Conditional Distributional Treatment Effects: Doubly Robust Estimation and Testing

Abstract

Beyond conditional average treatment effects, treatments may impact the entire outcome distribution in covariate-dependent ways, for example, by altering the variance or tail risks for specific subpopulations. We propose a novel estimand to capture such conditional distributional treatment effects, and develop a doubly robust estimator that is minimax optimal in the local asymptotic sense. Using this, we develop a test for the global homogeneity of conditional potential outcome distributions that accommodates discrepancies beyond the maximum mean discrepancy (MMD), has provably valid type 1 error, and is consistent against fixed alternatives -- the first test, to our knowledge, with such guarantees in this setting. Furthermore, we derive exact closed-form expressions for two natural discrepancies (including the MMD), and provide a computationally efficient, permutation-free algorithm for our test.

Figures (5)

• Figure 1: Simple setting where the conditional average treatment effect is null (left) even though there is DTE heterogeneity (right). In more detail: (left) Scatter plot of $X$ and $Y(a)$, $a\in \{0,1\}$, with: $X, Y(0)\sim\mathrm{Unif}[-1, 1]$ independently and $Y(1)\,|\, X$ a $\mathrm{Unif}[-.5,.5]$ distribution if $X>0$ and a $\mathrm{Unif}([-1,-.5]\cup [.5,1])$ if $X<0$. (right) Proposed witness function for conditional DTE.
• Figure 2: Type 1 error and power at $\alpha=0.05$ across sample sizes and nuisance misspecification regimes. (Left) Scenario satisfying asymptotic guarantees (the product of nuisance estimation errors is $o_p(n^{-1/2})$). (Right) Robustness checks under model misspecification. The proposed tests benefit from double robustness of the estimator; type 1 error is closer than baseline to the nominal level under propensity misspecification; under outcome misspecification, type 1 error is inflated but stable, while power increases with sample size.
• Figure 3: Estimated witness functions with 95% uniform confidence bands for two household profiles (rows) across three wealth outcomes (columns; in $1k$). Shaded regions indicate statistical significance. While Profile 1 exhibits a distributional shift along the first axis, Profile 2 shows no detectable effect.
• Figure 4: Empirical MSE of the SCoDiTE estimator $\bar{\psi}_n$ under the global null, across sample sizes and model misspecificatiom regimes. (Left three panels) The MSE decays sharply to zero when at least one of the nuisance models is correctly specified. (Rightmost panel) The MSE decays at a much slower rate when both propensity and outcome models are simultaneously misspecified.
• Figure 5: Type 1 error and power at $\alpha=0.05$ across sample sizes when propensity scores are known. (Left) The outcome model is correctly specified. (Right) The outcome model is misspecified. Since the propensity scores are known, the product of nuisance estimation errors is $o_p(n^{-1/2})$ in both scenarios. Thus, in contrast to the baseline, type 1 error is controlled at the nominal level and power increases with sample size even under outcome misspecification.

Theorems & Definitions (45)

• Proposition 2.0: Equivalent null
• Lemma 2.0: Existence and form of the EIF
• Theorem 3.1: Weak convergence
• Theorem 3.2: Local asymptotic minimax optimality
• Theorem 3.3: Validity of the test in Alg. \ref{['alg:skcd']}
• Theorem 3.4: Uniform confidence band for the SCoDiTE
• Proposition 3.4: Closed-form MMD statistic from Alg. \ref{['alg:skcd']}
• Proposition 3.4: Closed-form Wald-type statistic from Alg \ref{['alg:skcd']}
• Proposition 4.0: Equivalent null
• proof