The exact multi-period mean-square forecast error for the first-order autoregressive model (Q1118311)

The finite-sample behaviour of the multi-period least-squares forecast is considered in the simple normal autoregressive model \(y_ t=\beta y_{t-1}+u_ t\) where \(| \beta | <1\). 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 \(\beta\) 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 \(| \beta |\) close to 1.