Anon

168 The Basics of financial economeTrics

A robust correlation coefficient is defined as

cS=+aX bY −−SaXbY

1

4

[( )(^22 )]

The robust correlation coefficient thus defined is not confined to stay in the inter-
val [–1,+1]. For this reason, the following alternative definition is often used:

r

SaXbYSaX bY

SaXbYSaX bY

=

+−−

++−

()()

()()

22

22

Applications

As explained in Chapter 3, regression analysis has been used to estimate the
market risk of a stock (beta) and to estimate the factor loadings in a factor
model. Robust regressions have been used to improve estimates in these
two areas.
Martin and Simin provide the first comprehensive analysis of the impact
of outliers on the estimation of beta.^5 Moreover, they propose a weighted
least squares estimator with data-dependent weights for estimating beta,
referring to this estimate as “resistant beta,” and report that this beta is a
superior predictor of future risk and return characteristics than the beta
calculated using the method of least squares described in Chapter 13. To
demonstrate the potential dramatic difference between the ordinary least
squares (OLS) beta and the resistant beta, the estimates of beta and the
standard error of the estimate for four companies reported by Martin and
Simin are shown as follows:^6

OLS Estimate Resistant Estimate

Beta Standard Error Beta Standard Error

AW Computer Systems 2.33 1.13 1.10 0.33

Chief Consolidated Mining Co. 1.12 0.80 0.50 0.26

Oil City Petroleum 3.27 0.90 0.86 0.47

Metallurgical Industries Co. 2.05 1.62 1.14 0.22

(^5) R. Douglas Martin and Timothy T. Simin, “Outlier Resistant Estimates of Beta,”
Financial Analysts Journal 59 (September–October 2003): 56–69.
(^6) Reported in Table 1 of the Martin-Simin study. Various time periods were used
from January 1962 to December 1996.