Abstract: We consider the estimation of a causal effect when the confounders are subject to missingness. We allow the missingness of the confounders to be nonignorable; that is, the missingness may depend on the missing confounders, conditional on the observed data. The identification has been discussed in the literature; however, few studies have focused on semiparametric causal inference with nonignorably missing confounders. To address this, we propose three semiparametric estimators: the inverse probability weighting (IPW), regression, and doubly robust (DR) estimators. The IPW and regression estimators require a correct specification of the propensity scores and the regression models for the confounders and outcome, respectively. Assuming the selection bias odds ratio function is always correctly specified, the DR estimator uses both sets of models and is consistent if either set of models, but not necessarily both, is correctly specified. We investigate the finite-sample performance of our proposed semiparametric estimators using simulation studies and apply our estimators to SO₂ emissions data.

Key words and phrases: Causal inference, doubly robustness, nonignorable missing, outcome-independent missingness, shadow variable.