The Annals of Statistics

Nonparametric model checks for time series

Hira L. Koul and Winfried Stute

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This paper studies a class of tests useful for testing the goodness-of-fit of an autoregressive model. These tests are based on a class of empirical processes marked by certain residuals. The paper first gives their large sample behavior under null hypotheses. Then a martingale transformation of the underlying process is given that makes tests based on it asymptotically distribution free. Consistency of these tests is also discussed briefly.

Article information

Ann. Statist., Volume 27, Number 1 (1999), 204-236.

First available in Project Euclid: 5 April 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M30: Spatial processes 62J02: General nonlinear regression

Marked empirical process $\psi$-residuals martingale transform tests autoregressive median function


Koul, Hira L.; Stute, Winfried. Nonparametric model checks for time series. Ann. Statist. 27 (1999), no. 1, 204--236. doi:10.1214/aos/1018031108.

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  • An,H.-S. and Bing,C. (1991). A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. Internat. Statist. Rev. 59 287-307.
  • Billingsley,P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Chang,N. M. (1990). Weak convergence of a self-consistent estimator of a survival function with doubly censored data. Ann. Statist. 18 391-404.
  • Chow,Y. S. and Teicher,H. (1978). Probability Theory: Independence, Interchangeability, Martingales. Springer, New York.
  • Delgado,M. A. (1993). Testing the equality of nonparametric curves. Statist. Probab. Lett. 17 199-204.
  • Diebolt,J. (1995). A nonparametric test for the regression function: asymptotic theory. J. Statist. Plann. Inference 44 1-17.
  • Durbin,J. (1973). Weak convergence of the sample distribution function when parameters are estimated. Ann. Statist. 1 279-290.
  • Durbin,J.,Knott,M. and Taylor,C. C. (1975). Components of Cram´er-von Mises statistics. II. J. Roy. Statist. Soc. Ser. B 37 216-237.
  • Hall,P. and Heyde,C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Hjellvik,V. and Tjøstheim,D. (1995). Nonparametric test of linearity for time series. Biometrika 82 351-368.
  • Hjellvik,V. and Tjøstheim,D. (1996). Nonparametric statistics for testing linearity and serial independence. Nonparametric Statist. 6 223-251.
  • Huber,P. J. (1981). Robust Statistics. Wiley, New York.
  • Khmaladze,E. V. (1981). Martingale approach in the theory of goodness-of-fit tests. Theory Probab. Appl. 26 240-257.
  • Khmaladze,E. V. (1988). An innovation approach to goodness-of-fit tests in Rm. Ann. Statist. 16 1503-1516.
  • Khmaladze,E. V. (1993). Goodness of fit problem and scanning innovation martingales. Ann. Statist. 21 798-829.
  • Koul,H. L. (1996). Asymptotics of some estimators and sequential residual empiricals in nonlinear time series. Ann. Statist. 24 380-404.
  • MacKinnon,J. G. (1992). Model specification tests and artificial regression. J. Econom. Literature XXX (March) 102-146.
  • McKeague,I. W. and Zhang,M. J. (1994). Identification of nonlinear time series from first order cumulative characteristics. Ann. Statist. 22 495-514.
  • Nikabadze,A. and Stute,W. (1996). Model checks under random censorship. Statist. Probab. Lett. 32 249-259.
  • Ozaki,T. and Oda,H. (1978). Non-linear time series model identification by Akaike's information criterion. In Proceedings of the IFAC Workshop on Information and Systems. Pergamon Press, New York.
  • Rao,K. C. (1972). The Kolmogorof, Cram´er-von Mises chi-squares statistics for goodness-of-fit tests in the parametric case (Abstract). Bull. Inst. Math. Statist. 1 87.
  • Robinson,P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185-207.
  • Robinson,P. M. (1984). Robust nonparametric autoregression. Robust and Nonlinear Time Series Analysis. Lecture Notes in Statist. 26 247-255. Springer, Berlin.
  • Roussas,G. G. and Tran,L. T. (1992). Asymptotic normality of the recursive estimate under dependence condition. Ann. Statist. 20 98-120.
  • Stute,W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641.
  • Stute,W.,Thies,S. and Zhu,L. X. (1998). Model checks for regression: an innovation process approach. Unpublished manuscript.
  • Su,J. Q. and Wei,L. J. (1991). A lack-of-fit test for the mean function in a generalized linear model. J. Amer. Statist. Assoc. 86 420-426.
  • Tjøstheim,D. (1986). Estimation in nonlinear time series models I: stationary series. Stochastic Process. Appl. 21 251-273.
  • Tong,H. (1990). Non-linear Time Series Analysis: A Dynamical Approach. Oxford Univ. Press.
  • Truong,Y. K. and Stone,C. J. (1992). Nonparametric function estimation involving time series. Ann. Statist. 20 77-97.
  • Wheeden,R. L. and Zygmund,A. (1977). Measure and Integral: An Introduction to Real Analysis. Dekker, New York.