A note on the coefficient of determination in regression models with infinite-variance variables
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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- Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2007.
"Learning, Structural Instability, and Present Value Calculations,"
Taylor & Francis Journals, vol. 26(2-4), pages 253-288.
- Pesaran, M.H. & Pettenuzzo, D. & Timmermann, A., 2006. "Learning, Structural Instability and Present Value Calculations," Cambridge Working Papers in Economics 0602, Faculty of Economics, University of Cambridge.
- M. Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2006. "Learning, Structural Instability and Present Value Calculations," CESifo Working Paper Series 1650, CESifo Group Munich.
- M. Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2006. "Learning, structural instability and present value calculations," Computing in Economics and Finance 2006 529, Society for Computational Economics.
- Pesaran, Mohammad Hashem & Pettenuzzo, Davide & Timmermann, Allan, 2006. "Learning, structural instability and present value calculations," Discussion Paper Series 1: Economic Studies 2006,27, Deutsche Bundesbank, Research Centre.
- Hashem Pesaran & Davide Pettenuzzo & Allan Timmermann, 2006. "Learning, Structural Instability and Present Value Calculations," IEPR Working Papers 06.42, Institute of Economic Policy Research (IEPR).
- 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.
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