Nonparametric specification for non-stationary time series regression. (English) Zbl 1400.62205

Summary: We investigate the behavior of the generalized likelihood ratio test (GLRT) [J. Fan et al., Ann. Stat. 29, No. 1, 153–193 (2001; Zbl 1029.62042)] for time varying coefficient models where the regressors and errors are non-stationary time series and can be cross correlated. It is found that the GLRT retains the minimax rate of local alternative detection under weak dependence and non-stationarity. However, in general, the Wilks phenomenon as well as the classic residual bootstrap are sensitive to either conditional heteroscedasticity of the errors, non-stationarity or temporal dependence. An averaged test is suggested to alleviate the sensitivity of the test to the choice of bandwidth and is shown to be more powerful than tests based on a single bandwidth. An alternative wild bootstrap method is proposed and shown to be consistent when making inference of time varying coefficient models for non-stationary time series.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62G08 Nonparametric regression and quantile regression
62G09 Nonparametric statistical resampling methods


Zbl 1029.62042


fda (R)
Full Text: DOI arXiv Euclid


[1] An, H. and Cheng, B. (1991). A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. International Statistical Review 59 287-307. · Zbl 0748.62049
[2] Brown, J.P., Song, H. and McGillivray, A. (1997). Forecasting UK house prices: A time varying coefficient approach. Economic Modeling 14 529-548.
[3] Cai, Z. (2007). Trending time-varying coefficient time series models with serially correlated errors. J. Econometrics 136 163-188. · Zbl 1418.62306
[4] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1-37. · Zbl 0871.62080
[5] Dahlhaus, R. (2009). Local inference for locally stationary time series based on the empirical spectral measure. J. Econometrics 151 101-112. · Zbl 1431.62362
[6] Dette, H. (1999). A consistent test for the functional form of a regression based on a difference of variance estimators. Ann. Statist. 27 1012-1040. · Zbl 0957.62036
[7] Dette, H. and Hetzler, B. (2007). Specification tests indexed by bandwidths. Sankhyā 69 28-54. · Zbl 1193.62062
[8] Dette, H., Preuss, P. and Vetter, M. (2011). A measure of stationarity in locally stationary processes with applications to testing. J. Amer. Statist. Assoc. 106 1113-1124. · Zbl 1229.62119
[9] Dette, H. and Spreckelsen, I. (2003). A note on a specification test for time series models based on spectral density estimation. Scand. J. Stat. 30 481-491. · Zbl 1035.62095
[10] Dette, H. and Spreckelsen, I. (2004). Some comments on specification tests in nonparametric absolutely regular processes. J. Time Series Anal. 25 159-172. · Zbl 1051.62043
[11] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66 . London: Chapman & Hall. · Zbl 0873.62037
[12] Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 1031-1057. · Zbl 1098.62077
[13] Fan, J. and Jiang, J. (2005). Nonparametric inferences for additive models. J. Amer. Statist. Assoc. 100 890-907. · Zbl 1117.62328
[14] Fan, J. and Jiang, J. (2007). Nonparametric inference with generalized likelihood ratio tests. TEST 16 409-444. · Zbl 1131.62035
[15] Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153-193. · Zbl 1029.62042
[16] Fan, Y. and Li, Q. (1999). Central limit theorem for degenerate \(U\)-statistics of absolutely regular processes with applications to model specification testing. J. Nonparametr. Stat. 10 245-271. · Zbl 0974.62044
[17] Gersch, W. and Kitagawa, G. (1985). A time varying AR coefficient model for modelling and simulating earthquake ground motion. Earthquake Engineering & Structural Dynamics 13 243-254.
[18] Hjellvik, V., Yao, Q. and Tjøstheim, D. (1998). Local polynomial estimation of conditional quantities with application to linearity testing. J. Statist. Plann. Inference 68 295-321. · Zbl 0942.62051
[19] Hong, Y. and Lee, Y. (2009). A loss function approach to model specification testing and its relative efficiency to the GLR test. Unpublished manuscript.
[20] Hoover, D.R., Rice, J.A., Wu, C.O. and Yang, L.P. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika 85 809-822. · Zbl 0921.62045
[21] Horowitz, J.L. and Spokoiny, V.G. (2001). An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 599-631. · Zbl 1017.62012
[22] Ingster, Y.I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I. Math. Methods Statist. 2 85-114. · Zbl 0798.62057
[23] Kitagawa, G. and Gersch, W. (1985). A smoothness priors time-varying AR coefficient modeling of nonstationary covariance time series. IEEE Trans. Automat. Control 30 48-56. · Zbl 0554.62079
[24] Lehmann, E.L. (2006). On likelihood ratio tests, 2nd ed. In Optimality. Institute of Mathematical Statistics Lecture Notes-Monograph Series 49 1-8. Beachwood, OH: IMS. · Zbl 1268.62023
[25] Müller, H.G. (2007). Comments on: Nonparametric inference with generalized likelihood ratio tests. TEST 16 450-452.
[26] Nason, G.P., von Sachs, R. and Kroisandt, G. (2000). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 271-292.
[27] Neumann, M.H. and von Sachs, R. (1997). Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 38-76. · Zbl 0871.62081
[28] Ombao, H., von Sachs, R. and Guo, W. (2005). SLEX analysis of multivariate nonstationary time series. J. Amer. Statist. Assoc. 100 519-531. · Zbl 1117.62407
[29] Orbe, S., Ferreira, E. and Rodriguez-Poo, J. (2005). Nonparametric estimation of time varying parameters under shape restrictions. J. Econometrics 126 53-77. · Zbl 1334.62064
[30] Orbe, S., Ferreira, E. and Rodriguez-Poo, J. (2006). On the estimation and testing of time varying constraints in econometric models. Statist. Sinica 16 1313-1333. · Zbl 1109.62117
[31] Paparoditis, E. (2000). Spectral density based goodness-of-fit tests for time series models. Scand. J. Stat. 27 143-176. · Zbl 0940.62084
[32] Paparoditis, E. (2009). Testing temporal constancy of the spectral structure of a time series. Bernoulli 15 1190-1221. · Zbl 1200.62049
[33] Paparoditis, E. (2010). Validating stationarity assumptions in time series analysis by rolling local periodograms. J. Amer. Statist. Assoc. 105 839-851. · Zbl 1392.62275
[34] Ramsay, J.O. and Silverman, B.W. (2005). Functional Data Analysis , 2nd ed. Springer Series in Statistics . New York: Springer. · Zbl 1079.62006
[35] Robinson, P.M. (1989). Nonparametric estimation of time-varying parameters. In Statistical Analysis and Forecasting of Economic Structural Change (P. Hackl, ed.) 164-253. Berlin: Springer.
[36] Sergides, M. and Paparoditis, E. (2009). Frequency domain tests of semi-parametric hypotheses for locally stationary processes. Scand. J. Stat. 36 800-821. · Zbl 1224.62070
[37] Stock, J.H. and Watson, M.W. (1998). Median unbiased estimation of coefficient variance in a time-varying parameter model. J. Amer. Statist. Assoc. 93 349-358. · Zbl 0906.62093
[38] Wu, W.B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150-14154. · Zbl 1135.62075
[39] Wu, W.B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statist. Sinica 21 1397-1413. · Zbl 1251.60029
[40] Zhang, C. and Dette, H. (2004). A power comparison between nonparametric regression tests. Statist. Probab. Lett. 66 289-301. · Zbl 1102.62049
[41] Zhang, C.M. (2003). Adaptive tests of regression functions via multiscale generalized likelihood ratios. Canad. J. Statist. 31 151-171. · Zbl 1040.62035
[42] Zhou, Z. and Wu, W.B. (2009). Local linear quantile estimation for nonstationary time series. Ann. Statist. 37 2696-2729. · Zbl 1173.62066
[43] Zhou, Z. and Wu, W.B. (2010). Simultaneous inference of linear models with time varying coefficients. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 513-531.
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