Statistical Inference in Causal Partial Identification under Smooth Densities

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, which precludes the asymptotic normality and hinders statistical inference. In this paper, we exploit the smoothness in the marginal densities of covariates and potential outcomes, and develop a wavelet-based primal approach for COT that attains a faster convergence rate. Moreover, for quadratic cost functions, we establish a stability result for COT and prove asymptotic normality of the proposed estimator, enabling valid statistical inference for the PI set. Empirically, we validate the estimation and inference performance of our approach through numerical experiments in comparison with existing benchmarks.