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Encyclopedia of Financial Models II by Frank J. Fabozzi

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Robust Estimates of Betas and Correlations

THOMAS K. PHILIPS, PhD

Regional Head of Investment Risk and Performance, BNP Paribas Investment Partners

Abstract: The Theil-Sen estimator is an exceptionally simple and robust linear regression estimator, affording estimates of slope and intercept that are virtually identical to their ordinary least squares counterparts in the absence of outliers, but which do not change appreciably in the presence of outliers. In fact, with univariate data, it improves on ordinary least squares in almost every way imaginable, and it is therefore a striking fact that this remarkable estimator is not universally known and used. It can be used to derive robust estimates of beta and the correlation coefficient that are virtually identical to their classical counterparts when asset returns are normally distributed, and which are significantly more robust when asset returns are highly skewed or contaminated with outliers.

Point estimates of betas and correlations are most often obtained using ordinary least squares (OLS) and the standard maximum likelihood estimator, respectively. While these estimators are clearly optimal when asset returns are normally distributed, and when we hold no view on their prior distribution, they can be far from optimal when these conditions are not satisfied. In this entry, a novel explanation of OLS is provided and is then used to motivate a robust univariate regression algorithm due to Theil (1950) and Sen (1968). This ...

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