Semiparametric IV Estimation with Parameter Dependent Instruments

A well-known result in the method of moments literature is that the efficient instruments for the estimation of a model are functions of the conditional expectation of its gradient. Some recent studies have suggested the nonparametric estimation of these instruments when they are of unknown functional form. When these instruments in turn depend on the unknown parameters it has been suggested that these be replaced by preliminary consistent estimates. It is shown here that solving the sample moment equations simultaneously over the instruments and the residuals of the model will generally produce the same asymptotic efficiency and avoid the disadvantages inherent with the use of preliminary estimates.