参数单指标分位数自回归模型的诊断检验
|
摘要
本文研究参数单指标时间序列分位数自回归模型有效性的检验问题.当分位数回归变量的维数较大时,现有的检验方法将面临"维数灾难"问题.为了解决这个问题,本文基于残差经验过程,利用降维思想构造统计量,它有效地适应于参数单指标时间序列分位数自回归模型.本文提出Khmaladze鞅转换方法来替代经验过程,并构造检验统计量,证明所构造的检验统计量能够渐近收敛到分布自由的标准Brown运动.模拟研究和实际数据分析的结果表明,本文所提方法在参数单指标分位数自回归模型的检验中优于已有的检验方法.
In this paper, we study the nonparametric tests for the validity of the parametric single-index time series quantile autoregression with a given parametric link function. When the dimension of the quantile regressors is large, the existing tests will face the "curse of dimensionality" problem. To solve this problem, we utilize a dimension reduction idea, which perfectly adapts to parametric single-index time series, based on certain empirical processes marked by the residuals. We use the Khmaladze transformation to replace the underlying test process by its martingale part, and then the resulting test statistic is an asymptotically distribution-free test related to a standard Brownian motion. The simulation results and a real data example show that the proposed method performs better than the existing methods in checking parametric single-index quantile autoregression.
In this paper, we study the nonparametric tests for the validity of the parametric single-index time series quantile autoregression with a given parametric link function. When the dimension of the quantile regressors is large, the existing tests will face the "curse of dimensionality" problem. To solve this problem, we utilize a dimension reduction idea, which perfectly adapts to parametric single-index time series, based on certain empirical processes marked by the residuals. We use the Khmaladze transformation to replace the underlying test process by its martingale part, and then the resulting test statistic is an asymptotically distribution-free test related to a standard Brownian motion. The simulation results and a real data example show that the proposed method performs better than the existing methods in checking parametric single-index quantile autoregression.
引文
1 Koenker R,Bassett G Jr.Regression quantiles.Econometrica,1978,46:33-50
2 Cai Z.Regression quantiles for time series.Econometric Theory,2002,18:169-192
3 Koenker R,Xiao Z J.Quantile autoregression.J Amer Statist Assoc,2006,101:980-1006
4 Li G D,Li Y,Tsai C L.Quantile correlations and quantile autoregressive modeling.J Amer Statist Assoc,2015,110:246-261
5 Wu T Z,Yu K,Yu Y.Single-index quantile regression.J Multivariate Anal,2010,101:1607-1621
6 Kong E,Xia Y C.A single-index quantile regression model and its estimation.Econometric Theory,2012,28:730-768
7 Charlier I,Paindaveine D,Saracco J.Conditional quantile estimation through optimal quantization.J Statist Plann Inference,2015,156:14-30
8 Ma S,He X.Inference for single-index quantile regression models with profile optimization.Ann Statist,2016,44:1234-1268
9 Zheng J X.A consistent nonparametric test of parametric regression models under conditional quantile restrictions.Econometric Theory,1998,14:123-138
10 Koul H L,Stute W.Nonparametric model checks for time series.Ann Statist,1999,27:204-236
11 Horowitz J L,Spokoiny V G.An adaptive,rate-optimal test of linearity for median regression models.J Amer Statist Assoc,2002,97:822-835
12 He X M,Zhu L X.A lack-of-fit test for quantile regression.J Amer Statist Assoc,2003,98:1013-1022
13 Escanciano J C,Velasco C.Specification tests of parametric dynamic conditional quantiles.J Econometrics,2010,159:209-221
14 Galvao A F,Kato K,Montes-Rojas G,et al.Testing linearity against threshold effects:Uniform inference in quantile regression.Ann Inst Statist Math,2014,66:413-439
15 Koenker R,Xiao Z J.Inference on the quantile regression process.Econometrica,2002,70:1583-1612
16 McCullagh P,Nelder J A.Generalized Linear Models,2nd ed.London:Chapman and Hall,1989
17 Fahrmeir L,Tutz G.Multivariate Statistical Modeling Based on Generalized Linear Models.New York:SpringerVerlag,1994
18 Tsiatis A A.A note on a goodness-of-fit test for the logistic regression model.Biometrika,1980,67:250-251
19 Stute W,Zhu L X.Model checks for generalized linear models.Scand J Statist,2002,29:535-545
20 Li,G R,Peng H,Dong K,et al.Simultaneous confidence bands and hypothesis testing for single-index models.Statist Sinica,2014,24:937-955
21 Whang Y J.Smoothed empirical likelihood methods for quantile regression models.Econometric Theory,2005,22:173-205
22 Bierens H J,Ginther D K.Integrated conditional moment testing of quantile regression models.Empir Econom,2001,26:307-324
23 Stute W.Nonparametric model checks for regression.Ann Statist,1997,25:613-641
24 Xiao Z J.Time series quantile regressions.Handbook of Statist,2012,30:213-257
25 Mukherjee K.Asymptotics of quantiles and rank scores in nonlinear time series.J Time Ser Anal,1999,20:173-192
26 Xia Q,He K,Niu C.A model-adaptive test for parametric single-index time series models.J Time Ser Anal,2017,38:981-999
27 Stute W,Presedo-Quindimil M,González-Manteiga W,et al.Modelchecks of higher order time series.Statist Probab Lett,2006,76:1385-1396
28 Khmaladze E V.A martingale approach in the theory of goodness-of-fit tests.Theory Probab Appl,1981,26:240-257
29 Hall P,Heyde C C.Martingale Limit Theory and Its Application.New York:Academic Press,1980
30 Billingsley P.Convergence of Probability Measures,2nd ed.New York:John Wiley&Sons,1999
2 Cai Z.Regression quantiles for time series.Econometric Theory,2002,18:169-192
3 Koenker R,Xiao Z J.Quantile autoregression.J Amer Statist Assoc,2006,101:980-1006
4 Li G D,Li Y,Tsai C L.Quantile correlations and quantile autoregressive modeling.J Amer Statist Assoc,2015,110:246-261
5 Wu T Z,Yu K,Yu Y.Single-index quantile regression.J Multivariate Anal,2010,101:1607-1621
6 Kong E,Xia Y C.A single-index quantile regression model and its estimation.Econometric Theory,2012,28:730-768
7 Charlier I,Paindaveine D,Saracco J.Conditional quantile estimation through optimal quantization.J Statist Plann Inference,2015,156:14-30
8 Ma S,He X.Inference for single-index quantile regression models with profile optimization.Ann Statist,2016,44:1234-1268
9 Zheng J X.A consistent nonparametric test of parametric regression models under conditional quantile restrictions.Econometric Theory,1998,14:123-138
10 Koul H L,Stute W.Nonparametric model checks for time series.Ann Statist,1999,27:204-236
11 Horowitz J L,Spokoiny V G.An adaptive,rate-optimal test of linearity for median regression models.J Amer Statist Assoc,2002,97:822-835
12 He X M,Zhu L X.A lack-of-fit test for quantile regression.J Amer Statist Assoc,2003,98:1013-1022
13 Escanciano J C,Velasco C.Specification tests of parametric dynamic conditional quantiles.J Econometrics,2010,159:209-221
14 Galvao A F,Kato K,Montes-Rojas G,et al.Testing linearity against threshold effects:Uniform inference in quantile regression.Ann Inst Statist Math,2014,66:413-439
15 Koenker R,Xiao Z J.Inference on the quantile regression process.Econometrica,2002,70:1583-1612
16 McCullagh P,Nelder J A.Generalized Linear Models,2nd ed.London:Chapman and Hall,1989
17 Fahrmeir L,Tutz G.Multivariate Statistical Modeling Based on Generalized Linear Models.New York:SpringerVerlag,1994
18 Tsiatis A A.A note on a goodness-of-fit test for the logistic regression model.Biometrika,1980,67:250-251
19 Stute W,Zhu L X.Model checks for generalized linear models.Scand J Statist,2002,29:535-545
20 Li,G R,Peng H,Dong K,et al.Simultaneous confidence bands and hypothesis testing for single-index models.Statist Sinica,2014,24:937-955
21 Whang Y J.Smoothed empirical likelihood methods for quantile regression models.Econometric Theory,2005,22:173-205
22 Bierens H J,Ginther D K.Integrated conditional moment testing of quantile regression models.Empir Econom,2001,26:307-324
23 Stute W.Nonparametric model checks for regression.Ann Statist,1997,25:613-641
24 Xiao Z J.Time series quantile regressions.Handbook of Statist,2012,30:213-257
25 Mukherjee K.Asymptotics of quantiles and rank scores in nonlinear time series.J Time Ser Anal,1999,20:173-192
26 Xia Q,He K,Niu C.A model-adaptive test for parametric single-index time series models.J Time Ser Anal,2017,38:981-999
27 Stute W,Presedo-Quindimil M,González-Manteiga W,et al.Modelchecks of higher order time series.Statist Probab Lett,2006,76:1385-1396
28 Khmaladze E V.A martingale approach in the theory of goodness-of-fit tests.Theory Probab Appl,1981,26:240-257
29 Hall P,Heyde C C.Martingale Limit Theory and Its Application.New York:Academic Press,1980
30 Billingsley P.Convergence of Probability Measures,2nd ed.New York:John Wiley&Sons,1999