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

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  • Kurz-Kim, Jeong-Ryeol
  • Loretan, Mico

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.

Suggested Citation

  • Kurz-Kim, Jeong-Ryeol & Loretan, Mico, 2014. "On the properties of the coefficient of determination in regression models with infinite variance variables," Journal of Econometrics, Elsevier, vol. 181(1), pages 15-24.
  • Handle: RePEc:eee:econom:v:181:y:2014:i:1:p:15-24
    DOI: 10.1016/j.jeconom.2014.02.004
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    Keywords

    Coefficient of determination; α-stable distributions; Signal to noise ratio; Density transformation theorem; Monte Carlo simulation; Fama–MacBeth regression; CAPM;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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