Quasi-Maximum Likelihood based Model Selection Procedures for Binary Outcomes

Unmeasured covariate is one of the important problems in the causal inference. Even if there are some unmeasured covariates, a two stage residual inclusion (2SRI) estimator can estimate an unbiased estimate for causal effects even when there are nonlinear outcomes such as binary outcomes, however, we need to specify not only a correct outcome model but also a correct treatment model. Therefore, to detect a correct model is an important process when use a 2SRI. In this paper, I propose some model selection procedures and confirm their properties. The proposed model selection procedures are based on a Quasi-maximum likelihood (QML) estimator which has the similar features as a 2SRI. I prove that a proposed BIC-type model selection procedure has a model selection consistency, and confirm properties of the proposed model selection procedures through simulation datasets. Simulation results show that these procedures have the same properties as ordinary ones. Also, it is shown that a 2SRI does not work when use model selection procedures in the sense that the true second model cannot be selected.