When Do Covariates Matter? And Which Ones, and How Much?

Authors often add covariates to a base model sequentially either to test a particular coefficient’s “robustness” or to account for the “effects” on this coefficient of adding covariates. This is problematic, due to sequence-sensitivity when added covariates are intercorrelated. Using the omitted variables bias formula, I construct a conditional decomposition that accounts for various covariates’ role in moving base regressors’ coefficients; I also provide a consistent covariance formula. I illustrate this conditional decomposition with NLSY data in an application that exhibits sequence-sensitivity. Related extensions include IV, the fact that my decomposition nests the Oaxaca-Blinder decomposition, and a Hausman-test result.