Multifold Predictive Validation in ARMAX Time Series Models

Abstract

This article presents a new procedure for multifold predictive validation in time series. The procedure is based on the so-called “filtered residuals,” in-sample prediction errors evaluated in such a way that they are similar to out-of-sample ones. The filtered residuals are obtained from parameters estimated by eliminating from the estimation process the estimated innovations at the points to be predicted. Thus, instead of using the deletion of observations to validate the predictions, as in classical cross-validation, the procedure is based on deletion of the estimated innovations. It is proved that the filtered residuals are uncorrelated, up to terms of small order, with the in-sample innovations, a property shared with the out-of-sample residuals. The parameters needed for computing the filtered residuals can be obtained by estimating a model with innovational outliers at the points to be predicted. The proposed multifold predictive validation is asymptotically equivalent to an efficient model selection procedure. Some Monte Carlo evidence of the performance of the procedure is presented, and the application is illustrated in an example.