Efficient Estimation of Simultaneous Equations with Auto-Regressive Errors by Instrumental Variables

Abstract

T HE purpose of this paper is to point out how the efficient instrumental-variables technique discussed by Brundy and Jorgenson (1971) can be modified to take into account auto-regressive properties of the error terms. The limited-information and full-information estimators proposed in this paper are consistent and have the same asymptotic distributions as the limited-information and full-information maximum likelihood estimators, respectively. The full-information estimation of simultaneous equations models with auto-regressive errors has been discussed by Sargan (1961), Hendry (1971), Chow and Fair (1973), and Dhrymes (1971). Sargan originally proposed the full-information maximum likelihood estimation of such models, and Hendry and Chow and Fair have recently developed computationally feasible methods for obtaining the maximum likelihood estimates. Hendry considered only the case of completely unrestricted auto-regressive coefficient matrices (i.e., no zero elements), whereas Chow and Fair considered the case of restricted auto-regressive coefficient matrices as well. Dhrymes has recently proposed the three-stage least squares estimator of simultaneous equations models with auto-regressive errors. Dhrymes also considered only the case of completely unrestricted auto-regressive coefficient matrices. The limited-information estimation of simultaneous equations models with auto-regressive errors has been discussed by Sargan (1961), Amemiya (1966), and Fair (1970), among others. Sargan proposed the limited-information maximum likelihood estimation of such models, and Amemiya and Fair considered various two-stage least souares estimators of such models. Most of the work on limited-information estimators has been concerned with the case of diagonal auto-regressive coefficient matrices. Brundy and Jorgenson's criticism of the twoand three-stage least squares estimators, namely, that the first stage involves estimating reduced form equations with a very large number of variables included in them, holds even more so for models with auto-regressive errors. For these models, the reduced form equations include not only all of the predetermined variables in the system but also all of the lagged endogenous and lagged predetermined variables. In fact, one of the main purposes of the work by Fair (1970) was to suggest ways in which the number of variables used in the first stage regressions of two-stage least squares might be decreased with perhaps small loss of asymptotic efficiency. The advantage of the instrumental-variables techniques proposed in the Brundy-Jorgenson paper and in this paper is that the first stage regressions need not be run.