Goodness-of-Fit Tests for Linear and Nonlinear Time Series Models

In this article we study a general class of goodness-of-fit tests for a parametric conditional mean of a linear or nonlinear time series model. Among the properties of the proposed tests are that they are suitable when the conditioning set is infinite-dimensional; that they are consistent against a broad class of alternatives, including Pitman's local alternatives converging at the parametric rate n−1/2, with n the sample size; and that they do not need to choose a lag order depending on the sample size or to smooth the data. It turns out that the asymptotic null distributions of the tests depend on the data generating process, so a new bootstrap procedure is proposed and theoretically justified. The proposed bootstrap tests are robust to higher-order dependence, particularly to conditional heteroscedasticity of unknown form. A simulation study compares the finite-sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling. Finally, an application to an economic price series highlights the merits of our approach.

[1]  B. LeBaron,et al.  A test for independence based on the correlation dimension , 1996 .

[2]  Winfried Stute,et al.  Bootstrap Approximations in Model Checks for Regression , 1998 .

[3]  Changbao Wu,et al.  Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis , 1986 .

[4]  H. Koul.  Weighted Empirical Processes in Dynamic Nonlinear Models , 2002 .

[5]  Regina Y. Liu.  Bootstrap Procedures under some Non-I.I.D. Models , 1988 .

[6]  Herman J. Bierens,et al.  Asymptotic Theory of Integrated Conditional Moment Tests , 1997 .

[7]  J. Carlos Escanciano,et al.  Generalized spectral tests for the martingale difference hypothesis , 2006 .

[8]  Rohit S. Deo,et al.  Spectral Tests of the Martingale Hypothesis Under Conditional Heteroscedasticity , 2000 .

[9]  Yongmiao Hong,et al.  Generalized Spectral Tests for Conditional Mean Models in Time Series with Conditional Heteroscedasticity of Unknown Form , 2005 .

[10]  Jianqing Fan,et al.  Goodness-of-Fit Tests for Parametric Regression Models , 2001 .

[11]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[12]  Robert M. de Jong,et al.  THE BIERENS TEST UNDER DATA DEPENDENCE , 1996 .

[13]  Maxwell B. Stinchcombe,et al.  CONSISTENT SPECIFICATION TESTING WITH NUISANCE PARAMETERS PRESENT ONLY UNDER THE ALTERNATIVE , 1998, Econometric Theory.

[14]  Winfried Stute,et al.  Nonparametric model checks for regression , 1997 .

[15]  Miguel A. Delgado,et al.  Distribution Free Goodness-of-Fit Tests for Linear Processes , 2005, math/0603043.

[16]  Juan Carlos Escanciano,et al.  Model Checks Using Residual Marked Empirical Processes , 2004 .

[17]  P. Billingsley,et al.  Convergence of Probability Measures , 1969 .

[18]  Alan David Hutson,et al.  Resampling Methods for Dependent Data , 2004, Technometrics.

[19]  E. Mammen.  Bootstrap and Wild Bootstrap for High Dimensional Linear Models , 1993 .

[20]  Adam Jakubowski.  On Limit Theorems for Sums of Dependent Hilbert Space Valued Random Variables , 1980 .

[21]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[22]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[23]  Hira L. Koul,et al.  Nonparametric model checks for time series , 1999 .

[24]  H. Bierens.  Consistent model specification tests , 1982 .

[25]  E. Khmaladze,et al.  Martingale Approach in the Theory of Goodness-of-Fit Tests , 1982 .

[26]  Herold Dehling,et al.  Empirical Process Techniques for Dependent Data , 2002 .

[27]  W. Härdle,et al.  Bootstrap Methods for Time Series , 2003 .

[28]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[29]  Julio Rodríguez,et al.  A Powerful Portmanteau Test of Lack of Fit for Time Series , 2002 .

[30]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[31]  Jianqing Fan,et al.  Nonlinear Time Series : Nonparametric and Parametric Methods , 2005 .

[32]  P. Hall,et al.  Martingale Limit Theory and Its Application , 1980 .

[33]  Bärbel Finkenstädt,et al.  Nonlinear Dynamics in Economics , 1995 .

[34]  E. Giné,et al.  Bootstrapping General Empirical Measures , 1990 .

[35]  Tae-Hwy Lee,et al.  DIAGNOSTIC CHECKING FOR THE ADEQUACY OF NONLINEAR TIME SERIES MODELS , 2003, Econometric Theory.

[36]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[37]  R. L. Eubank,et al.  Testing Goodness-of-Fit in Regression Via Order Selection Criteria , 1992 .

[38]  Yongmiao Hong,et al.  Hypothesis Testing in Time Series via the Empirical Characteristic Function: A Generalized Spectral Density Approach , 1999 .

[39]  E. Mammen,et al.  Bootstrap of kernel smoothing in nonlinear time series , 2002 .

[40]  Myron N. Chang.  Weak Convergence of a Self-Consistent Estimator of the Survival Function with Doubly Censored Data , 1990 .

[41]  C. Heyde,et al.  Quasi-likelihood and its application , 1997 .

[42]  Efstathios Paparoditis,et al.  Spectral Density Based Goodness-of-Fit Tests for Time Series Models , 2000 .

[43]  J. Zinn,et al.  Consistent specification tests for semiparametric/nonparametric models based on series estimation methods , 2003 .

[44]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .