The Optimal Construction of Instruments in Nonlinear Regression: Implications for GMM Inference

Interpreted as an instrumental variables estimator, nonlinear least squares constructs its instruments optimally from the explanatory variables using the nonlinear specification of the regression function. This has implications for the use of GMM estimators in nonlinear regression models, including systems of nonlinear regressions, where the explanatory variables are exogenous or predetermined and so serve as their own instruments, and where the restrictions under test are the only source of overidentification. In such situations the use of GMM test criteria involves a suboptimal construction of instruments; the use of optimally constructed instruments leads to conventional non-GMM test criteria. These implications are illustrated with two empirical examples, one a classic study of models of the short-term interest rate.