The exact multi-period mean-square forecast error for the first-order autoregressive model

Abstract We study the exact finite-sample behaviour of the mean-square forecast error ( MSFE ) of multi-period least-squares forecasts in the normal autoregressive model y t = α + β y t - 1 + u t . We obtain necessary and sufficient conditions for the existence of the MSFE and give an exact expression which we use to obtain numerical results for both the stationary and the fixed start-up model. We conclude, inter alia, that the behaviour of the MSFE in the model with intercept can be very different from that in the model without intercept, especially when β is close to unity.