Direct instrumental nonparametric estimation of inverse regression functions

This paper treats the estimation of the inverse g−1 of a monotonic function g satisfying E[Y−g(X)|W]=0 where (X,W) is continuously distributed. Using instrumental restrictions, many parameters of interest in econometrics can be expressed as inverses of functions satisfying such a conditional moment. As far as I know, consistent estimators are available only if g(X)=E[Y|X], which precludes endogenous models. This, and other technical concerns, motivates a methodology for estimating g−1 in one step from the data when W contains components excluded from X. The presented estimator achieves this objective by identifying g−1 as the solution of a nonlinear integral equation whose sample analog can be estimated nonparametrically. The existence of a ‘well-behaved’ joint density function for (X,W) produces an ill-posed problem, namely the solution is discontinuous in the data. To solve this, the estimator is regularized with the Tikhonov technique. If g−1 is smooth in a certain sense, then the MSE rate of convergence in a global metric is equal to ϑ(n−r∕(2r+1)) where ϑ is a continuous function with ϑ(0)=0, and r>1 denotes the number of partial derivatives for the joint density of (X,W) with respect to its second argument. The analytical expression of ϑ depends on a link function characterizing the smoothness of g−1. This result holds for a broad class of link functions belonging to a suitable space.