基于TVS-MHAR模型金融市场高频多元波动率的预测
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摘要
本文基于Kalli和Griffin(2011)的时变稀疏模型和多元HAR模型,构建了具有时变稀疏性的多元HAR模型(TVS-MHAR),并利用中国上证综指、沪深300期货和国债期货的五分钟高频数据,对金融市场的已实现波动率矩阵进行预测.本文通过Cholesky转换方法保证预测波动率矩阵的正定性.通过对不同多元波动率模型的预测结果进行数值比较和经济比较,本文发现,本文构建的TVS-MHAR模型无论对于短期预测、中期预测还是长期预测都具有最高的预测精度和最大的投资改善.同时,时变多元波动率模型可以获得比固定参数模型更好的预测效果,高频数据模型比低频数据模型获得更大的投资改善.
We develop a multivariate HAR model with time-varying sparsity,or TVS-MHAR by combining the multivariate HAR model with the time-varying sparsity structure of Kalli and Griffin(2011).We forecast the covariance matrix of China financial markets by utilizing the high frequency data from Shanghai Stock Exchange(SHSE) Composite index,China stock index 300(CSI300) futures and China Treasury futures.We employ the Cholesky decomposition approach to guarantee the positive definiteness of the forecast volatility matrices.And then,we compare the forecast performance of the proposed TVS-MHAR model with other multivariate volatility forecast models in literatures based on both the statistical loss functions and the economic evaluation criterions.The results suggest the TVS-MHAR model performs the best for the out-of-sample forecasts and has the greatest economic gains among all the forecast models.In addition,the time-varying multivariate volatility forecast model performs better as compared with the fixed-parameter models and the models based on high frequency data have more economic gains as compared with the models based on the low-frequency data.
We develop a multivariate HAR model with time-varying sparsity,or TVS-MHAR by combining the multivariate HAR model with the time-varying sparsity structure of Kalli and Griffin(2011).We forecast the covariance matrix of China financial markets by utilizing the high frequency data from Shanghai Stock Exchange(SHSE) Composite index,China stock index 300(CSI300) futures and China Treasury futures.We employ the Cholesky decomposition approach to guarantee the positive definiteness of the forecast volatility matrices.And then,we compare the forecast performance of the proposed TVS-MHAR model with other multivariate volatility forecast models in literatures based on both the statistical loss functions and the economic evaluation criterions.The results suggest the TVS-MHAR model performs the best for the out-of-sample forecasts and has the greatest economic gains among all the forecast models.In addition,the time-varying multivariate volatility forecast model performs better as compared with the fixed-parameter models and the models based on high frequency data have more economic gains as compared with the models based on the low-frequency data.
引文
[1]Bollerslev T.Modelling the coherence in short-run nominal exchange rates:A multivariate generalized ARCH model[J].Review of Economics&Statistics,1990,72(3):498-505.
[2]Bauwens L,Rombouts J V K.Multivariate GARCH models:A survey[J].Journal of Applied Econometrics,2006,21(1):79-109.
[3]Asai M,McAleer M,Yu J.Multivariate stochastic volatility:A review[J].Econometric Reviews,2006,25(2-3):145-175.
[4]Andersen T G,Bollerslev T,Diebold F X,et al.Modeling and forecasting realized volatility[J].Econometrica,2003,71(2):579-625.
[5]Andersen T G,Terasvirta T.Realized volatility[M]//Handbook of Financial Time Series,2009,71(2):555-575.
[6]Barndorff-Nielsen O E,Hansen P R,Lunde A,et al.Multivariate realised kernels:Consistent positive semidefinite estimators of the covariation of equity prices with noise and non-synchronous trading[J].Journal of Econometrics,2011,162(2):149-169.
[7]Corsi F.A simple approximate long-memory model of realized volatility[J].Journal of Financial Econometrics,2009,7(2):174-196.
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[9]Chiriac R,Voev V.Modelling and forecasting multivariate realized volatility[J].Journal of Applied Econometrics,2011,26(6):922-947.
[10]Gribisch B.A latent dynamic factor approach to forecasting multivariate stock market volatility[J].Empirical Economics,2013:1-31.
[11]Cech F,Barunik J.On the modelling and forecasting of multivariate realized volatility:Generalized heterogeneous autoregressive(GHAR)model[J].Journal of Forecasting,2016,36(2):181-206.
[12]陈浪南,杨科.中国股市高频波动率的特征、预测模型以及预测精度比较[J].系统工程理论与实践,2013,33(2):296-307.Chen L N,Yang K.High-frequency volatility features,forecast model and performance evaluation[J].Systems Engineering—Theory&Practice,2013,33(2):296-307.
[13]杨继平,冯毅俊,王辉.基于结构转换PTTGARCH模型沪深股市波动率的估计[J].系统工程理论与实践,2016,36(9):2205-2215.Yang J P,Feng Y J,Wang H.Estimation of volatility of CSI 300 index based on regime switching PTTGARCH model[J].Systems Engineering—Theory&Practice,2016,36(9):2205-2215.
[14]施雅丰,艾春荣.中国股市波动率的广义周内特征及其预测模型[J].系统工程理论与实践,2016,36(8):1918-1927.Shi Y F,Ai C R.A volatility model for Chinese stock market with generalized day-of-the-week effect[J].Systems Engineering—Theory&Practice,2016,36(8):1918-1927.
[15]赵树然,姜亚萍,任培民.高频波动率矩阵估计的比较分析——基于有噪非同步的金融数据[J].中国管理科学,2015,23(10):19-29.Zhao S R,Jiang Y P,Ren P M.Comparing estimators of the high-frequency volatility matrix in the presence of non-synchronous trading and market microstructure noise[J].Chinese Journal of Management and Science,2015,23(10):19-29.
[16]刘丽萍.多维金融高频协方差阵预测模型的比较分析[J].数学的实践与认识,2014(12):203-214.Liu L P.The comparative analysis of multidimensional financial high frequency covariance matrix's prediction model[J].Mathematics in Practice and Theory,2014(12):203-214.
[17]赵树然,张燕燕,任培民.异质金融市场高频期货交叉套期保值问题研究[J].系统工程理论与实践,2016,36(9):2189-2204.Zhao S R,Zhang Y Y,Ren P M.Study on high-frequency futures cross-hedging problem driven by heterogeneous financial market[J].Systems Engineering—Theory&Practice,2016,36(9):2189-2204.
[18]Kalli M,Griffin J E.Time-varying sparsity in dynamic regression models[J].Journal of Econometrics,2011,178(2):779-793.
[19]Engle R.Dynamic conditional correlation:A simple class of multivariate generalized autoregressive conditional heteroskedasticity models[J].Journal of Business&Economic Statistics,2002,20(3):339-350.
[20]Longertaey J,Spencer M.RiskMetricsTM—Technical document[J].New York:Morgan Guaranty Trust Company of New York,1996.
[21]Markowitz H.Portfolio selection[J].Journal of Finance,1952,7(1):77-91.
[22]Barndorff-Nielsen O E,Shephard N.Estimating quadratic variation using realized variance[J].Journal of Applied econometrics,2002,17(5):457-477.
[23]Bessembinder H,Seguin P J.Price volatility,trading volume,and market depth:Evidence from futures markets[J].The Journal of Financial and Quantitative Analysis,1993,28(1):21-39.
[24]Lucia J J,Pardo A.On measuring speculative and hedging activities in futures markets from volume and open interest data[J].Applied Economics,2010,42(12):1549-1557.
[25]Pitt M K,Chatfield C,Walker S G.Constructing first order stationary autoregressive models via latent processes[J].Scandinavian Journal of Statistics,2002,29(4):657-663.
[26]Pitt M K,Walker S G.Constructing stationary time series models using auxiliary variables with applications[J].Journal of the American Statistical Association,2005,100(470):554-564.
[27]Gourieroux C,Jasiak J.Autoregressive gamma processes[J].Journal of Forecasting,2006,25(2):129-152.
[28]Friihwirth-Schnatter S.Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering[J].Statistics and Computing,1994,4(4):259-269.
[29]Carter C K,Kohn R.On Gibbs sampling for state space models[J].Biometrika,1994,81(3):541-553.
[30]Griffin J E,Brown P J.Inference with normal-gamma prior distributions in regression problems[J].Bayesian Analysis,2010,5(1):171-188.
[31]Hansen P R,Lunde A,Nason J M.The model confidence set[J].Econometrica,2011,79(2):453-497.
[32]Memmel C,Kempf A.Estimating the global minimum variance portfolio[J].Social Science Electronic Publishing,2006,58(4):332-348.
1.Parzen核的构建公式为:k(x)={1-6x~2+6x~3 0<=x<=1/2 {2(1-x)~3 1/2<=x<=1 {0 x>1
2.m×m矩阵A的Frobenius模定义为||A||_F~2=∑_(i,j)|a_(i,j)|~2.
[2]Bauwens L,Rombouts J V K.Multivariate GARCH models:A survey[J].Journal of Applied Econometrics,2006,21(1):79-109.
[3]Asai M,McAleer M,Yu J.Multivariate stochastic volatility:A review[J].Econometric Reviews,2006,25(2-3):145-175.
[4]Andersen T G,Bollerslev T,Diebold F X,et al.Modeling and forecasting realized volatility[J].Econometrica,2003,71(2):579-625.
[5]Andersen T G,Terasvirta T.Realized volatility[M]//Handbook of Financial Time Series,2009,71(2):555-575.
[6]Barndorff-Nielsen O E,Hansen P R,Lunde A,et al.Multivariate realised kernels:Consistent positive semidefinite estimators of the covariation of equity prices with noise and non-synchronous trading[J].Journal of Econometrics,2011,162(2):149-169.
[7]Corsi F.A simple approximate long-memory model of realized volatility[J].Journal of Financial Econometrics,2009,7(2):174-196.
[8]Bauer G H,Vorkink K.Forecasting multivariate realized stock market volatility[J].Journal of Econometrics,2011,160(1):93-101.
[9]Chiriac R,Voev V.Modelling and forecasting multivariate realized volatility[J].Journal of Applied Econometrics,2011,26(6):922-947.
[10]Gribisch B.A latent dynamic factor approach to forecasting multivariate stock market volatility[J].Empirical Economics,2013:1-31.
[11]Cech F,Barunik J.On the modelling and forecasting of multivariate realized volatility:Generalized heterogeneous autoregressive(GHAR)model[J].Journal of Forecasting,2016,36(2):181-206.
[12]陈浪南,杨科.中国股市高频波动率的特征、预测模型以及预测精度比较[J].系统工程理论与实践,2013,33(2):296-307.Chen L N,Yang K.High-frequency volatility features,forecast model and performance evaluation[J].Systems Engineering—Theory&Practice,2013,33(2):296-307.
[13]杨继平,冯毅俊,王辉.基于结构转换PTTGARCH模型沪深股市波动率的估计[J].系统工程理论与实践,2016,36(9):2205-2215.Yang J P,Feng Y J,Wang H.Estimation of volatility of CSI 300 index based on regime switching PTTGARCH model[J].Systems Engineering—Theory&Practice,2016,36(9):2205-2215.
[14]施雅丰,艾春荣.中国股市波动率的广义周内特征及其预测模型[J].系统工程理论与实践,2016,36(8):1918-1927.Shi Y F,Ai C R.A volatility model for Chinese stock market with generalized day-of-the-week effect[J].Systems Engineering—Theory&Practice,2016,36(8):1918-1927.
[15]赵树然,姜亚萍,任培民.高频波动率矩阵估计的比较分析——基于有噪非同步的金融数据[J].中国管理科学,2015,23(10):19-29.Zhao S R,Jiang Y P,Ren P M.Comparing estimators of the high-frequency volatility matrix in the presence of non-synchronous trading and market microstructure noise[J].Chinese Journal of Management and Science,2015,23(10):19-29.
[16]刘丽萍.多维金融高频协方差阵预测模型的比较分析[J].数学的实践与认识,2014(12):203-214.Liu L P.The comparative analysis of multidimensional financial high frequency covariance matrix's prediction model[J].Mathematics in Practice and Theory,2014(12):203-214.
[17]赵树然,张燕燕,任培民.异质金融市场高频期货交叉套期保值问题研究[J].系统工程理论与实践,2016,36(9):2189-2204.Zhao S R,Zhang Y Y,Ren P M.Study on high-frequency futures cross-hedging problem driven by heterogeneous financial market[J].Systems Engineering—Theory&Practice,2016,36(9):2189-2204.
[18]Kalli M,Griffin J E.Time-varying sparsity in dynamic regression models[J].Journal of Econometrics,2011,178(2):779-793.
[19]Engle R.Dynamic conditional correlation:A simple class of multivariate generalized autoregressive conditional heteroskedasticity models[J].Journal of Business&Economic Statistics,2002,20(3):339-350.
[20]Longertaey J,Spencer M.RiskMetricsTM—Technical document[J].New York:Morgan Guaranty Trust Company of New York,1996.
[21]Markowitz H.Portfolio selection[J].Journal of Finance,1952,7(1):77-91.
[22]Barndorff-Nielsen O E,Shephard N.Estimating quadratic variation using realized variance[J].Journal of Applied econometrics,2002,17(5):457-477.
[23]Bessembinder H,Seguin P J.Price volatility,trading volume,and market depth:Evidence from futures markets[J].The Journal of Financial and Quantitative Analysis,1993,28(1):21-39.
[24]Lucia J J,Pardo A.On measuring speculative and hedging activities in futures markets from volume and open interest data[J].Applied Economics,2010,42(12):1549-1557.
[25]Pitt M K,Chatfield C,Walker S G.Constructing first order stationary autoregressive models via latent processes[J].Scandinavian Journal of Statistics,2002,29(4):657-663.
[26]Pitt M K,Walker S G.Constructing stationary time series models using auxiliary variables with applications[J].Journal of the American Statistical Association,2005,100(470):554-564.
[27]Gourieroux C,Jasiak J.Autoregressive gamma processes[J].Journal of Forecasting,2006,25(2):129-152.
[28]Friihwirth-Schnatter S.Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering[J].Statistics and Computing,1994,4(4):259-269.
[29]Carter C K,Kohn R.On Gibbs sampling for state space models[J].Biometrika,1994,81(3):541-553.
[30]Griffin J E,Brown P J.Inference with normal-gamma prior distributions in regression problems[J].Bayesian Analysis,2010,5(1):171-188.
[31]Hansen P R,Lunde A,Nason J M.The model confidence set[J].Econometrica,2011,79(2):453-497.
[32]Memmel C,Kempf A.Estimating the global minimum variance portfolio[J].Social Science Electronic Publishing,2006,58(4):332-348.
1.Parzen核的构建公式为:k(x)={1-6x~2+6x~3 0<=x<=1/2 {2(1-x)~3 1/2<=x<=1 {0 x>1
2.m×m矩阵A的Frobenius模定义为||A||_F~2=∑_(i,j)|a_(i,j)|~2.