Abstract: Multiple imputation is widely used to handle missing data. Although Rubin’s combining rule is simple, it is not clear whether the standard multiple imputation inference is consistent when coupled with the commonly used full-sample estimators. Here, we establish a unified martingale representation of multiple imputation for a wide class of asymptotically linear full-sample estimators. This representation invokes the wild bootstrap inference to provide a consistent variance estimation under a correct specification of the imputation models. As a motivating application, we use the proposed method to estimate the average causal effect (ACE) with partially observed confounders in a causal inference. Our framework applies to asymptotically linear ACE estimators, including the regression imputation, weighting, and matching estimators. Lastly, we extend the proposed method to include scenarios in which both the outcome and the confounders are subject to missingness, and when the data are missing not at random.

Key words and phrases: Causality, congeniality, influence function, martingale representation, weighted bootstrap.