An Asymptotically Pivotal Transform of the Residuals Sample Autocorrelations With Application to Model Checking

Abstract

We propose an asymptotically distribution-free transform of the sample autocorrelations of residuals in general parametric time series models, possibly nonlinear in variables. The residuals autocorrelation function is the basic model checking tool in time series analysis, but it is not useful when its distribution is incorrectly approximated because the effects of parameter estimation and/or higher-order serial dependence have not been taken into account. The limiting distribution of the residuals sample autocorrelations may be difficult to derive, particularly when the underlying innovations are uncorrelated but not independent. In contrast, our proposal is easily implemented in fairly general contexts and the resulting transformed sample autocorrelations are asymptotically distributed as independent standard normals when innovations are uncorrelated, providing an useful and intuitive device for time series model checking in the presence of estimated parameters. We also discuss in detail alternatives to the classical Box–Pierce test, showing that our transform entails no efficiency loss under Gaussianity in the direction of MA and AR departures from the white noise hypothesis, as well as alternatives to Bartlett’s Tp-process test. The finite-sample performance of the procedures is examined in the context of a Monte Carlo experiment for the new goodness-of-fit tests discussed in the article. The proposed methodology is applied to modeling the autocovariance structure of the well-known chemical process temperature reading data already used for the illustration of other statistical procedures. Additional technical details are included in a supplemental material online.