A distribution-free transform of the residuals sample autocorrelations with application to model checking

We propose an asymptotically distribution-free transform of the sample autocorrelations of residuals in general parametric time series models, possibly non-linear in variables. The residuals autocorrelation function is the basic model checking tool in time series analysis, but it is useless when its distribution is incorrectly approximated because the effects of parameter estimation or of unnoticed higher order serial dependence have not been taken into account. The limiting distribution of residuals sample autocorrelations may be difficult to derive, particularly when the underlying innovations are not independent. However, the transformation we propose is easy to implement and the resulting transformed sample autocorrelations are asymptotically distributed as independent standard normals, providing an useful and intuitive device for model checking by taking over the role of the standard sample autocorrelations. We also discuss in detail alternatives to the classical Box-Pierce and Bartlett's Tp-process tests, showing that our transform entails no efficiency loss under Gaussianity. The finite sample performance of the procedures is examined in the context of a Monte Carlo experiment for the two 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.