带厚尾噪声的TGARCH模型的估计及检验:一个统一的框架
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摘要
本文基于伪最大似然方法和t-标准化二次抽样(percentile-t subsample)bootstrap方法,研究了厚尾TGARCH(1,1)(threshold generalized autoregressive conditional heteroskedasticity(1,1))模型的估计和检验问题.此处,厚尾的含义是,TGARCH(1,1)模型噪声平方的分布位于指数为κ∈(1,2)的稳定分布的吸引场,即噪声不存在4阶矩.本文首先证明了,无论厚尾TGARCH(1,1)模型平稳与否,在一定正则性条件下,其ARCH(autoregressive conditional heteroskedasticity)和GARCH(generalized autoregressive conditional heteroskedasticity)系数的伪最大似然估计(QMLE)均具有相合性,其渐近分布位于指数为κ∈(1,2)的稳定分布的吸引场.然而,该模型位置参数的QMLE只有在平稳情形下才具有相合性.其次,基于上述渐近结果,本文结合t-标准化二次抽样bootstrap方法,给出了检验厚尾TGARCH(1,1)模型严平稳性和对称性的方法,克服了因QMLE的收敛速度和渐近分布依赖于未知尾指数而无法进行统计推断的困难,且该方法无论模型平稳与否均适用.最后,通过Monte Carlo随机试验考察了估计和检验方法的有限样本表现,并且基于本文的估计及检验方法对中国5年期国债期货收益率进行了实证分析.
This paper studies the estimation and tests of heavy-tailed TGARCH(1, 1) models based on quasimaximum likelihood estimation(QMLE) and percentile-t subsample bootstrap method, where heavy-tail means that the distribution of the square errors are in the domain of attraction of a stable law with infinite fourth moment.Under regular conditions, we first establish that the QMLE of ARCH and GARCH coefficients are consistent andtheir asymptotic distribution is non-Gaussian but is some stable law with exponent κ ∈(1, 2) despite of whether the heavy tailed TGARCH model is strictly stationary or not. However, the QMLE of the location parameter is consistent only when the model is strictly stationary. Then we propose tests for strict stationarity and symmetry for heavy-tailed TGARCH(1, 1) models based the above asymptotic results and percentile-t subsample bootstrap method to overcome the difficulty that the scale parameter and asymptotic distribution depend on the unknown exponent. These tests can be used in the universal parameter space. Finally, we conduct a simulation study through Monte Carlo methods to illustrate the performance of the proposed methods with finite sample sizes and then give an empirical analysis for 5-year China Treasury Bond Futures to illustrate the application of our methods.
This paper studies the estimation and tests of heavy-tailed TGARCH(1, 1) models based on quasimaximum likelihood estimation(QMLE) and percentile-t subsample bootstrap method, where heavy-tail means that the distribution of the square errors are in the domain of attraction of a stable law with infinite fourth moment.Under regular conditions, we first establish that the QMLE of ARCH and GARCH coefficients are consistent andtheir asymptotic distribution is non-Gaussian but is some stable law with exponent κ ∈(1, 2) despite of whether the heavy tailed TGARCH model is strictly stationary or not. However, the QMLE of the location parameter is consistent only when the model is strictly stationary. Then we propose tests for strict stationarity and symmetry for heavy-tailed TGARCH(1, 1) models based the above asymptotic results and percentile-t subsample bootstrap method to overcome the difficulty that the scale parameter and asymptotic distribution depend on the unknown exponent. These tests can be used in the universal parameter space. Finally, we conduct a simulation study through Monte Carlo methods to illustrate the performance of the proposed methods with finite sample sizes and then give an empirical analysis for 5-year China Treasury Bond Futures to illustrate the application of our methods.
引文
1 Engle R F.Autoregressive conditional heteroscedasticity with estimates of variance of UK inflation.Econometrica,1982,50:987-1008
2 Bollerslev T.Generalized autoregressive conditional heteroskedasticity.J Econometrics,1986,31:307-327
3 Weiss A.Asymptotic theory for ARCH models:Estimation and testing.Econometric Theory,1986,2:107-131
4 Lee S W,Hansen B E.Asymptotic theory for the GARCH(1,1)quasi-maximum likelihood estimator.Econometric Theory,1994,10:29-52
5 Lumsdaine R L.Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1)and covariance stationary GARCH(1,1)models.Econometrica,1996,64:575-596
6 Berkes I,Horv′ath L,Kokoszka P.GARCH processes:Structure and estimation.Bernoulli,2003,9:201-227
7 Francq C,Zako¨?an J M.Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes.Bernoulli,2004,10:605-637
8 Hall P,Yao Q.Inference in ARCH and GARCH models with heavy-tailed errors.Econometrica,2003,71:285-317
9 Mikosh T,Straumann D.Stable limits of martingale transforms with application to the estimation of GARCH parameters.Ann Statist,2006,34:493-522
10 Peng L,Yao Q.Least absolute deviation estimation for ARCH and GARCH models.Biometrika,2003,90:967-975
11 Loretan M,Phillips P C B.Testing the covariance stationarity of heavy-tailed time series.J Empir Finance,1994,1:211-248
12 Hwang S Y,Baek J S,Park J A,et al.Explosive volatilities for threshold-GARCH processes generated by asymmetric innovations.Statist Probab Lett,2010,80:26-33
13 St?aric?a C,Herzel S,Nord T.Why does the GARCH(1,1)model fail to provide sensible longer-horizon volatility forecasts?Http://econwpa.repec.org/eps/em/papers/0508/0508003.pdf,2005
14 Jensen S T,Rahbek A.Asymptotic normality of the QMLE estimator of ARCH in the nonstatinary case.Econometrica,2004,72:641-646
15 Jensen S T,Rahbek A.Asymptotic inference for nonstationary GARCH.Econometric Theory,2004,20:1203-1226
16 Chan N H,Ng C T.Statistical inference for non-stationary GARCH(p,q)models.Electron J Stat,2009,3:956-992
17 Linton O,Pan J Z,Wang H.Estimation for a non-stationary semi-strong GARCH(1,1)model with heavy-tailed errors.Econometric Theory,2010,26:1-28
18 Pan B,Chen M.Quasi-maximum exponential likelihood estimation for a nonstationary GARCH(1,1)model.Comm Statist Theory Methods,in press,2014
19 Francq C,Zako¨?an J M.Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models.Econometrica,2012,80:821-861
20 Glosten L R,Jaganathan R,Runkle D.On the relation between the expected values and the volatility of the nominal excess return on stocks.J Finance,1993,48:1779-1801
21 Li C W,Li W K.On a double-threshold autoregressive heteroscedastic time series model.J Appl Econometrics,1996,11:253-274
22 Pan J Z,Wang H,Tong H.Estimation and tests for power-transformed and threshold GARCH models.J Econometrics,2008,142:352-378
23 Hamadeh T,Zako¨?an J M.Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes.J Statist Plann Inference,2011,141:488-507
24 Wang H,Pan J.Restricted normal mixture QMLE for non-stationary TGARCH(1,1)models.Sci China Math,2014,57:1341-1360
25 Wang H,Pan J.Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1)models.Statist Probab Lett,2014,91:117-123
26 Francq C,Zako¨?an J M.Inference in non stationary asymmetric GARCH models.Ann Statist,2013,41:70-98
27 Mittnik S,Rachev S T.Stable Paretian Models in Finance.New York:Wiley,2000
28 Polzehl J,Spokoiny V.Varying coefificent GARCH versus local constant volatility modeling:Comparison of the predictive power.Http://hdl.handle.net/10419/25116,2006
29 Hall P.Asymptotic properties of the bootstrap for heavy-tailed distributions.Ann Probab,1990,18:1342-1360
30 Mammen E.When Does Bootstrap Work?Asymptotic Results and Simulations.New York:Springer,1992
31 Bickel P,Gotze F,Van Zwet W R.Resampling fewer than n observations:Gains,losses,and remedies for losses.Statist Sinica,1995,7:1-31
32 Politis D,Romano J P,Wolf M.Subsampling.New York:Springer,1999
33 Feller W.An Introduction to Probability Theory and Its Applications.New York:Wiley,1971
34 Straumann D.Estimation in Consitionally Heteroscedastic Time Series Models.New York:Springer,2005
35 Mokkadem A.Propri′et′es de m′elange des processus autor′egressifs polynomiaux.Ann Inst H Poincar′e Probab Statist,1990,26:219-260
2 Bollerslev T.Generalized autoregressive conditional heteroskedasticity.J Econometrics,1986,31:307-327
3 Weiss A.Asymptotic theory for ARCH models:Estimation and testing.Econometric Theory,1986,2:107-131
4 Lee S W,Hansen B E.Asymptotic theory for the GARCH(1,1)quasi-maximum likelihood estimator.Econometric Theory,1994,10:29-52
5 Lumsdaine R L.Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1)and covariance stationary GARCH(1,1)models.Econometrica,1996,64:575-596
6 Berkes I,Horv′ath L,Kokoszka P.GARCH processes:Structure and estimation.Bernoulli,2003,9:201-227
7 Francq C,Zako¨?an J M.Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes.Bernoulli,2004,10:605-637
8 Hall P,Yao Q.Inference in ARCH and GARCH models with heavy-tailed errors.Econometrica,2003,71:285-317
9 Mikosh T,Straumann D.Stable limits of martingale transforms with application to the estimation of GARCH parameters.Ann Statist,2006,34:493-522
10 Peng L,Yao Q.Least absolute deviation estimation for ARCH and GARCH models.Biometrika,2003,90:967-975
11 Loretan M,Phillips P C B.Testing the covariance stationarity of heavy-tailed time series.J Empir Finance,1994,1:211-248
12 Hwang S Y,Baek J S,Park J A,et al.Explosive volatilities for threshold-GARCH processes generated by asymmetric innovations.Statist Probab Lett,2010,80:26-33
13 St?aric?a C,Herzel S,Nord T.Why does the GARCH(1,1)model fail to provide sensible longer-horizon volatility forecasts?Http://econwpa.repec.org/eps/em/papers/0508/0508003.pdf,2005
14 Jensen S T,Rahbek A.Asymptotic normality of the QMLE estimator of ARCH in the nonstatinary case.Econometrica,2004,72:641-646
15 Jensen S T,Rahbek A.Asymptotic inference for nonstationary GARCH.Econometric Theory,2004,20:1203-1226
16 Chan N H,Ng C T.Statistical inference for non-stationary GARCH(p,q)models.Electron J Stat,2009,3:956-992
17 Linton O,Pan J Z,Wang H.Estimation for a non-stationary semi-strong GARCH(1,1)model with heavy-tailed errors.Econometric Theory,2010,26:1-28
18 Pan B,Chen M.Quasi-maximum exponential likelihood estimation for a nonstationary GARCH(1,1)model.Comm Statist Theory Methods,in press,2014
19 Francq C,Zako¨?an J M.Strict stationarity testing and estimation of explosive and stationary generalized autoregressive conditional heteroscedasticity models.Econometrica,2012,80:821-861
20 Glosten L R,Jaganathan R,Runkle D.On the relation between the expected values and the volatility of the nominal excess return on stocks.J Finance,1993,48:1779-1801
21 Li C W,Li W K.On a double-threshold autoregressive heteroscedastic time series model.J Appl Econometrics,1996,11:253-274
22 Pan J Z,Wang H,Tong H.Estimation and tests for power-transformed and threshold GARCH models.J Econometrics,2008,142:352-378
23 Hamadeh T,Zako¨?an J M.Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes.J Statist Plann Inference,2011,141:488-507
24 Wang H,Pan J.Restricted normal mixture QMLE for non-stationary TGARCH(1,1)models.Sci China Math,2014,57:1341-1360
25 Wang H,Pan J.Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1)models.Statist Probab Lett,2014,91:117-123
26 Francq C,Zako¨?an J M.Inference in non stationary asymmetric GARCH models.Ann Statist,2013,41:70-98
27 Mittnik S,Rachev S T.Stable Paretian Models in Finance.New York:Wiley,2000
28 Polzehl J,Spokoiny V.Varying coefificent GARCH versus local constant volatility modeling:Comparison of the predictive power.Http://hdl.handle.net/10419/25116,2006
29 Hall P.Asymptotic properties of the bootstrap for heavy-tailed distributions.Ann Probab,1990,18:1342-1360
30 Mammen E.When Does Bootstrap Work?Asymptotic Results and Simulations.New York:Springer,1992
31 Bickel P,Gotze F,Van Zwet W R.Resampling fewer than n observations:Gains,losses,and remedies for losses.Statist Sinica,1995,7:1-31
32 Politis D,Romano J P,Wolf M.Subsampling.New York:Springer,1999
33 Feller W.An Introduction to Probability Theory and Its Applications.New York:Wiley,1971
34 Straumann D.Estimation in Consitionally Heteroscedastic Time Series Models.New York:Springer,2005
35 Mokkadem A.Propri′et′es de m′elange des processus autor′egressifs polynomiaux.Ann Inst H Poincar′e Probab Statist,1990,26:219-260