Limit distributions of prediction errors when the parameters are estimated are obtained for a general class of nonstationary processes that includes supercritical branching processes and explosive autoregressive processes as typical examples. The estimated prediction errors are used to construct new tests of fit whose limit distributions are also derived.
"Asymptotic Distributions of Prediction Errors and Related Tests of Fit for Nonstationary Processes." Ann. Statist. 15 (1) 46 - 58, March, 1987. https://doi.org/10.1214/aos/1176350252