May 2007

A Note on the Coefficient of Determination in Models with Infinite Variance Variables

Jeong-Ryeol Kurz-Kim and Mico Loretan


Since the seminal work of Mandelbrot (1963), alpha-stable distributions with infinite variance have been regarded as a more realistic distributional assumption than the normal distribution for some economic variables, especially financial data. After providing a brief survey of theoretical results on estimation and hypothesis testing in regression models with infinite-variance variables, we examine the statistical properties of the coefficient of determination in models with alpha-stable variables. If the regressor and error term share the same index of stability alpha<2, the coefficient of determination has a nondegenerate asymptotic distribution on the entire [0, 1] interval, and the density of this distribution is unbounded at 0 and 1. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable. In contrast, if the indices of stability of the regressor and error term are unequal, the coe¢ cient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability. In an empirical application, we revisit the Fama-MacBeth two-stage regression and show that in the in…nite-variance case the coefficient of determination of the second-stage regression converges to zero in probability even if the slope coe¢ cient is nonzero.

Full paper (screen reader version)

Keywords: Regression models, alpha-stable distributions, infinite variance, coefficient of determination, Fama-MacBeth regression, Monte Carlo simulation, signal-to-noise ratio, density transformation theorem

PDF: Full Paper

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Last Update: October 19, 2020