Modelling Price Volatility in the Hong Kong Property Market Sherry Z. Zhou and Helen X. H. Bao * Department of Management Sciences, City University of Hong Kong, Hong Kong. Department of Land Economy, University of Cambridge, U.K. Abstract The property market in Hong Kong has been playing an essential role in the political, social and economic lives in this vibrant city. The active transactions and volatile price movement in the Hong Kong property market also make it a hot spot for real estate investors worldwide. Understanding the volatility of this market is important to guide government policy making and investment decisions. Using data collected between 1986 and 2006, this paper investigates the return and volatility in the residential, office and retails properties in Hong Kong. We aim to explore the volatility patterns and compare their difference for various properties in Hong Kong. It is found that all properties show strong auto- and cross-correlations, indicating that the sectors relate to each other closely. All three sectors have higher volatility in market booms around 1997 and 2004. However, residential property sectors behave more like a financial product, with high persistence in shocks to volatility. We also found that within-sector conditional correlations are greater than cross-sector correlations. Both retail and office properties have greater correlations with residential sectors. Keywords: Price volatility, GARCH, copula, dynamic conditional correlation. JEL Classification: C32, R31, R33 Corresponding author (email: zhefzhou@cityu.edu.hk)
Introduction There has been growing interests in investing in real estate due to its ability to hedge inflation, diversify and balance investment portfolio, reduce risk, and generating a stable flow of income (Hudson-Wilson, et al. 2005; Hudson-Wilson, Fabozzi & Gordon, 2003). The benefits of increasing real estate exposure in portfolio are well-documented (see, for example, Maroney & Naka, 2006; Stevenson, 2004; and Ling & Naranjo, 2002). However, managing real estate poses challenges to portfolio managers due to the illiquidity issue and the difficulty to diversify across property types. The introduction of Real Estate Investment Trusts (REITs) alleviates the illiquidity problem by securitizing real assets. It also offers diversification opportunities. Some REITs are specialized in real estate in certain regions (i.e., Japan property market) or sectors (i.e., office buildings). Investors can choose and combine different REITs to reduce risk. Some REITs also diversify across regions and sectors. In these cases a sound understanding of inter-relationship among real estate return across different market segmentations is essential for individual, institutional investors and REITs managers to make their investment decisions. This is an even more eminent issue in the fast-growing Asian real estate market, where REITs are still at its infant stage. There is a need for research on the inter-relationship of returns and volatility among real estate sectors and geographic regions. The setting of this paper is in Hong Kong where real estate plays an essential role in the regional economy. The IPO of the first Hong Kong REITs, Link REIT, is in November 2005. By January 2007 there are six REITs in Hong Kong, covering real estate primarily in the office and retail sectors. The short history of REITs in this market does not support the analysis of risk and return correlation across these six REITs. Nevertheless, such a study will help Hong Kong REITs managers to diversify their investment across sectors, and institutional investors in optimizing their portfolio by including different REITs. The objective of this paper is to study the inter-relationship among the returns and volatility in the office, retail and residential property market in Hong Kong. By unveiling the dynamic of the Hong Kong property market, the study casts light on opportunities to diversify investment portfolio. There has been long lasting interests in studying the relationship between real estate and other financial assets (e.g., stocks and bonds) as well as the relationship within real estate sectors. For example, Cotter and Stevenson (2006) investigate the risk and
return among REITs sub-sectors in the U.S. market; Bond, et al (2003) shows the differences of real estate returns across 14 countries. There are also studies on correlation between securitized real estate markets in different regions (see, for example, Michayluk, Wilson & Zuibruegg, 2006). Much of the existing literature focuses on the U.S. market, largely due to the availability of data and the sophistication of the market. Although there has been evidences of the time-varying nature of the correlation among real estate sub-markets (see, for example, Glascock, Liu & So, 2000), this is usually overlooked either due to the use of cross-sectional data or the adoption of static model specifications. This paper adds to the literature by conducting a time series analysis of the inter-relationship across the three property sectors in Hong Kong using a vector autoregressive (VAR) model for conditional mean and generalized autoregressive conditional heteroscedasticity (GARCH) model. The dynamic conditional correlation (DCC) model by Engle (2002) is adopted for our GARCH model specification. The methodology allows us to correctly identify the time-varying volatility and correlation among the three property submarkets in Hong Kong. The reminder of the paper is structured as follows. The analytical framework is introduced in Session 2, followed by a description of data and the three property sectors in Hong Kong in Session 3. The empirical findings are presented and discussed in Session 4. The last session concludes. Methodology Denote the vector of property return series as r t = {r kt }, where r kt = log kt logp kt-1 and P kt is the property price index for series k at time t. The model structure employed in this study is r t = μ t + ε t μ t E(r t F t-1 ) H t Cov(ε t F t-1 ) whereμ t is the conditional mean of r t given the past information F t-1, ε t is the shock of the series at time t, and H t is the conditional covariance matrix of ε t given F t-1. It is assumed that ε t follows a multivariate normal distribution with mean zero and covariance matrix H t. H t is required to be a k k positive-definite matrix. To conduct the empirical analysis,
r t is characterized by an VAR (Vector AutoRegressive) process in the mean and a MGARCH (Multivariate Generalized AutoRegressive Conditional Heteroscedasticity) process in the variance. We use Akaike information criterion (AIC) and its the variant, or BIC, to identify the VAR order in the mean. For a given vector time series, the order p is selected such that AIC(or BIC)(p)= min 0 i m AIC(or BIC)(i), where m is a predetermined positive integer. As generalizations of univariate volatility models, many multivatiate GARCH models have been proposed. Bollerslev, Engle, and Wooldrige (1988) extend the exponentially weighted moving-average approach to the diagonal VEC model, DVEC(l,s), l H = A + A ( ε ε ) + B H t 0 i t i t i j t j i= 1 j= 1 s where l and s are non-negative integers, A i and B j are symmetric matrices, and denotes element-by-element multiplication. This method lowers the dimension of unknown parameters; however, it may not provide a positive-definite covariance matrix. To guarantee the positive-definite constraint, Engle and Kroner (1995) adopt the BEKK model, l H = AA + A ( ε ε )A + B H B t i t i t i i j t j j i= 1 j= 1 s where A is a lower triangular matrix and A i and B j are k k matrices of parameters to be estimated. This model allows for dynamic dependence between the volatility series and also reveals how the conditional covariance matrix changes according to new information arrival (Engle 2004). However, the parameters in A i and B j do not have direct interpretations about the lagged values of volatilities. Moreover, the number of parameters increases rapidly with l and s, which causes the curse of dimension (Tsay 2005). Therefore, the full BEKK model is usually applied to low dimensional cases. To reduce the number of volatility equations, Bollerslev (1990) considers the case in which the correlation coefficient is time-invariant. Theoretically, H t is decomposed as 2 D t RD t, where D = ( ) t diag H t and R is the time-invariant correlation matrix. This model connects the conditional covariance and conditional variance through the conditional correlation matrix. However, its major drawback is that the correlation coefficient tends to change over time in practice. For instance, Tse (2000) shows that the correlations among national stock returns in Singapore, Japan, and Hong Kong are time varying.
To describe the time-varying correlations, several approaches have been developed. What will be used in this study is the Dynamic Conditional Correlation (DCC) model, proposed by Engle (2002). The model specification for DCC(1,1) is 2 2 2 ( 1 β ) D + α ε + β D, i = 1 k Di,t = α i i i i i t i i t,...,, D t = diag(d 1,t,, d k,t ), 1, 1, 1 z t = D t ε t, (2.1) ( 1 a b) a 1 1 b 1 Q = Q + (z z' ) + Q, t t t t R t 1/ 2 = diag( Qt ) Qtdiag( Qt ) 1/ 2 H t = D t R t D t, where D is the unconditional variance of residual series ε i, Q t is a positive-definite i matrix,q is the unconditional covariance matrix of z t, and α i, β i, a, and b are non-negative scalar parameters satisfying α i + β i (0,1) and a + b (0,1). The DCC model can be estimated in two steps. In the first step, univariate GARCH models are fitted for each residual series; while during the second stage, the standardized residuals are used to estimate the correlation parameters. More formally, the log-likelihood function of the entire model can be expressed as the sum of a volatility part and a correlation part: L( θ, φ) = L ( θ) + L ( θ, φ) V C where L V ( θ )is the sum of individual GARCH likelihoods, and 1 R z R z z z 1 C(, ) = (log t + t t t t t) 2 t L θφ is used to estimate the correlation parameters. The estimation procedure is to find ˆ θ = arg max{ ( θ )} L V in the first step and then solve for the maximum likelihood, max{ ( ˆ θ, φ )}, in the second stage by taking ˆ θ as given (Engle 2002). Thanks to this estimation approach, DCC model avoids the curse-of-dimension problem usually faced by the full BEKK model and can be applied to handle high-dimensional cases. φ L C
Data and the Hong Kong Property Market The data are obtained from the Rating and Valuation Department (RVD), HKSAR. We consider residential, office and retail properties in our analysis. Monthly price indices of the three sectors are used. The sampling period is from 1993 to 2006. The characteristics of and price index series used for each sector are discussion in the following sessions. The base period of the indices is January 1999. Residential property market Hong Kong s residential property market consists of condominiums primarily. Due to the high population density and limited land supply in the region, residential housing units are typically smaller than 100 squared metres. There are only less than 5% residential properties have more than 100 squared metres of saleable size, which is generally considered as luxury properties. The RVD publishes seven different price indices for residential properties based on the saleable area of the units. We use two price indices in our analysis, namely, the one of class A, B, and C (apartments with saleable area smaller than 100 square metres), and the one of class D and E (apartments with saleable area equals to or greater than 100 square metres). We denote the two series Residential ABC and Residential DE in this study. The two price indices represent the price movement of regular and luxury apartments respectively. The time series plots of the two price indices are given in Figure 2(a). The two indices move closely between 1994 and 1999. There has been an increasing level of discrepancy between the price level of the two types of properties, with the larger units enjoying more appreciation in the recent years. Office property market Offices in Hong Kong are classified by the RVD into three categories. Grade A offices are modern with high quality finishes; Grade B offices have ordinary design with good quality finishes; whilst Grade C office are plain with basic finishes. The price indices of these three grades are depicted in Figure 2(b). The price indices indicate the office price level in Hong Kong is generally influenced by
the regional economic development. The impact from the 1997 Asia Financial Crisis and the 2003 SARS outbreak are obvious in Figure 2(b). Moreover, offices price has a more volatile pattern compared with residential property price. The fluctuation between 1995 and 1997 is especially conspicuous, partly due to the fact that the fast economic growth during the period of time is largely driven by financial related activities. The competition among companies to obtain office space during this period results in both a high level of office rent and volatile price changes. After 1999 the price trend of office property has been moving closely with residential property price (see Figure 2(a) and Figure 2(b) ). Overall Grade A offices enjoys the highest price level, followed by Grade B and Grade C office properties. High quality is obviously the determining factor of offices rent. The gap among the three types of offices has been widening from 2003 onward. Figure 2(b) suggests that Grade C offices claims a much lower rent than Grade B and Grade A offices recently, whilst the difference between Grade A and Grade B offices rent remains relatively small. Retail property market According to the definition provided by RVD, retail property price index covers retail and other premises designed or adapted for commercial use, with the exception of purpose-built offices. The price index of retail property is given in Figure 2(c). The price indices of overall residential and office property market are also shown in Figure 2(c) for comparison purpose. All three indices use January 1999 as the base period. In comparison the office property enjoys the highest rate of appreciation after the economic turning point in 2004, followed closely by retail properties. The recovery of the residential property market is relatively slow. In general the three indices follow each other closely during the period from 1998 to 2002 when the market was suffering from the 1997 Asian Financial Crisis aftermath. There are two periods of high fluctuations over the entire sample range. The first phase is around 1997, resulted from the real estate bubbles in Hong Kong and the Asian Financial Crisis. From year 2003 on, we observe another wave of significant turbulence in Hong Kong property market. This is due to the impact of the 2003 SARS outbreak and a series of favourable policies issued by the China government to boast Hong Kong economy.
Preliminary statistics Table 1 lists the descriptive statistics of the monthly returns. Luxury residential properties (e.g., Class D and Class E units) have the highest average monthly return with a small standard deviation during our sampling period. On the other hand, the returns of the office properties are negative for Grade B and Grade C. The standard deviation of monthly returns is also very high for offices. The negative average return is largely due to the high price level before 1997. This indicates that luxury residential property is the safest and most profitable to invest in Hong Kong, whilst office property is the most risky sector. Table 2 provides cross-correlation matrices of the six return series. To make explanation easier, we use the simplified notations proposed by Tiao and Box (1981). A cross-correlation matrix consists of just three symbols +, -, and., which mean that the corresponding correlation coefficient is significantly positive, negative, or non-significant at 5% level respectively. Table 2 shows significant cross-correlations for all the return series. Besides, we use the multivariate portmanteau test to further check the auto- and cross-correlations in the vector series r t = {r kt }, k = 1,, 6, with the test statistic denoted by Q(m) for the first m cross-correlation matrices of r t. This test is an extension of the univariate Ljung-Box statistic and is proposed by Hosking (1980, 1981) and Li and McLeod (1981) 1. For the six return series, we have Q(5) = 513.41881, with the p-value very close to zero. Hence, significant auto- and cross-correlations present among the return series. The use of vector autoregressive model VAR(p) for the mean is then justified. In sum, the price indices do not seem to follow a uniform pattern within each property sector nor among the three sectors. The relationship among the three property markets seems to change over time (see Figure 2). The VAR-GARCH model proposed in Session two will help to reveal the inter-relationship among the returns in residential, office and retail property market in Hong Kong. Empirical Analysis and discussions The measurements AIC and BIC are used to determine order p of the vector autoregressive model, as mentioned in the previous section. Table 3 shows the results of 1 For details of the test, please refer to Hosking (1980, 1981) and Li and McLeod (1981).
these two statistics for various orders. AIC suggests a VAR(3) model while BIC supports a VAR(1). Applying the multivariate portmanteau test mentioned above to the two models, we have Q(10) = 538.086 (< 0.01) and Q(10) = 355.53 (0.512) for VAR(1) and VAR(3), respectively, where the number in parentheses denotes the corresponding p-value. Based on the test results, the mean equation is adequate at the 5% level of significance for VAR(3) model. Therefore, VAR(3) is selected for the mean. Multivariate GARCH is applied to model the residuals left from the conditional mean such that the volatility of each series is analyzed and the co-movement among series is described. When exploring the residuals of VAR(3) model, we find that all series except office A property price index have GARCH effect. Therefore, we construct a VAR(3) model for all the six return series but modelling the residuals using MGARCH for the remaining five series excluding office A property return. Due to the small sample size and high dimension characteristic related to our problem, we choose DCC technique for volatility analysis. Hence, VAR(3)-DCC(1,1) is applied to the study. Test results are shown in Table 4. Since office A return series has no GARCH effect, it only serves as an exploratory variable in VAR modelling and DCC model is built for residuals associated with the other five series. Applying the multivariate Ljung-Box statistics mentioned above to the standardized residuals, we have Q(6) = 112.825 (0.983). for the squared standardized residuals, we obtain that Q 2 (6) = 163.059 (0.173). Therefore, no significant serial correlations or conditional heteroscedasticities remain in the residuals of the fitted model, indicating the model is adequate. We will then make the discussions based on the VAR(3)-DCC(1,1) model. Conditional returns of property price indices Part (a) of Table 4 presents the parameter estimates for the VAR(3) model. Φ 1, Φ 2, and Φ 3 are the AR coefficient matrices at lags 1 to 3 of r t = {r kt }, respectively. In this analysis, k = 6 with 1 to 6 representing returns of retail property, residential ABC property, residential DE property, office B property, office C property, and office A property, respectively. The results show that the property returns present strong autocorrelations and cross-correlations. Each return series has impact on all the other property returns, indicating a close relationship among the various properties. In comparison with lags 2 and 3, more coefficients at lag 1 are statistically significant, indicating that lag 1 terms have more important impact on the property returns. Thus
when analyzing the auto- and cross-correlations of the return series, we focus on the parameter estimates at lag 1. We observe that the impact of a property on its own is negative with residential ABC property as the only exception. This implies a self-correction characteristic of the property returns. In contrast, the impact of cross properties is positive, indicating an imitation effect among different property returns. Therefore, the submarkets are not isolate; instead, they affect each other and learn from each other. In Hong Kong, residential ABC property can be considered as an economical real estate asset. Its price and return is relatively low and the price fluctuation range is smaller than others, confirmed by the smallest variance and low mean shown in Table 1. Hence, the return of last period serves as a good signal for the return of this period. Therefore, we observe a significant positive relationship between residential ABC property return at lag 1 and itself. Conditional volatilities of property price indices Part (b) of Table 4 summarizes the parameter estimates α 1i and β 1i of each individual GARCH(1,1) model for series i, where i = 1,,5 representing retail property, residential ABC property, residential DE property, office B property, and office C property, respectively. The constant term in each equation is very close to zero, indicating that in the long run, the expected variance of all the property returns tend to be zero. However, the conditional volatility is time-varying. From Table 4 we observe that the estimated parameters of the conditional variance equation are positive and statistically significant at the 10% level except β 11 and β 14. This result confirms the characteristics of heteroscedasticities in all the return series. The sum α 1i + β 1i is very close to one in most cases except the office B property, indicating a characteristic of volatility clustering. The persistence of shocks to volatility is clearly in evidence in Figure 2. In Hong Kong, residential property is more like an investment product than a consumer product. Hence, they are in line with a financial time series in terms of the great volatility persistence. Among the three office properties, office A has the highest grade, indicating its most reliable quality and highest price. This sub-sector presents a relatively stable pattern. On the other hand, the grades of offices B and C properties are lower, implying their qualities are less reliable. In this case, their returns demonstrate a more volatile pattern. Therefore, we observe that office A return series shows no GARCH effect, while B and C property exhibit heteroscedasticity. Moreover, office C shows long persistence of shocks to volatility.
When observing the time plots of conditional volatility, as shown in Figure 2, we notice that there are mainly two periods of high volatilities, being around 1997 and 2004 respectively. These two periods are the market booms in Hong Kong real estate, with characteristics of high price and active transactions. Meanwhile, the risk is also higher. As a result, all the monthly returns demonstrate relatively high volatility during this period. The Asian Financial Crisis occurred in 1997, leading to the market recession in which we see low price, low return, and low volatility. Since 2004, the market begins to revive. However, it has not returned to the same prosperity phase of 1997. Therefore, both the return and volatility are still much lower than the 1997 peak. For the residential sector, we find that the second boom is not as obvious as other sectors. This is because after a hard time of the market recession, investors and consumers need a long period to rebuild their confidence. Before the Asian Financial Crisis, residential ABC property presents a higher volatility than its DE counterpart. However, during the market recession and recovery phases residential DE property exhibits a little lower volatility most of the time. A possible explanation for this behaviour is that residential DE property, a type of high price investment product, shows notable speculation nature compared with residential ABC property. Therefore, during the market boom, it demonstrates high return and high volatility. In contrast, during the market session the volatility become lower than ABC property due to the far decreased transactions. When analyzing the office sector, we find that in general office B property exhibits higher conditional variance than its C counterpart. However, monthly returns of these two types are about the same, although the price of office B is higher than C property. This is because the difference between B and A, or B and C, is not quite obvious sometime, resulting in a wider price fluctuation range for C property. Hence, the conditional volatility of office C returns is also larger. Conditional correlations of property price indices To explore the correlations among property return series, we focus on the parameters a and b in model (2.1). It is shown in part (b) of Table 4 that both parameters are statistically significant at 10% level. Therefore, the adoption of dynamic conditional correlation model is justified. Figure 4 presents the time-varying correlation coefficients of pairwise property return series. The general pattern is that the correlation coefficients gradually increasing in the market boom before mid-1997, followed by relatively big fluctuations during the Asian Financial
Crisis. After the crisis, the market entered its recession phase, when the pairwise correlations remain a rather stable pattern. From 2003, Hong Kong real estate market started to recover. The correlations of return series still keep relatively stable, only with a bit more fluctuations than the recession period. During the market boom around 1997, Hong Kong real estate was in its prosperity phase. All property sectors are active and exhibit higher correlations among each other. Between 1997 and 1998, Asian Financial Crisis brought destructive shocks to the property market. The high fluctuations reflect the turbulent and panic state of the Hong Kong people and the entire market. During the recession period, transactions were inactive and the market almost hibernated. Since the market was in a stagnant but stable stage, the conditional correlations among property sectors present steady patterns. The current property market is in a transition phase, turning from the recession period to a healthy growth period. Correspondingly, the associations among various properties exhibit greater fluctuations than the time 1999-2003; however, they keep stable in general. Upon comparison in Figure 4, we observe that the average correlation between residential ABC and DE properties is the highest, about 0.7. The correlation between office B and C returns is 0.3 on average, ranking the 4 th position next to the additional two pairs retail and residential DE, and residential DE and office B. Therefore, the correlations within sectors are overall higher than cross-sector correlations. As a type of investment products with similar natures, residential ABC and DE properties exhibit the strongest association in their movements. When comparing the cross-sector associations, we observe that the retail property has greater correlation coefficients with residential properties than with office properties. Similarly, office properties also correlate more with residential properties. This is because the residential property sector plays a dominant role in Hong Kong real estate market. Centa-City Property Index (CCI), established for Hong Kong residential property market, serves as a guide of the price trend of the entire Hong Kong real estate market. Hence, both retail and office property sectors relate closer to the residential properties. Conclusions This paper conducts empirical analysis on the time-varying volatility and correlations of various property sectors in Hong Kong using the dynamic conditional correlations approach. It is found that all properties show strong auto- and cross-correlations, indicating that the sectors relate to each other closely. For the conditional volatilities,
different properties exhibit a similar pattern, with higher volatility in market booms around 1997 and 2004. However, residential property sectors behave more like a financial product, with high persistence in shocks to volatility. In terms of the pairwise relationship, within-sector conditional correlations are greater than cross-sector correlations. As to the cross-sector relationship, both retail and office properties have greater correlations with residential sectors.
Table 1 Descriptive Statistics of the six monthly return series Mean Standard Deviation Minimum Maximum Retail 0.0010 0.0190-0.0675 0.0469 Residential ABC 0.0001 0.0132-0.0554 0.0402 Residential DE 0.0015 0.0157-0.0463 0.0524 Office A 0.0002 0.0282-0.0787 0.0826 Office B -0.0003 0.0325-0.1482 0.0804 Office C -0.0006 0.0238-0.0823 0.0572
Table 2 Cross-Correlation Matrices of the six monthly return series (a) Cross-Correlation Matrices Lag 1 Lag 2 Lag 3-0.049 0.395 0.224 0.309 0.179 0.195 0.122 0.308 0.249 0.164 0.121 0.130 0.060 0.219 0.271 0.093 0.108 0.149 0.331 0.484 0.461 0.270 0.183 0.155 0.264 0.282 0.312 0.254 0.181 0.132 0.132 0.171 0.189 0.167 0.016 0.155 0.324 0.517 0.403 0.408 0.292 0.228 0.235 0.349 0.301 0.194 0.111 0.169 0.077 0.182 0.218 0.053 0.095 0.113 0.289 0.378 0.400-0.139 0.016 0.102 0.153 0.175 0.126 0.193 0.242 0.101 0.127 0.171 0.243 0.082 0.071 0.083 0.145 0.314 0.258 0.241-0.305 0.067 0.118 0.144 0.080 0.076 0.217 0.190 0.143 0.026 0.091 0.001-0.075-0.145 0.181 0.252 0.222 0.135 0.077-0.204 0.140 0.194 0.295 0.129 0.277 0.036 0.038 0.237 0.050 0.275-0.097 0.086 (b) Simplified Notation. + + + + +. + + +... + +... + + + + + + + + + + +.. + + +. + + + + + + + + + + +. +. + +... + + +.... +. + +.. + +.... + + + -..... + +...... + + +.. -. + +. +.. +. +..
Table 3 Order Selection Criteria for VAR (p) Model p 1 2 3 4 5 6 AIC -5185.821-5183.263-5185.823-5132.832-5107.112-5050.337 BIC -5060.030-4956.231-4864.432-4724.869-4621.436-4497.083 Table 4 Parameter Estimations for model VAR(3)-DCC(1,1) (a) Parameter estimations for VAR(3) in the mean Φ 1-0.2614** 0.3772** -0.1876** 0.0337 0.0270 0.2032** (0.0598) (0.0843) (0.0838) (0.0349) (0.0583) (0.0401) 0.1530** 0.3558** -0.0126-0.0035-0.0024 0.0997** (0.0353) (0.0537) (0.0515) (0.0192) (0.0304) (0.0275) 0.2229** 0.5118** -0.2682** 0.0808** 0.0026 0.1814** (0.0324) (0.0558) (0.0586) (0.0184) (0.0322) (0.0279) 0.1258 0.7132** 0.2482* -0.6096** 0.0591 0.3303** (0.0996) (0.1318) (0.1269) (0.0576) (0.0637) (0.0766) 0.1086* 0.4371** -0.1080 0.1159** -0.3167** 0.1086** (0.0600) (0.1204) (0.1037) (0.0452) (0.0713) (0.0470) Φ 2-0.1655** 0.3371** 0.0217-0.0371 0.1287** 0.0574 (0.0671) (0.1397) (0.1416) (0.0383) (0.0448) (0.0437) 0.1142** -0.1068 0.0026-0.0134 0.0548* 0.0599** (0.0430) (0.0866) (0.0832) (0.0259) (0.0280) (0.0269) 0.1295** -0.0949-0.0221 0.0519* 0.1083** 0.0210 (0.0480) (0.0893) (0.0955) (0.0277) (0.0284) (0.0285) 0.1874* -0.0031-0.2964-0.0636 0.2288** 0.1470 (0.0999) (0.1989) (0.1877) (0.0664) (0.0772) (0.0622) 0.0557-0.0707 0.0692 0.0954* -0.1992** 0.1322** (0.0790) (0.1348) (0.1247) (0.0489) (0.0621) (0.0447) Φ 3-0.2203** (0.0575) -0.1318 (0.1154) 0.2622** (0.0922) -0.1051** (0.0297) 0.1141** (0.0431) -0.0307 (0.0346) 0.0064-0.3498** 0.1472** -0.0390 0.0454 0.0272
(0.0425) (0.0566) (0.0582) (0.0255) (0.0286) (0.0227) -0.0985** (0.0474) -0.3557** (0.0549) 0.2866** (0.0655) -0.0141 (0.0258) 0.0181 (0.0305) -0.0956** (0.0231) 0.2663** (0.0891) -0.4839** (0.1647) 0.0281 (0.1442) -0.0656 (0.0565) -0.1019 (0.0698) 0.0146 (0.0540) 0.0846 (0.0723) 0.2121* (0.1148) -0.1967** (0.0965) -0.1161** (0.0465) -0.1339** (0.0613) 0.2446** (0.0481) Notes: Φ 1, Φ 2, and Φ 3 are the AR coefficient matrices at lags 1 to 3 of r t = {r kt }, respectively. In this analysis, k = 6 with 1 = retail property return, 2 = residential ABC property return, 3 = residential DE property return, 4 = office B property return, 5 = office C property return, and 6 = office A property return. Parameters marked with ** are statistically significant at 5% level of significance. Parameters marked with * are statistically significant at 10% level. (b) Parameter estimations for DCC(1,1) in the mean α 01 6.9820e-05* (4.1016e-05) α 11 0.7996** (0.3384) β 11 0.1367 (0.1199) α 02 4.3702e-06 (3.5445e-06) α 12 0.1057** (0.0376) β 12 0.8575** (0.0536) α 03 1.7313e-06 (1.5499e-06) α 13 0.0567** (0.0210) β 13 0.9223** (0.0280) α 04 4.5240e-04** (1.8618e-04) α 14 0.2148* (0.1105) β 14 0.0212 (0.3401) α 05 2.7136e-05* (1.4717e-05) α 15 0.2115** (0.0715) β 15 0.7111** (0.0843) a 0.0519** (0.0254) b 0.4604* (0.2546) Note: α 0i, α 1i, and β 1i are the coefficients of each individual GARCH(1,1) equation, where i = 5 with 1 = retail property return, 2 = residential ABC property return, 3 = residential DE property return, 4 = office B property return, and 5 = office C property return. Numbers in the parentheses are the standard errors. Parameters marked with ** are statistically significant at 5% level of significance. Parameters marked with * are statistically significant at 10% level.
Figure 2 Time plots of price indices from 1993 to 2006 (a) Residential property price indices Index 200 180 160 140 120 100 80 60 40 20 0 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Time Residential ABC Residential DE (b) Office property price index 300 250 200 Index 150 100 50 0 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Time Office Grade A Office Grade B Office Grade C
(c) Comparison among the three types of properties 300 250 200 Index 150 100 50 0 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 Jan-05 Time Residential Office Retail
Figure 3 Time plots of monthly returns for six properties from 1993 to 2006 (a) Monthly log returns for retail property price index (b) Monthly log returns for residential ABC and DE property price indices (c) Monthly log returns for office A property price index
(d) Monthly log returns for office B and C property price indices Figure 4 Time plots of the volatilities from the fitted DCC(1,1) model (a) Retail property
(b) Residential ABC property and Residential DE property (c) Office B property and Office C property Figure 5 Time-varying correlation coefficients from the fitted DCC(1,1) model (a) Retail property and Residential ABC property
(b) Retail property and Residential DE property (c) Retail property and Office B property (d) Retail property and Office C property (e) Residential ABC property and Residential DE property
(f) Residential ABC property and Office B property (g) Residential ABC property and Office C property (h) Residential DE property and Office B property (i) Residential DE property and Office C property
(j) Office B property and Office C property
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