Optimal Instrumental Variable Estimation for Linear Models With Stochastic Regressors Using Estimating Functions
A. C. Singh, R. Prabhakar Rao · Lecture notes-monograph series · 1997
In the usual Gauss-Markov (GM) framework of structural linear models, the GM estimators of the regression parameters become inconsistent if at least one of the regressors is correlated with the model error.The reason for this is that the transformation matrix in the GM estimating equation, which transforms the data to the parameter space (this happens to coincide with the design matrix .X"), cannot be regarded as conditionally fixed.Using a generalization of the method of estimating function of Godambe and Thompson (1989) to structural models, it is shown that an asymptotically consistent and optimal (in a restricted sense) estimator can be obtained by replacing the transfromation matrix X by E C (X), the linear regression of X on a given set of conditioning variables; the optimality is restricted in that it depends on the conditioning set.The matrix E C (X) can be viewed as a working (because of restricted optimality) transformation matrix with the desirable property of being uncorrelated with the model error but correlated with X.Although finding an unrestricted optimal transformation matrix is not generally feasible in practice, it is shown using the estimating function framework that a lower bound to the asymptotic covariance can be found.This bound is then used to propose a measure of asymptotic efficiency of the estimator.It is observed that the concept of a working transformation matrix is equivalent to that obtained from the method of instrumental variables.Through examples from different areas of modelling such as simultaneous equations, latent variables, and measurement errors, it is illustrated that the structural model estimating function provides a unifying principle which recovers existing results as well as leads to new results.