The Estimation of Beta with Leptocurtic Data

Antti Kanto, Eero Kasanen, and Vesa Puttonen
Helsinki School of Economics

Abstract

The estimation of the beta coefficient in the Capital Asset Pricing Model (CAPM) is considered. In the usual setup, with the normality assumption, the OLS-estimate of beta equals Cov Var . However, most empirical studies confirm that the error distribution is non-normal having very thick tails. In this paper, we consider a larger family of distributions, which allows for thick tails. We start with members of the Karl Pearson family of distributions satisfying

where P(x) and Q(x) are polynomials. This family includes a broad class of well-known distributions. As mentioned, the normality assumption leads to the usual estimator of beta, which is the ML-estimator.

The estimation is carried out in two phases: In the first phase, the shape parameters of the distribution, i.e., the coefficients of the polynomials are estimated. They are treated as nuisance parameters and estimated using the moment method, which leads to a very rapid algorithm, although the estimates are not optimal. In the second phase, the beta coefficient is estimated. It can be shown that the ML estimates are obtained by using weighted least squares with weigths given by phase 1. The procedure is then iterated until convergence.

In our empirical data from the Finnish stock market, the normality assumption is clearly rejected in all cases. The estimated beta coefficients do not differ significantly from OLS betas in most cases. However, the number of cases where the difference is clear is not small.

We test the practical relevance of the non-normality on beta estimates by ranking Finnish mutual funds based on Jensen-alfas derived from OLS-betas and Pearson-betas.