Analyzing cross-validation for forecasting with structural instability

When forecasting with economic time series data, researchers often use a restricted window of observations or downweight past observations in order to mitigate the potential effects of parameter instability. In this paper, we study the problem of selecting a window for point forecasts made at the end of the sample. We develop asymptotic approximations to the sampling properties of window selection methods, and post-window selection point forecasts, where there is local parameter instability of various sorts. We examine risk properties of point forecasts made after cross-validation to select the window, and compare this approach to some alternative methods of selecting the window. We also propose a quasi-Bayesian form of cross-validation that we find to have good risk properties.