Estimating Derivatives of Function-Valued Parameters in a Class of Moment Condition Models

TL;DR

This paper introduces a general method for estimating derivatives of function-valued parameters in models defined by moment conditions, applicable to various conditional distribution models like quantile and distribution regression.

Contribution

It provides an explicit derivative expression using the Implicit Function Theorem and develops a sample-based estimator employing local linear smoothing, applicable to diverse economic models.

Findings

Explicit derivative formula derived from moment conditions.

Estimator successfully applied to conditional density and quantile effects.

Method applicable to structural auction models in economics.

Abstract

We develop a general approach to estimating the derivative of a function-valued parameter $\theta_o(u)$ that is identified for every value of $u$ as the solution to a moment condition. This setup in particular covers many interesting models for conditional distributions, such as quantile regression or distribution regression. Exploiting that $\theta_o(u)$ solves a moment condition, we obtain an explicit expression for its derivative from the Implicit Function Theorem, and estimate the components of this expression by suitable sample analogues, which requires the use of (local linear) smoothing. Our estimator can then be used for a variety of purposes, including the estimation of conditional density functions, quantile partial effects, and structural auction models in economics.