Outlier-Resistant Estimates of Beta

Depending on their location, outliers in returns can substantially bias ordinary least-squares estimates of beta. We introduce a new beta estimate that is resistant to outliers that cause the most bias in OLS estimates but produces estimates similar to OLS for outlier-free data. The outlier-resistant beta is an intuitively appealing weighted least-squares estimate with data-dependent weights. We show that the resistant beta is a better predictor of future risk and return characteristics than is the OLS beta in the presence of outliers and is, therefore, a valuable complement to the OLS beta. Our analysis reveals that small companies' betas are most susceptible to outliers. Recent surveys show that many analysts continue to use the capital asset pricing model and that most of them purchase betas from commercial providers, which invariably use a raw or adjusted ordinary least-squares estimate of beta. The sanctified use of OLS is justified by the fact that the OLS beta is statistically the best estimate of the linear model parameters under idealized assumptions. In practice, however, one of the ways these assumptions fail is associated with the occurrence of a small fraction of exceptionally large or small returns—that is, outliers. We show by using several examples that outliers can, depending on their location in the equity-market-returns space, substantially bias OLS estimates of beta. Furthermore, the weekly returns for 8,314 companies from the CRSP database that had at least two years of returns in the period January 1992 through December 1996 contained many examples in which the deletion of a few outliers, sometimes even a single outlier, dramatically affected the OLS beta. The vast majority of commercial providers do nothing to deal with outliers; the few that do deal with this problem use some form of outlier treatment without a solid statistical rationale. We deal with the vulnerability of the OLS beta to outliers by introducing a new beta estimate that is resistant to the types of outliers that cause the most bias in OLS estimates but that produces estimates similar to OLS for outlier-free data. The outlier-resistant beta is an intuitively appealing weighted-least-squares estimate with data-dependent weights. It has several advantages over other commonly used “robust” techniques. The outlier-resistant beta applied to the CRSP database shows that the absolute value of the difference between the resistant and OLS betas is greater than 0.5 for 13 percent of the companies and that this difference is considerably larger than 1.0 for 3.2 percent of the companies. Such extreme sensitivity of the OLS beta to outliers results in misleading interpretations of the risk and return characteristics of a company. This study shows that outlier distortion of the OLS beta is primarily a small-firm effect (i.e., there is a monotonic relationship between the median market capitalization of companies and the absolute difference between the resistant and OLS betas). Furthermore, the resistant beta has superior performance relative to the OLS beta for predicting future betas when influential outliers are present but suffers (at most) only a slight degradation in performance when no influential outliers are present. The new resistant beta will serve well as an important complement to the OLS beta and should be applied in the following ways: The two betas in close agreement signal the absence of distortion of the OLS beta by influential outliers; in such cases, the OLS beta is an accurate representation of risk and return. When the two betas substantially differ, as measured by an appropriate statistical test or perhaps by a subjective value that the user of the beta deems financially relevant, outliers have considerable influence on the OLS beta, so the OLS beta is not to be trusted. In these cases, the returns should be checked for the presence of outliers and their possible causes (e.g., an inadvertent lack of adjustment for a split or a reverse split in the data or, in fact, unusual conditions of the company relative to the market). We do not recommend blindly replacing the OLS beta with the resistant beta, but we emphasize that the resistant beta, which excludes at most a small fraction of outliers, accurately describes the “typical” risk and return characteristics of a company as represented by the bulk of the returns. Outliers are never predictable based on return data alone, so the resistant beta will provide a more reliable forecast of the predictable risk and return characteristics of a company as represented by the nonoutlier portion of the future returns than will the OLS beta.