Covariate balancing estimation and model selection for difference-in-differences approach

Remarkable progress has been made in difference-in-differences (DID) approaches to causal inference that estimate the average effect of a treatment on the treated (ATT). Of these, the semiparametric DID (SDID) approach incorporates a propensity score analysis into the DID setup. Supposing that the ATT is a function of covariates, we estimate it by weighting the inverse of the propensity score. In this study, as one way to make the estimation robust to the propensity score modeling, we incorporate covariate balancing. Then, by attentively constructing the moment conditions used in the covariate balancing, we show that the proposed estimator is doubly robust. In addition to the estimation, we also address model selection. In practice, covariate selection is an essential task in statistical analysis, but even in the basic setting of the SDID approach, there are no reasonable information criteria. Here, we derive a model selection criterion as an asymptotically bias-corrected estimator of risk based on the loss function used in the SDID estimation. We show that a penalty term can be derived that is considerably different from almost twice the number of parameters that often appears in AIC-type information criteria. Numerical experiments show that the proposed method estimates the ATT more robustly compared with the method using propensity scores given by maximum likelihood estimation, and that the proposed criterion clearly reduces the risk targeted in the SDID approach in comparison with the intuitive generalization of the existing information criterion. In addition, real data analysis confirms that there is a large difference between the results of the proposed method and those of the existing method.