New in-sample prediction errors in time series with applications

^aThis article introduces two new types of prediction errors in time series: the filtered prediction errors and the deletion prediction errors. These two prediction errors are obtained in the same sample used for estimation, but in such a way that they share some common properties with out of sample prediction errors. It is proved that the filtered prediction errors are uncorrelated, up to terms of magnitude order O(T^-2), with the in sample innovations, a property that share with the out-of-sample prediction errors. On the other hand, deletion prediction errors assume that the values to be predicted are unobserved, a property that they also share with out-of-sample prediction errors. It is shown that these prediction errors can be computed with parameters estimated by assuming innovative or additive outliers, respectively, at the points to be predicted. Then the prediction errors are obtained by running the procedure for all the points in the sample of data. Two applications of these new prediction errors are presented. The first is the estimation and comparison of the prediction mean squared errors of competing predictors. The second is the determination of the order of an ARMA model. In the two applications the proposed filtered prediction errors have some advantages over alternative existing methods.