Exact and approximate multi-period mean-square forecast errors for dynamic econometric models

Both future disturbances and estimated coefficients contribute to the uncertainty in model-based ex ante forecasts, but only the first source is usually taken into account when calculating confidence intervals for practical applications. Schmidt (1974) and Baillie (1979) provide an easily computable second-order approximation to the mean-square forecast error (MSFE) for linear dynamic systems which recognizes both sources of uncertainty. To assess the accuracy of their approximation, and thus its usefulness, we compare it with three sets of estimates of the exact MSFE for the univariate AR(l) process: Monte Carlo estimates for OLS, analytically based values for OLS, and Monte Carlo estimates for maximum likelihood. We find that the Schmidt-Baillie formula is a good approximation to the exact MSFE, and that it helps explain why the exact MSFE can decrease as the forecast horizon increases. In fact, for dynamics typical to econometric models, the MSFE often has a maximum at a forecast horizon of one to twelve periods, i.e., at horizons that are of principal concern to forecasters and policy makers.