Median-based Splitting Rules for Causal Trees and Forests

Abstract

Heavy-tailed and skewed outcomes are common in the randomized experiments and observational studies used to estimate heterogeneous treatment effects, yet the mean-squared-error criterion that guides splitting in honest causal trees is sensitive to the extreme values they generate. Building on the causal forest framework (Athey and Imbens, 2016; Wager and Athey, 2018), we introduce the Median Squared Deviation (MSD) criterion, which replaces the leafwise difference in means in the honest splitting objective with the Hodges--Lehmann location estimator while leaving honest leaf estimation and forest inference unchanged. Two further median-based rules, the Median Absolute Deviation (MAD) and the Least Median of Squares (LMS), serve as robust baselines. We evaluate the criteria in a simulation study covering precision, bias, and confidence interval coverage. MSD restricts its robustness to split selection and lowers the error of conditional average treatment effect estimates under heavy-tailed and skewed outcomes. Further, we re-visit two empirical applications: the first analyzes the electoral effects of a Mexican conditional cash transfer program on precinct-level observations, while the second application studies antiretroviral treatments in HIV-positive adults.