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

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

Abstract

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. --

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Bibliographic Info

Paper provided by Deutsche Bundesbank, Research Centre in its series Discussion Paper Series 1: Economic Studies with number 2007,10.

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Date of creation: 2007
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Handle: RePEc:zbw:bubdp1:5574

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Related research

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

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References

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  1. 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, Research Centre.
  2. Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2007. "Learning, Structural Instability, and Present Value Calculations," Econometric Reviews, Taylor & Francis Journals, Taylor & Francis Journals, vol. 26(2-4), pages 253-288.
  3. repec:rus:hseeco:318682 is not listed on IDEAS
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Cited by:
  1. 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, Elsevier, vol. 17(2), pages 180-194, March.
  2. Richard M. Levich & Valerio Poti, 2008. "Predictability and 'Good Deals' in Currency Markets," NBER Working Papers 14597, National Bureau of Economic Research, Inc.

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