Best Mean Square Prediction for Moving Averages

Best mean square prediction for moving average time series models is generally non-linear prediction, even in the invertible case. Gaussian processes are an exception, since best linear prediction is always best mean square prediction. Stable numerical recursions are proposed for computation of residuals and evaluation of unnormalized conditional distributions in invertible or non-invertible moving average models, including those with distinct unit roots. The conditional distributions allow evaluation of the best mean square predictor via computation of a low-dimensional integral. For finite, discrete innovations, the method yields best mean square predictors exactly. For continuous innovations, an importance sampling scheme is proposed for numerical approximation of the best mean square predictor and its prediction mean square error. In numerical experiments, the method accurately computes best mean square predictors for cases with known solutions. The approximate best mean square predictor dominates the best linear predictor for out-of-sample forecasts of monthly US unemployment rates.