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

The finite-sample behaviour of the multi-period least-squares forecast is considered in the simple normal autoregressive model yt = βyt–1 + ut where ‖β‖ < 1. Necessary and sufficient conditions are established for the existence of the forecast bias and the mean-square forecast error (MSFE) and an exact expression for the MSFE is given. Exact numerical results are obtained for both the stationary and the fixed start-up case. Our main conclusions are that for small values of β the MSFE is a decreasing function of the number of forecast periods, and that the behaviour of the MSFE in the stationary and the fixed start-up case is very similar, except for values of ‖β‖ close to 1.

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