The Influence Function of Semiparametric Estimators

TL;DR

This paper characterizes the influence function of semiparametric estimators, showing it as a Gateaux derivative, and applies this to sensitivity analysis and bias correction in policy evaluation.

Contribution

It generalizes the influence function concept to semiparametric estimators dependent on nonparametric first steps, providing explicit formulas and applications.

Findings

Explicit influence functions derived for various first steps.

Demonstrated no sensitivity to endogeneity in a gasoline demand case.

Generalized bias correction formulas for policy analysis.

Abstract

Many useful parameters depend on nonparametric first steps. Examples include games, dynamic discrete choice, average exact consumer surplus, and treatment effects. Often estimators of these parameters are asymptotically equivalent to a sample average of an object referred to as the influence function. The influence function is useful in local policy analysis, in evaluating local sensitivity of estimators, constructing debiased machine learning estimators, in efficiency comparisons, and in formulating primitive regularity conditions for asymptotic normality, We show that the influence function is a Gateaux derivative with respect to a smooth deviation evaluated at a point mass. This result generalizes the classic Von Mises (1947) and Hampel (1974) calculation to estimators that depend on smooth nonparametric first steps. We give explicit influence functions for first steps that satisfy exogenous or endogenous orthogonality conditions. We use these results to generalize the omitted variable bias formula for regression to policy analysis for and sensitivity to structural changes. We apply this analysis and find no sensitivity to endogeneity of average equivalent variation estimates in a gasoline demand application.