The causal interpretation of estimated associations in regression models

A common causal identification strategy in political science is selection on observables. This strategy assumes one observes a set of covariates that is, after statistical adjustment, sufficient to make treatment status as-if random. Under adjustment methods such as matching or inverse probability weighting, coefficients for control variables are treated as nuisance parameters and are not directly estimated. This is in direct contrast to regression approaches where estimated parameters are obtained for all covariates. Analysts often find it tempting to give a causal interpretation to all the parameters in such regression models—indeed, such interpretations are often central to the proposed research design. In this paper, we ask when we can justify interpreting two or more coefficients in a regression model as causal parameters. We demonstrate that analysts must appeal to causal identification assumptions to give estimates causal interpretations. Under selection on observables, this task is complicated by the fact that more than one causal effect might be identified. We show how causal graphs provide a framework for clearly delineating which effects are presumed to be identified and thus merit a causal interpretation, and which are not. We conclude with a set of recommendations for how researchers should interpret estimates from regression models when causal inference is the goal.