliptical, see, e.g. Embrechts et al. 1999). However, abundant evidence has accumulated that the dependence between many important asset returns are non Gaussian (see, e.g. Erb et al. 1994; Ang and Chen 2002; Ang and Bekaert 2002; and Bae et al. 2003). One prominent example is where two assets returns exhibit greater correlation during market downturns than during market upturns. Second, ρ is defined to measure the linear association for the entire range of data. It makes no distinction between extreme and usual realizations. It assigns equal weights to extreme observations and all other observations. As such, it may not be an accurate measure of dependence if extreme returns present different patterns of dependence from the rest of the sample. However, it is well understood that the dependence among extreme returns is of crucial importance to portfolio managers especially during crisis periods when a lot of extreme returns occur. In the field of risk management, two extreme risk measures, namely valueatrisk (VaR) and expected shortfall (ES) have been widely used by banks and financial institutions. VaR measures the maximum loss on an investment over a specified time horizon at a given confidence level while ES measures the average loss when the loss exceeds VaR. For portfolios, the estimates of both measures are largely dependent on how the dependence structure is modeled. For example, Tsafack (2009) show that for equity and bond portfolios a multivariate symmetric model like DCC would underestimate VaR and ES if asymmetric tail dependence were present in the data. Finally, ρ, though it measures the degree of dependence, does not account for the structure of dependence. As research in multivariate theory has demonstrated, it is possible to construct multivariate distributions with identical ρ but otherwise completely different dependence structures. For example, Patton (2006) illustrated that, given ρ=0.5, two random variables can display a variety of contour plots of their bivariate distributions, pointing to the existence of different dependence structures. Dependence structure, as well recognized in the literature (e.g. Sun et al. 2009), can influence the diversification benefits. Previous studies focus on REITs diversification value owing to their low correlation with equities. Actually, an asset’s value to portfolio is not just from correlation influence, also from asset’s own volatility and returns. Only looking at asset correlation to determine an asset’s portfolio value is dangerous, this study focus on Asia REITs’ overall systematic risk to global equity in terms of beta measure – combining asset’s relative volatility and correlation. Besides, Previous studies discover that interasset correlation increase during financial crisis period. That implies diversification value will be overemphasized if we do not test if a certain asset still have some degree of independence property especially during financial crisis period. All of Taiwan REITs are commercetype (vs. residential type) and main cash income is from commercial building rental, which is thought as sensitive to business cycle. But underlying assets of Taiwan REITs are located as Taiwan, countryspecific and firmspecific factors might support Taiwan REITs are less sensitive to global financial crisis. Whether Asia REITs can provide wealth protection during recent global financial crisis period is an interesting question to be answered. Simple linear correlation analysis cannot fully and correctly answer this question, the study use multivariate timevarying correlation, volatilities, beta measures and Copula function approach to support my findings. To overcome the drawbacks of linear correlation, recent studies in finance have highlighted the use of copulas to model dependence. Simply speaking, a copula is a function that links together univariate distribution functions to form a multivariate distribution function. Copulas are capable of modeling any type of multivariate distributions including nonelliptical where the use of linear correlation is completely untenable. This is the primary motivation behind the use of copulas. Interested readers may refer to Joe (1997) and Nelson (2006) for comprehensive reviews of the copula theory. In addition, copulas can fully specify the dependence between two or more variables: they reveal not only the strength of dependence but also any specific aspect of the dependence structure, in particular, the tail dependence. Tail dependence, as its name suggests, refers to the strength of dependence among the tails of financial distributions. Because extreme values reside in the tails, tail dependence is particularly suitable to the investigation of crossmarket linkages during stressful times. Recent empirical studies have demonstrated an increase in the frequency and magnitude of joint extreme movements across asset markets (Longin and Solnik 2001; Hartmann et al. 2004). The more markets crash simultaneously, the more in danger are even widely diversified portfolios. The ongoing crisis has well exemplified this point. Tail dependence eludes the correlation analysis but can be estimated using copulas. Copulas contain all information about the joint behavior of the distribution tails. They can accommodate a variety of tail behaviors, ranging from tail dependence to tail independence, and also allow for asymmetric dependence between upper and lower tails. Tail dependence has been increasingly studied in the general finance literature but few in the REITs literature. Only two studies su as Goorah (2007) and Zhou & Gao (2010) have touched on real estate tail dependence. To our best knowledge, there is no studies on Asia REITs tail dependence on global equity using copula functions. Despite Goorah (2007)’s pioneering efforts, the study has some limitations. First, its analysis is restricted to U.S. and U.K. real estate intramarkets linkage. This restriction is problematic considering that intermarket linkages between REITs and global equities are more meaning for diversificationseeking global investors. Second, it uses a tdistribution to model the marginal distribution of REIT returns. Though an improved choice over the Gaussian one, the tdistribution still belongs to the elliptical family. Any misspecifications in the marginal distributions will automatically lead to misspecifications in the copulas, which in turn cause imprecise estimation of tail dependence. Our study aims to overcome these limitations. We use REITs weekly return data on seven REITs markets, namely, Japan, Hong Kong, Singapore, Taiwan, South Korea, Malaysia, and Thailand. The data sample spans from 07/12/2006 to 04/28/2010 with a total of 195 observations. We filter each series of returns by ARMA(1,1)GJRGARCH(1,1) to get i.i.d. (i.e. identically and independently distributed) residuals. QQ plots show that the residuals are not normally or t distributed. Hence, we adopt the semiparametric method of Danielsson and de Vries (2000) to construct the marginal distributions of the residuals. Under this method, the interior of the marginal distribution is modeled by a nonparametric Gaussian kernel while the tails are fitted by parametric generalized Pareto distributions (GPD). As a nonparametric method, Gaussian kernel has the advantage of avoiding making any specific assumptions and letting the data determine the shape of the distributions. However, being dataintensive, Gaussian kernel tends to perform poorly when applied to the tails where only a small number of extreme values can be found. To model the distributions of the tails, we resort to the extreme value theory (EVT). EVT has shown that a welldefined tail follows a specific distribution called generalized Pareto distribution (GPD). Because of this, we fit the tails by GPD. As such, the semiparametric method combines the advantages of the nonparametric kernel and the statistical vigor of EVT. It avoids the assumption of elliptical distributions in which normal and t distributions are special cases. Once we construct the marginal distributions, we use the symmetrized Joe Clayton (SJC) copula proposed by Patton (2006) to estimate the tail dependences between Asia REITs and global equity. The choice of the SJC copula is motivated by its flexibility of modeling tail dependence: it can accommodate different dependence structures ranging from tail dependence to tail independence for both the upper and lower tails; it can capture asymmetric tail dependence but still nests symmetric tail dependence as a special case. However, being flexible comes with a cost, that is, the SJC may not be parsimonious. It contains many parameters but in some cases only a subset turns out to be significant. This would undermine the quality of parameter estimations. However, this side effect is overwhelmed by the enormous flexibility of the SJC in the estimation of tail dependence especially for a large selection of markets. We apply the SJC copula to the marginal distributions constructed by the semiparametric method. We perform pair wise analyses for three pairs: Taiwan REITs vs. global equity, Asia REITs vs. global, and Asia REITs (ex. Japan) vs. global equity. Our main findings can be summarized as follows: (1) Asia REITs provide diversification potential to global equity investors in term of their low and nontrend correlations in crisis period. (2) Comparing with Taiwan REITs and AsiaREITs), Asia (ex.Japan) REITs with lower beta provide the best diversification value at recent global financial crisis. Only using the correlation coefficients, portfolio managers might make wrong asset selection decisions. Our findings have important implications for portfolio tail diversifications, portfolio selections, portfolio risk management and hedging strategies. The remainder of the paper is laid out as follows. The following section reviews the relevant literature on REIT linkages. Section 3 details the data analyzed and reports basic summary statistics. In addition, the methodological approach is discussed and diagnostic results on the model are reported. Section 4 presents the main empirical analysis and results with regard to both the multivariate GARCH model and the univariate GARCHcopula model. The final section provides concluding comments. 2. Literature Review The empirical evidence with regard to linkages in returns has often been inconsistent. In contrast the relationships in volatility have been generally more consistent and often more intuitive in nature. While the majority of studies (such as Li and Wang (1995), Liu et al. (1990) , Mei and Lee (1994) , Ling and Naranjo (1999) , Oppenheimer and Grissom (1998), Glascock et al. (2000)) have broadly found evidence that the two markets are heavily linked, with REITs heavily dependent on movements in the mainstream market, there are a number of papers (such as Wilson and Okunev (1996) Okunev and Wilson (1997). Okunev and Wilson (1997) use a nonlinear integration test to examine the relationship between REITs and the S&P 500 Composite. The results show that while the two markets may be related in a nonlinear fashion, the level of deviations between the two can be extensive, with the degree of mean reversion quite slow. In addition, Clayton and MacKinnon (2001) find that the sensitivity of REIT returns to the Stock Market declined significantly in the 1990s. In contrast to this literature the empirical evidence with regard to relationships in volatility has been more intuitive. A large literature has provided evidence of volatility spillovers. Studies such as Hamao et al. (1990); Lin et al. (1994) and Bekeart and Harvey (1997) all document the presence of spillovers in international capital markets. Further studies have also examined the foreign exchange markets such as Melvin and Melvin (2003) and Huang and Yang (2002). About REITs studies, Winniford (2003) and Najand and Lin (2004) both provide further evidence concerning the dynamics of daily volatility in the REIT sector. Stevenson (2002) examines volatility spillovers with regard to monthly REIT returns. The results with regard to the interlinkages with the equity market assets are intuitive, with small cap and value stocks being far more influential on REITs than the large cap S&P 500 Composite Index. While this is consistent with the literature to have highlighted the similarities between REITs and value stocks in the mid and small cap range, such as Chiang and Lee (2002), a recent paper by Cotter and Stevenson (2004) shows that these findings are sensitive to the frequency of data analyzed. The authors hypothesize that the more intuitive results found in Stevenson (2002) which illustrate more fundamental linkages are to some extent masked when analyzing higher frequency data, with market sentiment is strong. A number of other recent papers have also analyzed aspects of REIT volatility, however, they have largely concentrated upon the determinants of volatility and not the interlinkages with other asset classes. This paper will analyse the situation with Asia REITs linkages to global equity. Lee and Stevenson (2005) and McFall Lamm (2003) have analyzed the benefits of adding REITs into mixedasset portfolios. Their findings show that REITs attractiveness as a diversification asset increase as the holding period increases. In addition, their diversification qualities span the entire efficient frontier, providing return enhancement properties at the lower end of the frontier, switching to risk reduction qualities at the top end of the frontier. REITs returns typically have a mean and a standard deviation between that of bonds and stocks. Ziobrowski (1997) discover that REITs have negative skewness and positive excess kurtosis. Brooks and Kat(2002), Brunel (2004) argue that including REITs into portfolios very often leads to lower skewness and higher excess kurtosis , which is the exact opposite of what investors prefer. Therefore, relying on the mean and the standard deviation only is dangerous. Chen, Hsieh, Vines, and Chiou(1998) found that the crosssection of REIT returns is better explained by stock market beta and by FamaFrench (1992) companyspecific variables (i.e. company size and booktomarket) than by macroeconomic variables. Liow and Sim(2006) argue that Asian markets are generally aggressive with higher systematic and idiosyncratic risks are compared and contrasted with the USA and UK markets. Höcht, Ng, Wolf, and Zagst (2008) find that REITs are mainly used for diversification and added to optimal portfolio at comparably lower rates. Similarly, Lee and Stevenson (2005) find that REITs provide for return enhancement properties at the lower end of the efficient frontier and for risk reduction at the top end of the efficient frontier. Hartzell and Titman( 2009) discover that REITs that diversify by investing in more locations tend to be valued lower than REITs with a tighter geographical focus. Capturing comovement between financial asset returns with linear correlation has been the staple approach in modern finance since the birth of Harry Markowitz’s portfolio theory. Linear correlation is the appropriate measure of dependence if asset returns follow a multivariate normal (or elliptical) distribution. However, the statistical analysis of the distribution of individual asset returns frequently finds fat tails, skewness, and other nonnormal features. If the normal distribution is not adequate, then it is not clear how to appropriately measure the dependence between multiple asset returns. Fortunately, the theory of copulas provides a flexible methodology for the general modeling of multivariate dependence. Copulas were initially used by statisticians for nonparametric estimation and measure of dependence of random variables (see Genest and Rivest, 1993). Their application to financial and economic problems is a new and fastgrowing field of interest. Here, the use of this concept is essentially motivated by the fact that it allows to separate the features due to each marginal distribution from the dependence effect between all variables. This separation allows them to estimate the model in two steps. In the first step, they estimate the marginal parameters and use them in the estimation of the correlation parameters in a second step. Copulas offer a tool to generalize this separation while extending the linear concept of correlation to nonlinear dependence. Since the bivariate normal distribution does not adequately describe the joint behavior of returns, the correlation coefficient may not be the proper measure of dependence the tails of the marginal distribution of asset returns can be successfully modeled using the generalized Pareto distribution (GPD). As demonstrated in Carmona (2004), standard nonparametric techniques based on the empirical distribution function can be used to model the center of the distribution. This modeling strategy leads to a semiparametric model (parametric in the tails and nonparametric in the center) for the marginal distribution. 3. Research Data and Econometric Methodologies Research Data This study considers REITs in six Asia countries and global equity index. Weekly data are used after considering weakness of daily data argued by Cotter and Stevenson (2004) and avoiding synchronization problems using daily data when comparing with global data. Weekly return data of each asset is calculated as log(p_{t}/p_{t1}) . Daily price data of global equity index (proxied by AC world equity index) and equallyweighted REITs indexes of seven Asia countries such as Taiwan, Japan, Hong Kong, Singapore, Thailand, South Korea, and Malaysia are from Datastream database. The sample period is from 2006/7/12 to 2010/4/28, total 195 weeks. After check the return pattern of this period, we can just divided into three nearly equal length of period to represent beforecrisis, duringcrisis, and aftercrisis sub period : Beforecrisis:2006/7/12 to 2007/10/10; crisisperiod:2007/10/17 to 2009/1/21; and aftercrisis period:2009/2/4 to 2010/4/28.. Descriptive statistics of the respective series are outlined in Table 2. It shows that Asia REITs are quite volatile and large minimum negative weekly returns (except Malaysia) during the study period. The kurtosis numbers show fat tail exits at the all return series. Mean returns of Asia REITs are negative, but much smaller than global equity. Global bond seems to provide the best return/risk result, but this asset is not the main research subject in this paper. A little strange, Japan REITs, Hong Kong, and Singapore REITs have higher standard deviation than global equity. The possible reason is their higher international market integration (or contagion) efefcts. (Table 2 shown here) The unconditional linear correlation coefficients across various asset returns are shown at Table 3. Comapring with Malaysia REITs low correlation to global equity, Japan REITs show the high correlation (0.7) to global equity. Japan REITs, Hong Kong, and Singapore REITs have high correlation among them. (Table 3 shown here) Time series plots of weekly returns are given in Fig. 1. The global equity become more volatile after 2008 subprime crisis. But there is no clear trend for Various REITs after 2008. The volatility clusters and persistence seems to exist for all return series. The strong serial correlation of volatility indicates the existence of ARCH effects and validities the application of GARCH related processes. (Fig. 1 shown here) Formal diagnostics tests are performed and results are showed at Table 4. The significant results of test of JarqueBera normality tests, LjungBox autocorrelation test, and ARCH tests support ARMAGARCH model used at the next section. (Table 4 shown here) 