Subsample Least Squares Estimator for Heterogeneous Effects of Multiple Treatments with Any Outcome Variable

TL;DR

This paper introduces a robust subsample least squares estimator that accurately estimates heterogeneous treatment effects across various outcome types, overcoming biases of traditional OLS in non-constant effect scenarios.

Contribution

It proposes a modified subsample OLS method that is consistent for estimating overlap-weighted heterogeneous effects regardless of outcome type or model misspecification.

Findings

OLS is inconsistent with heterogeneous effects.

The proposed subsample OLS is robust and consistent.

Effective for various outcome variable types.

Abstract

For multiple treatments D=0,1,...,J, covariates X and outcome Y, the ordinary least squares estimator (OLS) of Y on (D1,...,DJ,X) is widely applied to a constant-effect linear model, where Dj is the dummy variable for D=j. However, the treatment effects are almost always X-heterogeneous in reality, or Y is noncontinuous, to invalidate such a linear model. The blind hope of practitioners is that the OLS "somehow" estimates a sensible average of the unknown X-heterogeneous effects. This paper shows that, unfortunately, the OLS is inconsistent unless all treatment effects are constant, because the estimand of the Dd-slope involves the X-heterogeneous effects of all treatments, not just Dd. One way to overcome this "contamination" problem is the OLS of Y on Dd-E(Dd|X, D=0,d) using only the subsample D=0,d, and this paper proposes a modified version of the subsample OLS that is robust to misspecifications of E(Dd|X, D=0,d). The robustified subsample OLS is proven to be consistent for an "overlap weight" average of the X-heterogeneous effect of Dd for any form of Y (continuous, binary, count, ...).