Transforming processes in inference: goodness-of-fit for AR(p) models

Abstract

We present a test of fit for the hypothesis that a time series comes form an AR(p) process, specially sensitive to alternative models AR(p+1), or ARMA(p,1). The testing procedure can be viewed as a particular example of a more general inferential procedure based on the study of certain random processes which summarize the information contained in the data, and certain transformations that make the alternatives noticeable. Transformed Empirical processes are a tool introduced long ago in Cabaña, A. (1996), Transformations of the empirical measure and Kolmogorov-Smirnov tests, Ann. Statist. 24, Cabaña, A. and Cabaña, E.M. (1997), Transformed Empirical Processes and Modified Kolmogorov-Smirnov Tests for multivariate distributions}, Ann. Statist. 25, in order to develop goodness of fit tests which are consistent against any alternative, designed to be efficient for local alternatives selected by users, and distribution free. The procedure is based on replacing the empirical process by a Transformed Empirical Process (TEP) (or a Transformed Estimated Empirical Process (TEEP)) in standard nonparametric methods. In particular, use the supremum of the absolute value of the TEP or the TEEP as test variable and obtain Kolmogorov-Smirnov type tests, or use quadratic functionals and get Watson type statistics. After that, we reproduced the same techniques using the process of accumulated residuals in order to check the validity of linear models, and time series. We shall briefly discuss the general procedure, and also an intuitive approach to the tests for time series. We compare the results obtained with commonly used Portmanteau tests, and show that we obtain better performance.