On the Effect of a Binary Treatment in the Presence of a Control Covariate

The nonparametric regression of a response variable on a binary treatment and a control covariate equals the nonparametric regression of the response on the control plus the conditional (given the control) linear regression of the resulting residual on the residual of the nonparametric regression of the treatment on the control. While similar decompositions are typically obtained using the technique of orthogonal projection applicable to random variables with finite second moments, this result holds if the response only has a finite first moment and allows the control to be completely arbitrary. Consequently, if the propensity score of the treatment is bounded away from zero and one, the treatment effect, after controlling for the covariate, equals the ratio of the conditional covariance of the response and the treatment to the conditional variance of the treatment. AMS 2020 subject classification: Primary 62G08, Secondary 62J05, 91F10, 62P20, 62P25