Often, we are interested in estimatingβfor several assets. For instance, sup- pose we are analyzingN stocks and let β1,β2, . . . ,βN denote their respective parameters in the market model (8.1). In such cases, we may use shrinkage estimation, following the general approach described in Section 6.5.
Thus, in shrinkage estimation, we combine a simple estimator of βi, such as the least-squares estimator, with an estimator based on assumptions regarding the parametersβ1,β2, . . . ,βN, by taking a weighted average of the two estimators.
Let ˆβ1,ˆβ2, . . . ,ˆβN denote the least-squares estimators of β1,β2, . . . ,βN, respectively. For the assumption-based estimators, we may consider the assumption that β1=β2=ã ã ã=βN ≡βfor some β. To estimate β, we may use the average of the least-squares estimators:
¯β= 1 N
N i=1
βˆi.
Then a shrinkage estimator ofβi is given by a weighted average of ˆβi and ¯β. When ˆβ1,ˆβ2, . . . ,ˆβN are approximately equal, we give more weight to ¯β. When ˆβ1,ˆβ2, . . . ,ˆβN do not follow the assumption of equal beta, in the sense that there is a great deal of variability in ˆβ1,ˆβ2, . . . ,βˆN, then more weight is given to ˆβi.
In order to choose the weights given to ˆβi and ¯β, we adapt the procedure used in Section 6.5 when estimating mean returns. Let SE(ˆβi) denote the standard error of ˆβi, as given in the output of thelmfunction, and let
SE2= 1 N
N i=1
SE(ˆβi)2.
Then a shrinkage estimator ofβi is given by ψ¯β+ (1−ψ)ˆβi
where
ψ= SE2 SE2+τ2β and
τ2β= 1 N
N ij=1
(ˆβi−¯β)2.
Example 8.8 One use for shrinkage estimation is in estimating the values ofβ for a number of assets for which it is reasonable to expect similar relationships with the market index. For example, here we consider the four airline stocks, American Airlines Group, Inc. (symbol AAL), Delta Air Lines, Inc. (DAL), Southwestern Airline Company (LUV), and United Continental Holdings, Inc.
(UAL).
Five years of monthly returns for the period ending December 31, 2014, were computed for these stocks. The results are stored in variables with the name matching the stock symbol; for example, the returns on Ameri- can Airlines stock are stored in the variableaal. The returns for all the four stocks are stored as a matrix in the variableair, which is similar to the vari- ablebig8. Estimates of beta for each of the four stocks may be computed as follows:
> air.mm<-lm(air~sp500)
> air.beta<-air.mm$coefficients[2,]
> air.beta
AAL DAL LUV UAL 0.610 0.825 1.016 0.679
Therefore, ¯βandτ2β may be calculated by
> beta.bar<-mean(air.beta)
> beta.bar [1] 0.782
> tausq.beta<-mean((air.beta-beta.bar)^2)
> tausq.beta [1] 0.0242
To compute SE(ˆβi) for each stock, we may use theapplyfunction. Note that the[2, 2]element of the component$coefficientsofsummaryapplied to the output from thelmfunction yields the standard error of ˆβ.
Define a function f.betaseby
> f.betase<-function(y){ summary(lm(y~sp500))$coefficients[2,2]}
Then the vector of standard errors of ˆβi can be computed by
> air.betase<-apply(air, 2, f.betase)
> air.betase
AAL DAL LUV UAL
0.5628 0.3405 0.2400 0.3879 and SE2 is given by
> sesq.bar<-mean(air.betase^2)
> sesq.bar [1] 0.160
It follows that the weightψused in the shrinkage estimator is given by
> air.psi<-sesq.bar/(sesq.bar+tausq.beta)
> air.psi [1] 0.869
and the shrinkage estimates of beta for the four airline stocks are given by
> air.psi*beta.bar + (1-air.psi)*air.beta AAL DAL LUV UAL
0.760 0.788 0.813 0.769
Recall that the least-squares estimates are given by AAL DAL LUV UAL
0.610 0.825 1.016 0.679
Note that the variation of ˆβ1,ˆβ2,ˆβ3,ˆβ4 is small relative to the standard errors of the ˆβj so that the shrinkage estimates are all relatively close to the
average of the ˆβj.
An important feature of the shrinkage estimator described previously is that the same weight ψ is used for each asset. However, in many cases, the standard error of ˆβi varies considerably for the different assets. For assets in which beta is estimated accurately, in the sense that SE(ˆβi) is relatively small, we may want to give more weight to ˆβi; on the other hand, for assets for which SE(ˆβi) is relatively large, it may be preferable to give relatively little weight to ˆβi.
Thus, it may be desirable to use asset-specific values ofψ,ψ1,ψ2, . . . ,ψN, particularly when SE(ˆβi),i= 1,2, . . . , N exhibit large variation. Let
ψi= SE(ˆβi)2 SE(ˆβi)2+τ2β.
Then an alternative shrinkage estimator ofβi is given by ψi¯β+ (1−ψi)ˆβi.
Example 8.9 Consider the four airline stocks analyzed in Example 8.8.
Recall that the excess return data for the four stocks are stored in the vari- ablesaal,dal,luv, andual; the data matrix for these four assets is stored in the variableair. The variablesp500contains the excess returns on the S&P 500 index.
Consider estimation ofβfor American Airlines stock. Using the results of thelmfunction applied to the market model for American Airlines,
> summary(lm(aal~sp500)) Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0459 0.0218 2.10 0.04 *
sp500 0.6095 0.5628 1.08 0.28
Residual standard error: 0.162 on 58 degrees of freedom Multiple R-squared: 0.0198, Adjusted R-squared: 0.00292 F-statistic: 1.17 on 1 and 58 DF, p-value: 0.283
we have that SE(ˆβi) = 0.5628; recall that hereτ2β is given by 0.0242. Hence, the weight for AAL is
0.56282
((0.5828)2+ 0.0242)= 0.9290.
It follows that the shrinkage estimate of beta for AAL is given by
(0.929)¯β+ (1−0.929)(0.6095) = (0.929)(0.7822) + (1−0.929)(0.6095) = 0.7699.
To calculate the shrinkage estimates ofβfor all four stocks, recall that the variableair.betasecontains the standard errors of ˆβi for the four stocks
> air.betase
AAL DAL LUV UAL
0.5628 0.3405 0.2400 0.3879
The vector of weights used in this procedure along with the vector of shrinkage estimates ofβfor all four stocks may now be computed as follows:
> psi.air<-(air.betase^2)/(tausq.beta + air.betase^2)
> psi.air
AAL DAL LUV UAL 0.929 0.827 0.704 0.861
> psi.air*beta.bar + (1-psi.air)*air.beta AAL DAL LUV UAL
0.770 0.790 0.851 0.768
Recall that the shrinkage estimates based on a global value forψare given by AAL DAL LUV UAL
0.760 0.788 0.813 0.769
The two sets of estimates are very similar. The greatest difference occurs for LUV; note that LUV has the largest value of ˆβi and the smallest value of
the standard error of ˆβi.
Adjusted Beta
When estimating βfor a large number of assets, a simpler type of shrinkage estimator is sometimes used. Since the value of beta for the entire market is, by definition, equal to 1, it is often reasonable to use 1 in place of ¯β. Using global values for the weights given to one and ˆβi leads to an estimator of the form
(1−k) +kˆβi (8.7)
for some constant k. Often k= 2/3 is used, yielding the estimator of βi
given by
ˆβi,adj=1 3+2
3
βˆi, (8.8)
which is known as adjusted beta. It is sometimes attributed to analysts at the brokerage firm Merrill Lynch, Pierce, Fenner & Smith, Inc. (Vasicek 1973); this type of adjusted beta is used most notably by the financial data firm Bloomberg L. P. (www.bloomberg.com) so it is sometimes referred to as
“Bloomberg adjusted beta.”
This estimator has the advantage of requiring only ˆβi in order to estimate βi. However, it has the drawbacks of always shrinking the estimates toward 1 and of always using the weights 1/3 and 2/3; these choices may not be appropriate in all cases.
Example 8.10 Stocks for firms represented in the S&P 100 index were con- sidered; these data were also analyzed in Example 8.5. For each of the 96 stocks, five years of monthly returns were analyzed for the period ending December 31, 2014.
For each stock, the least-squares estimate ˆβi was calculated along with the adjusted beta for stocki, ˆβi,adj. To measure the accuracy in these estimates as predictions of future beta values, they were compared with the least-squares estimates ofβbased on the 12 monthly returns in 2015, which we denote by ˆβ∗i,i= 1,2, . . . ,96.
The average error in the least-squares estimates is given by 1
96 96 i=1
|ˆβi−ˆβ∗i|= 0.369;
for adjusted beta, the average error is 1
96 96 i=1
|ˆβi,adj−ˆβ∗i|= 0.321.
Thus, use of adjusted beta reduces the error in predicting the estimates of βfor 2015 by about 13%.
For comparison, the shrinkage estimates of the βi were also calculated.
The average error in these estimates in predicting ˆβ∗i is 0.347 using a global value ofψ and 0.344 using asset-specific values of ψ. Thus, at least for this example, adjusted beta appears to be at least as successful as the shrinkage estimators in predicting future beta values.
It is worth noting that none of the estimators are particularly accurate in predicting future beta estimates. In evaluating these results, it is important to keep in mind that the future beta values used for comparison are estimates, with their own sampling variability, in addition to the sampling variability of the least-squares, adjusted, and shrinkage estimates of beta.