A Variance-Based Test for Heterogeneous Treatment Effects

This paper proposes a robust nonparametric hypothesis test for the existence of heterogeneous treatment effects. We focus on the variance of the Conditional Average Treatment Effect (CATE) as a natural omnibus parameter, where a non-zero variance implies the presence of relevant heterogeneity. Standard inference for this parameter faces a fundamental theoretical challenge. On one hand, evaluating variance components on the same sample leads to null degeneracy, where the asymptotic variance collapses to zero under the null hypothesis of homogeneity, invalidating standard Gaussian inference. On the other hand, decoupling the empirical processes via standard sample-splitting breaks the Neyman orthogonality of the doubly robust scores due to their nonlinear squared loss, which prevents the cancellation of first-order regularization biases. To resolve this challenge, we propose a novel Intra-Fold Sample-Splitting algorithm. By evaluating variance components on mutually disjoint subsamples while coupling them to identical out-of-fold nuisance estimators, our procedure achieves algebraic cancellation of the nuisance biases. We prove this restores consistency and asymptotic normality, and ensures Type I error control. Monte Carlo simulations demonstrate that the proposed test achieves superior size control relative to existing tests while maintaining high power. In an empirical application to the NSW job training program, the test detects significant heterogeneity that traditional nonparametric tests fail to uncover.