Conditional heteroskedasticity in stock returns evidence from stock markets of mainland china

CONDITIONAL HETEROSKEDASTICITY IN STOCK RETURNS: EVIDENCE FROM STOCK MARKETS OF MAINLAND CHINA YIN ZIHUI (Econ Dept, NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SOCIAL SCIENCES DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgements First of all, I would like to express my deepest gratitude to my supervisor, Professor Albert Tsui, Department of Economics, National University of Singapore. Without his consistent and patient guidance and illuminating instructions, my thesis could not progress to today’s level. Second, my thanks would go to my colleagues and friends at the Risk Management Institute, National University of Singapore. Without discussion with them, I could not discover so many interesting topics in Finance. Therefore, I would like to thank Professor Duan, Oliver Chen, Deng Mu, Sun Jie and Wang Shuo. Especially, Xiao Yong was so nice to offer me Kim’s book that truly helped me a lot. In addition, I am sincerely grateful to my friends at the Department of Economics, Pei Fei, Zhou Xiaoqing, Li Lei and Xu Wei. They encouraged me and accompanied me while I was writing my thesis. Last but not least, I owe my sincere and loving thanks to my parents. Thanks to their unconditional love, encouragements and support, I can always overcome the difficulties during my master program. i Table of Contents 1. Introduction…………………………………………………………………………1 2. Models………………………………………………………………………………8 2.1 ARMA(1,1)-GARCH(1,1) Model………………………………………………8 2.2 Bivariate VC-MGARCH(1,1) models…………………………………………13 2.3 Markov-switching variance models and Time-varying-parameter models with Markov-switching heteroskedasticity……………………………………..16 3. Data and Estimation Results………………………………………………………24 3.1 Data specification…………………………………………………………….. 24 3.2 ARMA(1,1)-GARCH(1,1) models…………………………………………….29 3.2.1 Estimation results of Shanghai Stock Market………………………….29 3.2.2 Estimation results of Shenzhen Stock Market………………………….34 3.3 Bivariate VC-MGARCH(1,1) Models………………………………………...38 3.3.1 Estimation Results of Shanghai Stock Market…………………………38 3.3.2 Estimation Results of Shenzhen Stock Market………………………...43 3.4 Markov-switching variance models and Time-varying-parameter models with Markov-switching heteroskedasticity……………………………………..46 3.4.1 Estimation Results of Shanghai Stock Market…………………………50 3.4.2 Estimation Results of Shenzhen Stock Market………………………...54 4. Conclusion…………………………………………………………………………58 Bibliography………………………………………………………………………….59 ii Summary Mainland China’s stock markets are becoming more mature and more integrated with the global financial markets. It is worth further exploring not only for investors, but also for policy makers. This thesis investigates various features of conditional heteroskedasticity of stock returns in Shanghai and Shenzhen. It consists of three parts: exploring a more appropriate model to fit the stock returns; studying the dynamics of conditional correlation of returns; and examining the possible regimes by using the Markov-Switching technique. Our findings are reported as follows: First, the fitted ARMA(1,1)-A-PARCH(1,1,1) model with the generalized error distribution is a relatively more suitable one. Second, we find that the conditional correlation between mainland China’s and the U.S. stock markets is quite low and highly volatile. Third, we apply a structure of Markov-switching in conditional heteroskedasticity to identify two discrete volatility regimes of China’s stock markets and its changing relationship with the U.S. market. iii List of Tables 1 Summary of the structure of conditional variances of GARCH-type models……...13 2 Summary Statistics for rit , i = sh, sz , sp …………………………………………...26 3 Unit root tests………………………………………………………………………29 4 QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under normal distribution (Shanghai)……………………………………………..31 5 QMLE of the GARCH models and BIC under Student’s t distribution (Shanghai)………………………………………….32 6 QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under the GED (Shanghai)………………………………………………………...33 7 QMLE of the GARCH models and BIC under normal distribution (Shenzhen)……………………………………………..35 8 QMLE of the GARCH models and BIC under Student’s t distribution (Shenzhen)…………………………………………36 9 QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under the GED (Shenzhen)………………………………………………………...37 10 Summary statistics of rsht and rspt ………………………………………………39 11 VC-MGARCH(1,1) and CC-MGARCH(1,1) for ε sht and ε spt …………………40 12 Diagnostic tests on the standardized residuals of VC-MGARCH(1,1) and CC-MGARCH(1,1) (SH and SP)………………………………………………….42 13 Summary statistics of rszt and rspt ………………………………………………43 i 14 VC-MGARCH(1,1) and CC-MGARCH(1,1) for ε szt and ε spt …………………44 15 Diagnostic tests on the standardized residuals of VC-MGARCH(1,1) and CC-MGARCH(1,1) (SZ and SP)………………………………………………….45 16 Test statistics of multiple structural changes……………………………………...47 17 Descriptive statistics for the period 01/03/2005 ~ 12/31/2008…………………...49 18 Estimation results of Model 1 and Model 2 for Shanghai stock market………….51 19 Diagnostic tests on the standardized residuals of Model 1 and Model 2 for Shanghai stock market…………………………………………………………51 20 Estimation results of Model 1 and Model 2 for Shenzhen stock market…………57 21 Diagnostic tests on the standardized residuals of Model 1 and Model 2 for Shenzhen stock market………………………………………………………...57 ii List of Figures 1 Plot of rsht …………………………………………………………………………...2 2 Plot of rszt …………………………………………………………………………...2 3 Conditional volatility of Shanghai stock market…………………………………...34 4 Conditional volatility of Shenzhen stock market…………………………………..38 5 Plot of conditional correlation between Shanghai and the U.S. stock markets…….41 6 Plot of conditional correlation between Shenzhen and the U.S. stock markets……46 7 Conditional variances of Model 1 for Shanghai stock market……………………..52 8 Conditional variances of Model 2 for Shanghai stock market……………………..54 9 Conditional variances of Model 1 for Shenzhen stock market…………………….56 10 Conditional variances of Model 2 for Shenzhen stock market…………………...56 i 1. Introduction Several far-reaching events occurred and shaped China’s stock markets during the period, 01/05/2000 ~ 12/31/2008. They include the “dot-com bubble”, China’s non-tradable shares reform and the global financial crisis. Figures 1 and 2 display daily returns of Shanghai and Shenzhen markets. Both figures reveal influence of these historical events. At the beginning of 2000, returns of both Shanghai and Shenzhen stock indices were suffering from sharp oscillation induced by the “dot-com bubble” that originated in the U.S. The bubble was caused by over-speculation on dot-com companies. After a temporary prosperity, mainland China’s stock markets entered into a long bear market phase. Until June 2005, it was the reform of non-tradable shares that improved liquidity and brought the markets back to the bull markets. Unfortunately, the sub-prime mortgage crisis in the U.S. exported contagious shocks to the global financial markets and triggered a chain of negative impacts on the real economy since July 2007. China’s stock markets were with no exemption. They were badly affected by the vicious shock, thereby exhibiting extreme instability and wild volatility. In this thesis, we examine various features of conditional heteroskedasticity in the daily returns of the Shanghai Stock Exchange Composite Index ( rsht ) and the Shenzhen Stock Exchange Component Index ( rszt ). 1 -8 -6 -4 -2 0 2 4 6 8 Figure 1: Plot of rsht 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 -10 -5 0 5 10 Figure 2: Plot of rszt 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 Although many studies have been conducted on those indices, our approach and updated data may shed some new light in exploring a more appropriate GARCH model. Based on the ARCH (Autoregressive Conditional Heteroskedasticity) model proposed by Engle (1982), where time-varying variances are conditional on past information and unconditional variances are constant, Bollerslev (1986) generalizes it to the GARCH (Generalized Autoregressive Conditional Heteroskedasticity) model that extends the lag structure as an ARMA process. In particular, GARCH(1,1) is often sufficient for most of financial series, thereby effectively reducing the long lag length in the ARCH model that may induce cumbersome computation. In the 2 subsequent years, the ARCH and GARCH models have been extended and their applications have expanded from macroeconomics to financial fields. For example, they are instrumental in option pricing, portfolio selection and risk management. However, both of the ARCH and GARCH models fail to incorporate leverage effect, a stylized fact of stock returns, in which negative shocks tend to have a larger impact on volatility in subsequent periods than positive shocks with the same magnitude. In addition, the strict positive restrictions on parameters may be difficult to implement. To tackle those problems, Nelson (1991) proposes the EGARCH (Exponential GARCH) model. This model successfully captures the asymmetric response to “good news” and “bad news” through interpolating absolute residuals into the conditional variances equation and relaxes the non-negativity constraints by taking the log form. A similar model, GJR-GARCH model developed by Glosten, Jagannathan, and Runkle (1993) treats asymmetric effect as a dummy variable and is also capable of capturing leverage effect. Xu (1999) discovers that the standard GARCH model outperforms the EGARCH and the GJR-GARCH models and leverage effect is insignificant in capturing volatility of Shanghai stock market from May 21, 1992 to July 14, 1995, attributed to immaturity of the market and strong governmental influence on it. Copeland and Zhang (2003) also find no evidence of leverage effect in mainland China’s stock markets when they adopt the EGARCH model to capture their volatility during the later period, Nov. 25, 3 1994 ~ Apr. 27, 2001. In our thesis, the data is updated to be Jan. 05, 2000 ~ Dec. 31, 2008. Ding, Granger and Engle (1993) cast doubts on the squared residuals and the linear specification in the standard GARCH model. As such, they impose a Box-Cox power transformation on the conditional volatility function and introduce a more general structure, A-PARCH (Asymmetric Power ARCH) model. Moreover, they estimate the general model fitted with the returns of S&P 500 index, and find that the power is 1.43, significantly different from 2. Brooks (2007) adopts the A-PARCH model to study the volatility of emerging equity markets and then to make comparison with the one of developed markets. He finds that the power of emerging stock markets falls within a wider range than the one of developed markets and different emerging markets have significantly different degrees of volatility asymmetries. In addition, we capture dynamics of conditional correlation between returns of China’s stock markets and those of the U.S. in a bivariate VC-MGARCH framework. Despite large number of parameters involved, it sheds some light in how the two markets are correlated and whether they can bring diversification to investors. Although direct generalizations from the univariate GARCH models are straightforward, for example VEC and DVEC models proposed by Bollerslev et al. (1988), their applications are limited by practical issues associated with cumbersome 4 computation and strong restrictions on parameters to guarantee positive definiteness of variance matrixes. Engle and Kroner (1995) develop the Baba-Engle-Kraft-Kroner (BEKK) model that automatically ensures positive definiteness and Bollerslev and Engle (1993) propose the factor model to simplify conditional variances; however, these models still suffer from one common drawback that their parameters are difficult to interpret. Based on the four-variable asymmetric GARCH fitted in the BEKK structure, Li (2007) concludes that no direct linkage exists between mainland China’s stock markets and the U.S. market, thereby furnishing portfolio investors with diversification benefits. To tackle the computational complexities associated with the direct generalizations, Bollerslev (1990) introduces the constant conditional correlation (CCC)-MGARCH model. Specifically, the univariate GARCH models are used to capture each returns series and then linked together by the conditional correlation matrix. It allows for more flexibility, and is easier to interpret. Tsui and Yu (1999) apply this model to capture conditional correlation between Shanghai and Shenzhen stock markets and conclude the constancy is rejected by the information matrix test. However, the assumption of constant conditional correlations seems unrealistic for most of financial series. Hence, Tse and Tsui (2002) develop a varying-correlation MGARCH (VC-MGARCH) model that assumes that the time-varying conditional-correlation matrix follows an ARMA(1,1) structure, which is similar to a dynamic conditional correlation (DCC-MGARCH) model proposed by Engle (2002). For further details on 5 the VC-MGARCH model, see Section 2.2. Moreover, we identify two discrete regimes for each stock market, relatively stable state and highly volatile state, and make probabilistic inference on the persistence of each state, following the methodology of Hamilton (1989). Girardin and Liu (2003) adopt the same technique to identify three regimes of Shanghai A-share market, consisting of a speculative market, a bull market and a bear market, based on weekly capital gains from early January 1995 to early February 2002. They argue that high capital gains derived from a short period of the bull market and extreme risks associated with the speculative market indicate a “Casino” characteristic of China’s stock markets. The time-varying-parameter models with Markov-switching heteroskedasticity proposed by Kim (1993) is capable of capturing the changing relationship between returns of China’s stock markets and those of the U.S. In his paper, he models quarterly M1 growth rate as a function of changes in the interest rate, the inflation rate, the detrended full employment budget surplus and the lagged M1 growth rate and concludes that U.S. monetary growth uncertainty is not only derived from heteroskedastic disturbances, but also subject to the learning process of agents. Relevant application of the methodology is conducted on business cycle (see Kim and Piger (2002)), inflation uncertainty (see Telatar and Telatar (2003)), and impact of political risk on volatility dynamics (see Fong and Koh (2002)), among others. Most 6 of these extensions are on macroeconomics; however, in our thesis, we expand it to financial returns. The rest of our thesis is organized as follows. Section 2 specifies three models, consisting of ARMA(1,1)-GARCH(1,1) models, bivariate VC-MGARCH(1,1) models, Markov-switching variance models and time-varying-parameter models with Markov-switching heteroskedasticity. Data and estimation results are reported in Section 3. Section 4 concludes with implication of our findings on equity investment. 7 2. Models 2.1 ARMA(1,1)-GARCH(1,1) Models In this section, we introduce several GARCH models to capture conditional heteroskedasticity of rsht and rszt and then select a better one based on Bayesian information criteria. Before proceeding to the specific models, Lagrange multiplier tests are conducted to test whether any ARCH effect exists in the series. The idea is to compare T ⋅ R 2 derived from equation (1) with the value of χ 2 (m) under the null hypothesis, where m ≈ ln(T ) , as suggested by simulation studies. aˆit2 = αˆ i 0 + αˆ i1aˆi2,t −1 + αˆi 2 aˆi2,t − 2 + K + αˆ im aˆi2,t − m (1) where aˆit = rit − mean( rit ) , i = sh, sz . The empirical results reported in Section 3.1 provide evidence of strong ARCH effect in those series at the 1% significance level. For simplicity, an ARMA(1,1) structure is used for conditional mean equation. Conditional mean equation: rit = ci + φi ri ,t −1 + θ iε i ,t −1 + ε it (2) The GARCH(1,1) proposed by Bollerslev (1986) is used for the conditional variance equation: 8 ε it = ηitσ it , ηit ~ i.i.d .(0,1) (3) σ it2 = α i 0 + α i1ε i2,t −1 + βi1σ i2,t −1 (4) where α i 0 > 0 , α i1 ≥ 0 , βi1 ≥ 0 and α i1 + βi1 < 1 . It can be shown that the unconditional variances are time invariant, providing σ i2 = αi0 1 − (α i1 + β i1 ) (5) However, equation (4) fails to incorporate leverage effect, a stylized fact of stock returns. To capture the asymmetric feature, we introduce three types of asymmetric GARCH models as follows. The Exponential GARCH (EGARCH) model suggested by Nelson (1991) relaxes the positive restrictions on parameters in the GARCH(1,1), through assuming hit = log σ it2 . hit = α i 0 + α i1 ε i ,t −1 + γ i1ε i ,t −1 σ i ,t −1 + βi1hi ,t −1 (6) where negative γ i1 denotes leverage effect through imposing a larger coefficient on negative ε i ,t −1 ; βi1 < 1 is required for covariance stationary. The GJR-GARCH(1,1) applies a dummy variable, N i ,t −1 , to differentiate positive and negative shocks. σ it2 = α i 0 + (α i1 − γ i1 N i ,t −1 )ε i2,t −1 + β i1σ i2,t −1 (7) 9 1 if ε i ,t −1 < 0 where N i ,t −1 =  0 if ε i ,t −1 ≥ 0 The A-PARCH(1,d,1) imposes Box-Cox power transformation on the conditional volatility function, thus allowing for more flexibility. σ itd = α i 0 + α i1 ( ε i ,t −1 + γ i1ε i ,t −1 ) d + βi1σ id,t −1 (8) Negative γ i1 denotes leverage effect. We artificially impose d = 1 and d = 2 at first and then conduct quasi-maximum likelihood estimation on d. The model fitted with d = 1 is more robust to extreme values than the one with d = 2 . The restriction, α i1 (1 + γ i21 ) + β i1 < 1 , is required for the A-PARCH(1,2,1) to ensure covariance stationary. Similarly, in the A-PARCH(1,1,1), E (σ t ) and E ( ε t ) are guaranteed for existence, providing 2 / π * α i1 + βi1 < 1 . Regarding the distribution of ηit in equation (3), we assume three variants. They are, namely, [a] normal distribution, [b] a Student’s t distribution, and [c] a generalized error distribution (GED). For practical purposes, Jarque-Bera test is usually applied to return series to test for normality. It is defined as follows: ˆ i − 3) 2 Ti ˆ 2 ( kurt ( skewi + ) 6 4 (9) 1 Ti (rit − ri )3 ∑ T T − ( 1) T i t =1 ˆ = i i Sample skewness: skew ⋅ i Ti − 2 1 Ti ( ∑ (rit − ri )2 )3/2 Ti t =1 (10) Jarque-Bera test statistics: JBi = 10 Ti (T + 1)Ti (Ti − 1) ˆ i= i Sample kurtosis: kurt ⋅ (Ti − 2)(Ti − 3) ∑ (r it − ri ) 4 t =1 Ti − (∑ (rit − ri ) ) 2 2 3(Ti − 1) 2 (Ti − 2)(Ti − 3) (11) t =1 For case [a], the log-likelihood function can be specified as: log Li = − Ti 1 Ti 1 Ti ε 2 log(2π ) − ∑ log σ it2 − ∑ it2 2 2 t =1 2 t =1 σ it (12) Under the Student’s t distribution as specified in case [b], the density function of ηit is: f (ηit ) = Γ[(ν i + 1) / 2] Γ(ν i / 2) (ν i − 2)π (1 + ηit2 νi − 2 ) − (ν i +1)/2 , ν i >2 (13) where ν i denotes degrees of freedom and Γ(⋅) is the standard gamma function. Its corresponding log-likelihood function can be derived from (13): 1   log Li = Ti  ln Γ ( (ν i + 1) / 2 ) − ln ( Γ(ν i / 2) ) − ln ( (ν i − 2)π )  2    ν +1    1 ε it2 − ∑  ( i ) ⋅ ln  1 + + ln(σ it2 )  2  t =1   2  (ν i − 2)σ it  2  Ti (14) For case [c], i.e. the generalized error distribution, the density function of ηit is: νi ν exp[−(1/ 2) ηit / λi ] f (ηit ) = i λi ⋅ 2(ν +1)/ν Γ(1/ ν i ) i i (15) where 0 < ν i < 2 denotes the density that presents fatter tails than normal density and 1/2  2−2/ν i Γ(1/ ν i )  λi =   .  Γ(3 / ν i )  The corresponding log-likelihood function derived from (15) is as below. 11  ν  1 Ti ν +1 ν log Li = Ti  ln( i ) − ( i ) ln 2 − ln ( Γ(1/ ν i ) )  − ∑ ηit / λi i νi  λi  2 t =1 (16) Diagnostic tests on standardized residuals of those models include the LM and Ljung-Box tests. The latter one is capable of detecting the existence of serial correlation in standardized residuals. The Q statistics is computed as: mik2 k =1 T − k n Qi (n) = Ti (Ti + 2)∑ Ti ∑ (aˆ 2 it where mik = (17) − σˆ i 2 )(aˆ i2,t − k − σˆ i 2 ) t = k +1 Ti ∑ (aˆ , aˆit is the standardized residuals, equal to 2 it − σˆ i ) 2 2 t =1 T εˆit , and σˆ i 2 = ∑ aˆit2 / Ti . σˆ it t =1 i Under the null hypothesis of no serial correlation in standardized residuals, Q statistics asymptotically follow a chi-square distribution. Regarding our case, n is selected to be 10. No ARCH effect left and no serial correlation can demonstrate the appropriateness of our models. Finally, we select a better model for each series based on the Bayesian information criterion (BIC). BICi = −2 ⋅ ln Li + k ln(Ti ) (18) where Li denotes the maximized value of the likelihood function; k is number of parameters to be estimated; and Ti represents number of observations. The smaller 12 the BIC is, the better the model is. The models fitted with Student’s t and generalized error distributions are expected to have smaller BIC than those fitted with the normal distribution. The asymmetric GARCH(1,1) models are also anticipated to outperform the standard GARCH(1,1). Table 1 summarizes the structure of conditional variances of various GARCH-type models. Table 1: Summary of the structure of conditional variances of GARCH-type models GARCH(1,1) σ it2 = α i 0 + α i1ε i2,t −1 + βi1σ i2,t −1 hit = α i 0 + α i1 EGARCH(1,1) ε i ,t −1 + γ i1ε i ,t −1 σ i ,t −1 + βi1hi ,t −1 where hit = log σ it2 σ it2 = α i 0 + (α i1 − γ i1 N i ,t −1 )ε i2,t −1 + β i1σ i2,t −1 , GJR-GARCH(1,1) A-PARCH(1,d,1) 1 if ε i ,t −1 < 0 where N i ,t −1 =  0 if ε i ,t −1 ≥ 0 σ itd = α i 0 + α i1 ( ε i ,t −1 + γ i1ε i ,t −1 ) d + βi1σ id,t −1 2.2 Bivariate VC-MGARCH(1,1) models In order to capture conditional correlation between returns of the Shanghai Stock Exchange Composite Index/the Shenzhen Stock Exchange Component Index and returns of the S&P 500 Index, we model time-varying conditional correlations in a bivariate GARCH(1,1) framework and follow the methodology proposed by Tse and Tsui (2002). Specifically, both of rsht / rszt and rspt are fitted by univariate standard 13 GARCH(1,1) model structures with normal distribution for simplicity, and their conditional correlation matrix is assumed to follow an ARMA(1,1) structure. It is expected to outperform the CC-MGARCH(1,1) model suggested by Bollerslev (1990) where conditional correlations are assumed to be constant. For computational simplicity and easy comparison between the VC-MGARCH(1,1) and the CC-MGARCH(1,1), we directly interpolate the mean of the return series into the conditional mean equation. The VC-MGARCH(1,1) for rsht (= r1t ) and rspt (= r2t ) is specified as follows. Conditional mean equation: rit = mean(rit ) + ε it , i = 1, 2 (19) where ε t = (ε1t , ε 2t ) ' Conditional variance equation: σ it2 = α i 0 + α i1ε i2,t −1 + βi1σ i2,t −1 (20) σ it can be utilized to construct a 2*2 diagonal matrix Dt , with σ 1t and σ 2t lying on the ξt = ( diagonal line. The standardized ξt residual equals to Dt−1ε t , ε1t ε 2t , ) ' ~ i.i.d .(0, Γt ) , where Γt (= {ρ12t } ) is the conditional correlation σ 1t σ 2t matrix of ε t . ρ12t is assumed to follow an ARMA(1,1) process. Conditional correlation equation: ρ12t = (1 − θ1 − θ 2 ) ρ12 + θ1 ρ12,t −1 + θ 2ψ 12,t −1 M where ρ12 is time-invariant, ψ 12,t −1 = ∑ξ ξ 1,t − h 2,t − h h =1 M (∑ ξ 2 1,t − h h =1 (21) , M )(∑ ξ 2 2,t − h 0 ≤ θ1 ,θ 2 ≤ 1 , ) h =1 14 θ1 + θ 2 ≤ 1 . M = 2 is imposed based on the discussion that M ≥ 2 is a necessary condition to guarantee Ψ t (= {ψ ijt } ) positive definite, see Tse and Tsui (2002). Since ε t , Dt and Γ t have already been derived, the conditional log-likelihood lt and the log-likelihood function of the sample l can be estimated through: 1 1 2 1 lt = − ln Γ t − ∑ ln σ it2 − ε t' Dt−1Γ t−1 Dt−1ε t 2 2 i =1 2 (22) T l = ∑ lt (23) t =1 Obviously, the CC-MGARCH(1,1) is nested within the VC-MGARCH(1,1) when imposing the restriction, θ1 = θ 2 = 0 . The likelihood ratio test is implemented to assess whether the VC-MGARCH(1,1) outperforms the CC-MGARCH(1,1). The test is described as follows: χ 2 = −2 ln L0 = −2(ln L0 − ln L1 ) L1 (24) where L0 denotes the maximized value of the likelihood function of the CC-MGARCH that imposes restriction on θ1 and θ 2 ; L1 represents the maximized value of the likelihood function of the VC-MGARCH that imposes no restriction on them. Under the null hypothesis H 0 : θ1 = θ 2 = 0 , the test statistic asymptotically follows a chi-square distribution with two degrees of freedom. Other diagnostic tests on standardized residuals, including the LM and Ljung-Box tests, are also conducted. The same procedure is applicable to the VC-MGARCH(1,1) for rszt and rspt . 15 2.3 Markov-switching variance models and Time-varying-parameter models with Markov-switching heteroskedasticity In this section, Markov-switching heteroskedasticity is utilized to model conditional heteroskedasticity of rsht and rszt , rather than GARCH heteroskedasticity as mentioned in Sections 2.1 and 2.2. The oscillatory behavior of time-varying volatility can be categorized into two distinct regimes: relatively stable state and highly volatile state. Considering difficulties associated with quantifying an unobserved and discrete state variable, we assume St to follow a two-state, first-order Markov chain with transition probabilities specified as: Pr[ St = 0 St −1 = 0] = p00 , Pr[ St = 1 St −1 = 0] = p01 ; (25) Pr[ St = 0 St −1 = 1] = p10 , Pr[ St = 1 St −1 = 1] = p11 where St = 0 stands for the relatively stable state; St = 1 represents the highly volatile state; and p00 + p01 = 1 , p10 + p11 = 1 . High p00 indicates the stock market is highly persistent in the relatively stable state, with small probability of shifting to the highly volatile state. Contrary to that, high p11 denotes the stock market is always unstable, with small probability of transferring to the relatively stable state. If p00 and p11 are quite low, it shows the stock market is frequent in regime shifting. We introduce Markov-switching variance models (Model 1) and time-varying-parameter models with Markov-switching heteroskedasticity (Model 2) based on Kim (1993). The Markov-switching variances are specified as follows: 16 ε it ~ N (0, σ is2 ), σ is2 = σ i20 + (σ i21 − σ i20 ) St , t t (26) where i = sh, sz , ε it is assumed to follow normal distribution for simplicity, σ i20 denotes the variances when China’s stock markets are relatively stable, and σ i21 represents the variances when they are suffering from huge shocks. Specifically, σ i20 is smaller than σ i21 . In this Section, we derive correlation between China’s and the U.S. stock markets from the coefficient of rsp ,t −1 in the conditional mean equation. That is different from what we do in Section 2.2 and can provide us with a more thorough understanding of the connection between the two markets. In Model 1, the coefficient of rsp ,t −1 is assumed to be time-invariant. In Model 2, it is assumed to follow an AR(1) process to capture uncertainty induced by the dynamics of linkage between the stock markets. Kim imposes a random walk specification on βit in equation (29) to represent regime changes that only occur when new information is accessable, as suggested by Engle and Watson (1987). Distinct from that, we estimate α i on βi ,t −1 and anticipate it is positive. Details of Model 1 and Model 2 are specified as follows, i = sh, sz : Model 1: Markov-switching variance models rit = ci + φi ri ,t −1 + θ i ε i ,t −1 + ai rsp ,t −1 + ε it , ε it ~ N (0, σ is2 ), t σ = σ + (σ i21 − σ i20 ) St , 2 ist 2 i0 (28) Pr[ S t = 1 St −1 = 1] = p11 , Pr[ St = 0 S t −1 = 0] = p00 17 Model 2: Time-varying-parameter models with Markov-switching heteroskedasticity rit = ci + φi ri ,t −1 + θ iε i ,t −1 + β it rsp ,t −1 + ε it , β it = α i β i ,t −1 + vit , vit ~ N (0, σ v2i ), ε it ~ N (0, σ is2 ), (29) t σ is2 = σ i20 + (σ i21 − σ i20 ) S t , t Pr[ S t = 1 S t −1 = 1] = p11 , Pr[ S t = 0 S t −1 = 0] = p00 Modeling rsht as a function of rsh ,t −1 and rsp ,t −1 may cast doubt on why rsz ,t −1 is not included. The reason is that when we additionally interpolate rsz ,t −1 , its coefficient is not only insignificant, but results in a smaller t-statistic of the coefficient of rsp ,t −1 . Hence, only rsp ,t −1 is included in the conditional mean equation for Shanghai stock market and also for Shenzhen stock market. However, simply interpolating full sample of rsp ,t −1 into the conditional mean equation may induce multiple structural changes. To identify the existence of such a problem and corresponding number of breaks, we follow the efficient algorithm developed by Bai and Perron (2002) to perform several tests. We fit our regression as a pure structural change model that allows for all the coefficients to be time varying through treating p = 0 , accompanied with three changing variables c , ri ,t −1 and rsp ,t −1 . The maximum number of breaks allowed is set at three with ε = 0.2 . First, we apply one test to check the null hypothesis of no structural break against the alternative hypothesis of one break, of two breaks and of three breaks. Similarly, 18 another test is adopted to check the null hypothesis of l structural breaks against the alternative hypothesis of l + 1 breaks. The double maximum tests, i.e. the UDmax and WDmax tests, are more flexible, because they allow the breaks to be an unknown number rather than a specific one in the alternative hypothesis. BIC suggested by Yao (1998), modified Schwarz criterion (LWZ) developed by Liu et al. (1997) and sequential method proposed by Bai and Perron (2002) can be implemented to determine number of breaks in the regression. The corresponding results are reported in Section 3.3. Regardless of the number of breaks in the conditional mean equation, we mainly concentrate on the latest period because it is the most representative one for the relationship between rit and rspt . To conduct quasi-maximum likelihood estimation of Model 1, we adopt the filter developed by Hamilton (1989), and follow the related algorithm suggested by Kim (1993). Three probabilities are instrumental for estimation: [a] prediction probabilities, Pr[ St = j yt −1 ] , [b] filtered probabilities, Pr[ St = j yt ] , and [c] smoothed probabilities, Pr[ St = j yT ] (t = 1, 2,K , T ) , where j = 0,1 . The computational procedure for rsht is described as follows. Before iteration, initial values are required to be imposed on π 0 (= Pr[ S0 = 0 y0 ]) , π 1 (= Pr[ S0 = 1 y0 ]) and on the log-likelihood function, where yt represents information set up to time t. 19 1 − p11 , π 1 = 1 − π 0 , l (ψ ) = 0 . 2 − p00 − p11 (30) where l (ψ ) is the likelihood function, and ψ denotes all the eight parameters π0 = that will be estimated, consisting of c, φ ,θ , a, σ 0 , σ 1 , p00 , p11 . Based on the initial values, we can predict Pr[ St = j yt −1 ], j = 0,1 at the beginning of time t with the following relationships: 1 Pr[St = j yt −1 ] = ∑ Pr[ St = j, St −1 = i yt −1 ] i =0 (31) 1 = ∑ Pr[St = j St −1 = i]Pr[ St −1 = i yt −1 ], i =0 The density function, f (rsht yt −1 ) is: 1 1 f ( rsht yt −1 ) = ∑∑ f ( rsht ,St = j , St −1 = i yt −1 ) j =0 i =0 1 (32) 1 = ∑∑ f ( rsht St = j , St −1 = i, yt −1 ) Pr[ St = j , St −1 = i yt −1 ] j =0 i =0 where f ( rsht yt −1, St = j , St −1 = i ) = 1 2πσ S2t exp( − ε t2t −1 2σ S2t ). (33) f (rsht yt −1 ) is indispensable for probabilities updating and the maximum likelihood T estimation. The log-likelihood function is: ln L = ∑ ln{ f (rsht yt −1 )} (34) t =1 20 At the end of time t, the prediction probabilities Pr[ St = j yt −1 ] can be updated to the filtered probabilities Pr[ St = j yt ], j = 0,1 , provided with the additional information of rsht . 1 1 i =0 i =0 Pr[ St = j yt ] = ∑ Pr[ St = j , St −1 = i yt ] = ∑ Pr[St = j , St −1 = i yt −1 , rsht ] 1 f (rsht , St = j , St −1 = i yt −1 ) i=0 f ( rsht yt −1 ) =∑ (35) Finally, given full information, the smoothed probability Pr[ St = j yT ] (t = 1, 2,K , T ) can be estimated backward. 1 1 k =0 k =0 Pr[St = j yT ] = ∑ Pr[St = j, St +1 = k yT ] = ∑ Pr[ St +1 = k yT ] × Pr[ St = j St +1 = k , yT ] 1 = ∑ Pr[St +1 = k yT ] × Pr[St = j St +1 = k , yt ] (36) k =0 1 Pr[St +1 = k yT ] × Pr[St = j yt ] × Pr[ St +1 = k St = j ] k =0 Pr[St +1 = k yt ] =∑ For further information on detailed computation and the equivalent relationship between Pr[ St = j St +1 = k , yT ] and Pr[ St = j St +1 = k , yt ] , refer to Kim and Nelson (1999). For the sake of the application of the Kalman filter before the Hamilton filter, the 21 algorithm of Model 2 based on Kim (1993) is a little more complicated than the one employed in Model 1. To tackle the problem that βt is unobserved, the Kalman filter is adopted to make inference on βt based on yt −1 , denoted as βt(ti−, 1j ) , and its corresponding variance, denoted as Pt (ti−,1j ) , given St −1 = i and St = j , i = 0,1 , j = 0,1 . Initial values are imposed on β0i 0 and P0i0 . That is β 0i 0 = 0 , and P0i0 = 1 . βt(ti−, 1j ) = αβ ti−1 t −1 , (37) Pt (ti−,1j ) = α 2 Pt i−1 t −1 + σ v2 , (38) The conditional forecast error ε t(ti ,−j1) can be derived from the conditional mean equatioin. ε t(ti ,−j1) = rsht − c − φ rsh ,t −1 − θε t −1 − βt(ti−, 1j ) rsp ,t −1 , (39) Its conditional variance, f t (ti−,1j ) is specified as: f t (ti−,1j ) = Pt (ti−,1j ) rsp2 ,t −1 + σ 2j . (40) Moreover, a Kalman gain K t(i , j ) is required for βt(ti−, 1j ) and Pt (ti−,1j ) updating. K t(i , j ) = Pt (ti−,1j ) rsp ,t −1 / [ f t (ti−,1j ) ], (41) βt(ti , j ) = βt(ti−, 1j ) + K t(i , j )ε t(ti ,−j1) , (42) Pt (ti , j ) = (1 − K t(i , j ) rsp ,t −1 ) Pt (ti−,1j ) (43) 22 However, βt(ti , j ) and Pt (ti , j ) possibily induce cumbersome calculation. For computational simplicity, we implement the approximateions suggested by Kim (1993) to collapse βt(ti , j ) and Pt (ti , j ) to βt jt and Pt tj , j = 0,1 . 1 βt jt = ∑ Pr[S t i =0 , Pr[ St = j yt ] 1 Pt tj = = j , St −1 = i yt ]βt(ti , j ) ∑ Pr[S t i =0 (44) = j , St −1 = i yt ]{Pt (ti , j ) + ( β t jt − βt(ti , j ) )( βt jt − βt(ti , j ) )' } (45) Pr[ St = j yt ] where Pr[ St = j , St −1 = i yt ] and Pr[ St = j yt ] can be derived in the same way as described in Model 1. The log-likelihood function of Model 2, ln L is computed as: f (rsht yt −1, St = j , St −1 = i ) = 1 1 2π f t (ti−,1j ) exp{− (ε t(ti −, j1) ) 2 2 f t (ti−,1j ) } (46) 1 f (rsht yt −1 ) = ∑∑ f ( rsht ,St = j , St −1 = i yt −1 ) j =0 i =0 1 1 (47) = ∑∑ f (rsht St = j , St −1 = i, yt −1 ) Pr[ St = j , St −1 = i yt −1 ] j =0 i =0 T ln L = ∑ ln{ f (rsht yt −1 )} . (48) t =1 The LM and Ljung-Box tests are applied to assess the appropriateness of Model 1 and Model 2. 23 3. Data and Estimation Results 3.1 Data specification Our sample data is drawn from Yahoo. Finance, consisting of daily returns of the Shanghai Stock Exchange Composite Index, the Shenzhen Stock Exchange Component Index and the S&P 500 Index. The Shanghai Stock Exchange Composite Index launched on July 15, 1991 is a whole market index, including all listed A-shares and B-shares traded at the Exchange. A-shares are traded in RMB, while B-shares are traded in U.S. dollars at the Shanghai Stock Exchange and in Hong Kong dollars at the Shenzhen Stock Exchange. The index is compiled using Paasche weighted formula and it converts prices of B-shares denominated in U.S. dollars into RMB1. Current Index= Current total market cap of constitutents *Base Value total market capitalization of all stocks traded on Dec. 19, 1990 Total market cap=∑(price*share issued) (49) Base Value=100 The Shenzhen Stock Exchange Component Index is calculated similarly as the Composite Index and all prices of B-shares are converted into RMB. In lieu of covering all the tradable and non-tradable shares at the Shenzhen Exchange, the Component Index only selects 40 representative listing companies’ tradable shares to 1 The corresponding exchange rate should be the middle price of US dollars on the last trading day of each week provided by China Foreign Exchange Trading Center. 24 track the market’s performance, thereby minimizing the inaccuracy induced by non-tradable shares. Current Index= Current total market cap of 40 representative constitutents *Base Value total market capitalization of 40 shares traded on July 20, 1994 Total market cap=∑(price*number of tradable shares) (50) Base Value=1000 The S&P 500 Index, initially published in 1957, is one of the most widely quoted and tracked market-value weighted indices, representing prices of 500 stocks actively traded in either New York Stock Exchange or NASDAQ. It is more sensitive to stocks with higher market capitalization (=share prices*number of shares outstanding). Since March 2005, it has implemented the policy that only actively traded public shares (float weighted) are considered for calculation of market capitalization. In subsequent sections, the Shanghai Stock Exchange Composite Index, the Shenzhen Stock Exchange Component Index and the S&P 500 Index are abbreviated by sh, sz and sp respectively. The daily returns of those indices, rit , are computed as: rit = ln( Pit ) *100 Pit −1 (51) where i = sh, sz , sp , Pit stands for the close price of each index adjusted for dividends and splits at date t. Data period is 01/05/2000 ~ 12/31/2008. Table 2 describes some summary statistics for rit . 25 Table 2: Summary Statistics for rit , i = sh, sz, sp rsht rszt rspt Mean 0.0111 0.0275 -0.0194 Std. dev 1.6520 1.8340 1.3590 Minimum -9.2562 -12.1000 -9.4700 Maximum 9.4010 11.6300 10.9600 Skewness -0.0102 -0.0952 -0.1220 Kurtosis 7.9360 8.2210 11.9600 2337 2244 2261 2552.0874* 7568.8045* 134.2592* 673.7485* (A) Descriptive statistics No. of obs. (B) Jarque-Bera test for normality Jarque-Bera 2372.4928* (C) LM test for ARCH effect LM(10) 155.7168* (*: at the 1% significance level) All returns distributions are left-skewed and highly leptokurtic, especially rspt has the highest kurtosis. Attributed to these characteristics, the Jarque-Bera test statistics reject the null hypothesis of normal distribution at the 1% significance level. The high Lagrange multiplier test statistics indicate strong ARCH effects of these series. Before proceeding to the specific models, we check whether these series are stationary, 26 employing the Augmented Dickey-Fuller (ADF) test. This test allows rt to have a more general ARMA(p, q) structure rather than a simple AR(p) dynamics. In addition, the Efficient Modified PP test proposed by Ng and Perron (2001) overcomes the shortcoming of the ordinary PP test suggested by Phillips & Perron (1988). The ADF test is described as follows. p ∆rt = β ' Dt + π rt −1 + ∑ ϕ j ∆rt − j + ε t (52) j =1 where ε t is homoskedastic. Dt is specified with two cases: only intercept and both intercept and time trend terms. It is to investigate whether rit follows an I(1) structure under the null hypothesis against the alternative hypothesis of an I(0) process through testing whether the t-statistic value of π is significantly different from 0. In the ADF test, selection of the lag length p is important because on one hand, it ensures ε t to be serially uncorrelated; on the other hand, it determines the power of the ADF test. Based on the standard suggested by Schwert (1989), we can identify the T 1/4 upper bound of pi . Regarding rsht , it has pmax = [12*( ) ] = 26 and p = 15 is 100 determined to be the effective number through making the backward selection proposed by Ng and Perron(1995). The procedure is as follows. First, it is to check whether tϕ26 is larger than 1.6. In our case, it is smaller than 1.6. Therefore, we proceed to decrease the lag length one by one until the t-statistic of the coefficient of the last lagged differenced term is larger than 1.6, i.e. tϕ15 > 1.6 . The test results 27 reported in Table 3 reject the null hypothesis that rsht follows an I(1) process at the 1% significance level under both cases. However, the above procedure of selecting the effective pi on rszt and rspt seemingly loses effect, because none of their t-statistic values is larger than 1.6. To tackle the problem, we select pi equal to pmax and corresponding results demonstrate both of them are stationary. The PP test proposed by Phillips & Perron (1988) adjusts serial correlation directly with the modified statistics, Z t , which allows ε t to be heteroskedastic. Furthermore, the PP test is convenient to apply since it is not required to select an appropriate lag length. However, one shortcoming is that it may have severe size distortion when the autoregressive root is close to unity and the moving-average coefficient is a large negative number. See Schwert (1989), Ng and Perron (2001). Hence, we adopt the Efficient Modified PP test2, rather than the ordinary PP test. Ng and Perron (2001) employ Generalized Least Squares (GLS) detrending to improve the power of the PP test and select the truncation lag based on the modified Akaike Information Criterion (MAIC). The results of efficient modified PP tests are consistent with the conclusion derived from the ADF tests and support rit follows an I(0) process. 2 To prevent size distortion that the ordinary PP tests may have when the AR root is close to unity and MA coefficient is largely negative, we adopted Efficient Modified PP tests, which indicate the returns series are stationary. 28 Table 3: Unit root tests rsht : With intercept With intercept and time trend rszt : With intercept With intercept and time trend rspt : With intercept With intercept and time trend ADF test Efficient Modified PP test -10.89* -6.30* -10.90* -7.99* -10.19* -7.23* -10.19* -9.02* -12.59* -6.86* -12.60* -7.35* (*: at the 1% significance level) 3.2 ARMA(1,1)-GARCH(1,1) models 3.2.1 Estimation results of Shanghai Stock Market In this section, we fit the ARMA(1,1)-GARCH(1,1) models with rsht . The quasi-maximum likelihood estimation of each GARCH(1,1) model, diagnostic tests on standardized residuals and BIC under normal, Student’s t and the generalized error distributions are reported in Tables 4, 5 and 6 respectively. From Table 5, the models fitted with Student’s t distribution are improper as all estimated coefficients of rsh ,t −1 are larger than 1, which cannot guarantee weakly stationary. For Tables 4 and 6, all the parameters satisfy the restrictions imposed in 29 Section 2.1 and are significant at the 1% level except the constant terms in the conditional mean equations. Specifically, the significant estimation of γ sh1 indicates strong leverage effect of Shanghai stock market, contrary to the conclusion made based on the data before 2000. The small LM test statistics provide no evidence of ARCH effect. However, in Table 4, some Q(10) are significant at the 5% level, indicating the existence of serial correlation in the standardized residuals. Therefore, normal distribution is also inappropriate for our models. When comparing BIC reported in Tables 4-6, models fitted with the GED outperform those under the other two distributions. In addition, all asymmetric GARCH models have smaller BIC than the standard GARCH(1,1). This is consistent with the observations made in Section 2.1. Finally, the A-PARCH(1,1) model fitted with the GED is selected as a more appropriate one for Shanghai stock market. 30 -0.9867* -0.0002 -0.9739* -0.0006 -0.9873* -0.0001 (0.0004) (0.0100) (0.0134) -0.9851* 0.9917* -0.0001 (0.0129) (0.0004) (0.0100) 0.9925* (0.0097) (0.0007) (0.0058) 0.9930* (0.0140) (0.0004) (0.0109) 0.9919* (0.0117) (0.0007) (0.0076) 0.9908* (0.0041) 0.0241* (0.0042) 0.0298* (0.0029) 0.0198* (0.0042) 0.0292* (0.0045) -0.9747* -0.0772* (0.0042) 0.0279* -0.0001 0.5332 α sh 0 (0.6840) -0.5125 0.0434 θ sh (0.0411) (0.6932) φsh csh 0.9200* β sh1 0.9894* 0.9217* 0.9361* 0.9206* 0.9297* (0.0042) (0.0049) 0.0725* (0.0044) (0.0035) 0.0671* (0.0036) (0.0031) 0.0724* (0.0052) (0.0035) 0.0407* (0.0064) (0.0017) 0.1204* (0.0041) (0.0035) 0.0722* α sh1 (0.0446) -0.2415* (0.0356) -0.2140* (0.0527) -0.3903* (0.0091) -0.0578* (0.0550) -0.3458* N.A. γ sh1 (0.1327) 1.4838* 2 1 N.A. N.A. N.A. d 4.5460 4.0113 5.9160 4.0354 6.8485 4.2225 LM(10) (Note: *: at the 1% significance level; **: at the 5% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL Q2(10) BIC 17.9665 4.7855 8423.0710 18.8174** 4.2112 8421.8830 19.3858** 6.2942 8419.0660 18.8791** 4.2375 8421.6960 19.1273** 7.3415 8418.4460 18.7358** 4.3866 8435.6550 Q(10) Table 4: QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under normal distribution (Shanghai) 31 -0.9917* -0.0001 -0.9910* -0.0001 -0.9917* -0.0001 (0.0002) (0.0016) (0.0027) -0.9907* 1.0030* -0.0002 (0.0025) (0.0002) (0.0015) 1.0028* (0.0025) (0.0002) (0.0015) 1.0032* (0.0025) (0.0002) (0.0015) 1.0028* (0.0025) (0.0002) (0.0016) 1.0032* (0.0116) 0.0400* (0.0176) 0.0613* (0.0106) 0.0381* (0.0177) 0.0614* (0.0154) -0.9909* -0.1287* (0.0168) 0.0536* -0.0002 -0.9918* α sh 0 (0.0024) 1.0032* 0.0000 θ sh (0.0002) (0.0015) φsh csh 0.8868* β sh1 0.9791* 0.8753* 0.8916* 0.8754* 0.8916* (0.0182) (0.0142) 0.1266* (0.0233) (0.0159) 0.1227* (0.0175) (0.0136) 0.1293* (0.0204) (0.0159) 0.0731* (0.0289) (0.0060) 0.2213* (0.0237) (0.0152) 0.1224* α sh1 (Note: *: at the 1% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL (0.0835) -0.3228* (0.0620) -0.2290* (0.0840) -0.3495* (0.0325) -0.1126* (0.0812) -0.3065* N.A. γ sh1 2 1 N.A. N.A. N.A. d 3.4481 (0.2930) 3.3158 (0.2882) 3.3255 (0.2925) 3.3119 (0.2978) 3.3622 (0.2812) 3.1983 ν (0.2090) (0.3026) 1.1879* Table 5: QMLE of the GARCH models and BIC under Student’s t distribution (Shanghai) 8095.5420 8098.5710 8088.6060 8098.5780 8091.4970 8106.1210 BIC 32 -0.9937* 0.0001 -0.9927* 0.0001 -0.9938* 0.0001 (0.0001) (0.0018) (0.0027) -0.9942* 0.0001 0.9993* (0.0028) (0.0001) (0.0019) 0.9990* (0.0028) (0.0001) (0.0018) 0.9994* (0.0029) (0.0001) (0.0019) 0.9990* (0.0029) (0.0002) (0.0018) 0.9995* (0.0121) 0.0388* (0.0146) 0.0495* (0.0098) 0.0330* (0.0148) 0.0506* (0.0136) -0.9924* -0.1135* (0.0131) 0.0402* 0.0002 -0.9944* α sh 0 (0.0027) 0.9992* 0.0001 θ sh (0.0001) (0.0019) φsh csh 0.8973* β sh1 0.9812* 0.8840* 0.9061* 0.8859* 0.8973* (0.0149) (0.0147) 0.1080* (0.0158) (0.0144) 0.0958* (0.0126) (0.0117) 0.1026* (0.0168) (0.0146) 0.0580* (0.0221) (0.0058) 0.1810* (0.0148) (0.0135) 0.0944* α sh1 (Note: *: at the 1% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL (0.0896) -0.2774* (0.0723) -0.2261* (0.0953) -0.3327* (0.0265) -0.0886* (0.0943) -0.2982* N.A. γ sh1 0.9531 (0.0342) 0.9521 (0.0338) 0.9577 (0.0342) 0.9532 (0.0344) 0.9558 (0.0343) 0.9417 ν (0.2348) (0.0341) 1.2961* 2 1 N.A. N.A. N.A. d 3.6223 4.7592 3.7873 4.8093 3.7056 4.5336 LM(10) Table 6: QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under the GED (Shanghai) Q2(10) BIC 33 17.3848 3.7075 8052.4220 18.2622 4.6666 8050.4260 16.8578 3.9296 8046.5050 18.2246 4.6969 8050.6000 16.9654 3.8757 8047.7820 17.4670 4.6612 8054.8150 Q(10) Figure 3 depicts conditional volatility of the A-PARCH(1,1,1) fitted with the GED. Its oscillatory behavior is consistent with the bull and bear market phase of Shanghai stock market we discussed before. Figure 3: Conditional volatility of Shanghai stock market GARCH Volatility 2.5 2.0 1.5 1.0 Conditional SD 3.0 3.5 volatility 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 3.2.2 Estimation results of Shenzhen Stock Market The quasi-maximum likelihood estimation of each GARCH(1,1) model and BIC conducted on rszt fitted with normal, Student’s t and the generalized error distributions are summarized in Tables 7, 8 and 9 respectively. When the error terms are normally distributed, all estimates of AR(1) and MA(1) parameters are insignificant at the 5% level. Under Student’s t distribution, all estimates of φsz are found to be larger than one. As such, the GED is a more appropriate choice for rszt . Moreover, all its parameters satisfy the restrictions imposed in Section 2.1 and diagnostic tests on standardized residuals confirm accuracy of the models fitted with the GED. 34 0.0324* 0.0133* 0.0350* (0.0154) (0.5402) (0.5506) (0.0044) 0.0184* -0.4838 0.5077 -0.0035 (0.0065) (0.0376) (0.7416) (0.7373) 0.2763 0.0021 -0.2485 (0.0030) (0.0461) (0.9359) (0.9327) 0.2757 0.0419 -0.2559 (0.0062) (0.0310) (0.7929) (0.7934) 0.0598 -0.0004 -0.0321 (0.0050) 0.1672 (0.0401) (0.8541) (0.8528) -0.1438 -0.0785* 0.0374* 0.0340 0.1126 α sz 0 (0.0063) -0.0870 0.0242 θ sz (0.0337) (0.8591) (0.8582) φsz csz 0.9082* β sz1 0.9921* 0.9150* 0.9446* 0.9133* 0.9363* (0.0048) (0.0054) 0.0726* (0.0049) (0.0403) 0.0788* (0.0033) (0.0030) 0.0654* (0.0058) (0.0043) 0.0551* (0.0064) (0.0017) 0.1186* (0.0048) (0.0044) 0.0839* α sz1 (Note: *: at the 1% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL (0.0358) -0.1888* (0.0277) -0.1584* (0.0372) -0.1840* (0.0080) -0.0491* (0.0383) -0.1778* N.A. γ sz1 Table 7: QMLE of the GARCH models and BIC under normal distribution (Shenzhen) (0.1308) 1.3169* 2 1 N.A. N.A. N.A. d 8537.3960 8543.1560 8531.9010 8542.9120 8532.3010 8550.4250 BIC 35 -0.9900* -0.0004 -0.9889* -0.0004 -0.9900* -0.0004 (0.0003) (0.0022) (0.0035) -0.9896* 1.0010* -0.0004 (0.0033) (0.0003) (0.0021) 1.0013* (0.0037) (0.0003) (0.0023) 1.0007* (0.0033) (0.0003) (0.0021) 1.0013* (0.0035) (0.0003) (0.0022) 1.0010* (0.0096) 0.0288* (0.0150) 0.0482* (0.0088) 0.0273* (0.0150) 0.0482* (0.0141) -0.9895* -0.1038* (0.0154) 0.0496* -0.0004 -0.9907* α sz 0 (0.0030) 1.0015* -0.0003 θ sz (0.0003) (0.0019) φsz csz 0.9029* β sz1 0.9864* 0.9049* 0.9160* 0.9049* 0.9147* (0.0146) (0.0123) 0.0999* (0.0149) (0.0132) 0.0874* (0.0141) (0.0119) 0.0989* (0.0146) (0.0132) 0.0603* (0.0232) (0.0045) 0.1686* (0.0157) (0.0138) 0.0916* α sz1 (0.0800) -0.1925** (0.0599) -0.1694* (0.0840) -0.1948** (0.0216) -0.0592* (0.0825) -0.2056** N.A. γ sz1 3.9405 (0.3765) 3.9046 (0.3762) 3.8972 (0.3764) 3.9040 (0.3830) 3.9404 (0.3684) 3.8704 ν (0.2193) (0.3811) 1.1320* 2 1 N.A. N.A. N.A. d (Note: *: at the 1% significance level; **: at the 5% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL Table 8: QMLE of the GARCH models and BIC under Student’s t distribution (Shenzhen) 8283.2460 8286.8440 8275.6160 8286.8440 8277.0330 8287.6320 BIC 36 -0.9852* -0.9883* 0.0001 -0.9860* 0.0000 0.9969* (0.0003) (0.0033) 0.0000 (0.0003) (0.0032) (0.0135) 0.0386* (0.0078) 0.0214* (0.0135) 0.0386* (0.0129) -0.1014* (0.0139) 0.0416* α sz 0 (0.0051) (0.0079) -0.9880* 0.0201** (0.0050) -0.9883* 0.0001 0.9970* (0.0060) (0.0003) (0.0040) 0.9955* (0.0050) (0.0003) (0.0032) 0.9970* (0.0067) (0.0003) (0.0045) 0.9946* 0.0000 -0.9881* (0.0054) 0.9964* 0.0001 θ sz (0.0003) (0.0035) φsz csz 0.9046* β sz1 0.9892* 0.9107* 0.9235* 0.9108* 0.9274* (0.0116) (0.0104) 0.0849* (0.0121) (0.0116) 0.0787* (0.0107) (0.0096) 0.0891* (0.0132) (0.0116) 0.0534* (0.0191) (0.0043) 0.1554* (0.0130) (0.0124) 0.0869* α sz1 (0.0839) -0.1930** (0.0638) -0.1775* (0.0837) -0.2046** (0.0194) -0.0557* (0.0847) -0.2094** N.A. γ sz1 1.0538 (0.0373) 1.0405 (0.0367) 1.0564 (0.0373) 1.0405 (0.0371) 1.0528 (0.0386) 1.0435 ν (0.2267) (0.0372) 1.0842* 2 1 N.A. N.A. N.A. d sz (Note: *: at the 1% significance level; **: at the 5% significance level. Standard errors are in parentheses.) A-PARCH(1,d,1) A-PARCH(1,2,1) A-PARCH(1,1,1) GJR-GARCH(1,1) EGARCH(1,1) GARCH(1,1) MODEL 6.1130 6.6213 5.7034 6.6205 6.1846 6.3228 LM(10) Table 9: QMLE of the GARCH models, diagnostic tests of standardized residuals and BIC under the GED (Shenzhen) Q2(10) BIC 37 11.6724 6.3716 8264.5040 13.0791 6.8842 8267.3340 11.4482 5.9273 8257.2000 13.0741 6.8820 8267.3360 11.7372 6.4653 8257.9040 12.4671 6.3614 8268.5060 Q(10) In addition, the significant estimates of γ sz1 indicate strong leverage effect of Shenzhen stock market. Based on the BIC reported in Tables 7-9, the models under the GED are superior to those under the normal and Student’s t distributions. In particular, the A-PARCH(1,1,1) model fitted with the GED has the smallest BIC value, indicating it is a better one for rszt . Figure 4: Conditional volatility of Shenzhen stock market GARCH Volatility 2.5 2.0 1.5 1.0 Conditional SD 3.0 3.5 4.0 volatility 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 3.3 Bivariate VC-MGARCH(1,1) Models 3.3.1 Estimation Results of Shanghai Stock Market We re-shuffle the returns data in this section, because rsht and rspt have different trading days and only the common terms are selected for research. The effective number of observations is reduced from 2337 to 2252. Table 10 reports the descriptive statistics. 38 Table 10: Summary statistics of rsht and rspt Sample: 01/05/2000 ~ 12/31/2008 Mean Std. Dev Min. Median Max. Skewness Kurtosis rsht 0.0138 1.6530 -9.2560 0 9.4010 0.0050 8.0500 rspt -0.0139 1.3360 -9.4700 0.0417 10.9600 0.0114 11.6400 Jarque-Bera test for normality rsht 2392.9752* rspt 7004.6696* rspt 670.1745* LM(10) test for ARCH effect rsht 144.0358* (*: at the 1% significance level) Based on the conditional mean equation, ε sht and ε spt can be derived from: ε sht = rsht − 0.0138 ; ε spt = rspt + 0.0139 . (53) Quasi-maximum likelihood estimation results of the VC-MGARCH(1,1) and its special case, the CC-MGARCH(1,1) for ε sht and ε spt are reported in Table 11. 39 Table 11: VC-MGARCH(1,1) and CC-MGARCH(1,1) for ε sht and ε spt α0 α1 β1 θ1 θ2 ρ A: VC-MGARCH(1,1) model 0.0233* 0.0681** 0.9261** (0.0122) (0.0192) (0.0209) 0.9383** 0.0102 0.0109 0.0114* 0.0792** 0.9146** (0.0705) (0.0090) (0.0286) (0.0059) (0.0127) (0.0141) SH SP B: CC-MGARCH(1,1) model 0.0247** 0.0709** 0.9227** (0.0038) (0.0042) (0.0036) SH 0.0022 N.A. 0.0119** 0.0810** 0.9119** (0.0025) (0.0091) (0.0098) N.A. (0.0213) SP (*: at the 10% significance level; **: at the 1% significance level. Standard errors are in parentheses.) Both VC-MGARCH(1,1) and CC-MGARCH(1,1) models satisfy the restrictions imposed on the GARCH(1,1) model, i.e. α1 > 0 , 0 < β1 < 1 and α1 + β1 < 1 . The insignificant constant term in the conditional correlation equation in the VC-MGARCH(1,1) is consistent with the insignificant correlation in the CC-MGARCH(1,1) where the correlation is assumed to be time-invariant. That implies there is no signification linkage between Shanghai and the U.S. stock markets when assuming the correlation is time-invariant. However, this assumption 40 is quite unreasonable. From Table 11, the conditional correlation significantly follows an AR(1) process. The LM and Ljung-Box tests statistics for the VC-MGARCH(1,1) and the CC-MGARCH(1,1) are summarized in Table 12. They indicate no evidence of ARCH effect or serial correlation in both the standardized residuals and the standardized squared residuals. However, the large likelihood ratio test statistic demonstrates the VC-MGARCH(1,1) model indeed outperforms the CC-MGARCH(1,1) one at any conventional level of significance. Figure 5 displays the conditional correlation between Shanghai and the U.S. stock markets. Although the linkage between the two markets is quite low, even though the peak is still less than 0.10, there is still some connection between the two markets, attributed to the integration of global financial markets. Additionally, the correlation varies remarkably, with the presence of an upward or downward tendency even during a short period. That phenomenon can be explained by the immaturity of Shanghai stock market and governmental influence on the market. The low and highly volatile connection may bring diversification benefits to equity portfolio. Figure 5: Plot of conditional correlation between Shanghai and the U.S. markets 41 Std.dev 0.0022 SP 1.0010 1.0030 -0.0446 SP 1.0030 1.0050 (*: at the 1% significance level) Likelihood ratio test: 47.5884* -0.0186 SH B: CC-MGARCH(1,1) model -0.0045 SH A: VC-MGARCH(1,1) model Mean -6.7570 -5.3490 -6.6490 -5.3920 Min 3.3680 6.9310 3.4480 6.8120 Max -0.2841 0.0487 -0.2803 0.0535 skewness 4.3010 6.8610 4.3000 6.8240 kurtosis 9.1447 6.4453 9.1447 6.6713 LM(10) 16.5896 17.9222 15.9108 17.8404 Q(10) 9.5556 6.6215 9.2004 6.8490 Q2(10) Table 12: Diagnostic tests on the standardized residuals of VC-MGARCH(1,1) and CC-MGARCH(1,1) (SH and SP) 42 3.3.2 Estimation Results of Shenzhen Stock Market The characteristics of common terms between rszt and rspt are summarized in Table 13. The effective number of observations is 2172. Table 13: Summary statistics of rszt and rspt Sample: 01/05/2000 ~ 12/31/2008 Mean Std. Dev Min. Median Max. Skewness Kurtosis rszt 0.0346 1.8340 -12.1000 0.0000 11.6300 -0.0955 8.3420 rspt -0.0164 1.3550 -9.4700 0.0421 10.9600 0.0161 11.3900 Jarque-Bera test for normality rszt 2585.8819* rspt 6370.5689* rspt 644.3075* LM(10) test for ARCH effect rszt 120.8229* (*: at the 1% significance level) ε szt and ε spt can be derived from: ε szt = rszt − 0.0346 ; ε spt = rspt + 0.0164 . (54) Table 14 reports results of quasi-maximum likelihood estimation on ε sht and ε spt . The diagnostic tests on standardized residuals are reported in Table 15. Both two models show no sign of ARCH effect or serial correlation. The likelihood ratio test statistic indicates that the VC-MGARCH(1,1) is superior to the CC-MGARCH(1,1) at 43 any conventional level of significance. That is consistent with our previous conclusion. Table 14: VC-MGARCH(1,1) and CC-MGARCH(1,1) for ε szt and ε spt α0 α1 θ1 θ2 ρ 0.9185*** 0.0264 0.0078 (0.0825) (0.0253) (0.0099) β1 A: VC-MGARCH(1,1) model 0.0339** 0.0817*** 0.9124*** SZ (0.0168) 0.0115* (0.0241) (0.0244) 0.0843*** 0.9104*** SP (0.0060) (0.0138) (0.0151) B: CC-MGARCH(1,1) model 0.0381*** 0.0890*** 0.9044*** SZ (0.0063) (0.0051) 0.0231 (0.0044) N.A. 0.0119*** 0.0860*** 0.9084*** N.A. (0.0216) SP (0.0026) (0.0098) (0.0101) (*: at the 10% significance level; **: at the 5% significance level; ***: at the 1% significance level. Standard errors are in parentheses.) 44 Std.dev -0.0020 SP 1.0010 1.0030 -0.0375 SP 1.0000 1.0040 (*: at the 1% significance level) Likelihood ratio test: 33.8035* -0.0075 SZ B: CC-MGARCH(1,1) model -0.0124 SZ A: VC-MGARCH(1,1) model Mean -6.2490 -4.5560 -6.2210 -4.6300 Min 3.3750 6.6450 3.4340 6.6920 Max -0.2616 0.2071 -0.2641 0.2103 Skewness 4.0890 6.6410 4.1000 6.6150 Kurtosis 7.3781 8.0594 7.4111 7.9344 LM(10) 18.9729 19.1349 17.8046 19.3120 Q(10) Table 15: Diagnostic tests on the standardized residuals of VC-MGARCH(1,1) and CC-MGARCH(1,1) (SZ and SP) 7.5793 7.8829 7.5907 7.7742 Q2(10) 45 The conditional correlation between Shenzhen and the U.S. stock markets is plotted in Figure 6. Its shape is similar to Figure 5. It does not have a definite correlation pattern between the two markets. Furthermore, the magnitude of correlation is especially low and highly unstable, even lower than that of Shanghai and the U.S. stock markets. Figure 6: Plot of conditional correlation between Shenzhen and the U.S. stock markets 3.4 Markov-switching variance models and Time-varying-parameter models with Markov-switching heteroskedasticity In this section, we interpolate rsp ,t −1 into the conditional mean equation that may induce multiple structural changes. We follow the methodology proposed by Bai and Perron (2002) and summarize corresponding specification, tests and number of breaks in Table 16. 46 Table 16: Test statistics of multiple structural changes Specifications zt = {1, rsh,t −1 , rsp ,t −1} q=3 p=0 h = 450 M = 3, ε = 0.2 Tests 1 break against 0 2 against 0 3 against 0 2 against 1 3 against 2 11.4539 9.5802 7.6428 6.4411 3.7284 UDmax WDmax 11.4539 11.6362 Number of breaks selected Sequential : 0 LWZ : 0 BIC : 0 Specifications zt = {1, rsz ,t −1 , rsp ,t −1} q=3 p=0 h = 434 M = 3, ε = 0.2 Tests 1 break against 0 2 against 0 3 against 0 2 against 1 3 against 2 8.9663 6.4733 6.8632 6.0461 2.0347 UDmax WDmax 8.9663 10.1367 Number of breaks selected Sequential : 0 LWZ : 0 BIC : 0 To both of the equations, all the test statistics are insignificant at the 5% level, 47 indicating that no break is introduced when rsp ,t −1 is interpolated. This conclusion is confirmed by the BIC, modified Schwarz criterion (LWZ) and sequential method because all of them select zero break. Although there is no detection of structural break in the conditional mean equation, we still divide the full sample into two sub-samples with periods 01/05/2000 ~ 12/31/2004 and 01/03/2005 ~ 12/31/2008, in order to differentiate impacts of the burst of “dot-com bubble” and that of the current financial crisis. In particular, during the second period, China’s stock market becomes more mature as a result of the non-tradable shares reform and gradual integration with the global financial market. Therefore, we anticipate a closer and positive relationship between rsht and rsp ,t −1 or between rszt and rsp ,t −1 . Table 17 summarizes descriptive statistics for the common terms of rsht and rspt , and of rszt and rspt during the latter period. It provides evidence of ARCH effect and heavier outlier. However, for simplicity, normal distribution is assumed and Markov-switching heteroskedasticy is adopted to capture the ARCH effect. 48 -0.0168 0.0827 -0.0232 SP SZ SP 1.4640 2.3100 1.4140 2.0110 Std. Dev Min. -9.4700 -12.1000 -9.4700 -9.2560 (*: at the 1% significance level) 0.0361 SH Mean 10.9600 11.6300 10.9600 9.0340 Max. -0.0906 -0.3650 -0.1037 -0.2747 Skewness 16.1100 6.1650 17.0100 6.0470 Kurtosis Table 17: Descriptive statistics for the period 01/03/2005 ~ 12/31/2008 918 918 998 998 No. of Obs. 6575.3599* 403.5385* 8163.7654* 398.6278* Jarque-Bera 324.7310* 29.8632* 354.7106* 52.1799* LM(10) 49 3.4.1 Estimation Results of Shanghai Stock Market All estimates reported in Table 18 are significant at the 5% or 1% level, thereby providing some justification of two discrete regimes and Markov-switching heteroskedasticity. However, estimates of φ and θ are quite small, since they are near to unity as discussed in Section 3.2. It may be explained by the significantly positive relationship between Shanghai and the U.S. stock markets interpolated into the equation, denoted as a , which weakens the impact of rsh,t −1 on rsht . However, a is still relatively small, implying low level of linkage between the two markets and diversification benefits for portfolio investment, consistent with our previous conclusion in Section 3.3. Diagnostic tests for both models are summarized in Table 19. The LM and Ljung-Box tests indicate that no ARCH effect is left and no serial correlation exists in either the standardized residuals or in the standardized squared residuals. 50 (0.0687) 0.1903* 0.1211** (0.0405) (0.0754) (0.0411) 0.1891* 0.1099** (0.0731) -0.1910** (0.0773) -0.1941* θ N.A. (0.0493) 0.2400** a (0.0497) 1.0557** (0.0474) 1.0935** σ0 (0.1278) 2.6995** (0.1170) 2.7362** σ1 (0.0118) 0.9653** (0.0110) 0.9695** p00 0.7306 0.7266 Model 1 Model 2 Mean 0.7257 0.7230 Std. dev. 0.0004 0.0003 Min. 6.9300 6.6970 Max. 2.2940 2.2890 Skewness 12.5500 12.3300 Kurtosis 10.0724 2.3725 LM(10) 8.0844 5.7149 Q(10) 0.1363* N.A. σv 3.3508 2.4218 (0.0278) 0.9562** N.A. α 1971.7478 1969.5008 Likelihood value (0.0539) Q2(10) (0.0180) 0.9542** (0.0155) 0.9603** p11 Table 19: Diagnostic tests on the standardized residuals of Model 1 and Model 2 for Shanghai stock market (*: at the 5% significance level; **: at the 1% significance level; Standard errors are in parentheses.) Model 2 Model 1 φ c Table 18: Estimation results of Model 1 and Model 2 for Shanghai stock market 51 In Model 1, low volatility stays at the level of 1.0935, while high volatility maintains at the level of 2.7362. Both regimes are highly persistent implied by large values of p00 and p11 , meaning when Shanghai stock market persists at one state, either a relatively stable or a highly volatile state, it is hard to shift to the other regime. It is an interesting result, seemingly inconsistent with the reality. It may be explained by the highly diversified global financial market, so only rsp ,t −1 interpolated in the conditional mean equation is possibly not sufficient. However, we can observe the highly persisent characteristic during some period. For example, in 2008, the volatility always stays at the high level, whereas at the end of 2006, it remains quite stable. The duration of high volatility is expected to be 25 days on average, while the low volatility state is expected to last for 33 days, a little longer. Figure 7 displays condtional variances of Model 1. Figure 7: Conditional varaices of Model 1 for Shanghai stock market In Model 2, the time-varying relationship between rsht and rsp ,t −1 is characterized as an AR(1) process, compensating for the shortcoming that Model 1 assumes the 52 relationship is constant and fails to consider the uncertainty induced by the changing linkage pattern between Shanghai and the U.S. stock markets. All estimates of parameters reported in Table 18 are significant at the 1% level, except for σ v which is significant at the 5% level. However, it is still better than Kim’s results (1993) when he quantifies U.S. monetary growth uncertainty, in which all the σ vi are insignificant. Moreover, the AR(1) process is stationary, distinct from Kim’s random walk specification. Attributed to part of conditional volatility captured by βt , σ 0 and σ 1 are lower than those in Model 1. In Figure 8, the lower line depicts uncertainty induced by the time-varying coefficient and the higer line describes total conditional variances. It is apparent that during 2005 and 2006, the relationship between the two stock markets almost remains time invariant and total conditional variances are almost identical to those in Model 1. However, since 2007, the conditional variances captured by the changing coefficients have increased gradually, and oscillated particularly sharply at the end of 2008, signaled by the crash of several large U.S.-based financial institutions. At that time, the total variance also peaks at over 16. The likelihood value in Model 2 is larger than in Model 1. This confirms that Model 2 outperforms Model 1 as the changing coefficient has a significant implication for the conditional variances in 2007 and 2008. 53 Figure 8: Conditional variances of Model 2 for Shanghai stock market 3.4.2 Estimation Results of Shenzhen Stock Market With respect to βt at Shenzhen stock market, a minor adjustment is made, because when βt is estimated based on, βt = α 0 + α1βt −1 + vt , vt ~ N (0, σ v2 ), (55) α1 is insignificant, while σ v is significant at the 1% significance level. To tackle the problem, a constant term is interpolated into the AR(1) structure. The adjusted Model 2 for rszt is specified as: rszt = c + φ rsz ,t −1 + θεt −1 + βt rsp,t −1 + ε t , βt = α0 + α1βt −1 + vt , vt ~ N (0,σ v2 ), εt ~ N (0,σ s2 ), t (56) σ s2 = σ 02 + (σ12 − σ 02 )St , t Pr[St = 1 St −1 = 1] = p11 ,Pr[St = 0 St −1 = 0] = p00 Table 20 reports the corresponding quasi-maximum likelihood estimation results. Almost all estimates of parameters are significant at the 5% or 1% significance level, except AR(1) polynomial to describe βt . Hence, the time-varying parameter actually 54 follows: βt = 0.3441 + vt , vt ~ N (0, 0.45332 ) (0.1631) (0.1147) (56) Shenzhen stock market also has the highly persistent characteristic, with the probabilities that are even higher than those of Shanghai stock market. Nevertheless, the probability of persisting in the relatively stable regime is still higher than the one of persisting in the highly volatile regime. In Model 1, the low volatility state is expected to last for 86 days for average, whereas the corresponding period for the highly volatile state is 55 days. In Model 2, the corresponding probabilities and expected duration are lower. It is expected to stay at the relatively stable regime for 74 days, while 42 days for the highly volatile one. The explanation can refer to the discussion in Section 3.4.1. Diagnostic tests conducted on standardized residuals of both models are summarized in Table 21. No ARCH effect is left and no serial correlation exists in either the standardized residuals or in the standardized squared residuals, implied by the LM and Ljung-Box tests. Contrary to the case of Shanghai stock market where Model 2 is better than Model 1, to Shenzhen stock market, Model 1 is superior to Model 2, confirmed by the larger likelihood value of Model 1. The insignificant AR(1) polynomial that describes time-varying relationship between the two markets can provide some support of this conclusion. 55 Figure 9 depicts the conditional variances captured by Model 1 for Shenzhen stock market, which has the rough shape as shown in Figure 7, but with larger values. Figure 10 depicts the conditional variances captured by Model 2, contributed by Markov-switching heteroskedasticity and the time-varying parameter. It is apparent that the time-varying characteristic only converges for almost 60 days to the end of 2008; however, for other periods, it is almost zero, thereby providing another evidence that Model 1 outperforms Model 2. Figure 9: Conditional variances of Model 1 for Shenzhen stock market Figure 10: Conditional variances of Model 2 for Shenzhen stock market ( Lower line: conditional variances captured by the time-varying parameter; Higher line: total conditional variances) 56 (0.0596) 0.2600** 0.1018* (0.0506) (0.0624) (0.0495) 0.2398** 0.1045* (0.0595) -0.2552** (0.0632) -0.2451** θ N.A. (0.0589) 0.2843** a (0.0637) 1.4161** (0.0587) 1.4760** σ0 (0.1504) 3.1107** (0.1357) 3.1493** σ1 (0.0073) 0.9864** (0.0062) 0.9884** p00 p11 -0.0133 -0.0156 Model 1 Model 2 Mean 1.0150 1.0160 Std. dev. -4.1270 -4.0430 Min. 6.5340 6.5140 Max. 0.0150 0.0140 Skewness 5.8150 5.8150 Kurtosis 4.1902 3.6477 LM(10) 9.4344 9.3076 Q(10) Table 21: Diagnostic tests on the standardized residuals of Model 1 and Model 2 for Shenzhen stock market (0.0125) 0.9760** (0.0096) 0.9819** (*: at the 5% significance level; **: at the 1% significance level; Standard errors are in parentheses.) Model 2 Model 1 φ c Table 20: Estimation results of Model 1 and Model 2 for Shenzhen stock market 4.0879 3.5609 Q2(10) (0.1147) 0.4533** N.A. σv 1960.1412 1963.6127 N.A. α1 57 (0.5312) -0.1840 Likelihood value (0.1631) 0.3441* N.A. α0 4. Conclusion This thesis studies several aspects of conditional heteroskedasticity in daily returns of Shanghai and Shenzhen stock markets. First, we find that leverage effect is significant in both markets during the period, 01/05/2000 ~ 12/31/2008 and the ARMA(1,1)-A-PARCH(1,1) model fitted with the generalized error distribution is more suitable for capturing conditional volatility in mainland China’s stock markets. Second, based on bivariate VC-MGARCH(1,1) models, we find that the conditional correlation between mainland China’s and the U.S. stock markets is quite low and highly volatile. It may provide investors with opportunity of diversification. Third, we find that uncertainty derived from time-varying relationship between Shanghai and the U.S. stock markets is more significant than that between Shenzhen and the U.S. stock markets. In addition, mainland China’s stock markets are highly persistent, thereby reducing potential benefits induced by actively trading. 58 Bibliography Bai, J.S., and Perron, P. 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(2003), Modeling Financial Time Series with S-PLUS, New York: Springer. 63 [...]... high capital gains derived from a short period of the bull market and extreme risks associated with the speculative market indicate a “Casino” characteristic of China s stock markets The time-varying-parameter models with Markov-switching heteroskedasticity proposed by Kim (1993) is capable of capturing the changing relationship between returns of China s stock markets and those of the U.S In his paper,... fitted with the returns of S&P 500 index, and find that the power is 1.43, significantly different from 2 Brooks (2007) adopts the A-PARCH model to study the volatility of emerging equity markets and then to make comparison with the one of developed markets He finds that the power of emerging stock markets falls within a wider range than the one of developed markets and different emerging markets have... different degrees of volatility asymmetries In addition, we capture dynamics of conditional correlation between returns of China s stock markets and those of the U.S in a bivariate VC-MGARCH framework Despite large number of parameters involved, it sheds some light in how the two markets are correlated and whether they can bring diversification to investors Although direct generalizations from the univariate... are suffering from huge shocks Specifically, σ i20 is smaller than σ i21 In this Section, we derive correlation between China s and the U.S stock markets from the coefficient of rsp ,t −1 in the conditional mean equation That is different from what we do in Section 2.2 and can provide us with a more thorough understanding of the connection between the two markets In Model 1, the coefficient of rsp ,t... the appropriateness of Model 1 and Model 2 23 3 Data and Estimation Results 3.1 Data specification Our sample data is drawn from Yahoo Finance, consisting of daily returns of the Shanghai Stock Exchange Composite Index, the Shenzhen Stock Exchange Component Index and the S&P 500 Index The Shanghai Stock Exchange Composite Index launched on July 15, 1991 is a whole market index, including all listed A-shares... Markov-switching variance models and Time-varying-parameter models with Markov-switching heteroskedasticity In this section, Markov-switching heteroskedasticity is utilized to model conditional heteroskedasticity of rsht and rszt , rather than GARCH heteroskedasticity as mentioned in Sections 2.1 and 2.2 The oscillatory behavior of time-varying volatility can be categorized into two distinct regimes:... Bivariate VC-MGARCH(1,1) models In order to capture conditional correlation between returns of the Shanghai Stock Exchange Composite Index/the Shenzhen Stock Exchange Component Index and returns of the S&P 500 Index, we model time-varying conditional correlations in a bivariate GARCH(1,1) framework and follow the methodology proposed by Tse and Tsui (2002) Specifically, both of rsht / rszt and rspt are... A-shares are traded in RMB, while B-shares are traded in U.S dollars at the Shanghai Stock Exchange and in Hong Kong dollars at the Shenzhen Stock Exchange The index is compiled using Paasche weighted formula and it converts prices of B-shares denominated in U.S dollars into RMB1 Current Index= Current total market cap of constitutents *Base Value total market capitalization of all stocks traded on Dec... models, Markov-switching variance models and time-varying-parameter models with Markov-switching heteroskedasticity Data and estimation results are reported in Section 3 Section 4 concludes with implication of our findings on equity investment 7 2 Models 2.1 ARMA(1,1)-GARCH(1,1) Models In this section, we introduce several GARCH models to capture conditional heteroskedasticity of rsht and rszt and... p00 indicates the stock market is highly persistent in the relatively stable state, with small probability of shifting to the highly volatile state Contrary to that, high p11 denotes the stock market is always unstable, with small probability of transferring to the relatively stable state If p00 and p11 are quite low, it shows the stock market is frequent in regime shifting We introduce Markov-switching ... we find that the conditional correlation between mainland China s and the U.S stock markets is quite low and highly volatile Third, we apply a structure of Markov-switching in conditional heteroskedasticity. .. daily returns of Shanghai and Shenzhen markets Both figures reveal influence of these historical events At the beginning of 2000, returns of both Shanghai and Shenzhen stock indices were suffering... dynamics of conditional correlation between returns of China s stock markets and those of the U.S in a bivariate VC-MGARCH framework Despite large number of parameters involved, it sheds some light in