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Identification in Linear Simultaneous Equations Models with Covariance Restrictions: An Instrumental Variables Interpretation

Jerry A. Hausman
William E. Taylor

Abstract

Necessary and sufficient conditions for identification with linear coefficient and covariance restrictions are developed in a limited information context. For the limited information case, covariance restrictions aid identification if and only if they imply that a set of endogenous variables is predetermined in the equation of interest (generalizing the idea of recursiveness). Under full information, covariance restrictions imply that residuals from other equations are predetermined in a particular equation and, under certain conditions, can aid in identification. Sufficient conditions for identification are obtained for the hierarchical system in which the identification of a particular equation does not depend upon the identifiability of higher-numbered equations. In the general case, the FIML first order conditions show that if the system of equations is identifiable as a whole, covariance restrictions cause residuals to behave as instruments. In both limited and full information settings, the link between identification and estimation is worked out: restrictions useful for identification yield instruments required for estimation.