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.
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.
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.
[5] The form of the optimal instruments was characterized by Lars Peter Hansen,[6] and results for nonparametric estimation of optimal instruments are provided by Newey.
[7] A result for nearest neighbor estimators was provided by Robinson.
[8] The technique of optimal instruments can be used to show that, in a conditional moment linear regression model with iid data, the optimal GMM estimator is generalized least squares.
The task is to choose z to minimize the asymptotic variance of the resulting GMM estimator.
If the data are iid, the asymptotic variance of the GMM estimator is where