Outlier Resistant Estimates of Beta

R. Douglas Martin and Timothy T. Simin R. DOUGLAS MARTIN is a Professor of Statistics at the University of Washington and Chief Scientist at Insightful Corporation, both in Seattle, WA. (Insightful Corporation, 1700 Westlake Ave., N., Suite 500, Seattle, WA 98109-3044, Phone: 206/283-8802, Email: doug@statsci.com). TIMOTHY T. SIMIN is a Professor of Finance in the Department of Finance at The Pennsylvania State University. (Department of Finance, Smeal College of Business Administration, 609 Business Administration Building, University Park, PA 16802, E-mail: tsimin@psu.edu). The authors wish to thank Vladimir Atanasov, Gordon Hanka, Harold Mulherin, Jeff Wang, Eric Zivot, and an anonymous referee for helpful discussions and suggestions during the preparation of this paper. Edward Glidewell provided excellent assistance in this research. We also thank Andrew Lo for his comments and encouragement towards submission.

Outlier Resistant Estimates of Beta

Depending on their location, outliers in returns can substantially bias ordinary least squares (OLS) 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 the OLS beta in the presence of outliers, and is therefore a valuable complement to the OLS beta. Our analysis reveals that small firm betas are most susceptible to outliers. The authors wish to thank Vladimir Atanasov, Gordon Hanka, Harold Mulherin, Jeff Wang, Eric Zivot, and an anonymous referee for helpful discussions and suggestions during the preparation of this paper. Edward Glidewell provided excellent assistance in this research. We also thank Andrew Lo for his comments and encouragement towards submission.

1

1. INTRODUCTION

While academics continue to debate its relevance, the capital asset pricing model

(CAPM) beta is undoubtedly the best-known and most widely used estimator for expected

returns and measure of market risk. Widespread use of the CAPM beta is documented in

several surveys of industry participants. Graham and Harvey (2001) report 73.5% of

respondents always or almost always use the CAPM when estimating the cost of equity

capital. In a survey of financial analysts, Block (1999) presents evidence that more than 30%

of respondents view the CAPM as an important valuation model. While Gitman and

Mercurio (1982) report that only slightly more than fifty percent of managers familiar with

the technique use the CAPM, Bruner, Eades, Harris and Higgins (1998) report that the

CAPM is the dominant model used to estimate the cost of equity, and also note that beta

estimates used in practice are drawn primarily from the published sources of commercial

providers.

A review of a variety of commercial providers reveals beta is invariably estimated

using ordinary least squares (OLS) to fit a single factor market model

1t m, t tR R , t ,...,T= α + ⋅β + ε = (1)

to the time series of returns for a particular equity Rt and a market proxy Rm,t. Some

publishers report the raw OLS estimate β , while others report an adjusted version

35 67 ˆ. .β = + ⋅β% designed to correct for the tendency of β to revert toward one over time.

Since our findings apply to both raw and adjusted OLS betas, we use the term “OLS beta” to

encompass both methods. Other variations among published betas included differing return

horizons, the time span of data used in estimation, and use of raw versus excess returns.

The sanctified use of OLS for fitting the market model (1) is justified by the fact that

OLS estimates of α and β are statistically the best estimate of the linear model parameters

under idealized assumptions.1 Unfortunately, these assumptions often fail in practice, and

one of the ways in which they fail is associated with the occurrence of a small fraction of

exceptionally large or small returns called outliers. In practice outliers occur in both equity

1 The idealized assumptions are that the paired returns (Rm,t, Rt) conform to the linear model relationship (1), with zero mean errors tε that are Gaussian and uncorrelated with the market returns.

2

returns Rt and market returns Rm,t, with considerably greater frequency in equity returns than

in market returns. Outliers can occur because an individual equity or the market (or both)

makes an unusual move, because a split or reverse split occurs and is not removed from the

data, or because of a data error. Outliers occur more frequently for firms with small rather

than large market capitalization, an empirical regularity that is clearly revealed by our study.

We find many examples where the deletion of a very small number of outliers,

sometimes even a single outlier, dramatically changes the value of the OLS beta. In several

cases the change is considerably larger than one. Such extreme sensitivity of the OLS beta to

outliers can result in quite misleading interpretations of the risk and return characteristics of a

firm. We argue here that it is highly desirable not to rely solely on OLS betas, but to also

compute resistant betas that are not much influenced by a very small fraction of outliers.

The main goals of this paper are to introduce a new resistant estimator of beta that has

an intuitive outlier-trimming property, and establish the viability of the resulting resistant

beta for routine use by commercial providers and other practitioners. The performance of the

resistant beta is for all practical purposes identical to that of the OLS beta when the data

contains no outliers, but differs considerably from the OLS beta by down weighting or

outright rejecting outlying returns pairs (Rm,t , Rt) that have substantial influence on the OLS

beta. The resistant beta also has well-established statistical resistance properties that are

ultimately the foundation upon which practice should be built.

As motivation, we document in Section 2 how only a small fraction of commercial

providers utilize any method for dealing with outliers, none of which are based on solid

statistical rationalization. In Section 3 we point out that in the context of fitting a market

model (1) outliers are a two-dimensional phenomenon, which invalidates using one-

dimensional approaches such as trimming or Winsorizing equity returns and market returns

separately. In Section 4 we describe the proposed resistant beta and provide examples where

the OLS and resistant betas differ quite substantially. Section 5 reports the comparative

behavior of the OLS and resistant betas using weekly equity returns from the Center for

Research on Stock Prices (CRSP) database. In the same section we show that outlier

distortion of the OLS beta is primarily a small-firm effect. In Section 6 we evaluate the new

resistant beta’s ability to predict future beta. Concluding comments are in Section 7.

3

2. CURRENT COMMERCIAL PRACTICE

To gain a feel for the current standards applied in practice we conducted a survey of

what commercial providers of beta do to deal with outliers. The check involved reviewing

web sites and published information on beta calculations, and many telephone interviews

with employees knowledgeable about the process used in computing betas. Exhibit 1 below

summarizes our findings, namely the overwhelming majority of providers do nothing to

mitigate the influence of outliers, Barra and Ibbotson being the exceptions. Except for Barra,

all the vendors report OLS betas based on the single factor model (1).

Provider No Outlier 1-D Outlier Other Barra X Bloomberg X Dow Jones X Ibbotson X X Merril Lynch X Standard and Poor’s X Value Line X Vestek X Wilshire X

Exhibit 1. OUTLIER TREATMENT BY PROVIDERS OF BETA

Ibbotson Associates indirectly addresses the outlier issue in computing OLS betas,

stating in their online information: “If an OLS Beta estimate is greater than five in absolute

value, it is discarded.” This practice reflects an apparent awareness by Ibbotson that outliers

sometimes greatly distort the OLS beta in a positive direction. However, the procedure does

not protect against less extreme distortions of the OLS beta by an outlier, e.g., a single outlier

in returns can easily move the OLS beta from one to three.

Barra is the only provider we surveyed that employs a statistical method to deal with

outliers. Specifically, Barra uses a one-dimensional Winsorization method on each of the

factor exposures separately (but not on the equity returns) in their multi-factor model

approach to calculating risk models and betas (Connor and Herbert, 1999). We argue below

why one-dimensional treatments such as Winsorization are inadequate methods of dealing

4

with outliers. Our arguments raise the question of whether the Barra one-dimensional outlier

treatment is as effective as would be an extension of our resistant beta method to their multi-

factor model. However, instead of focusing on the Barra methodology, we concentrate on

the use of OLS betas from the single factor market model and establish that our proposed

resistant beta offers a clear improvement relative to the majority of the providers surveyed.

3. OUTLIER INFLUENCE CAN RESULT IN MISLEADING OLS BETAS

Estimation of beta is an exercise in fitting a straight line to the scatter plot of equity

returns versus market returns. The OLS estimate fits the straight line by minimizing the sum-

of-squared errors

( )2

1

T

t m,tt

R R=

−α − ⋅β∑ (2)

with respect to α and β . Because errors are penalized quadratically, outliers can make

exceptionally large contributions to the sum-of-squared errors, and so have the potential to

exert considerable influence on the OLS fit. Furthermore the influence of an outlying pair

(Rm,t , Rt) on the OLS fit depends on the two-dimensional location of the pair. We illustrate

this by means of a number of examples, with a single outlier pair (Rm,t , Rt) in different types

of locations. The conclusions drawn apply equally well to larger sized data sets with a very

small fraction of outliers in roughly the same location.

Different Outlier Locations and their Influence on OLS Beta

Exhibit 2, panel (a) shows a prototype scatter plot consisting of twenty pairs of

normally distributed returns free of outliers, with the OLS straight line fit superimposed and

the OLS beta of .96 displayed. The other five plots in Exhibit 2 contain the same data as in

panel (a) plus an additional outlier in different positions in each of the plots. These types of

outliers can and do occur in practice and their sizes are moderate relative to what sometimes

occurs with real returns. The solid lines in panels (b) through (f) are the OLS fits, with

corresponding values of beta displayed in the plot, and for comparison purposes we

superimpose the OLS fit for the outlier-free data of panel (a) with a dashed line.

Panels (b) and (c) illustrate that there are certain two-dimensional outlier positions

that result in essentially no influence on the OLS beta. In panel (b) the outlier is an unusually

5

large equity return at a time when the market return is very near the mean of the market

returns. In this case the outlier exerts no tilting effect on the OLS line fit, and hence no

influence on the OLS beta (the outlier does influence the intercept estimate, biasing it

positively, but that is not our concern here). The outlier in panel (c) is much like an unusually

large downward movement of the market return coupled with a downward movement of the

equity return. In this case the outlier is consistent with the OLS fit and has no influence on

beta. This type of outlier in fact results in increased precision in estimating beta, i.e., the

standard error of the OLS beta is greatly reduced by such an outlier, and in this regard it is a

“good” outlier. We have found that this type of two-dimensional outlier occurs fairly

frequently at the time of the crash of 1987, as illustrated in our examples in Section 5.

The outlier in panels (d) through (f) has substantial influence on the OLS beta. The

standard error of the OLS beta in panel (a) is .13 implying that the single outlier in each of

the last three panels has caused a change of at least three standard deviations in the OLS beta.

The outlier in panel (d) is an outlying equity return, coupled with the largest, none-the-less

non-outlier, market return in the data. This outlier exerts considerable influence resulting in

a positively biased value of 1.38 for the OLS beta, as compared with .96 when the outlier is

removed. The outlier of panel (e) is an outlying market return but is not an outlying equity

return, and in this case the outlier exerts considerable influence resulting in a negatively

biased OLS beta value of .42, as compared with .96 when the outlier is removed.

Because the outlier pairs (Rm,t , Rt) in panels (c) and (e) are market returns outliers,

i.e., they are outliers in the one-dimensional “independent” or “predictor” variable

dimension, they have the potential to cause considerable bias by exerting leverage on the

OLS straight line fit. Such outliers are often called leverage points in the Econometrics and

Statistics literature.2 Panel (c) illustrates that a leverage point does not necessarily result in

bias, while panel (e) illustrates a leverage point that does cause bias.

Panel (f) represents a particularly important case in that neither the equity return

outlier nor the market return outlier, but the pair (Rm,t , Rt) is clearly a two-dimensional

outlier that is well separated from the bulk of the data. That is, considering the stock returns

alone Rt is not an outlier, it is just the smallest return. Likewise, considering only the market

2 Further discussion of outliers with leverage may be found for example in Judge et. al. (1988) and Cook and Weisberg (1982).

6

returns Rm,t is not an outlier, just the largest return. Yet this outlier pair exerts considerable

influence on the OLS beta, resulting in a negatively biased OLS beta value of .54 as

compared with .96 when the outlier is removed.

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 0.96

( a )

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 0.96

( b )

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 0.98

( c )

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 1.38

( d )

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 0.42

( e )

Market Returns

Stoc

k R

etur

ns

-0.2 0.0 0.2

-0.4

0.0

0.4

BETA = 0.54

( f )

Exhibit 2. OUTLIER INFLUENCES ON OLS BETAS

The bias impact of the outlier in the final three panels is dramatic. As a consequence

the OLS beta does not provide a good description of the risk and return relationship for the

outlier-free portion of the data (the vast majority), nor does it adequately describe the

7

relationship between equity and market returns for the outlier. Since the OLS beta fails to

describe either the bulk of the data or the outliers in many of the cases where outliers are

present, it is highly desirable to have a resistant beta that at least fits the bulk of the data well,

providing a good description of the typical risk and return relationship as a complement to

the OLS beta. Because outliers are virtually never predictable in practice it is only the bulk

of the data that is at all reliably predictable, and since beta is used in practice as a predictor of

future risk and return, a resistant beta that fits the bulk of the data well seems quite desirable.

Why One-Dimensional Outlier Treatment is Not Enough

As noted above, one-dimensional treatments are the only statistical methods for

dealing with outliers being employed by any of the commercial beta providers of our survey.

By “one-dimensional” outlier treatment, we mean treatment of outliers in equity returns and

market returns separately, followed by computation of an OLS beta. The typical treatment

consists of either “rejecting” or “Winsorizing” outliers. The former consists of removing

data points that are outliers, while the latter consists of altering outliers by bringing them in

to specified values. The best-known rejection method is the classical three-sigma edit rule,

which is known to be quite inferior to alternative resistant rejection rules (see Hampel, 1986).

A common Winsorization method of mitigating the influence of outliers is to use resistant

estimates of the mean and standard deviation to “bring in” outliers as follows: values that are

a distance greater than 5.2 times the median absolute deviation about the median (MADM)

from the sample median are brought in to a distance 5.2 times MADM from the sample

median.3 For example, this is the method used by Barra to mitigate the effect of outliers in

factor exposures in their multi-factor models (Connor and Herbert (1999).

It should be clear from the discussion of the previous section that neither of the one-

dimensional outlier treatments above will be adequate for all the types of two-dimensional

return pair outliers that occur in practice. On the one hand, it is quite inappropriate to modify

outliers of the type displayed in Exhibit 2, panel (c), where the outlier conforms perfectly

well to the market model (1). On the other hand, one-dimensional outlier treatment will do

nothing at all to help the situation displayed in Exhibit 2, panel (f). What is required to deal

3 This method is resistant because it uses the sample median instead of the sample mean as the “center” of the data, and MADM instead of the sample standard deviation as the “scale” of the data.

8

with all of these situations in a uniform and effective manner is a resistant two-dimensional

straight-line fitting method such as the one we propose in the next section.

4. THE NEW RESISTANT BETA

Our proposed resistant beta is a form of weighted least squares (WLS) with data-

dependent weights. A classical WLS estimate of beta is obtained by minimizing

( )2

1

T

t t m,tt

w R R=

⋅ −α − ⋅β∑ (3)

with weights wt that are fixed and independent of the data. Differentiation of (3) with respect

to α and β leads to the classical WLS equations for the estimates ˆˆ , α β :

( )1

0T

t t m, tt

ˆˆw R R =

⋅ −α − ⋅β =∑ (4)

( )1

0T

t t t m, tt

ˆˆR w R R =

⋅ ⋅ − α − ⋅β =∑ . (4’)

The OLS estimate is obtained when the weights are equal to one.

We obtain resistant estimates ˆˆ , α β by replacing the fixed weights above with data-

dependent weights wt computed as follows:

( )t tˆw w u= (5)

where

t m, tt

ˆˆR Ru

s−α − ⋅β

= (6)

are the standardized residuals (errors in the fit), and the weight function w has the shape

shown in Exhibit 3. The s in the denominator is a robust scale estimate for the residual, i.e.,

a robust replacement for the residuals standard error estimate, determined by the overall

robust regression algorithm referenced in the Appendix. The smooth transitions of the

weight function from a value of one to a value of zero at the cutoff points –2.7 and +2.7

insure that the resistant beta is not overly sensitive to small changes in the data, e.g., due to

rounding.

9

0.0

0.2

0.4

0.6

0.8

1.0

-5.0 -2.5 0.0 2.5 5.0

Exhibit 3. WEIGHT FUNCTION FOR RESISTANT BETA

With the weight function of Exhibit 3, the above WLS equations with data-dependent

weights result in the following intuitively appealing rejection rule: all return pairs (Rm,t , Rt )

whose standardized residuals ut are larger than 2.7 in magnitude at the solution ˆˆ , α β are in

effect removed from the data set. The solid line in Exhibit 5 below is the resistant straight-

line fit. The dotted lines, parallel to the fit, are spaced at a distance 2.7 $s above and below

the fit, corresponding to standardized residuals ut that are at distance 2.7 from the fit. All

data points above the top dotted line and below the bottom dotted line are considered outliers

and deleted. We use an optimization approach to estimate the data-dependent weights and

nonlinear equations (3) and (4) that is implemented in the statistical package S-PLUS (1999),

described briefly in the Appendix.

Statistical Properties of the Resistant Beta

The resistant beta has the following two key statistical properties:

(a) When there are no outliers, the resistant beta performs essentially as well as the OLS

beta in the following statistical sense: when the market model (1) holds with

10

Gaussian errors the resistant beta has a standard deviation of only about 5% greater

than that of OLS beta.

(b) Arbitrary deviations from the market model (1) in the form of a small fraction of

returns outlier pairs (Rm,t , Rt ) can cause arbitrarily large bias in the OLS beta, and the

resistant beta minimizes this bias in an optimal manner.

The second property is a recent theoretical “bias robustness” result from the statistical

literature due to Yohai and Zamar (1997). The first property is in effect a very small

insurance “premium” paid in the form of a slightly increased standard deviation of the

resistant beta relative to the OLS beta when the returns are Gaussian, in return for optimally

controlling beta bias due to outliers.

The 5% in (a) above is a consequence of using the cutoff values –2.7 and +2.7 in

Exhibit 3. The 5% loss of efficiency represents a compromise between achieving a resistant

beta that is close to the OLS beta when there are no outliers, i.e., a small insurance premium,

while at the same time achieving effective trimming of bias inducing outliers when they are

present. Decreasing this premium, by increasing the cutoff points of the weight function,

makes the standard deviation of the resistant beta less then 5% greater the OLS beta, but only

at the expense of potentially worse beta bias due to outliers. Conversely, better bias control

is achievable by decreasing the cutoff points, but only at the expense of an increased

insurance premium in the form of a standard deviation for the resistant beta that is greater

than 5% larger than that of the OLS beta. While the procedure allows users the flexibility to

increase or decrease the weight function cutoff points, we stand by our recommended values

of –2.7 and 2.7.

One additional and important statistical property: The resistant beta estimate is

approximately normally distributed with a standard error that can be conveniently estimated

from the data. In the examples of the next section we use the standard errors produced by the

S-PLUS (1999) implementation of the robust regression.

Some Striking Examples

The following four examples are among many we have found where there is a

strikingly large difference between the OLS and resistant beta due to outlier-induced

11

distortion of the OLS beta. Exhibit 4 displays the scatter plots of monthly equity returns for

four small firms versus returns for the value weighted NYSE/AMEX/NASDAQ composite as

the market proxy. All returns are in excess of the one-month T-bill rate from the CRSP

database. The return series vary in length but all fall in the time interval from January 1962

to December 1996.

The OLS beta and our proposed resistant beta estimates are the slopes of the dashed

and solid straight line fits respectively.

-20 -10 0 10

0

200

400

600

MARKET RETURN

EQU

ITY

RET

UR

N

A W COMPUTER SYSTEMS INC.

ROBUSTOLS

1.12.33

(0.33)(1.13)

-20 -10 0 10

0

100

200

300

400

500

MARKET RETURN

EQU

ITY

RET

UR

N

OIL CITY PETROLEUM INC.

ROBUSTOLS

0.863.27

(0.47)(0.9)

-20 -10 0 10

0

100

200

300

MARKET RETURN

EQU

ITY

RET

UR

N

CHIEF CONSOLIDATED MNG CO

ROBUSTOLS

0.51.12

(0.26)(0.8)

-20 -10 0 10

0

500

1000

1500

MARKET RETURN

EQU

ITY

RET

UR

N

METALLURGICAL INDUSTRIES INC.

ROBUSTOLS

1.142.05

(0.22)(1.62)

Exhibit 4. OLS AND RESISTANT BETAS FOR FOUR SMALL FIRMS The outliers have considerable influence on the OLS beta causing it to be a relatively poor fit

to the bulk of the data. The resistant beta is not much influenced by the outliers and provides

a good fit to the bulk of the data, which is an important feature of a good resistant estimate.

The unadjusted OLS and resistant beta estimates of Exhibit 4 are assembled in Panel

A of Exhibit 5, and corresponding adjusted estimates are provided in Panel B. In each of the

12

four examples the unadjusted and the adjusted OLS betas are substantially greater than one,

and the values are all considerably larger than the corresponding resistant betas.

Panel A: Unadjusted Beta Estimates, (std. errors)

A W Computer Systems Inc.

Chief Consolidated Mining Co.

Oil City Petroleum Inc.

Metallurgical Industries Inc.

OLS 2.33 (1.13)

1.12 (0.80)

3.27 (0.90)

2.05 (1.62)

ROBUST 1.10 (0.33)

0.50 (0.26)

0.86 (0.47)

1.14 (0.22)

Panel B: Adjusted Beta Estimates, (std. errors)

A W Computer Systems Inc.

Chief Consolidated Mining Co.

Oil City Petroleum Inc.

Metallurgical Industries Inc.

OLS 1.89 (0.76)

1.08 (0.25)

2.52 (0.60)

1.70 (1.09)

ROBUST 1.07 (0.22)

0.67 (0.17)

0.91 (0.31)

1.09 (0.15)

Panel C: Unadjusted Beta Estimates with Oct. 1987 Deleted, (std. errors)

A W Computer Systems Inc.

Chief Consolidated Mining Co.

Oil City Petroleum Inc.

Metallurgical Industries Inc.

OLS 2.27 (1.24)

1.14 (0.96)

3.67 (1.01)

2.11 (1.71)

ROBUST 0.84 (0.36)

0.28 (0.30)

0.68 (0.47)

1.15 (0.24)

Panel D: Adjusted Beta Estimates with Oct. 1987 Deleted, (std. errors)

A W Computer Systems Inc.

Chief Consolidated Mining Co.

Oil City Petroleum Inc.

Metallurgical Industries Inc.

OLS 1.85 (0.83)

1.09 (0.64)

2.79 (0.68)

1.74 (1.15)

ROBUST 0.89 (0.24)

0.49 (0.53)

0.79 (0.31)

1.10 (0.16)

Exhibit 5. OLS AND RESISTANT BETAS (UNADJUSTED AND ADJUSTED) Notice that in these examples positive bias in the OLS estimates is caused by a tendency for

large positive outliers to occur when the market returns are more or less positive. This is a

tendency we have noticed for many of the equities with outliers in our study.

The standard error estimates provided in parentheses in Exhibit 5 are much larger for

the OLS beta estimates than for the resistant beta estimates. This is because the standard

error estimate for the OLS beta involves the sum-of-squared residuals, and the latter can be

considerably inflated by the presence of outliers. On the other hand, the resistant estimate

13

down-weights outliers as described previously, and hence yields an estimate of standard

errors that is insensitive to outliers. Notice that in Panels A and B the OLS beta is always

larger than the resistant beta by at least three times the standard error of the resistant beta,

i.e., the differences between OLS and resistant betas are all judged highly significant using

the latter as the benchmark.

In all four examples the crash of October 1987 (the data point in the lower left corner

of each graph) is a leverage point, i.e., an outlier in the one-dimensional market returns axis.

This leverage point outlier might substantially contribute to increasing the value of the OLS

beta over that of the resistant beta. However, Panels C and D of Exhibit 5 show quite clearly

that the OLS beta is relatively unaffected by the October 1987 outlier (and the same is true of

the resistant beta). It is the other outliers that are causing the OLS beta to be substantially

larger than the resistant beta. In these examples the crash of October 1987 represents a non-

influential leverage point because the equity return for that month is consistent with the

straight-line fit, and therefore does not adversely influence the fit. Of course this will not

necessarily be the case for all equities.

Interpretation of Resistant versus OLS Betas

The presence of outlier induced positive bias in the OLS betas mistakenly portrays

three of the four firms in Exhibit 5 as moderately to highly aggressive assets in the sense that

they have somewhat or substantially higher levels of risk and expected return than the

market. This does not sufficiently characterize true nature of these firms. First of all, the

behavior of the overwhelming bulk of the data suggests levels of risk and return that are no

more than 14% above those of the market, and are as much as 36% below those of the

market. Secondly, the returns giving rise to the unusually large values of OLS betas

represent a very small number of unusually large positive returns that tend to occur when the

market returns are more or less positive, with no corresponding negative returns in evidence.

On the basis of the returns data alone it is hardly a complete summary to state, as

suggested by the OLS beta, that the firms are more risky and yield a higher return than the

market. It is more accurate to say that for the most part, the firms risk and return

characteristics are represented by the resistant beta, and that in addition there are a very small

fraction of unusually large positive returns with no unusually negative returns in evidence.

14

5. THE CRSP DATA SET COMPARISONS

We compare the behavior of the OLS and resistant beta estimates using weekly

returns for firms listed on the NYSE, AMEX, and NASDAQ exchanges from the CRSP

database, and using the NYSE/AMEX/DASDAQ composite as the market proxy. All returns

are in excess of the one-month T-bill rate from the CRSP database. To be included the firm

must have all data for at least two-years in the period 1/1992 through 12/1996.

Exhibit 6 displays binned counts of the absolute differences between the OLS and

corresponding resistant beta, for both unadjusted betas β and adjusted betas given by

35 67 ˆ. .β = + ⋅β% . The count for each bin is the number of firms for which the absolute

difference between the OLS and resistant beta estimates fall in that cell. Many of the betas

for the resistant beta are fairly close to the OLS beta. But significant fractions of the firms

have OLS and resistant betas that differ by moderate to large amounts. For example, 26% of

the firms have unadjusted absolute differences larger than 0.3. Such differences are likely to

be financially significant to many investors. The use of adjusted betas shrinks the difference

between OLS and resistant betas by the factor .67 which clearly ameliorates but does not

solve the problem. Exhibit 6 shows that 15% of the firms still exhibit adjusted beta absolute

differences larger than 0.3.

∆ = |βOLS – βRES| N = 8314 Market Capitalization Unadjusted Adjusted Mean Median

0.0+ to 0.1 3133 4235 1222.18 152.84 0.1+ to 0.2 1910 1898 637.20 92.49 0.2+ to 0.3 1099 925 330.72 73.65 0.3+ to 0.4 677 456 176.76 54.48 0.4+ to 0.5 430 279 185.14 45.02 0.5+ to 0.6 270 181 112.29 34.11 0.6+ to 0.7 205 107 90.53 29.73 0.7+ to 0.8 142 66 74.67 27.35 0.8+ to 0.9 111 35 91.96 30.07 0.9+ to 1.0 72 27 50.14 12.80

> 1.0+ 265 105 39.43 13.32

Exhibit 6. ABSOLUTE DIFFERENCES BETWEEN OLS AND RESISTANT BETAS

15

Exhibit 6 also displays mean and median summary statistics on market capitalization

(price times shares outstanding) in millions of dollars for each absolute difference bin for the

OLS and resistant betas. For each firm, the time series average over the sample period of the

year-end price/share times the number of shares is used as the firm's market capitalization

rate. Both the price and the shares outstanding are the values closest to the end of the year.

It is clear from the table that both the mean and median firm size decreases

approximately monotonically as the absolute difference between the OLS and resistant beta

increases. This behavior is consistent with the findings of Knez and Ready (1997) in

comparing OLS and resistant straight line fits to equity returns versus firm size, namely

influential outliers in returns occur primarily for smaller sized firms.

6. PREDICTIVE PERFORMANCE OF OLS AND RESISTANT BETAS

Predictive power is a natural measure to use in evaluating the relative utility of the

OLS and resistant betas. The prediction root-mean-squared-error (RMSE) is a commonly

used measure of the quality of a predictor. We compare the OLS and resistant betas as

predictors of future beta in terms of their RMSE’s for both one-week ahead predictions and

one-year ahead predictions.

Performance of OLS and Resistant Betas for Predicting One-Week Future Betas

Exhibit 7 displays the scatter plot of stock returns versus market returns and the time

series of OLS and resistant betas for Bristol Myers Squibb Co., SASOL Ltd., Shared

Technologies Inc., and Environmental Tectonics Corp.. Exhibit 8 provides the RMSE’s for

these firms. We estimate the OLS and resistant betas using a 104-week (two-year) moving

window. Forecast errors are the difference between the current beta and the one-week (year)

ahead beta.

16

-0.05 0.0 0.05

0.05

0.0

0.05

Bristol Myers Squibb Co. vs. Market

0.6

0.8

1.0

1.2

1/2/90 11/2/93 9/2/97

Ols beta, Resistant beta

-0.05 0.0 0.05-0.2

-0.1

0.0

0.1

0.2

0.3 SASOL Ltd. vs. Market

-0.20.00.20.40.60.81.0

1/2/90 11/13/93 12/2/97

Ols beta, Resistant beta

-0.06 -0.02 0.02

-0.4

-0.2

0.0

0.2

0.4

0.6Shared Technologies Inc. vs. Market

-1

0

1

2

3

4

1/2/90 12/6/94

Ols beta, Resistant beta

-0.05 0.0 0.05-0.20.00.20.40.60.81.01.2

Environmental Tectonics Corp. vs. Market

-3

-2

-1

0

1

2

1/2/90 11/9/93 1/13/98

Ols beta, Resistant beta

Exhibit 7. OLS and Resistant Betas for Bristol Myers Squibb Co., SASOL Ltd., Shared Technologies Inc., Environmental Tectonics Corp.

17

These four firms were chosen to illustrate the expected range of relative behavior of the OLS

and Resistant betas depending upon whether or not the scatter plot of equity returns versus

market returns contains influential outliers.

While the returns for Bristol Myers Squibb Co. contain no dominant outliers, the

scatter plot of these returns versus market returns reveals three moderate sized outliers. The

latter have little influence as is reflected in the time series plot of the OLS and resistant betas.

The time series of the two betas are virtually indistinguishable except for short periods of

time during 1995 and 1998. The similarity of the two estimates is reflected in the nearly

identical one-week ahead RMSE’s of Exhibit 8, i.e. there is only a 5% difference between the

unadjusted RMSE’s. This type of relative behavior is expected when there is little influence

on the OLS beta due to outliers, i.e., the data is essentially Gaussian and consequently the

two beta estimates are highly correlated.4

The returns for SASOL Ltd. have at least one dominant outlier, and the corresponding

scatter plot shows two moderate sized outliers. The time series plots of the OLS and resistant

betas are less consistent indicating that the outliers in the scatter plot have some degree of

influence. Furthermore, the RMSE’s reveal that the resistant beta is a noticeably (13.5%)

better predictor than the OLS beta.

The returns for Shared Technologies Inc. contain one dominant positive outlier at

December 31st of 1991 and two substantial negative outliers at December 10th and December

17th of 1991. These three outliers are among the largest outliers in the corresponding scatter

plot. There are also a number of moderate sized positive outliers scattered in subsequent

values of the stock returns, and these also appear in the scatter plot. The time series plots of

the OLS and resistant beta reveal a dramatic difference between the two estimates over the

time period when the estimation window contains the two large positive outliers. After that

period the two estimates are highly correlated, but the resistant estimate has a consistently

smaller value. This is evidently due to a positive bias in the OLS beta due to the moderate

sized positive outliers scattered throughout the remainder of the returns series evidenced in

4 The resistant beta used in this paper has a fairly high variance efficiency of .85 at the Gaussian model, while the OLS estimate has efficiency 1 at the Gaussian model. It is known from statistical theory that the correlation between the two estimates is the square root of the efficiency of the resistant estimate. In this case the correlation is therefore .92.

18

the scatter plot. For this firm the RMSE’s of the OLS beta are substantially higher than those

of the resistant beta: 45.5% higher for both the unadjusted and the adjusted beta.

Panel A: Unconditional RMSE’s Unadjusted Beta Adjusted Beta OLS Res. OLS/Res OLS Res. OLS/ResBristol Myers Squibb All data 0.021 0.020 1.050 0.014 0.013 1.037 SASOL All data 0.057 0.050 1.135 0.038 0.034 1.134 Shared Technologies All data 0.172 0.118 1.455 0.115 0.079 1.455 Environmental Tectonics All data 0.164 0.088 1.873 0.110 0.059 1.874

Panel B: Conditional RMSE’s Unadjusted Beta Adjusted Beta OLS Res. OLS/Res OLS Res. OLS/ResShared Technologies W/ outlier 0.274 0.178 1.537 0.183 0.119 1.537 No outlier 0.089 0.074 1.203 0.060 0.050 1.203 Environmental Tectonics W/ outlier 0.276 0.087 3.161 0.185 0.059 3.159 No outlier 0.073 0.082 0.895 0.049 0.055 0.894

Exhibit 8. RMSE’s for OLS and Resistant Betas

Environmental Tectonics Corp. provides an even more dramatic example of the

improvement in RMSE obtainable by using the resistant beta. For this firm there is one very

dominant outlier in the returns and in the scatter plot. The resistant beta differs radically

from the OLS beta during the time interval when the outlier is in the estimation window. At

other times the two betas are in close agreement. This single influential outlier has inflated

the RMSE of the OLS beta by 87% for both unadjusted and adjusted beta.

It should be noted that the RMSE’s above are unconditional in that they measure the

average prediction error across the history of the returns. Conditional on the period when a

dominant outlier is within the estimation window, the behavior of the OLS beta is much

worse relative to the resistant beta than is indicated by the unconditional RMSE’s reported in

Panel A of Exhibit 8. Panel B of Exhibit 8 displays conditional RMSE’s for the two firms

whose returns contained highly influential outliers. The row labeled “W/ outlier” lists the

RMSE’s conditioned on the estimator moving 104-week interval containing at least one

19

outlier. The row labeled “W/out outlier” contains the RMSE’s conditioned on the estimator

moving 104-week interval not containing an outlier.

Conditioned on the estimation window containing one or more outliers, the OLS beta

RMSE’s are much worse than when using all the data, and are much greater relative to the

resistant beta than when all the data is used. For example in the case of the unadjusted beta

for Environmental Technologies Corp., the conditional OLS beta RMSE is .276 as compared

with .164 when all the data is used, while the resistant beta has virtually the same value of

.0875 in both cases. For this firm, the conditional RMSE of the OLS beta is 216% greater

than the RMSE of the resistant beta, whereas for all the data the RMSE of the OLS beta is

only 87% greater than that of the resistant beta. Similar results hold for the adjusted betas.

The results for Shared Technologies are similar with somewhat less extreme differences.

Furthermore, conditioned on the estimation window not containing highly influential outliers,

the OLS and resistant betas not surprisingly yield rather similar RMSE’s, with the resistant

beta a little worse than the OLS beta for Environmental Technologies Corp., and a little

better for Shared Technologies, Inc.

In order to provide a somewhat broader indication of the relative performance of the

OLS and resistant betas, we selected a larger group of 58 firms based on four categories of

absolute differences (ADELTA) between OLS and resistant beta: ADELTA very near zero,

very near .5, very near 1, and greater than 2. For this group of 58 firms we used all the data

available during the time period Jan. 1990 through Dec. 1998. All firms had weekly returns

for at least two years, and over half of them had weekly returns for more than four years. We

ended up with 27 firms with ADELTA very near zero, 13 with ADELTA very near .5, 17

with ADELTA very near 1, and 3 with ADELTA greater than 2. We then computed the

RMSE’s of the OLS and resistant unadjusted betas for these firms over the time period

indicated. The ratios of these numbers are displayed in the histogram of Exhibit 9.

The distribution of the RMSE ratios is quite skewed to the right, indicating that

resistant beta has a smaller RMSE than the OLS beta by substantial amounts more than half

of the time. There are only 18 firms with RMSE ratios less than one, while the remaining 40

have RMSE ratios larger than one. This skewness becomes even more evident by noting that

only one firm has an RMSE ratio less than .75, while there are 16 firms with RMSE ratios

larger than the symmetrically located value of 1.25.

20

0

2

4

6

1.0 1.5 2.0 2.5

RMSERatio of OLS beta RMSE to Resistant beta RMSE

Exhibit 9. Ratios of OLS Beta RMSE to Resistant Beta RMSE

A more informative picture of the relative behavior of the two estimators emerges

from the distribution of RMSE ratio’s conditioned on values of the absolute differences

(ADELTA) in the OLS and resistant beta estimates as in Exhibit 10.

0.000 to 0.001

0.496 to 0.504

0.956 to 3.317

1.0 1.5 2.0 2.5

RMSE RATIO

ADEL

TA

Exhibit 10. Distribution of OLS/Resistant Beta RMSE’s Conditioned on ADELTA

21

This exhibit displays box plots5 of the distribution of RMSE ratio’s, conditioned on three

value ranges of ADELTA, those very near zero, those very near one-half, and those greater

or equal to .956. The larger the value of ADELTA, the more the conditional distribution of

RMSE ratios is positively shifted to values greater than one.

Performance of OLS and Resistant Betas for Predicting One-Year Future Betas

For consumers using beta to make long term capital budgeting decisions it will be

more interesting to know the relative performance of the OLS and resistant betas as

predictors of beta for a longer period in the future than one week. As an indication that our

results for one-week ahead prediction hold up for longer periods in the future, Exhibit 11

shows comparative results for the OLS and resistant betas as predictors of one-year ahead

betas (as computed by the same moving 104 week window one year later).

Bristol Myers Squibb Co. OLS Resistant OLS/Res

Unadjusted Beta 0.1975 0.2592 0.7620 Adjusted Beta 0.1323 0.1737 0.7617 SASOL Ltd.

OLS Resistant OLS/ResUnadjusted Beta 0.3524 0.3094 1.1390 Adjusted Beta 0.2361 0.2073 1.1389 Shared Technologies Inc. OLS Resistant OLS/ResUnadjusted Beta 1.6673 0.7739 2.1544 Adjusted Beta 1.1171 0.5185 2.1545 Environmental Tectonics Corp. OLS Resistant OLS/ResUnadjusted Beta 1.5893 0.7318 2.1718 Adjusted Beta 1.0648 0.4903 2.1717

Exhibit 11: Comparative RMSE’s of Betas as One-Year Ahead Predictors 5 The boxes represent the interquartile range of the data (the range of the data between the 25th and 75th quartiles. The ends of the whiskers are located at the most extreme data points falling within a distance of 1.5* the interquartile range from the quartiles. Points outside the whiskers are considered outliers.

22

The results are for the same four firms discussed earlier in this section. Note that for

this one-year forecast horizon the ratio’s of RMSE’s for adjusted betas are virtually the same

as for unadjusted betas. While the OLS beta has a 24% lower RMSE than the resistant beta

for Bristol Myers Squibb Co., it is 14% worse for SASOL Ltd., 115% worse for Shared

Technologies and 117% worse for Environmental Tectonics Corp. These results are fairly

consistent with those of Exhibit 9. The main difference is that for the Bristol Myers Squibb

Co. one-year predictions in Exhibit 11, the resistant beta is somewhat worse than the OLS

beta whereas in Exhibit 9 it was slightly better.

The way to view the above behavior is as follows. When the data does not contain

influential outliers, the OLS beta is the most precise estimate and the resistant estimate will

tend to have a little less precision. But, when influential outliers are present, the OLS beta

can be an exceedingly poor estimate while the resistant beta is a good estimate, as has been

amply demonstrated. Use of a resistant beta in effect entails payment of a small “insurance

premium” in terms of inflated prediction RMSE when the returns series is outlier free in

return for protection in the form of substantial performance improvement relative to using the

OLS beta when the series contains highly influential outliers. While the these results are not

conclusive with respect to the entire population of equity returns, they are highly indicative

of superior overall performance of the resistant beta and set the stage for an exhaustive study

across all U.S. equities.

7. CONCLUDING COMMENTS

Our investigation has shown quite emphatically that outliers can have substantial

biasing influence on OLS betas, and that this bias can cause the OLS beta to be a quite

misleading measure of risk and return. The new resistant beta is not much influenced by

outliers and accurately describes the risk and return characteristics of the bulk of the data.

Furthermore the resistant beta was shown to have superior performance to the OLS beta for

predicting future betas when influential outliers are present, while suffering at most only a

slight degradation in performance when no influential outliers are present.

We believe that the new resistant beta will serve well as an important complement

to the OLS beta in the following manner. When the two betas are in close agreement, it

signals the absence of distortion of the OLS beta by influential outliers, and in such cases the

23

classical interpretation of the OLS beta is appropriate. When the two betas are substantially

different, it indicates that outliers have likely substantially influenced the OLS beta and that

the OLS beta is not to be trusted. These two alternatives can be determined by comparing the

absolute difference of the two betas with some threshold value DELTA. This value could be

determined on the basis of a statistical significance as in Martin and Simin (1999) or perhaps

by a subjective value that seems financially relevant by the user of the beta.

Once a value of DELTA has been set, the resistant beta can be used to routinely

screen for firms whose OLS beta is influenced by outliers. When the absolute difference

between the two betas is less than DELTA, the supplier can just provide the OLS beta.

When this difference is greater than DELTA, the supplier should provide both betas. This

alone would have the beneficial effect of alerting unsuspecting consumers with a warning

that influential outliers are likely present in the data. Furthermore, such warnings should

trigger follow-up checks for data errors and their correction and further investigation into the

cause and meaning of the outliers in the event that the data is determined to be error-free.

The routine provision of such information would add value to the beta providers’ service.

As an example the referee pointed out that outliers can be artificially created by not

properly accounting for stock splits when computing returns. Given that CRSP devotes

considerable effort to producing clean data, it is not surprising that none of the outliers in the

examples of Exhibit 4 and Exhibit 8 are associated with splits. Even so, a large value of

DELTA could be used to indicate such data errors and it is a fortunate benefit of our

proposed resistant beta that it eliminates large outliers of this kind. The resistant beta

therefore has an automatic data-cleaning feature with respect to such errors.

Note that we are not recommending the resistant beta as a blind replacement for the

OLS beta. However, we do reiterate that the resistant beta accurately describes the “typical”

risk and return characteristics of a firm as represented by the bulk of the returns, excluding at

most a small fraction of outliers. It is our contention that outliers are never predictable based

on returns data alone. This leads us to conjecture that the resistant beta will provide a more

reliable forecast of the predictable risk and return characteristics of a firm, as represented by

the non-outlier portion of the future returns, than the OLS beta. Further research is needed to

confirm or reject this conjecture.

24

Finally, our study begs the question of what causes the outliers and whether or not

they can be modeled with some degree of predictive power using exogenous data. For

example, perusal of the time series of returns for many small size firms gives the impression

that outliers tend to occur in the vicinity of calendar quarters, indicating they may often be

associated with corporate announcements. Such questions are in need of further study.

25

BIBLIOGRAPHY

Block, S. B., 1999, A Study of Financial Analysts: Practice and Theory, Financial Analysts Journal, July/August, 86-95.

Bruner, R. F., K. M. Eades, R. S. Harris, and R. C. Higgins, 1998, Best Practices in

Estimating the Cost of Capital: Survey and Synthesis, Financial Management 27, 13-28.

Connor and Herbert, 1999. Horizon Newsletter article entitled “Estimation of the European

Equity Model”, ( No. 169). Cook, R. D. and S. Weisberg, 1982. “Residuals and Influence in Regression.” Chapman and

Hall, New York. Gitman, L. J. and V. A. Mercurio, 1982. Cost of Capital Techniques Used by Major U.W.

Firms: Survey and Analysis of Fortunes 1000, Financial Management Winter, 21-29. Graham, J. R. and C. R. Harvey, 2001. The Theory and Practice of Corporate Finance:

Evidence from the Field, Journal of Financial Economics 60: 187-243. Grinold R. C. and R. N. Kahn, 1995. Active Portfolio Management: Quantitative Theory and

Applications , New York, McGraw-Hill. Hampel, F. R., 1986. Robust statistics: the approach based on influence functions, Wiley

series in probability and mathematical statistics. Probability and mathematical statistics, New York, Wiley.

Judge, G. G., R. C. Hill, W. E. Griffiths, H. Lütkepohl, and T. C. Lee. 1988. “Introduction to

the Theory and Practice of Econometrics.” Second edition. John Wiley & Sons, Inc., New York. (Reviewer: Kajal Lahiri)

Knez, P. J. and M. J. Ready. 1997. “On the Robustness of Size and Book-to-Market in Cross-

Sectional Regressions.” Journal of Finance, vol. 52, no. 4 (September):1355-82. Martin, R. D. and Simin, T. 1999. “Robust Estimation of Beta”, Technical Report No. 350,

Dept. of Statistics, Univ. of Washington, (March). S-PLUS 2000. 1999. Guide to Statistics, Vol I, Insightful Corporation, Seattle, WA. Yohai, V. J., and R. H. Zamar, 1997. “Optimal Locally Robust M-estimates of Regression.”

Journal of Statistical Planning and Inference, vol. 64, no. 2 (November):309-324.

26

APPENDIX

The Analytic Expressions for the Weight Function w

( ) 2 4 6

1 1 81 944 2 133 476 030 1 8 2 7

0 2 7

u .w u . . u . u . u . u .

u .

≤= − + ⋅ − ⋅ + ⋅ ≤ ≤ ≥

Computing the Resistant Beta

Equations (4) and (4’) for the resistant beta can be obtained by minimizing the objective

function 1

nt m,t

i

R Rs=

−α − ⋅β ρ

∑ with respect to α and β , where s is a robust scale measure

for the residuals in the numerator. The function ρ is given be the formula

( )

2

2 4 6 8

0 5 1 81 452 972 533 079 004 1 8 2 72 633 2 7

. u u .u . . u . u . u . u . u .

. u .

⋅ ≤ρ = − ⋅ + ⋅ − ⋅ + ⋅ ≤ ≤ ≥

The weight function ( )w u is given by ( ) ( )w u u / r= ψ where ( ) ( )u u′ψ = ρ is the result of

differentiating ρ in the above minimizaton problem.

We use the computation algorithm for this minimization that has been implemented in

the statistical modeling system S-PLUS (1999) to compute the resistant betas. See “Chapter

9: Robust Regression” in the S-PLUS (1999) Guide to Statistics, Vol I., for further details.