On nearly assumption-free tests of nominal confidence interval coverage for causal parameters estimated by machine learning

TL;DR

This paper introduces assumption-free tests called AFECTs that assess the coverage accuracy of confidence intervals for causal parameters estimated by machine learning, addressing bias issues without relying on smoothness or sparsity assumptions.

Contribution

The paper develops AFECTs, a novel class of tests that can verify and bound the coverage of confidence intervals for causal effects without assumptions on nuisance parameters.

Findings

AFECTs can falsify the null hypothesis of small bias.

They provide upper bounds on true coverage of confidence intervals.

The tests are valid without smoothness or sparsity assumptions.

Abstract

For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hat{\psi}_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal $(1 - \alpha)$ Wald confidence interval $\hat{\psi}_{1} \pm z_{\alpha / 2} \widehat{\mathsf{se}} [\hat{\psi}_{1}]$ may still undercover even in large samples, because the bias of $\hat{\psi}_{1}$ may be of the same or even larger order than its standard error of order $n^{-1/2}$. In this paper, we introduce essentially assumption-free tests that (i) can falsify the null hypothesis that the bias of $\hat{\psi}_{1}$ is of smaller order than its standard error, (ii) can provide an upper confidence bound on the true coverage of the Wald interval, and (iii) are valid under the null under no smoothness/sparsity assumptions on the nuisance parameters. The tests, which we refer to as \underline{A}ssumption \underline{F}ree \underline{E}mpirical \underline{C}overage \underline{T}ests (AFECTs), are based on a U-statistic that estimates part of the bias of $\hat{\psi}_{1}$.