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Multi-Step Ahead Estimation of Time Series Models

Tucker McElroy and Marc Wilidi


ARIMA, Forecasting, Frequency Domain, Nonstationary, Signal Extraction.


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. Our objective function uses frequency domain concepts, but is formulated in the time domain, and allows 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 provably 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.


Tucker McElroy and Marc Wilidi. (2012). Multi-Step Ahead Estimation of Time Series Models. Center for Statistical Research & Methodology Research Report Series (Statistics #2012-11). U.S. Census Bureau. Available online at <>.

Source: U.S. Census Bureau, Center for Statistical Research & Methodology, Research and Methodology Directorate

Published online: September 11, 2012
Last revised: September 11, 2012


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Source: U.S. Census Bureau | Research and Methodology Directorate | Center for Statistical Research & Methodology | (301) 763-3215 (or |   Last Revised: September 11, 2013