Statistical Inference in Causal Partial Identification with Smooth Densities

This study tackles the challenge of statistical inference in causal partial identification, a problem arising from the unobservability of potential outcomes. Traditional methods struggle in high-dimensional settings, often relying on dual formulations that require strong smoothness assumptions on potentials, which are difficult to verify and computationally intensive. To address this, the authors introduce a novel wavelet-based primal estimator that exploits the smoothness of marginal densities, enabling efficient density estimation without solving complex optimization problems. This approach is particularly effective under quadratic cost functions, where the authors establish a stability result for the conditional optimal transport (COT) problem, leading to the proof of asymptotic normality of the estimator. The theoretical results demonstrate near-parametric convergence rates, significantly faster than existing methods, especially in high-dimensional covariate and outcome spaces. Numerical experiments on simulated and real datasets confirm the estimator’s superior accuracy and coverage, with errors reduced by over 30% and confidence intervals achieving 95% coverage. These advances have profound implications for causal inference, allowing practitioners to quantify uncertainty more reliably in complex, high-dimensional scenarios. The methodology bridges the gap between optimal transport theory and causal analysis, providing a scalable, theoretically grounded inference framework. Looking ahead, the authors suggest extending their approach to non-smooth densities, non-quadratic costs, and large-scale applications, promising a versatile tool for future causal and statistical research.