Optimal instruments

rdf:langString Optimal instruments

rdf:langString In statistics and econometrics, optimal instruments are a technique for improving the efficiency of estimators in conditional moment models, a class of semiparametric models that generate conditional expectation functions. To estimate parameters of a conditional moment model, the statistician can derive an expectation function (defining "moment conditions") and use the generalized method of moments (GMM). However, there are infinitely many moment conditions that can be generated from a single model; optimal instruments provide the most efficient moment conditions. As an example, consider the nonlinear regression model where y is a scalar (one-dimensional) random variable, x is a random vector with dimension k, and θ is a k-dimensional parameter. The conditional moment restriction is consistent with infinitely many moment conditions. For example: More generally, for any vector-valued function z of x, it will be the case that . That is, z defines a finite set of orthogonality conditions. A natural question to ask, then, is whether an asymptotically efficient set of conditions is available, in the sense that no other set of conditions achieves lower asymptotic variance. Both econometricians and statisticians have extensively studied this subject. The answer to this question is generally that this finite set exists and have been proven for a wide range of estimators. Takeshi Amemiya was one of the first to work on this problem and show the optimal number of instruments for nonlinear simultaneous equation models with homoskedastic and serially uncorrelated errors. The form of the optimal instruments was characterized by Lars Peter Hansen, and results for nonparametric estimation of optimal instruments are provided by Newey. A result for nearest neighbor estimators was provided by Robinson.