A New Framework for Estimation of Unconditional Quantile Treatment Effects: The Residualized Quantile Regression (RQR) Model

The identification of unconditional quantile treatment effects (QTE) has become increasingly popular within social sciences. However, current methods to identify unconditional QTEs of continuous treatment variables are incomplete. Contrary to popular belief, the unconditional quantile regression model introduced by Firpo, Fortin, and Lemieux (2009) does not identify QTE, while the propensity score framework of Firpo (2007) allows for only a binary treatment variable, and the generalized quantile regression model of Powell (2020) is unfeasible with high-dimensional fixed effects. This paper introduces a two-step approach to estimate unconditional QTEs. In the first step, the treatment variable is regressed on the control variables using an OLS. Then, in the second step, the residuals from the OLS are used as the treatment variable in a quantile regression model. The intuition is that the first step decomposes the variance of the treatment variable into a piece explained by the observed control variables and a residual piece independent of the controls. Since the control variables purge the treatment of confounding, they are redundant in the second step. Therefore, our RQR approach circumvents the problem that the inclusion of controls together with the treatment variable in CQR changes the interpretation of the treatment coefficients. Unlike much of the literature on quantile regression, this two-step residualized quantile regression framework is easy to understand, computationally fast, and can include high-dimensional fixed effects.