On the properties of the coefficient of determination in regression models with infinite variance variables

We examine the asymptotic properties of the coefficient of determination, R2, in models with α-stable random variables. If the regressor and error term share the same index of stability α<2, we show that the R2 statistic does not converge to a constant but has a nondegenerate distribution on the entire [0,1] interval. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable, and we show that the density function is unbounded at 0 and 1. If the indices of stability of the regressor and error term are unequal, we show that the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability, irrespective of the value of the slope coefficient. In an empirical application, we revisit the Fama and MacBeth (1973) two-stage regression and demonstrate that in the infinite-variance case the R2 statistic of the second-stage regression converges to 0 in probability even if the slope coefficient is nonzero. We deduce that a small value of the R2 statistic should not, in itself, be used to reject the usefulness of a regression model.