Recursive and Non-Recursive Generalized Least-Squares Methods for Estimation of Time Series Models with Exogenous Variables

Abstract

We present in this paper two generalized least-squares (GLS) methods for estimating regression coefficients of time series models with exogenous variables. The non-recursive GLS method is a generalization of the GLS method suggested by Cochrane and Orcutt (1949). The proposed GLS method consists of a sequence of four linear regressions. A first regression is fitted and provides residuals. These residuals are modeled as an autoregressive process and are used in a second regression (or autoregression) for obtaining estimators of autoregressive coefficients. These estimators are used to generate transformed endogenous and exogenous variables. A third regression makes use of the lagged values of these transformed variables to estimate the regression coefficients. The estimators of the regression coefficients are used to determine the true residuals which are modeled as an ARMA process which is finally used for obtaining the estimators of autoregressive and moving average parameters. The second GLS method is a recursive version of the first GLS method where the estimators are updated at each time point on receipt of the additional observations. The Simulation results based on different model structures with varying numbers of observations are used to compare the performance of our methods with that of exact maximum likelihood (EML) estimates.