Covariate balancing propensity score for a continuous treatment: Application to the efficacy of political advertisements

Propensity score matching and weighting are popular methods when estimating causal effects in observational studies. Beyond the assumption of unconfoundedness, however, these methods also require the model for the propensity score to be correctly specified. The recently proposed covariate balancing propensity score (CBPS) methodology increases the robustness to model misspecification by directly optimizing sample covariate balance between the treatment and control groups. In this paper, we extend the CBPS to a continuous treatment. We propose the covariate balancing generalized propensity score (CBGPS) methodology, which minimizes the association between covariates and the treatment. We develop both parametric and nonparametric approaches and show their superior performance over the standard maximum likelihood estimation in a simulation study. The CBGPS methodology is applied to an observational study, whose goal is to estimate the causal effects of political advertisements on campaign contributions. We also provide open-source software that implements the proposed methods.