A Theory of Dichotomous Valuation with Applications to Variable Selection

An econometric or statistical model may undergo a marginal gain if we admit a new variable to the model, and a marginal loss if we remove an existing variable from the model. We quantify the value of a variable to the model by its expected marginal gain and expected marginal loss in all modeling scenarios. Assuming the equality of opportunity among all candidate variables, we derive a few formulas which evaluate the overall performance of each variable in potential modeling scenarios. However, the value is not symmetric to marginal gain and marginal loss; thus, we introduce an unbiased solution. Simulation studies show that our bias-adjusted approaches significantly outperform a few variable selection methods used in practice. The tools employed are concepts from cooperative game theory, and the results explore several novel features of the Shapley value. We even bridge the Shapley value and the Banzhaf value with binomial distributions.