Statistical Inference in Causal Partial Identification with Smooth Densities

TL;DR

This paper introduces a wavelet-based method for causal inference using optimal transport, leveraging smooth densities to achieve asymptotic normality and enable valid statistical inference in high-dimensional settings.

Contribution

It develops a novel wavelet-based primal approach for conditional optimal transport with multivariate data, establishing asymptotic normality under smoothness assumptions.

Findings

The method achieves accurate estimation of PI sets in simulations.

It provides valid confidence intervals for causal quantities.

Outperforms existing benchmarks in numerical experiments.

Abstract

Many causal quantities are only partially identifiable due to the inherent missingness of potential outcomes, and the associated partial identification (PI) sets can be obtained by solving an optimal transport (OT) problem. Covariates often provide additional information about the potential outcomes and thus yield tighter PI sets, which can be obtained via conditional optimal transport (COT). However, COT-based PI set estimators are susceptible to the curse of dimensionality in the covariates and outcomes, which precludes the asymptotic normality and hinders statistical inference. In this paper, we exploit smoothness in the marginal densities of covariates and potential outcomes and develop a wavelet-based primal method for COT with multivariate outcomes and covariates. Moreover, for quadratic cost functions, we establish a stability result for COT and prove asymptotic normality of the proposed estimator. This characterization of the asymptotic distribution enables valid statistical inference for the partial identification set. Empirically, we validate the estimation and inference performance of our approach through numerical experiments in comparison with existing benchmarks.