JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol. 20 No. 9 Sep Copula Copula. EIS-ML Copula. Copu

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1 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 20 No 9 Sep Copula Copula EIS-ML Copula Copula Copula Copula Copula Copula F830 9 A Black 1 GARCH Christie 2 SV Yu 9 SV leverage effect Asai McAleer 10 SV S&P500 TOPIX Ning 25 Copula S&P Lai Sheu Bandi Renò 26 S&P500 Veraart A Veraart L YJC M QG SQRW025ZD xywu aufe@ gmail com

2 9 Copula 71 Bretó 28 SV S&P500 Copula 29 GJR-GARCH-M Copula Patton 33 Copula 30 Hafner Reznikova 34 Copula Hafner Manner 35 Copula 31 Okimoto 36 Chollete 37 Garcia Tsafack 38 Copula Manner Reznik- 32 SV ova 39 Copula Hafner Manner 35 Manner Reznikova 39 Copula Copula Copula Copula 3 Copula Copula Copula 2 Copula efficient importance samplingbased maximum likelihood EIS-ML Copula Copula 1 Copula Copula Copula 2 3 Ning 25 Copula Ning 25 Copula Copula Ning 25 Copula Copula Copula Ning 25 Copula 4 Xu Li 40

3 x t y t x t-1 y t H x t-1 y t Sklar Copula C H x t -1 y t = C u t -1 v t θ 1 Copula 1 u t -1 = F x t -1 v t = G y t x t -1 y t θ Copula 2 Copula 0 1 Copula Copula Hafner Manner 3 5 Patton 42 x t -1 y t Copula Fermanian Scaillet x t -1 y t Copula λ L λ U λ L = lim Pr G y t u F x t -1 u 2 1 u 0 + C u u θ = lim 2 u 0 + u λ U = lim Pr G y t > u F x t -1 > u u u + C u u θ = lim 3 u u λ L λ U 0 1 λ L λ U 0 1 x t -1 y t λ L λ U = 0 x t-1 y t Copula 1 G^ y = 1 y Copula T + 1 T t < y 7 t = 1 Copula 1 Copula T Glivenko-Cantelli Copula F^ xg^ y Copula F xg y Copula Copula Hafner Manner 3 5 Copula θ 2 Joe 41 IFM inference function for margins F xg y rescaled empirical cumulative distribution function F xg y 1 F^ x = T-1 1 x T + 1 t < x 6 θ t = Ψ h t h t = α + βh t -1 + ν t 5 Ψ Copula 6 7 Copula h t AR β < 1 AR 1 5 t Copula Copula SV t = 0 ARMA p q - GARCH

4 9 Copula 73 k t h t h t -1 a t = SV p h t h t -1 Θ exp { - 1 Copula GMM 2 a 1 th 2 t - 2a 2 t h t } QML EMM MCMC EIS a t = a 1 t a 2 t ' ML EIS m t h t h t -1 a t μ at = σ 2 μ t σ a t ( + a Copula σ 2 2 ) 2 t σ 2 t a t = a t 1 t σ 2 t Copula Copula ML L U V Θ = EIS-ML T-1 c u t v t+1 h t p h t h t-1 Θ [ t =0 m t h t h t-1 a t ] Copula T-1 m t h t h t-1 a t dh 14 Copula Copula t =0 Θ = α β ν ' U = u 0 u T-1 ' L U V ΘEIS V = v 1 v T ' H = h 0 h T-1 ' Copula L U 槇 V Θ = 1 L U V Θ = p U V H Θ dh 8 S T-1 c u t v t+1 h s p h s h s t-1 Θ [ t t S s =1 t =0 m t h s t h s t-1 a t ] 15 p U V H ΘU V H h s t EIS m t h t h s t -1 a t p U V H Θ = T-1 c u t v t+1 h t p h t h t-1 Θ t =0 9 c u t v t +1 h t Copula p h t h t -1 Θ T-1 c u t v t+1 h t p h t h t-1 Θ t =0 m t h t h t-1 a t { α μ t = 1 - β t =0 { ν 2 = T-1 c u t v t+1 h t χ t h t-1 a t σ 2 t = 1-β 2 t =0 t =0 exp{ a 1 th 2 t - 2a 2 t h t } α+βh t-1 t 1 ν 2 t 1 = T-1 c u t v t+1 h t p h t h t-1 Θ 10 t =0 k t h t h t-1 a t /χ t h t-1 a t = χ 0 a 0 T-1 c u t v t+1 h t χ t+1 h t a t+1 8 t =0 exp{ a 1 th 2 t - 2a 2 t h t } EIS 16 8 Richard χ Zhang 4 5 T h T-1 a T 1 Richard EIS m t h t h t -1 a t = k Zhang 4 5 t h t h t -1 a t 11 χ t h t -1 a t EIS a t χ t h t -1 a t = k t h t h t -1 a t dh t 12 a^ t c^t =arg min a t c t { ln[ c u S t v t+1 h s t χ t+1 h s t a^ t+1 - ] s = 1 k t h t h t -1 a t a t c EIS t - ln [ exp ( a 1 t h s t 2-2a 2 t h s 2 t ) ] } EIS EIS a t EIS 11 17

5 h s t p h t h s t -1 Θ 17 ln c u t v t+1 h s t +ln χ t+1 h s t a^ t+1 = c t a 1 t h s t 2 +a 2 t h s t + u s t s = 1 S 18 u s t EIS 1 p h t h s t-1 Θ T-1 t =0 h s t T-1 t = t = T -1 0 EIS a^ t T-1 t = 0 3 EIS m h t h s t -1 a^ t T-1 t = 0 h s T-1 t = 0 t Windows a^ t T-1 t = EIS m h t h s t -1 a^ t T-1 t = 0 EIS-ML h s t T-1 t = 0 15 RMSE L 槇 EIS-ML Copula Copula EIS-ML Θ^ = arg max Θ ln L U 槇 V Θ 3 2 EIS-ML Copula α = β = 0 98 ν = 0 15 Copula Copula survival Copula Copula Copula survival Copula 1 45 T = EIS-ML 100 RMSE EIS-ML MATLAB Copula survival Copula 1 Table 1 Simulation results RMSE α β ν α survival β ν EIS S = RMSE 4 SZSE 5 min ~ SSE

6 9 Copula 75 η 2 1 = var r 2 t 0 η 2 2 = var RV t η 12 = cov r 2 t 0 RV t μ 0 μ 1 μ 2 r 2 t 0 + RV t r 2 t 0 RV t 5 min Hansen Lunde 49 RV * t 21 RV t = n r 2 t i 19 i = SSE SZSE n r t i = ln P t i - ln DR DR t = ln P t - ln P t -1 P t t P t i -1 t i ln Barndorff-Nielsen Shephard 4 8 RV * 2 2 RV integrated volatility IV < 0 > 3 RV t p t IV t = σ 2 s ds n 20 Jarque-Bera t -1 σ 2 t RV > 0 Jarque-Bera 10 Ljung- 4 9 Hansen Lunde Box ARCH RV * t = ω 1 r 2 t 0 + ω 2 RV t 21 r t 0 = ln P t-1 n - ln P t 0 ω 1 = ARMA p q - GARCH φ μ 0 ω μ 2 = φ μ 0 μ 2 2η μ 1 μ 2 η 12 φ = 1 μ 2 μ 2 2η μ 2 1η 2 2-2μ 1 μ 2 η 12 1 SSE DR ln RV * Fig 1 Time series of daily returns DR and log realized volatilities ln RV * of SSE index

76 2017 9 2 SZSE DR ln RV * Fig 2 Time series of daily returns DR and log realized volatilities ln RV * 2 Table 2 Descriptive statistics of SZSE index Jarque-Bera Ljung-Box 10 ARCH 10 SSE DR 0 000

7 SZSE DR ln RV * Fig 2 Time series of daily returns DR and log realized volatilities ln RV * 2 Table 2 Descriptive statistics of SZSE index Jarque-Bera Ljung-Box 10 ARCH 10 SSE DR ln RV * SZSE DR ln RV * DR ln RV * p 4 2 AIC AR 1 SSE SZSE BDS K-S ARMA 2 2ARMA 2 1 SSE BDS Kolmogorov-Smirnov K-S U V 3 3 BDS K-S SZSE 5% GARCH U V Copula 4

8 9 Copula Copula λ L = ω2-1 /θ 1 λ U = 1 - ω 2-1 /θ 2 25 Copula 4 2 Copula Copula C GM u v θ 1 θ 2 = ω C SG u v θ ω C Gum u v θ 2 22 C CM u v θ 1 θ 2 = ω C Clay u v θ ω C SC u v θ 2 23 C Gum C SG C Clay C SC Copula survival Copula Copula survival Copula ω Copula surviv- θ 2 al Copula Copula Copula Copula 3 Table 3 Tests of the marginal distributions λ L = ω /θ 1 λ U = 1 - ω /θ 2 24 Copula 4 4 Copula ω θ 1 SSE SZSE U V U V BDS K-S BDS K-S p 4 Copula Table 4 Mixture Copula results Copula θ 1 θ 2 ω Log-lik AIC BIC SSE * * SZSE * * Log-lik AIC BIC * 5% 4 4 Copula 4 3 Copula Copula

9 Copula survival Copula Copula survival Copula AIC BIC Copula 2 Copula AIC EIS-ML EIS BIC Copula 5 5 survival Copula Copula Copula Copula survival Copula Patton 33 Copula 6 Copula survival Copula ML K-S A-D 5 survival Copula Copula Copula λ U = /θ Copula 0 10 /0 11 SSE /SZSE survival Copula AR 1 β 1 λ U = 2-1 /θ 0 01 /0 02 SSE / Copula Copula Copula Copula Copula Copula S = 200 EIS 5 5 Copula 5 Copula K-S Anderson-Darling A-D Copula AIC BIC K-S A-D SZSE 1% ~ 11% Copula survival Copula slev 50 Ning 25 Boller- AIC Copula BIC Copula /23 65 Copula Copula Copula / survival Copula Copula /18 63 Copula AIC BIC / survival Copula 7 1% Copula Copula Copula Copula AIC BIC EIS common random numbers CRNs EIS 200 EIS 5 Copula Copula ARMA 1 10 θ t = Ψ h t h t = α + βh t-1 + v 1 10 u 10 t-1-j - v t-j LR = - 2 LL res - LL ur LL res Copula LL ur Copula Hafner Manner 35 LR 1% 9 99 j =1

10 9 Copula 79 5 Table 5 Estimation results Copula θ α β ν Log-lik AIC BIC K-S A-D survival survival survival survival survival survival SSE * * * * * * * * * * SZSE * * * * * * * * * * * Log-lik AIC BIC * 5% K-S A-D K-S A-D p Copula Copula survival Copula particle filter ~ ~ ~

11 Copula survival Copula Fig 3 Filtered time-varying leverage effects for SSE index Top stochastic Copula model Bottom stochastic survival Copula model 4 Copula survival Copula Fig 4 Filtered time-varying leverage effects for SZSE index Top stochastic Copula model Bottom stochastic survival Copula model 5 Copula Copula survival Copula Copula

12 9 Copula 81 EIS-ML Copula Copula Copula Copula survival Copu- la Copula 1 Black F Studies of stock price volatility changes C / / Proceedings of the Business and Economic Statistical Section American Statistical Association Christie A A The stochastic behavior of common stock variances Value leverage and interest rate effects J Journal of Financial Economics Engle R Ng V Measuring and testing the impact of news in volatility J Journal of Finance Glosten L Jagannathan R Runkle D On the relation between the expected value and volatility of nominal excess returns on stocks J Journal of Finance Harvey A C Shephard N Estimation of an asymmetric stochastic volatility model for asset returns J Journal of Business and Economic Statistics Jacquier E Polson N Rossi P Bayesian analysis of stochastic volatility models with fat tails and correlated errors J Journal of Econometrics Li Q Yang J Hsiao C et al The relationship between stock returns and volatility in international stock markets J Journal of Empirical Finance Yu J On leverage in a stochastic volatility model J Journal of Econometrics Yu J A semiparametric stochastic volatility model J Journal of Econometrics Asai M McAleer M Dynamic asymmetric leverage in stochastic volatility models J Econometric Reviews Asai M McAleer M Alternative asymmetric stochastic volatility models J Econometric Reviews Lai Y S Sheu H J On the importance of asymmetries for dynamic hedging during the subprime crisis J Applied Financial Economics Levy J Wu Hengyu Zhu Fumin Wen Jinming Option pricing based on conditional infinite pure jump Levy processes with leverage effect J Journal of Management Sciences in China in Chinese 14 Yeh Y H Lee T S The interaction and volatility asymmetry of unexpected returns in the greater China stock markets J Global Finance Journal / J Lu Rong Xu Longbing Asymmetric effects of policy information on China s stock markets J China Economic Quarterly in Chinese J Lu Rong Xu Longbing The asymmetry information effect on bull and bear stock markets J Economic Research Journal in Chinese J He Xingqiang Li Tao The asymmetric reactions of stock market under different phases An empirical study based on Shanghai Stock Exchange J Journal of Financial Research in Chinese

13 MS-TGARCH MCMC J Zhu Junjun Xie Shiyu Double asymmetry of volatility in the Chinese stock market and its explanations Estimation and a- nalysis of MS-TGARCH model based on MCMC method J Journal of Financial Research in Chinese 19 J Wang Chunfeng Jiang Xianglin Li Gang Estimating volatility of Chinese stock market by stochastic volatility model J Journal of Management Sciences in China in Chinese 20 SV J Meng Lifeng Zhang Shiyin He Xin Bayesian analysis of stochastic volatility model with leverage effect and its application J Systems Engineering in Chinese 21 SV J Meng Lifeng Zhang Shiying Nonlinear stochastic volatility model with leverage effect and its application J Journal of Systems & Management in Chinese J Huang Bo Gu Mengdi Li Zhan Stochastic volatility model with skew normal distribution and its empirical test J Journal of Management Sciences in China in Chinese 23 J Liu Qingfu Zhou Chengyuan Study on asymmetric effect of Chinese stock markets J Journal of Systems Engineering in Chinese 24 EIS J Wu Xinyu Zhou Hailin Wang Shouyang et al EIS-based maximum likelihood estimation of stochastic volatility model with leverage effect J Journal of Management Sciences in China in Chinese 25 Ning C Xu D H Wirjanto T Modelling leverage effect with Copulas and realized volatility J Finance Research Letters Bandi F M Renó R Time-varying leverage effects J Journal of Econometrics Veraart A Veraart L Stochastic volatility and stochastic leverage J Annals of Finance Bretó C On idiosyncratic stochasticity of financial leverage effects J Statistics and Probability Letters J Chen Langnan Huang Jiekun Empirical study of the asymmetric effects of volatility on Chinese stock market J Journal of Financial Research in Chinese 30 SV J Wu Qiquan Wang Chunfeng Fang Zhenming et al A research on time-varying parameters of SV model and the policy effects of Chinese stock market J Journal of Beijing Institute of Technology Social Science Edition in Chinese J Wang Liang Liu Jinquan Study on the asymmetric reactions of Chinese stock market J Industrial Technology & Economy in Chinese 32 J Wu Xinyu Zhou Hailin Wang Shouyang et al A threshold stochastic volatility model with double leverage and its empirical research J Journal of Management Sciences in China in Chinese 33 Patton A Modelling asymmetric exchange rate dependence J International Economic Review Hafner C M Reznikova O Efficient estimation of a semiparametric dynamic Copula model J Computational Statistics &

14 9 Copula 83 Data Analysis Hafner C M Manner H Dynamic stochastic Copula models Estimation inference and applications J Journal of Applied Econometrics Okimoto T New evidence of asymmetric dependence structures in international equity markets J Journal of Financial and Quantitative Analysis Chollete L Heinen A Valdesogo A Modeling international financial returns with a multivariate regime switching Copula J Journal of Financial Econometrics Garcia R Tsafack G Dependence structure and extreme comovements in international equity and bond markets with portfolio diversification effects J Journal of Banking and Finance Manner H Reznikova O A survey on time-varying Copulas Specification simulations and application J Econometric Reviews Xu D H Li Y Y Empirical Evidence of Leverage Effect in a Stochastic Volatility Model A Realized Volatility Approach R University of Waterloo Joe H Multivariate Models and Dependence Concepts M London Chapman & Hall Patton A J Copula-Based Models for Financial Time Series M / / Andersen T G et al Ed Handbook of Financial Time Series Berlin Heidelberg Springer-Verlag Fermanian J D Scaillet O Some statistical pitfalls in Copula modeling for financial applications C / / Klein E Ed Capital Formation Governance and Banking New York Nova Science Publishing Copula J Zhang Lianzeng Hu Xiang Comparison of parametric and semiparametric estimation methods for Copula J Statistical Research in Chinese 45 Richard J F Zhang W Efficient high-dimensional importance sampling J Journal of Econometrics Andersen T G Bollerslev T Answering the skeptics Yes standard volatility models do provide accurate forecasts J International Economic Review Andersen T G Bollerslev T Diebold F X et al The distribution of realized stock return volatility J Journal of Financial Economics Barndorff-Nielsen O E Shephard N Non-Gaussian Ornstein-Ulhlenbeck-based models and some of their uses in financial e- conomics J Journal of the Royal Statistical Society Series B Hansen P R Lunde A A realized variance for the whole day based on intermittent high-frequency data J Journal of Financial Econometrics Bollerslev T Litvinova J Tauchen G Leverage and volatility feedback effects in high-frequency data J Journal of Financial Econometrics Gordon N J Salmond D J Smith A F Novel approach to nonlinear /Non-Gaussian Bayesian state estimation J IEE Proceedings-F Time-varying leverage effects in Chinese stock markets Empirical analysis based on stochastic Copula models WU Xin-yu 1 REN Sen-chun 1 MA Chao-qun 2 WANG Shou-yang 3 1 School of Finance Anhui University of Finance and Economics Bengbu China 2 Business School Hunan University Changsha China 3 Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Abstract This paper constructs stochastic Copula models to study the time-varying leverage effects in Chinese

15 stock markets at the extremes It is well-known that volatility can not be directly observed in the financial markets To overcome this problem realized volatility is used to measure the latent volatility Then the efficient importance sampling-based maximum likelihood EIS-ML estimation is adopted to estimate the parameters of the stochastic Copula models The empirical results from data of Shanghai and Shenzhen stock markets demonstrate that the leverage effects in Chinese stock markets exhibit asymmetric features Specifically extremely low stock market returns tend to be associated with extremely large volatilities but extremely high stock market returns are not related to small volatilities Moreover the leverage effects in Chinese stock markets are found to be changing over time and exhibit similar variation trends in Shanghai and Shenzhen stock markets The stochastic Copula models are shown to outperform other Copula models including the static Copula models and time-varying Copula models Key words time-varying leverage effects tail dependence stochastic Copula models realized volatility maximum likelihood estimation A Copula survival Copula Copula C Gum u v θ = exp - - ln u θ + - ln v θ 1 /θ c Gum u v θ = - ln u θ -1 - ln v θ -1 exp - - ln u θ + - ln v θ 1 /θ uv - ln u θ + - ln v θ 1 /θ-2 - ln u θ + - ln v θ 1 /θ + θ - 1 θ 1 Copula 4 Ψ x = exp x + 1 Copula λ U = /θ λ L = 0 survival Copula C SG u v θ = C Gum 1 - u 1 - v θ + u + v - 1 c SG u v θ = c Gum 1 - u 1 - v θ survival Copula λ L = /θ λ U = 0 B Copula survival Copula Copula C Clay u v θ = u -θ + v -θ -1 /θ - 1 c Clay u v θ = u -1 -θ v -1 -θ u -θ + v -θ /θ 1 + θ θ 0 Copula 4 Ψ x = exp x Copula λ L = 2-1 /θ λ U = 0 survival Copula C SC u v θ = C Clay 1 - u 1 - v θ + u + v - 1 c SC u v θ = c Clay 1 - u 1 - v θ survival Copula λ U = 2-1 /θ λ L = 0

经济与管理耿庆峰 : 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角 87 Copula 模型均能较好地刻画金融市场间的动态关系, 但 Copula 模型效果要好于

经济与管理耿庆峰 : 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角 87 Copula 模型均能较好地刻画金融市场间的动态关系, 但 Copula 模型效果要好于第 19 卷第 6 期中南大学学报 ( 社会科学版 ) Vol.19 No.6 013 年 1 月 J. CENT. SOUTH UNIV. (SOCIAL SCIENCE) Dec. 013 我国创业板市场与中小板市场动态相关性实证研究基于方法比较视角耿庆峰 ( 闽江学院公共经济学与金融学