On the Bias in Estimates of Forecast Mean Squared Error

We examine the forecasting problem for a finite series of observations from either a nonseasonal or a seasonal autoregressive-moving average process. Four sources of bias in the usual estimator of forecast mean squared error are identified and analyzed for particular models using maximum likelihood and least squares parameter estimates. The usual estimator is found to be biased downwards, especially near the boundary of a stationary or invertible region, and the bias is severe for least squares estimators. An alternative estimator is proposed for the maximum likelihood case, which is shown generally to have reduced bias.