A Double Robust Approach for Non-Monotone Missingness in Multi-Stage Data

Multivariate missingness with a non-monotone missing pattern is complicated to deal with in empirical studies. The traditional Missing at Random (MAR) assumption is difficult to justify in such cases. Previous studies have strengthened the MAR assumption, suggesting that the missing mechanism of any variable is random when conditioned on a uniform set of fully observed variables. However, empirical evidence indicates that this assumption may be violated for variables collected at different stages. This paper proposes a new MAR-type assumption that fits non-monotone missing scenarios involving multi-stage variables. Based on this assumption, we construct an Augmented Inverse Probability Weighted GMM (AIPW-GMM) estimator. This estimator features an asymmetric format for the augmentation term, guarantees double robustness, and achieves the closed-form semiparametric efficiency bound. We apply this method to cases of missingness in both endogenous regressor and outcome, using the Oregon Health Insurance Experiment as an example. We check the correlation between missing probabilities and partially observed variables to justify the assumption. Moreover, we find that excluding incomplete data results in a loss of efficiency and insignificant estimators. The proposed estimator reduces the standard error by more than 50% for the estimated effects of the Oregon Health Plan on the elderly.