A Note on the Coefficient of Determination in Regression Models with Infinite-variance Variables
Author: Jeong-Ryeol Kurz-Kim
Publisher:
Published: 2007
Total Pages: 33
ISBN-13:
DOWNLOAD EBOOKSince 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 estimationand 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 non degenerate asymptotic distribution on the entire (0,1) interval, and the probability density function of this distribution is unbounded at 0 and 1. We provide closed form 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.