Causal Inference with High-dimensional Discrete Covariates

TL;DR

This paper analyzes the challenges of estimating causal effects with high-dimensional discrete covariates, establishing bounds on estimator errors, and proposing improved methods under additional structural assumptions.

Contribution

It provides theoretical bounds for common estimators, derives minimax lower bounds, and introduces new estimators that leverage structure for faster convergence.

Findings

Estimator mean squared error bounded by d^2/n^2 + 1/n

Minimax lower bound of order d^2/(n^2 log^2 n) + 1/n

Proposed estimators achieve faster convergence under additional assumptions

Abstract

When estimating causal effects from observational studies, researchers often need to adjust for many covariates to deconfound the non-causal relationship between exposure and outcome, among which many covariates are discrete. The behavior of commonly used estimators in the presence of many discrete covariates is not well understood since their properties are often analyzed under structural assumptions including sparsity and smoothness, which do not apply in discrete settings. In this work, we study the estimation of causal effects in a model where the covariates required for confounding adjustment are discrete but high-dimensional, meaning the number of categories $d$ is comparable with or even larger than sample size $n$. Specifically, we show the mean squared error of commonly used regression, weighting and doubly robust estimators is bounded by $\frac{d^2}{n^2}+\frac{1}{n}$. We then prove the minimax lower bound for the average treatment effect is of order $\frac{d^2}{n^2 \log^2 n}+\frac{1}{n}$, which characterizes the fundamental difficulty of causal effect estimation in the high-dimensional discrete setting, and shows the estimators mentioned above are rate-optimal up to log-factors. We further consider additional structures that can be exploited, namely effect homogeneity and prior knowledge of the covariate distribution, and propose new estimators that enjoy faster convergence rates of order $\frac{d}{n^2} + \frac{1}{n}$, which achieve consistency in a broader regime. The results are illustrated empirically via simulation studies.