Multi-Step Ahead Estimation of Time Series Models

We study the fitting of time series models via minimization of a multi-step ahead forecast error criterion that is based on the asymptotic average of squared forecast errors. As in Haywood and Tunnicliffe-Wilson (1997) our score function is formulated in the frequency domain, but our time series models are not limited to those with spectra linear in the parameters; our formulation includes all linear processes. Moreover, the variables of our objective function are identical with the model parameters, which improves the interpretability of results. Asymptotic normality of parameter estimates is derived, enabling comparison to the classical one-step ahead formulation. In particular, parameter estimates are consistent under a correct model specification. 1 Statistical Research Division, U.S. Census Bureau, 4600 Silver Hill Road, Washington, D.C. 20233-9100 2 Institute of Data Analysis and Process Design