Causal inference without complete information: A closed-form solution

Understanding the effect of certain factors or interventions is the objective of many researchers and decision-makers. However, when using quantitative analyses for this purpose, it is virtually impossible to include all relevant data, which often leads to biased coefficients that only indicate correlation rather than effect. To help overcome this limitation, we introduce relatively simple formulas (closed-form solutions) to estimate direct relationships between variables in the presence of unobserved factors (confounders) that simultaneously affect the two variables of interest. The estimated coefficient can be interpreted as causal when it is possible to rule out bidirectional relationships. Simulations show that our estimates match the unbiased coefficients up to 15 decimal places when the controls are independent of the omitted variables, significantly outperforming the respective Ordinary Least Squares (OLS) estimates in all scenarios considered. We also find that, even with moderate correlation between the controls and the unobserved variables (up to 0.40), our method leads to estimates closer to the actual causal coefficients as compared to the OLS estimates in more than 70% of the cases. A real-world dataset is used to illustrate the application of the method in the field of management.