  Estimating causal effects under exogeneity hinges on two key assumptions:
unconfoundedness and overlap. Researchers often argue that unconfoundedness is
more plausible when more 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
observational studies with high-dimensional covariates and formalize
curse-of-dimensionality argument, suggesting that these assumptions are
stronger than investigators likely realize. Our key innovation is to explore
how strict overlap restricts global discrepancies between the covariate
distributions in the treated and control populations. Exploiting results from
information theory, we derive explicit bounds on the average imbalance in
covariate means under strict overlap and show that these bounds become more
restrictive as the dimension grows large. We discuss how these implications
interact with assumptions and procedures commonly deployed in observational
causal inference, including sparsity and trimming.
