Specification in Simultaneous Linear Equations Models: The Relation Between A Priori Specifications and Resulting Estimators

The estimation of coefficients of a linear model is discussed, where we do not assume that the random vector X satisfies the model, but that the model is only a model of best fit. We consider both the situation without additional constraints on the coefficients and the situation with differentiable constraints. In both cases, the vector of parameters is a differentiable function of a covariance matrix, making the δ-method applicable. It is shown that the asymptotic distribution of the estimators is normal. Closed formulas for their asymptotic covariance matrices are derived.