Three-Stage Semi-Parametric Inference : Control Variables and Differentiability

We show the usefulness of the path-derivative calculations that were introduced in econometrics by Newey (1994) for multi-step semi-parametric estimators. These estimators estimate a finite-dimensional parameter using moment conditions that depend on nonparametric regressions on observed and estimated regressors that are estimated in the second and first step of the estimation procedure, respectively. Our earlier paper showed that Newey's calculations can be extended to three-step estimators. In the current paper we consider the control variable (CV) estimator and related statistics in semi-parametric econometric models with non-separable errors and regressors that are correlated with these errors. Non-separable econometric models with endogenous regressors are often identified by average moment restrictions that average over control variables, and these control variables are estimated in a first stage by (non)parametric regression. We study aspects of inference for such estimators where we focus on a finite-dimensional parameter vector or statistic. The asymptotic distribution and a closed-form expression for the asymptotic variance of the CV estimator were not available until now. Our path derivative calculations are much simpler than the derivation of the asymptotic distribution by a stochastic expansion that is particularly complicated for multi-step semiparametric estimators. We also consider just and over identification of the parameters. This allows us to propose a diagnostic test for overidentifying restrictions in models with non-separable errors and endogenous regressors. Finally, the path-derivative calculation breaks down if the moment condition is not differentiable. In an example we show that non-difierentiability is associated with irregular behavior of the estimator