基于门限预平均已调整多次幂变差的可积波动估计及其应用
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摘要
本文在市场微观结构噪声和跳跃下新提出一类可积波动估计.这些估计联合采用了预平均已调整多次幂变差估计和门限技术,分别消除噪声和跳跃的影响.我们同时给出这一估计的渐近性质,包括一致性和中心极限定理.蒙特卡罗模拟结果表明这一估计对噪声和Lévy跳跃稳健,并且相比预平均已调整多次幂变差(PMMV)估计(Vetter, 2008)具有更好的表现.在实证应用中,基于中国股市逐笔交易的高频数据估计出2015年股灾前后上证50成分股的连续波动,跳跃波动以及噪声波动,并且研究了它们之间相互关系.实证发现:1)噪声波动对潜在收益波动具有正向预测作用,但该预测作用主要针对连续波动,对跳跃波动预测作用并不显著;2)噪声波动具有很强的相依性,连续波动对其具有显著正向预测作用,而跳跃波动对其不存在显著的预测能力.本文研究结果表明噪声包含了有助于潜在价格波动预测的信息,不完全是"噪声",其包含的信息有待深入挖掘.
This paper develops a new class of estimators for integrated volatility in the presence of market microstructure noise and jumps. These estimators are based on the simultaneous use of preaveraged modulated multi-power variation estimation and the threshold technique, which serve to remove microstructure noise and jumps respectively. We also prove the asymptotic properties of the proposed estimators, including consistency and the associated central limit theorems. Monte Carlo simulations show that these estimators are robust to both microstructure noise and Lévy jumps and provide better performances than pre-averaged modulated multi-power variation(PMMV) estimation(Vetter, 2008). In the empirical applications, using the tick by tick high-frequency data from the Chinese stock market, we estimate continuous volatility, jump volatility and noise variance for SSE 50 component stocks during the stock market crash in 2015 based on our new estimators, and study the relationships among them. The main findings are summarized as follows: 1) noise volatility has significantly positive predictability for total volatility of underlying returns, however, such predictability holds mainly for continuous volatility rather than jump volatility; 2) continuous volatility, rather than jump volatility, has significantly positive predictability for noise volatility that exhibits strong dependence. The results imply that noise is not pure"noise",instead, it contains some useful information that helps predict volatility of the underlying asset price and such information merits further research.
This paper develops a new class of estimators for integrated volatility in the presence of market microstructure noise and jumps. These estimators are based on the simultaneous use of preaveraged modulated multi-power variation estimation and the threshold technique, which serve to remove microstructure noise and jumps respectively. We also prove the asymptotic properties of the proposed estimators, including consistency and the associated central limit theorems. Monte Carlo simulations show that these estimators are robust to both microstructure noise and Lévy jumps and provide better performances than pre-averaged modulated multi-power variation(PMMV) estimation(Vetter, 2008). In the empirical applications, using the tick by tick high-frequency data from the Chinese stock market, we estimate continuous volatility, jump volatility and noise variance for SSE 50 component stocks during the stock market crash in 2015 based on our new estimators, and study the relationships among them. The main findings are summarized as follows: 1) noise volatility has significantly positive predictability for total volatility of underlying returns, however, such predictability holds mainly for continuous volatility rather than jump volatility; 2) continuous volatility, rather than jump volatility, has significantly positive predictability for noise volatility that exhibits strong dependence. The results imply that noise is not pure"noise",instead, it contains some useful information that helps predict volatility of the underlying asset price and such information merits further research.
引文
[1]Vetter M. Estimation methods in noisy diffusion models[D]. Bochum:Ruhr-Universitat, 2008.
[2]陈海强,张传海.股指期货交易会降低股市跳跃风险吗?[J].经济研究,2015, 42(1):153-167.Chen H Q, Zhang C H. Does index futures trading reduce stock market jump risk?[J]. Economic Research Journal,2015, 42(1):153-167.
[3]唐文进,苏帆.极端金融事件对系统性风险的影响分析——以中国银行部门为例[J].经济研究,2017(4):17-33.Tang W J, Su F. An analysis of the effects of extreme financial events on systemic risk:Evidence from China's banking sector[J]. Economic Research Journal, 2017(4):17-33.
[4]Barndorff-Nielsen O E. Econometric analysis of realized volatility and its use in estimating stochastic volatility models[J]. Journal of the Royal Statistical Society:Series B(Statistical Methodology), 2002, 64(2):253-280.
[5]Andersen T G, Bollerslev T, Diebold F X, et al. Modeling and forecasting realized volatility[J]. Econometrica,2003, 71(2):579-625.
[6]Lee S, Hannig J. Detecting jumps from Levy jump diffusion processes[J]. Journal of Financial Economics, 2010,96:271-290.
[7]Barndorff-Nielsen O E, Shephard N. Power and bipower variation with stochastic volatility and jumps[J]. Journal of Financial Econometrics, 2004, 2(1):1-37.
[8]Mancini C. Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps[J].Scandinavian Journal of Statistics, 2009, 36(2):270-296.
[9]Andersen T G, Dobrev D, Schaumburg E. Jump-robust volatility estimation using nearest neighbor truncation[J].Journal of Econometrics, 2012, 169(1):75-93.
[10]Christensen K, Podolskij M. Realized range-based estimation of integrated variance[J]. Journal of Econometrics,2008, 141(2):323-349.
[11]唐勇,刘微.加权已实现极差四次幂变差分析及其应用[J].系统工程理论与实践,2013, 33(11):2766-2775.Tang Y, Liu W. Analysis of weighted realized range-based quadpower variation and its application[J]. Systems Engineering—Theory&Practice, 2013, 33(11):2766-2775.
[12]Corsi F, Pirino D, Reno R. Threshold bipower variation and the impact of jumps on volatility forecasting[J].Journal of Econometrics, 2010, 159(2):276-288.
[13]Vetter M. Limit theorems for bipower variation of semimartingales[J]. Stochastic Processes and their Applications,2010, 120(1):22-38.
[14]Hansen P R, Lunde A. Realized variance and market microstructure noise[J]. Journal of Business&Economic Statistics, 2006, 24(2):127-161.
[15]Bandi F M, Russell J R. Microstructure noise, realized variance, and optimal sampling[J]. The Review of Economic Studies, 2008, 75(2):339-369.
[16]Zhang L, Mykland P A, A(i|¨)t-Sahalia Y. A tale of two time scales:Determining integrated volatility with noisy high-frequency data[J]. Journal of the American Statistical Association, 2005, 100(472):1394-1411.
[17]Zhang L. Efficient estimation of stochastic volatility using noisy observations:A multi-scale approach[J]. Bernoulli,2006, 12(6):1019-1043.
[18]Barndorff-Nielsen O E, Hansen P R, Lunde A, et al. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise[J]. Econometrica, 2008, 76(6):1481-1536.
[19]Jacod J, Li Y, Mykland P A, et al. Microstructure noise in the continuous case:The pre-averaging approach[J].Stochastic Processes and their Applications, 2009, 119(7):2249-2276.
[20]Xiu D. Quasi-maximum likelihood estimation of volatility with high frequency data[J]. Journal of Econometrics,2010, 159:235-250.
[21]Bibinger M, Hautsch N, Malec P, et al. Estimation the quadeatic covariation matrix from noisy observations:Local method of moments and efficiency[J]. The Annals of Statistics, 2014, 42(4):1312-1346.
[22]Fan J, Wang Y. Multi-scale jump and volatility analysis for high-frequency financial data[J]. Journal of the American Statistical Association, 2007, 102(480):1349-1362.
[23]Podolskij M, Vetter M. Bipower-type estimation in a noisy diffusion setting[J]. Stochastic Processes and their Applications, 2009, 119(9):2803-2831.
[24]Hautsch N, Podolskij M. Preaveraging-based estimation of quadratic variation in the presence of noise and jumps:Theory, implementation, and empirical evidence[J]. Journal of Business&Economic Statistics, 2013,31(2):165-183.
[25]Jing B Y, Liu Z, Kong X B. On the estimation of integrated volatility with jumps and microstructure noise[J].Journal of Business&Economic Statistics, 2014, 32(3):457-467.
[26]马丹,尹优平.噪声、跳跃与高频价格波动——基于门限预平均实现波动分析[J],金融研究,2012, 382(4):124-139.Ma D, Yin Y P. Noise, jumps and high-frequency price volatility—Based on threshold pre-averaging realized variance analysis[J]. Journal of Financial Research, 2012, 382(4):124-139.
[27]Podolskij M, Vetter M. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[J]. Bernoulli, 2009, 15(3):634-658.
[28]Wang K, Liu J, Liu Z. Disentangling the effect of jumps on systematic risk using a new estimator of integrated co-volatility[J]. Journal of Banking&Finance, 2013, 37(5):1777-1786.
[29]Jacod J, Li Y, Zheng X. Statistical properties of microstructure noise[J]. Econometrica, 2017, 85(4):1133-1174.
[30]Christensen K, Oomen R C, Podolskij M. Fact or friction:Jumps at ultra-high frequency[J]. Journal of Financial Economics, 2014, 114(3):576-599.
[31]A(i|¨)t-Sahalia Y, Jacod J. Analyzing the spectrum of asset returns:Jump and volatility components in high frequency data[J]. The Journal of Economic Literature, 2012, 50:1007-1050.
[32]Andersen T G, Benzoni L, Lund J. An empirical investigation of continuous-time equity return models[J]. The Journal of Finance, 2002, 57(3):1239-1284.
[33]Jacod J, Mykland P. Microstructure noise in the continuous case:Approximate efficiency of the adaptive preaveraging method[J]. Stochastic Processes and their Applications, 2015, 125(8):2910-2936.
[34]Barndorff-Nielsen O E, Hansen P R, Lunde A, et al. Realized kernels in practice:Trades and quotes[J]. The Econometrics Journal, 2009, 12(3):C1-C32.
[35]A(i|¨)t-Sahalia Y, Yu J. High frequency market microstructure noise estimates and liquidity measures[J]. Annals of Applied Statistics, 2009, 3(1):422-457.
[36]陈国进,王占海.我国股票市场连续性波动与跳跃性波动实证研究[J].系统工程理论与实践,2010, 30(9):1554-1562.Chen G J, Wang Z H. Continuous volatility and jump volatility in China's stock market[J]. Systems Engineering—Theory&Practice,2010, 30(9):1554-1562.
[37]赵华,秦可佶.股价跳跃与宏观信息发布[J].统计研究,2014, 31(4):79-88.Zhao H, Qin K J. Jumps in stock Prices and macro information release[J]. Statistical Research, 2014, 31(4):79-88.
[38]Dovonon P, Goncalves S, Hounyo U, et al. Bootstrapping high-frequency jump tests[J]. Journal of the American Statistical Association, 2018. https://doi.org/10.1080/01621459.2018.1447485.
[39]Barndorff-Nielsen O E, Shephard N. Econometrics of testing for jumps in financial economics using bipower variation[J]. Journal of financial Econometrics, 2006, 4(1):1-30.
[40]Bajgrowicz P, Scaillet O, Treccani A. Jumps in high-frequency data:Spurious detections, dynamics, and news[J].Management Science, 2016, 62(8):2198-2217.
[41]Corsi F, RenòR. Discrete-time volatility forecasting with persistent leverage effect and the link with continuoustime volatility modeling[J]. Journal of Business&Economic Statistics, 2012, 30(3):368-380.
[42]Patton A J, Sheppard K. Good volatility, bad volatility:Signed jumps and the persistence of volatility[J]. Review of Economics and Statistics, 2015, 97(3):683-697.
[2]陈海强,张传海.股指期货交易会降低股市跳跃风险吗?[J].经济研究,2015, 42(1):153-167.Chen H Q, Zhang C H. Does index futures trading reduce stock market jump risk?[J]. Economic Research Journal,2015, 42(1):153-167.
[3]唐文进,苏帆.极端金融事件对系统性风险的影响分析——以中国银行部门为例[J].经济研究,2017(4):17-33.Tang W J, Su F. An analysis of the effects of extreme financial events on systemic risk:Evidence from China's banking sector[J]. Economic Research Journal, 2017(4):17-33.
[4]Barndorff-Nielsen O E. Econometric analysis of realized volatility and its use in estimating stochastic volatility models[J]. Journal of the Royal Statistical Society:Series B(Statistical Methodology), 2002, 64(2):253-280.
[5]Andersen T G, Bollerslev T, Diebold F X, et al. Modeling and forecasting realized volatility[J]. Econometrica,2003, 71(2):579-625.
[6]Lee S, Hannig J. Detecting jumps from Levy jump diffusion processes[J]. Journal of Financial Economics, 2010,96:271-290.
[7]Barndorff-Nielsen O E, Shephard N. Power and bipower variation with stochastic volatility and jumps[J]. Journal of Financial Econometrics, 2004, 2(1):1-37.
[8]Mancini C. Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps[J].Scandinavian Journal of Statistics, 2009, 36(2):270-296.
[9]Andersen T G, Dobrev D, Schaumburg E. Jump-robust volatility estimation using nearest neighbor truncation[J].Journal of Econometrics, 2012, 169(1):75-93.
[10]Christensen K, Podolskij M. Realized range-based estimation of integrated variance[J]. Journal of Econometrics,2008, 141(2):323-349.
[11]唐勇,刘微.加权已实现极差四次幂变差分析及其应用[J].系统工程理论与实践,2013, 33(11):2766-2775.Tang Y, Liu W. Analysis of weighted realized range-based quadpower variation and its application[J]. Systems Engineering—Theory&Practice, 2013, 33(11):2766-2775.
[12]Corsi F, Pirino D, Reno R. Threshold bipower variation and the impact of jumps on volatility forecasting[J].Journal of Econometrics, 2010, 159(2):276-288.
[13]Vetter M. Limit theorems for bipower variation of semimartingales[J]. Stochastic Processes and their Applications,2010, 120(1):22-38.
[14]Hansen P R, Lunde A. Realized variance and market microstructure noise[J]. Journal of Business&Economic Statistics, 2006, 24(2):127-161.
[15]Bandi F M, Russell J R. Microstructure noise, realized variance, and optimal sampling[J]. The Review of Economic Studies, 2008, 75(2):339-369.
[16]Zhang L, Mykland P A, A(i|¨)t-Sahalia Y. A tale of two time scales:Determining integrated volatility with noisy high-frequency data[J]. Journal of the American Statistical Association, 2005, 100(472):1394-1411.
[17]Zhang L. Efficient estimation of stochastic volatility using noisy observations:A multi-scale approach[J]. Bernoulli,2006, 12(6):1019-1043.
[18]Barndorff-Nielsen O E, Hansen P R, Lunde A, et al. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise[J]. Econometrica, 2008, 76(6):1481-1536.
[19]Jacod J, Li Y, Mykland P A, et al. Microstructure noise in the continuous case:The pre-averaging approach[J].Stochastic Processes and their Applications, 2009, 119(7):2249-2276.
[20]Xiu D. Quasi-maximum likelihood estimation of volatility with high frequency data[J]. Journal of Econometrics,2010, 159:235-250.
[21]Bibinger M, Hautsch N, Malec P, et al. Estimation the quadeatic covariation matrix from noisy observations:Local method of moments and efficiency[J]. The Annals of Statistics, 2014, 42(4):1312-1346.
[22]Fan J, Wang Y. Multi-scale jump and volatility analysis for high-frequency financial data[J]. Journal of the American Statistical Association, 2007, 102(480):1349-1362.
[23]Podolskij M, Vetter M. Bipower-type estimation in a noisy diffusion setting[J]. Stochastic Processes and their Applications, 2009, 119(9):2803-2831.
[24]Hautsch N, Podolskij M. Preaveraging-based estimation of quadratic variation in the presence of noise and jumps:Theory, implementation, and empirical evidence[J]. Journal of Business&Economic Statistics, 2013,31(2):165-183.
[25]Jing B Y, Liu Z, Kong X B. On the estimation of integrated volatility with jumps and microstructure noise[J].Journal of Business&Economic Statistics, 2014, 32(3):457-467.
[26]马丹,尹优平.噪声、跳跃与高频价格波动——基于门限预平均实现波动分析[J],金融研究,2012, 382(4):124-139.Ma D, Yin Y P. Noise, jumps and high-frequency price volatility—Based on threshold pre-averaging realized variance analysis[J]. Journal of Financial Research, 2012, 382(4):124-139.
[27]Podolskij M, Vetter M. Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps[J]. Bernoulli, 2009, 15(3):634-658.
[28]Wang K, Liu J, Liu Z. Disentangling the effect of jumps on systematic risk using a new estimator of integrated co-volatility[J]. Journal of Banking&Finance, 2013, 37(5):1777-1786.
[29]Jacod J, Li Y, Zheng X. Statistical properties of microstructure noise[J]. Econometrica, 2017, 85(4):1133-1174.
[30]Christensen K, Oomen R C, Podolskij M. Fact or friction:Jumps at ultra-high frequency[J]. Journal of Financial Economics, 2014, 114(3):576-599.
[31]A(i|¨)t-Sahalia Y, Jacod J. Analyzing the spectrum of asset returns:Jump and volatility components in high frequency data[J]. The Journal of Economic Literature, 2012, 50:1007-1050.
[32]Andersen T G, Benzoni L, Lund J. An empirical investigation of continuous-time equity return models[J]. The Journal of Finance, 2002, 57(3):1239-1284.
[33]Jacod J, Mykland P. Microstructure noise in the continuous case:Approximate efficiency of the adaptive preaveraging method[J]. Stochastic Processes and their Applications, 2015, 125(8):2910-2936.
[34]Barndorff-Nielsen O E, Hansen P R, Lunde A, et al. Realized kernels in practice:Trades and quotes[J]. The Econometrics Journal, 2009, 12(3):C1-C32.
[35]A(i|¨)t-Sahalia Y, Yu J. High frequency market microstructure noise estimates and liquidity measures[J]. Annals of Applied Statistics, 2009, 3(1):422-457.
[36]陈国进,王占海.我国股票市场连续性波动与跳跃性波动实证研究[J].系统工程理论与实践,2010, 30(9):1554-1562.Chen G J, Wang Z H. Continuous volatility and jump volatility in China's stock market[J]. Systems Engineering—Theory&Practice,2010, 30(9):1554-1562.
[37]赵华,秦可佶.股价跳跃与宏观信息发布[J].统计研究,2014, 31(4):79-88.Zhao H, Qin K J. Jumps in stock Prices and macro information release[J]. Statistical Research, 2014, 31(4):79-88.
[38]Dovonon P, Goncalves S, Hounyo U, et al. Bootstrapping high-frequency jump tests[J]. Journal of the American Statistical Association, 2018. https://doi.org/10.1080/01621459.2018.1447485.
[39]Barndorff-Nielsen O E, Shephard N. Econometrics of testing for jumps in financial economics using bipower variation[J]. Journal of financial Econometrics, 2006, 4(1):1-30.
[40]Bajgrowicz P, Scaillet O, Treccani A. Jumps in high-frequency data:Spurious detections, dynamics, and news[J].Management Science, 2016, 62(8):2198-2217.
[41]Corsi F, RenòR. Discrete-time volatility forecasting with persistent leverage effect and the link with continuoustime volatility modeling[J]. Journal of Business&Economic Statistics, 2012, 30(3):368-380.
[42]Patton A J, Sheppard K. Good volatility, bad volatility:Signed jumps and the persistence of volatility[J]. Review of Economics and Statistics, 2015, 97(3):683-697.