Addressing the omitted variables problem in a three-equation linear system

The omitted variables problem in regression analysis poses a major challenge for causal inference from observational data. This paper introduces a novel framework for addressing this issue within a three-equation linear system featuring shared omitted variables. As a baseline, we propose the Common Omission Solution (COS) estimator, assuming observed explanatory variables across equations are mutually uncorrelated, and establish its consistency and asymptotic normality. To accommodate more realistic settings with correlated regressors, we develop the Divide-and-Conquer COS (DC-COS) method. It uses clustering and trimming to construct approximately orthogonal data subsets, applies COS within each subset, and aggregates results via robust median pooling. Monte Carlo simulations confirm the estimators' effectiveness, and empirical applications illustrate their utility.