  Unmeasured covariates constitute one of the important problems in causal
inference. Even if there are some unmeasured covariates, some instrumental
variable methods such as a two-stage residual inclusion (2SRI) estimator, or a
limited-information maximum likelihood (LIML) estimator can obtain an unbiased
estimate for causal effects despite there being nonlinear outcomes such as
binary outcomes; however, it requires that we specify not only a correct
outcome model but also a correct treatment model. Therefore, detecting correct
models is an important process. In this paper, we propose two model selection
procedures: AIC-type and BIC-type, and confirm their properties. The proposed
model selection procedures are based on a LIML estimator. We prove that a
proposed BIC-type model selection procedure has model selection consistency,
and confirm their properties of the proposed model selection procedures through
simulation datasets.
