Improving risk equalization with constrained regression

Richard C. van Kleef, Thomas G. McGuire, René C. J. A. van Vliet, Wynand P. P. M. van de Ven

State-of-the-art risk equalization models undercompensate some risk groups and overcompensate others, leaving systematic incentives for risk selection. A natural approach to reducing the under- or overcompensation for a particular group is enriching the risk equalization model with risk adjustor variables that indicate membership in that group. For some groups, however, appropriate risk adjustor variables may not (yet) be available. For these situations, this paper proposes an alternative approach to reducing under- or overcompensation: constraining the estimated coefficients of the risk equalization model such that the under- or overcompensation for a group of interest equals a fixed amount. We show that, compared to ordinary least-squares, constrained regressions can reduce under/overcompensation for some groups but increase under/overcompensation for others. In order to quantify this trade-off two fundamental questions need to be answered: "Which groups are relevant in terms of risk selection actions?" and "What is the relative importance of under- and overcompensation for these groups?" By making assumptions on these aspects we empirically evaluate a particular set of constraints using individual-level data from the Netherlands (N = 16.5 million). We find that the benefits of introducing constraints in terms of reduced under/overcompensations for some groups can be worth the costs in terms of increased under/overcompensations for others. Constrained regressions add a tool for developing risk equalization models that can improve the overall economic performance of health plan payment schemes.