Article ID Journal Published Year Pages File Type
5096055 Journal of Econometrics 2014 10 Pages PDF
Abstract
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.
Related Topics
Physical Sciences and Engineering Mathematics Statistics and Probability
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