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


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


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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    References listed on IDEAS

    1. Fama, Eugene F & French, Kenneth R, 1992. " The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-465, June.
    2. Jagannathan, Ravi & Wang, Zhenyu, 1996. " The Conditional CAPM and the Cross-Section of Expected Returns," Journal of Finance, American Finance Association, vol. 51(1), pages 3-53, March.
    3. McCulloch, J Huston, 1997. "Measuring Tail Thickness to Estimate the Stable Index Alpha: A Critique," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 74-81, January.
    4. Loretan, Mico & Phillips, Peter C. B., 1994. "Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets," Journal of Empirical Finance, Elsevier, vol. 1(2), pages 211-248, January.
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    6. Carlo V. Fiorio & Vassilis A. Hajivassiliou & Peter C. B. Phillips, 2010. "Bimodal t-ratios: the impact of thick tails on inference," Econometrics Journal, Royal Economic Society, vol. 13(2), pages 271-289, July.
    7. Mittnik, Stefan & Rachev, Svetlozar T. & Kim, Jeong-Ryeol, 1998. "Chi-Square-Type Distributions For Heavy-Tailed Variates," Econometric Theory, Cambridge University Press, vol. 14(03), pages 339-354, June.
    8. Runde, Ralf, 1993. "A note on the asymptotic distribution of the F-statistic for random variables with infinite variance," Statistics & Probability Letters, Elsevier, vol. 18(1), pages 9-12, August.
    9. Robert Blattberg & Thomas Sargent, 2010. "Regression With Non-Gaussian Stable Disturbances: Some Sampling Results," World Scientific Book Chapters,in: Perspectives On Promotion And Database Marketing The Collected Works of Robert C Blattberg, chapter 1, pages 7-16 World Scientific Publishing Co. Pte. Ltd..
    10. Phillips, Peter C. B. & Loretan, Mico, 1991. "The Durbin-Watson ratio under infinite-variance errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 85-114, January.
    11. Kim, Jeong-Ryeol, 2003. "Finite-sample distributions of self-normalized sums," Discussion Paper Series 1: Economic Studies 2003,02, Deutsche Bundesbank.
    12. Fama, Eugene F & MacBeth, James D, 1973. "Risk, Return, and Equilibrium: Empirical Tests," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 607-636, May-June.
    13. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
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    15. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(01), pages 44-62, March.
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    1. repec:erh:journl:v:10:y:2018:i:1:p:14-23 is not listed on IDEAS

    More about this item


    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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