Multi-step-ahead estimation of time series models

We study the fitting of time series models via the minimization of a multi-step-ahead forecast error criterion that is based on the asymptotic average of squared forecast errors. Our objective function uses frequency domain concepts, but is formulated in the time domain, and allows the estimation of all linear processes (e.g., ARIMA and component ARIMA). By using an asymptotic form of the forecast mean squared error, we obtain a well-defined nonlinear function of the parameters that is proven to be minimized at the true parameter vector when the model is correctly specified. We derive the statistical properties of the parameter estimates, and study the asymptotic impact of model misspecification on multi-step-ahead forecasting. The method is illustrated through a forecasting exercise, applied to several time series.