Smoothed estimating equations for instrumental variables quantile regression

TL;DR

This paper introduces smoothed estimating equations for instrumental variables quantile regression, improving estimator accuracy, test power, and computational reliability through optimal bandwidth selection.

Contribution

It proposes a novel smoothing approach for estimating equations that reduces MSE, enhances test performance, and simplifies computation in IV quantile regression.

Findings

Smoothed estimating equations decrease asymptotic MSE.

Optimal bandwidth improves test size and power.

Simulation results confirm finite-sample advantages.

Abstract

The moment conditions or estimating equations for instrumental variables quantile regression involve the discontinuous indicator function. We instead use smoothed estimating equations (SEE), with bandwidth $h$. We show that the mean squared error (MSE) of the vector of the SEE is minimized for some $h>0$, leading to smaller asymptotic MSE of the estimating equations and associated parameter estimators. The same MSE-optimal $h$ also minimizes the higher-order type I error of a SEE-based $\chi^2$ test and increases size-adjusted power in large samples. Computation of the SEE estimator also becomes simpler and more reliable, especially with (more) endogenous regressors. Monte Carlo simulations demonstrate all of these superior properties in finite samples, and we apply our estimator to JTPA data. Smoothing the estimating equations is not just a technical operation for establishing Edgeworth expansions and bootstrap refinements; it also brings the real benefits of having more precise estimators and more powerful tests. Code for the estimator, simulations, and empirical examples is available from the first author's website.