GARCH模型的贝叶斯局部影响分析及其应用
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摘要
广义自回归条件异方差(GARCH)模型能够很好地刻画金融资产收益二阶矩的相依关系,因此在金融时间序列中受到了广泛的应用。在GARCH模型的框架下,本文利用贝叶斯局部影响分析来评价先验、个体观测和样本分布的微小扰动的影响,利用扰动模型来刻画不同类型的扰动形式。我们构建了扰动模型的贝叶斯扰动形式,计算其几何量来表征扰动模型的内部结构。基于几个目标函数,本文利用几个不同的局部影响测量来量化不同扰动的程度。数值模拟研究验证了所提方法的有限样本表现。对纽约证券交易所综合指数(NYSE)和标准普尔500指数的GARCH建模说明了所提方法在实例研究中的有效性。
With the good capacity of addressing the dependency of conditional second moments of returns on financial assets,the generalized autoregressive conditional heteroscedasticity(GARCH)model has been widely used in financial time series.This paper develops a Bayesian local influence analysis method for assessing the minor perturbations to the prior,individual observations,and the sampling distribution in GARCH model.A perturbation model is introduced to characterize the different perturbations.We construct a Bayesian perturbation manifold to the perturbation model and calculate its geometric quantities to characterize the intrinsic structure of the perturbation model.Several local influence measures are proposed to quantify the degree of various perturbations based on several objective functions.Several numerical studies are conducted to evaluate the finite sample performance of the proposed method.Two empirical studies involving GARCH modeling of the continuously compounded daily returns on the New York Stock Exchange(NYSE)composite index and the series of daily log-returns for the stock index S&P500 illustrate the effectiveness of the proposed method.
With the good capacity of addressing the dependency of conditional second moments of returns on financial assets,the generalized autoregressive conditional heteroscedasticity(GARCH)model has been widely used in financial time series.This paper develops a Bayesian local influence analysis method for assessing the minor perturbations to the prior,individual observations,and the sampling distribution in GARCH model.A perturbation model is introduced to characterize the different perturbations.We construct a Bayesian perturbation manifold to the perturbation model and calculate its geometric quantities to characterize the intrinsic structure of the perturbation model.Several local influence measures are proposed to quantify the degree of various perturbations based on several objective functions.Several numerical studies are conducted to evaluate the finite sample performance of the proposed method.Two empirical studies involving GARCH modeling of the continuously compounded daily returns on the New York Stock Exchange(NYSE)composite index and the series of daily log-returns for the stock index S&P500 illustrate the effectiveness of the proposed method.
引文
[1]Engle R F.Autoregressive conditional Ieteroscedasticity with estimates of the variance of United Kingdom inflation[J].Econometrica:Journal of the Econometric Society,1982,50(4):987-1007.
[2]Bollerslev T.Generalized autoregressive conditional heteroskedasticity[J].Journal of econometrics,1986,31(3):307-327.
[3]Hall P,Yao Q.Inference in ARCH and GARCH models with heavy-tailed errors[J].Econometrica,2003,71(1):285-317.
[4]刘亭,赵月旭.基于QR-t-GARCH(1,1)模型沪深指数收益率风险度量的研究[J].数理统计与管理,2018,37(3):533-543.
[5]于孝建,王秀花.基于混频已实现GARCH模型的波动预测与VaR度量[J].统计研究,2018,35(1):104-116.
[6]吴思鑫,冯牧,张虎,陈敏.基于高频数据的非平稳GARCH(1,1)模型的拟极大指数似然估计[J].中国科学(数学),2018,48(3):443-456.
[7]Vrontos I D,Dellaportas P,Politis D N.Full Bayesian inference for GARCH and EGARCH models[J].Journal of Business&Economic Statistics,2000,18(2):187-198.
[8]Cook R D.Assessment of local influence[J].Journal of the Royal Statistical Society(Series B:Methodological),1986,48(2):133-169.
[9]Lawrance A J.Regression transformation diagnostics using local influence[J].Journal of the American Statistical Association,1988,83(404):1067-1072.
[10]Russo C M,Paula G A,Aoki R.Influence diagnostics in nonlinear mixed-effects elliptical models[J].Computational Statistics&Data Analysis,2009.53(12):4143-4156.
[11]Xie F C,Lin J G,Wei B C.Influence analysis of additive mixed-effects nonlinear regression models via EM algorithm[J].Journal of Statistical Computation and Simulation,2010,80(10):1115-1129.
[12]Villegas C,Paula G A,Cysneiros F J A,Galea M.Influence diagnostics in generalized symmetric linear models[J].Computational Statistics&Data Analysis,2013,59:161-170.
[13]Cao C Z,Lin J G,Shi J Q.Diagnostics on nonlinear model with scale mixtures of skew-normal and first-order autoregressive errors[J].Statistics,2014,48(5):1033-1047.
[14]McCulloch R E.Local model influence[J].Journal of the American Statistical Association,1989,84(406):473-478.
[15]Zhu H,Ibrahim J G,Tang N.Bayesian influence analysis:A geometric approach[J].Biometrika,2011,98(2):307-323.
[16]Tang N S,Duan X D.Bayesian influence analysis of generalized partial linear mixed models for longitudinal data[J].Journal of Multivariate Analysis,2014,126:86-99.
[17]Ibrahim J G,Zhu H,Tang N.Bayesian local influence for survival models[J].Lifetime data analysis,2011,17(1):43-70.
[18]Bayesian T.analysis of ARMA-GARCH models:A Markov Chain sampling approach[J].Journal of Econometrics,2000,95(1):57-69.
[19]Zhang X,King M L.Influence diagnostics in generalized autoregressive conditional heteroscedasticity processes[J].Journal of Business&Economic Statistics,2005,23(1):118-129.
[2]Bollerslev T.Generalized autoregressive conditional heteroskedasticity[J].Journal of econometrics,1986,31(3):307-327.
[3]Hall P,Yao Q.Inference in ARCH and GARCH models with heavy-tailed errors[J].Econometrica,2003,71(1):285-317.
[4]刘亭,赵月旭.基于QR-t-GARCH(1,1)模型沪深指数收益率风险度量的研究[J].数理统计与管理,2018,37(3):533-543.
[5]于孝建,王秀花.基于混频已实现GARCH模型的波动预测与VaR度量[J].统计研究,2018,35(1):104-116.
[6]吴思鑫,冯牧,张虎,陈敏.基于高频数据的非平稳GARCH(1,1)模型的拟极大指数似然估计[J].中国科学(数学),2018,48(3):443-456.
[7]Vrontos I D,Dellaportas P,Politis D N.Full Bayesian inference for GARCH and EGARCH models[J].Journal of Business&Economic Statistics,2000,18(2):187-198.
[8]Cook R D.Assessment of local influence[J].Journal of the Royal Statistical Society(Series B:Methodological),1986,48(2):133-169.
[9]Lawrance A J.Regression transformation diagnostics using local influence[J].Journal of the American Statistical Association,1988,83(404):1067-1072.
[10]Russo C M,Paula G A,Aoki R.Influence diagnostics in nonlinear mixed-effects elliptical models[J].Computational Statistics&Data Analysis,2009.53(12):4143-4156.
[11]Xie F C,Lin J G,Wei B C.Influence analysis of additive mixed-effects nonlinear regression models via EM algorithm[J].Journal of Statistical Computation and Simulation,2010,80(10):1115-1129.
[12]Villegas C,Paula G A,Cysneiros F J A,Galea M.Influence diagnostics in generalized symmetric linear models[J].Computational Statistics&Data Analysis,2013,59:161-170.
[13]Cao C Z,Lin J G,Shi J Q.Diagnostics on nonlinear model with scale mixtures of skew-normal and first-order autoregressive errors[J].Statistics,2014,48(5):1033-1047.
[14]McCulloch R E.Local model influence[J].Journal of the American Statistical Association,1989,84(406):473-478.
[15]Zhu H,Ibrahim J G,Tang N.Bayesian influence analysis:A geometric approach[J].Biometrika,2011,98(2):307-323.
[16]Tang N S,Duan X D.Bayesian influence analysis of generalized partial linear mixed models for longitudinal data[J].Journal of Multivariate Analysis,2014,126:86-99.
[17]Ibrahim J G,Zhu H,Tang N.Bayesian local influence for survival models[J].Lifetime data analysis,2011,17(1):43-70.
[18]Bayesian T.analysis of ARMA-GARCH models:A Markov Chain sampling approach[J].Journal of Econometrics,2000,95(1):57-69.
[19]Zhang X,King M L.Influence diagnostics in generalized autoregressive conditional heteroscedasticity processes[J].Journal of Business&Economic Statistics,2005,23(1):118-129.