Overlap in Observational Studies with High-Dimensional Covariates
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Causal inference in observational settings typically rests on a pair of identifying assumptions: (1) unconfoundedness and (2) covariate overlap, also known as positivity or common support. Investigators often argue that unconfoundedness is more plausible when many covariates are included in the analysis. Less discussed is the fact that covariate overlap is more difficult to satisfy in this setting. In this paper, we explore the implications of overlap in high-dimensional observational studies, arguing that this assumption is stronger than investigators likely realize. Our main innovation is to frame (strict) overlap in terms of bounds on a likelihood ratio, which allows us to leverage and expand on existing results from information theory. In particular, we show that strict overlap bounds discriminating information (e.g., Kullback-Leibler divergence) between the covariate distributions in the treated and control populations. We use these results to derive explicit bounds on the average imbalance in covariate means under strict overlap and a range of assumptions on the covariate distributions. Importantly, these bounds grow tighter as the dimension grows large, and converge to zero in some cases. We examine how restrictions on the treatment assignment and outcome processes can weaken the implications of certain overlap assumptions, but at the cost of stronger requirements for unconfoundedness. Taken together, our results suggest that adjusting for high-dimensional covariates does not necessarily make causal identification more plausible.
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