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X-13ARIMA-SEATS Seasonal Adjustment Program

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Optimality of GLS for One-Step-Ahead Forecasting of regARIMA and Related Models When the Regression is Misspecified

David F. Findley(1)


We consider the modeling of a time series described by a linear regression component whose regressor sequence satisfies the generalized asymptotic sample second moment stationarity conditions of Grenander (1954). Similarly, the associated disturbance term is assumed to have sample second moments that converge with increasing series length, perhaps after differencings. The model's regression component is taken to be underspecified, due perhaps to simplifications, approximations, parsimony, etc. Also, the model's ARMA or ARIMA-type structure for the disturbance term need not be correct. Both Ordinary Least Squares and Generalized Least Squares estimates of the mean function are considered. An optimality property of GLS relative to OLS (and other alternatives) is obtained for one-step- ahead forecasting. Asymptotic bias characteristics of the regression estimates are shown to distinguish the forecasting performance. These results provide support for the application by Statistics Netherlands (Aelen, 2004) of regARIMA models with stochastic regressors, with coefficients estimated by the GLS procedure of X-12-ARIMA, to forecast/impute the net contribution of late-reporting firms to monthly economic time series from surveys.


Incorrect models ; Out-of-sample forecasting ; Real-time forecasting ; Model selection

(1) David F. Findley was the Senior Mathematical Statistican for Time Series and is now a consultant, U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

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Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or |  Last Revised: November 19, 2012