Smoothed Estimating Equations for Instrumental Variables Quantile Regression

The moment conditions or estimating equations for instrumental variables quantile regression involves the discontinuous indicator function. We instead use smoothed estimating equations, with bandwidth h. This is known to allow higher-order expansions that justify bootstrap refinements for inference. Computation of the estimator also becomes simpler and more reliable, especially with (more) endogenous regressors. We show that the mean squared error of the vector of estimating equations is minimized for some h > 0, which also reduces the mean squared error of the parameter estimators. The same h also minimizes higher-order type I error for a χ2 test, leading to improved size-adjusted power. Our plug-in bandwidth consistently reproduces all of these properties in simulations.