Estimation of Constrained Singular Seemingly Unrelated Regression Models

The topic of this paper is the problem of a singular disturbance covariance matrix in (seemingly unrelated) systems of linear regression equations. This singularity is considered as being caused by exact linear restrictions on the endogenous variables, adding-up to a predetermined aggregate. It is well known, that the estimation of such systems require the substitution of one equation by the "adding up condition" as proposed by Barten (1969), or, as alternatively proposed by Theil (1971), a modification of the GLS estimator. The consequently occuring question of the invariance of the parameter estimates to the choice of which equation is deleted has been discussed exhaustively by Powell (1969) for GLS and Barten (1969) for ML estimation. Dhrymes and Schwarz (1987a,b) pointed out the parallels between the different procedings and argued that the estimator of Theil (1971) fails to exist in most of the practically relevant constellations. The rank conditions given by Theil (1971) and the corresponding objections of Dhrymes and Schwarz (1987a,b) are substantially simplified and generalized. In the second section of this paper we introduce a general model, where we discuss all necessary preliminaries, assumptions and constraints and consider relevant model alternatives. Estimation topics are discussed in section three. In section four we derive less restrictive rank conditions than postulated by Theil (1971) in a more general framework as in the corresponding considerations of Dhrymes and Schwarz (1987a,b). The special case of identical regressors in every equation of the system and homoscedasticity is considererd in section five. Section six deals with the problem of the deletion choice. Strongly related to the work of Powell (1969) and Barten (1969) we show, that under consideration of the constraints, the estimator is invariant to the deletion choice and equivalent to the modified GLS estimator. In section seven we briefly discuss starting conditions for feasible GLS estimation.