A Useful Interpretation of R-Squared in Binary Choice Models (Or, Have We Dismissed the Good Old R-Squared; Prematurely)

Abstract

The discreditation of the Linear Probability Model (LPM) has led to the dismissal of the standard \(R^{2}\) as a measure of goodness-of-fit in binary choice models. It is argued that as a descriptive tool the standard \(R^{2}\) is still superior to the measures currently in use. In the LPM model \(R^{2}\) has a simple interpretation: it equals the difference between the average predicted probability in the two groups. It also measures the fraction of the explained part of the variance (SSR) due to the difference between the conditional means (SSB). Given \(R^{2}\) and the sample proportion \(P\) one can calculate the conditional means, \(\bar{P}_{0}\) and \(\bar{P}_{1}\). This interpretation still holds for non-linear cases when \(R\) is computed as the regression coefficient of the predicted value on the dependent binary variable: However, even if other definitions of \(R^{2}\) are used in this case (e. g., the share of the variance explained by the regression, or the correlation coefficient between true and predicted values), the measure is very close to \(\bar{P}_{1} - \bar{P}_{0}\).