  In observational causal inference, exact covariate matching plays two
statistical roles: (i) it effectively controls for bias due to measured
confounding; (ii) it justifies assumption-free inference based on randomization
tests. This paper shows that inexact covariate matching does not always play
these same roles. We find that inexact matching often leaves behind
statistically meaningful bias and that this bias renders standard randomization
tests asymptotically invalid. We therefore recommend additional model-based
covariate adjustment after inexact matching. In the framework of local
misspecification, we prove that matching makes subsequent parametric analyses
less sensitive to model selection or misspecification. We argue that gaining
this robustness is the primary statistical role of inexact matching.
