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A note on the coefficient of determination in regression models with infinite-variance variables


  • Loretan, Michael Stanislaus
  • Kurz-Kim, Jeong-Ryeol


Since Mandelbrot's seminal work (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 regression models with infinite-variance variables. These properties differ in several important aspects from those in the well-known finite variance case. In the infinite-variance case when the regressor and error term share the same index of stability, the coefficient of determination has a nondegenerate asymptotic distribution on the entire [0,1] interval, and the probability density function of this distribution is unbounded at 0 and 1. We provide closedform expressions for the cumulative distribution function and probability density function of this limit random variable. In an empirical application, we revisit the Fama-MacBeth two-stage regression and show that in the infinite variance case the coefficient of determination of the second-stage regression converges to zero asymptotically.

Suggested Citation

  • Loretan, Michael Stanislaus & Kurz-Kim, Jeong-Ryeol, 2007. "A note on the coefficient of determination in regression models with infinite-variance variables," Discussion Paper Series 1: Economic Studies 2007,10, Deutsche Bundesbank.
  • Handle: RePEc:zbw:bubdp1:5574

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    References listed on IDEAS

    1. Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2007. "Learning, Structural Instability, and Present Value Calculations," Econometric Reviews, Taylor & Francis Journals, vol. 26(2-4), pages 253-288.
    2. Taylor, Mark P. & Schmidt, Markus & Reitz, Stefan, 2007. "End-user order flow and exchange rate dynamics," Discussion Paper Series 1: Economic Studies 2007,05, Deutsche Bundesbank.
    3. repec:rus:hseeco:318682 is not listed on IDEAS
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    Cited by:

    1. Serttas, Fatma Ozgu, 2010. "Essays on infinite-variance stable errors and robust estimation procedures," ISU General Staff Papers 201001010800002742, Iowa State University, Department of Economics.
    2. Bouhaddioui, Chafik & Ghoudi, Kilani, 2012. "Empirical processes for infinite variance autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 107(C), pages 319-335.
    3. Levich, Richard M. & Potì, Valerio, 2015. "Predictability and ‘good deals’ in currency markets," International Journal of Forecasting, Elsevier, vol. 31(2), pages 454-472.
    4. Dufour, Jean-Marie & Kurz-Kim, Jeong-Ryeol, 2010. "Exact inference and optimal invariant estimation for the stability parameter of symmetric [alpha]-stable distributions," Journal of Empirical Finance, Elsevier, vol. 17(2), pages 180-194, March.

    More about this item


    Regression models; alpha-stable distributions; infinite variance; coefficient of determination; Fama-MacBeth regression; Monte Carlo simulation;

    JEL classification:

    • 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
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General

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