What to do with a mediating and confounding variable that is partly a mediator and partly a confounder in causal inference

What variable to control for is of great importance if the goal of analysis is causal inference using observational data. The literature in causal inference is unanimous that we should control for a confounder while we should not control for a mediator that comes in the middle of the causal path. However, suppose that one variable is simultaneously a confounder and a mediator. In other words, some observations in this variable are observed before the treatment variable, while other observations in this same variable are observed after the treatment variable. Let us call this variable a mediating confounder. If we control for a mediating confounder, there will be bias from mediation; if we do not control for a mediating confounder, there will be an omitted variable bias. Therefore, this seems to be an unsolvable problem. This article proposes a method of multiple imputation with cold deck imputation, where we regard a mediating confounder as a special type of missing data. The results of Monte Carlo simulations present the evidence that the proposed method is indeed superior to the existing methods. Real data on economic and political variables are used to illustrate the usefulness of the proposed method.