基于混频数据抽样的已实现波动率长记忆模型
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摘要
基于已实现GARCH模型和混频数据抽样(MIDAS)结构,提出了已实现混频数据抽样GARCH模型.该模型使用混频数据抽样结构从已实现测度中提取长短期波动率信息以提升模型对波动率的拟合和预测能力.基于指数和个股数据的实证分析表明,相比传统的已实现GARCH模型,新模型的样本内拟合能力更强,对长记忆性的捕捉更好.样本外结果表明,新模型显著提升了波动率的多步预测效果,并且改进效果随着预测期的延长而增强.
This paper proposed a realized MIDAS GARCH model based on the realized GARCH model and the mixed data sampling(MIDAS) regression structure. The model uses MIDAS structure to extract long and short term information from realized measures to improve the model's ability to fit and forecast volatility process. Empirical results based on indices and stocks data show that, compared with the classical realized GARCH model, the new model is better in in-sample data fitting and replicating long memory feature. The outof-sample forecasting results show that the new model significantly improves the multi-period out-of-sample volatility forecast. The improvement is more pronounced in longer horizons.
This paper proposed a realized MIDAS GARCH model based on the realized GARCH model and the mixed data sampling(MIDAS) regression structure. The model uses MIDAS structure to extract long and short term information from realized measures to improve the model's ability to fit and forecast volatility process. Empirical results based on indices and stocks data show that, compared with the classical realized GARCH model, the new model is better in in-sample data fitting and replicating long memory feature. The outof-sample forecasting results show that the new model significantly improves the multi-period out-of-sample volatility forecast. The improvement is more pronounced in longer horizons.
引文
[1]Engle R.Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation.Econometrica,1982,50(4):987-1007.
[2]Bollerslev T.Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics,1986,31(3):307-327.
[3]Bollerslev T.Glossary in ARCH(GARCH)Volatility and Time Series Econometrics,Oxford:Oxford University Press,2010:281-323.
[4]Engle R,Giampiero G.A multiple indicators model for volatility using intra-daily data.Journal of Econometrics,2006,131(1):3-27.
[5]Shephard N,Sheppard K,Realising the future:Forecasting with high-frequency-based volatility(HEAVY)models.Journal of Applied Econometrics,2010,25(2):197-231.
[6]Hansen P,Huang Z,Shek H.Realized GARCH:A joint model for returns and realized measures of volatility.Journal of Applied Econometrics,2012,27(6):877-906.
[7]Watanabe T.Quantile forecasts of financial returns using realized GARCH models.Japanese Economic Review,2012,63(1):68-80.
[8]Hansen P,Zhuo H.Exponential GARCH modeling with realized measures of volatility.Journal of Business&Economic Statistics,2016,34(2):269-287.
[9]Lunde A,Olesen V.Modeling and Forecasting the Volatility of Energy Forward Returns:Evidence from the Nordic Power Exchange.Denmark:Aarhus University,2013.
[10]王天一,黄卓.高频数据波动率建模:基于厚尾分布的realized GARCH模型.数量经济技术经济研究,2012(5):149-160.Wang T Y,Huang Z.High frequency volatility modelling:A fat-tail distribution based realized GARCH model.The Journal of Quantitative&Technical Economics,2012(5):149-160.(in Chinese)
[11]马锋,魏宇,黄登仕,等.隔夜收益率能提高高频波动率模型的预测能力吗.系统工程学报,2016,31(6):783-797.Ma F,Wei Y,Huang D S,et al.Can overnight returns improve the forecasting performance of high-frequency volatility models.Journal of Systems Engineering,2016,31(6):783-797.(in Chinese)
[12]黄友珀,唐振鹏,唐勇.基于藤copula-已实现GARCH的组合收益分位数预测.系统工程学报,2016,31(1):45-54.Huang Y P,Tang Z P,Tang Y.Portfolio quantile forecasts based on vine copula and realized GARCH.Journal of Systems Engineering,2016,31(1):45-54.(in Chinese)
[13]Bollerslev T,Chou R,Kroner K.ARCH modeling in finance:A review of the theory and empirical evidence.Journal of Econometrics,1992,52(1):5-59.
[14]Zumbach G.Volatility processes and volatility forecast with long memory.Quantitative Finance,2004,4(1):70-86.
[15]Taylor J.Consequences for option pricing of a long memory in volatility.Handbook of Financial Econometrics and Statistics,New York:Springer,2015:903-933.
[16]Chkili W,Hammoudeh S,Nguyen K.Volatility forecasting and risk management for commodity markets in the presence of asymmetry and long memory.Energy Economics,2014,41(1):1-18.
[17]曹广喜,曹杰,徐龙炳.双长记忆GARCH族模型的预测能力比较研究:基于沪深股市数据的实证分析.中国管理科学,2012,20(2):41-49.Cao G X,Cao J,Xu L B.Comparative research on forecasting ability of double-long-memory GARCH family models:Empirical analysis of Shanghai and Shenzhen stock markets.Chinese Journal of Management Science,2012,20(2):41-49.(in Chinese)
[18]庞淑娟,刘向丽,汪寿阳.中国期货市场高频波动率的长记忆性.系统工程理论与实践,2011,31(6):1039-1044.Pang S J,Liu X L,Wang S Y.Long memory of China futures markets volatility for high-frequency time series.Systems Engineering:Theory&Practice,2011,31(6):1039-1044.(in Chinese)
[19]Baillie T,Bollerslev T,Ole Mikkelsen H.Fractionally integrated generalized autoregressive conditional heteroskedasticity.Journal of Econometrics,1996,74(1):3-30.
[20]Engle R,Lee G.A long-run and short-run component model of stock return volatility//Cointegration,Causality,and Forecasting,Oxford:Oxford University Press,1999:475-497.
[21]Andersen T,Bollerslev T,Diebold F,et al.Modeling and forecasting realized volatility.Econometrica,2003,71(2):579-625.
[22]Ghysels E,Santa-Clara P,Valkanov R.There is a risk-return trade-off after all.Journal of Financial Economics,2005,76(3):509-548.
[23]Corsi F.A Simple Approximate long-memory model of realized volatility.Journal of Financial Econometrics,2009,7(2):174-196.
[24]徐剑刚,张晓蓉,唐国兴.混合数据抽样波动模型.数量经济技术经济研究,2007,24(11):77-85.Xu J G,Zhang X R,Tang G X.Mixed data sampling volatility model.The Journal of Quantitative&Technical Economics,2007,24(11):77-85.(in Chinese)
[25]瞿慧,李洁,程昕.HAR族模型与GARCH族模型对不同期限波动率的预测精度比较:基于沪深300指数高频价格的实证分析.系统工程,2015(3):25-31.Qu H,Li J,Cheng X.Comparison of HAR-family models’and GARCH-family models’forecasting preformances for volatilities of different terms:An empirical study using CSI300 index’s high-frequency prices.Systems Engineering,2015(3):25-31.(in Chinese)
[26]Barndorff-Nielsen O E,Shephard N.Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise.Econometrica,2008,76(6):1481-1536.
[27]Andersen T,Bollerslev T.Answering the skeptics:Yes,standard volatility models do provide accurate forecasts.International Ecnomic Review,1998,39(4):885-905.
[28]Hansen P,Lunde A.A realized variance for the whole day based on intermittent high-frequency data.Journal of Financial Econometrics,2005,3(4):525-554.
[29]Patton A.Volatility forecast comparison using imperfect volatility proxies.Journal of Econometrics,2011,160(1):246-256.
1虽然双参数Beta函数有更丰富的函数结构,但其参数估计稳定性较差,而且实证结果上与单参数Beta函数几乎没有差别,因此采用单参数Beta函数进行建模.
2 相同滞后阶的混频数据抽样回归实际上对应的是已实现测度的带参数约束的AR(22)模型.
3其中SPY是对应SP500指数的ETF,对整个市场有代表性. XOM为成分股中的大市值股票. IBM为成分股中的大权重股票. INTC, MSFT属于科技类股票. WMT代表的零售业相对周期性较弱.
4 作者同样试验过基于“开盘价-开盘价”收益率和5 min已实现方差(RV)的实证结果,结果并无明显差异.
5 收益率的单位为百分之一,已实现测度亦做相应的调整.
6由于本文使用的是日间收益率,因此估计得到的条件波动率为日间波动率.因其包含了隔夜收益率变化,理论上该条件波动率会比已实现核测度更大一点.
7理论ACF的计算方式基于式(8)和式(9),使用ARMA模型ACF计算公式计算.对于lnh而言,由于其没有当期冲击,计算公式做了相应调整.为节约空间这里仅给出指数ETF序列SPY的结果,其他序列的结果类似,如需要可向作者索取.
8“k天累计”存在高估低估相互抵消的状况,并不如直接考察“第k天”更准确.
9由于比较的两个模型都使用了该已实现测度,因此这样的选择并不偏向于某一个模型.使用5 min RV并不改变结果.
10关于类似调节的更多信息,可以参见文献[28].
11文献[29]指出的是稳健损失函数是MSE,由于RMSE更常用且是MSE的保序变换,本文汇报结果时使用的RMSE,计算统计显著性时基于MSE.常见的平均绝对误差(MAE)指标并不是稳健损失函数.因篇幅所限,本文仅选取了代表性的对称和非对称损失函数作为标准.
[2]Bollerslev T.Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics,1986,31(3):307-327.
[3]Bollerslev T.Glossary in ARCH(GARCH)Volatility and Time Series Econometrics,Oxford:Oxford University Press,2010:281-323.
[4]Engle R,Giampiero G.A multiple indicators model for volatility using intra-daily data.Journal of Econometrics,2006,131(1):3-27.
[5]Shephard N,Sheppard K,Realising the future:Forecasting with high-frequency-based volatility(HEAVY)models.Journal of Applied Econometrics,2010,25(2):197-231.
[6]Hansen P,Huang Z,Shek H.Realized GARCH:A joint model for returns and realized measures of volatility.Journal of Applied Econometrics,2012,27(6):877-906.
[7]Watanabe T.Quantile forecasts of financial returns using realized GARCH models.Japanese Economic Review,2012,63(1):68-80.
[8]Hansen P,Zhuo H.Exponential GARCH modeling with realized measures of volatility.Journal of Business&Economic Statistics,2016,34(2):269-287.
[9]Lunde A,Olesen V.Modeling and Forecasting the Volatility of Energy Forward Returns:Evidence from the Nordic Power Exchange.Denmark:Aarhus University,2013.
[10]王天一,黄卓.高频数据波动率建模:基于厚尾分布的realized GARCH模型.数量经济技术经济研究,2012(5):149-160.Wang T Y,Huang Z.High frequency volatility modelling:A fat-tail distribution based realized GARCH model.The Journal of Quantitative&Technical Economics,2012(5):149-160.(in Chinese)
[11]马锋,魏宇,黄登仕,等.隔夜收益率能提高高频波动率模型的预测能力吗.系统工程学报,2016,31(6):783-797.Ma F,Wei Y,Huang D S,et al.Can overnight returns improve the forecasting performance of high-frequency volatility models.Journal of Systems Engineering,2016,31(6):783-797.(in Chinese)
[12]黄友珀,唐振鹏,唐勇.基于藤copula-已实现GARCH的组合收益分位数预测.系统工程学报,2016,31(1):45-54.Huang Y P,Tang Z P,Tang Y.Portfolio quantile forecasts based on vine copula and realized GARCH.Journal of Systems Engineering,2016,31(1):45-54.(in Chinese)
[13]Bollerslev T,Chou R,Kroner K.ARCH modeling in finance:A review of the theory and empirical evidence.Journal of Econometrics,1992,52(1):5-59.
[14]Zumbach G.Volatility processes and volatility forecast with long memory.Quantitative Finance,2004,4(1):70-86.
[15]Taylor J.Consequences for option pricing of a long memory in volatility.Handbook of Financial Econometrics and Statistics,New York:Springer,2015:903-933.
[16]Chkili W,Hammoudeh S,Nguyen K.Volatility forecasting and risk management for commodity markets in the presence of asymmetry and long memory.Energy Economics,2014,41(1):1-18.
[17]曹广喜,曹杰,徐龙炳.双长记忆GARCH族模型的预测能力比较研究:基于沪深股市数据的实证分析.中国管理科学,2012,20(2):41-49.Cao G X,Cao J,Xu L B.Comparative research on forecasting ability of double-long-memory GARCH family models:Empirical analysis of Shanghai and Shenzhen stock markets.Chinese Journal of Management Science,2012,20(2):41-49.(in Chinese)
[18]庞淑娟,刘向丽,汪寿阳.中国期货市场高频波动率的长记忆性.系统工程理论与实践,2011,31(6):1039-1044.Pang S J,Liu X L,Wang S Y.Long memory of China futures markets volatility for high-frequency time series.Systems Engineering:Theory&Practice,2011,31(6):1039-1044.(in Chinese)
[19]Baillie T,Bollerslev T,Ole Mikkelsen H.Fractionally integrated generalized autoregressive conditional heteroskedasticity.Journal of Econometrics,1996,74(1):3-30.
[20]Engle R,Lee G.A long-run and short-run component model of stock return volatility//Cointegration,Causality,and Forecasting,Oxford:Oxford University Press,1999:475-497.
[21]Andersen T,Bollerslev T,Diebold F,et al.Modeling and forecasting realized volatility.Econometrica,2003,71(2):579-625.
[22]Ghysels E,Santa-Clara P,Valkanov R.There is a risk-return trade-off after all.Journal of Financial Economics,2005,76(3):509-548.
[23]Corsi F.A Simple Approximate long-memory model of realized volatility.Journal of Financial Econometrics,2009,7(2):174-196.
[24]徐剑刚,张晓蓉,唐国兴.混合数据抽样波动模型.数量经济技术经济研究,2007,24(11):77-85.Xu J G,Zhang X R,Tang G X.Mixed data sampling volatility model.The Journal of Quantitative&Technical Economics,2007,24(11):77-85.(in Chinese)
[25]瞿慧,李洁,程昕.HAR族模型与GARCH族模型对不同期限波动率的预测精度比较:基于沪深300指数高频价格的实证分析.系统工程,2015(3):25-31.Qu H,Li J,Cheng X.Comparison of HAR-family models’and GARCH-family models’forecasting preformances for volatilities of different terms:An empirical study using CSI300 index’s high-frequency prices.Systems Engineering,2015(3):25-31.(in Chinese)
[26]Barndorff-Nielsen O E,Shephard N.Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise.Econometrica,2008,76(6):1481-1536.
[27]Andersen T,Bollerslev T.Answering the skeptics:Yes,standard volatility models do provide accurate forecasts.International Ecnomic Review,1998,39(4):885-905.
[28]Hansen P,Lunde A.A realized variance for the whole day based on intermittent high-frequency data.Journal of Financial Econometrics,2005,3(4):525-554.
[29]Patton A.Volatility forecast comparison using imperfect volatility proxies.Journal of Econometrics,2011,160(1):246-256.
1虽然双参数Beta函数有更丰富的函数结构,但其参数估计稳定性较差,而且实证结果上与单参数Beta函数几乎没有差别,因此采用单参数Beta函数进行建模.
2 相同滞后阶的混频数据抽样回归实际上对应的是已实现测度的带参数约束的AR(22)模型.
3其中SPY是对应SP500指数的ETF,对整个市场有代表性. XOM为成分股中的大市值股票. IBM为成分股中的大权重股票. INTC, MSFT属于科技类股票. WMT代表的零售业相对周期性较弱.
4 作者同样试验过基于“开盘价-开盘价”收益率和5 min已实现方差(RV)的实证结果,结果并无明显差异.
5 收益率的单位为百分之一,已实现测度亦做相应的调整.
6由于本文使用的是日间收益率,因此估计得到的条件波动率为日间波动率.因其包含了隔夜收益率变化,理论上该条件波动率会比已实现核测度更大一点.
7理论ACF的计算方式基于式(8)和式(9),使用ARMA模型ACF计算公式计算.对于lnh而言,由于其没有当期冲击,计算公式做了相应调整.为节约空间这里仅给出指数ETF序列SPY的结果,其他序列的结果类似,如需要可向作者索取.
8“k天累计”存在高估低估相互抵消的状况,并不如直接考察“第k天”更准确.
9由于比较的两个模型都使用了该已实现测度,因此这样的选择并不偏向于某一个模型.使用5 min RV并不改变结果.
10关于类似调节的更多信息,可以参见文献[28].
11文献[29]指出的是稳健损失函数是MSE,由于RMSE更常用且是MSE的保序变换,本文汇报结果时使用的RMSE,计算统计显著性时基于MSE.常见的平均绝对误差(MAE)指标并不是稳健损失函数.因篇幅所限,本文仅选取了代表性的对称和非对称损失函数作为标准.
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