Direct autoregressive predictors for multistep prediction: Order selection and performance relative to the plug in predictors

A direct method for multistep prediction of a stationary time series consists of fitting a new autoregression for each lead time, h, by a linear regression procedure and to select the order to be fitted from the data. By contrast, a more usual 'plug in' method involves the least-squares fitting of an initial kth order autoregression; the multistep forecasts are then obtained from the model equation, but with the unknown future values replaced by their own forecasts. The asymptotic distributions of the direct and plug in estimates of the h-step prediction constants and their respective mean squared errors of prediction are derived for a finite autoregressive process; explicit asymptotic expressions for comparing the loss in predictive and parameter estimation efficiency due to using the direct method instead of the plug in method in this situation are also given. The finite sample behaviour of the prediction errors with these two methods is investigated by a simulation study.