A critical review of the use of R² in risk equalization research

Nearly all empirical studies that estimate the coefficients of a risk equalization formula present the value of the statistical measure R². The R²-value is often (implicitly) interpreted as a measure of the extent to which the risk equalization payments remove the regulation-induced predictable profits and losses on the insured, with a higher R²-value indicating a better performance. In many cases, however, we do not know whether a model with R² = 0.30 reduces the predictable profits and losses more than a model with R² = 0.20. In this paper we argue that in the context of risk equalization R² is hard to interpret as a measure of selection incentives, can lead to wrong and misleading conclusions when used as a measure of selection incentives, and is therefore not useful for measuring selection incentives. The same is true for related statistical measures such as the Mean Absolute Prediction Error (MAPE), Cumming's Prediction Measure (CPM) and the Payment System Fit (PSF). There are some exceptions where the R² can be useful. Our recommendation is to either present the R² with a clear, valid, and relevant interpretation or not to present the R². The same holds for the related statistical measures MAPE, CPM and PSF.