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Quantitative Finance, 2009, 113, iFirst

Robust estimation with flexible parametricdistributions: estimation of

utility stock betas

JAMES B. MCDONALDy, RICHARD A. MICHELFELDER*z andPANAYIOTIS THEODOSSIOUzx

yDepartment of Economics, Brigham Young University, Provo, UT, USA

zSchool of BusinessCamden, Rutgers University, 227 Penn Street, Camden, NJ 08102, USA

xCyprus University of Technology, Cyprus

(Received 26 June 2006; in final form 16 January 2009)

The distributions of stock returns and capital asset pricing model (CAPM) regression residualsare typically characterized by skewness and kurtosis. We apply four flexible probabilitydensity functions (pdfs) to model possible skewness and kurtosis in estimating the parametersof the CAPM and compare the corresponding estimates with ordinary least squares (OLS) andother symmetric distribution estimates. Estimation using the flexible pdfs provides moreefficient results than OLS when the errors are non-normal and similar results when the errorsare normal. Large estimation differences correspond to clear departures from normality. Ourresults show that OLS is not the best estimator of betas using this type of data. Our resultssuggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capitalfor public utility stocks.

Keywords: Robust estimation; Beta; Flexible distributions; Skewness; Kurtosis

1. Introduction and purpose

Consistent with the well-established literature on the

characteristics of the distributions of stock returns in

general, public utilities stock returns distributions have

thick tails (leptokurtosis) as well as skewness. Estimating

capital asset pricing model (CAPM) betas using ordinary

least squares (OLS) when the data (returns and regression

errors) are non-normal results in inefficient estimators.

Inefficient betas are prone to greater estimation error

as their distributions have larger dispersion. They aremore likely to be insignificant due to larger standard

errors. The major focus of this study is efficient robust

estimation with application to the CAPM for public

utilities. Its main motivation stems from the fact that

public utility regulators and utilities, in addition to

investors and stock analysts, regularly use CAPM betas

to estimate the cost of common equity for public utilities.

Harrington (1980) conducted two surveys on the use

of the CAPM for utility regulation and found that the

model had either been considered or was being used by

38 utility commissions. Cooley (1981) reviewed the use

of the CAPM in estimating the cost of equity capital for

public utility companies and concluded that its use

has not been merely nominal. In a review of surveys,

Cooley (1981) found that the Federal Communications

Commission and a minimum of 20 state utility commis-

sions had heard testimony involving the application of the

CAPM. Out of 54 jurisdictions surveyed in 1978, 16 rate

cases involve the use of the CAPM, and there were 12 more

the following year. A web search of the use of the CAPM in

public utility rate cases today will easily demonstrate its

widespread application. Bey (1983) found that the out-

comes of public utility rate cases had a tremendous impact

on financial health of both the consumers and the utility

companies. He concluded that the CAPM should be used

in such cases in the best possible manner.

Investor-owned public utilities are price and rate-

of-return regulated. Estimating the cost of common equity*Corresponding author. Email: [email protected]

Quantitative Finance

ISSN 14697688 print/ISSN 14697696 online 2009 Taylor & Francis

http://www.informaworld.comDOI: 10.1080/14697680902814241

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for setting the regulated utilitys allowed rate of return with

inefficiently estimated betas results in larger errors in the

pricing of electricity, and therefore creates inequitable shifts

of wealth between the regulated firms and consumers.

Moreover, more precise cost of capital estimates result in

less uncertainty in the regulatory electricity price setting

process and capital investment. In a risk-averse world, more

precise cost of capital estimates will have a significant

positive impact on the societal economic welfare. From theinvestors point of view, more accurate estimates of cost of

capital and portfolio inputs in general will lead to the

construction of more efficient portfolios. Siegal and

Woodgate (2007) and Klein and Bawa (1976) discuss the

impact of estimation error on the optimal portfolio choice

and performance.

The major sources of beta estimates to investors,

utilities and regulators come from investor information

services such as those provided by Value Line, Merrill

Lynch and Goldman Sachs. These beta estimates are

mainly based on the OLS estimation method and as such

they are likely to possess larger estimation error. In this

paper, we show that the use of flexible pdfs in regressionestimation leads to betas which are more efficient

in that they may possess smaller variances than those

associated with OLS. We evaluate the effectiveness

of several flexible parametric probability distributions

for estimating more efficient betas with quasi-maximum

likelihood estimation and compare them with OLS

and the generalized method of moments (GMM). These

pdfs include the skewed generalized T (SGT), the skewed

generalized error distribution (SGED), the skewed expo-

nential generalized beta of the second kind (SEGB2)

and the inverse hyperbolic sine distribution (IHS).

The first three pdfs have been developed in the last 10

years. The IHS was first introduced in 1949 but has

remained obscure until the recent interest in robustestimation and addressing non-normality in regression.

The flexible pdfs accommodate wide ranges of skewness

and kurtosis and therefore may result in more efficient

estimated betas when the data are non-normally dis-

tributed. Although the applications herein involve electric

utility stocks, the estimation methods universally apply

to all types of company stock CAPM parameters as their

stock returns pdfs typically have thick tails and skewness.

Thus, we suggest that the application of inefficient betas

may be a source of general equity market mis-pricing

and inefficiency as stock returns and their regression

errors are typically non-normal.

2. Empirical distributions of stock returns

Mandlebrot (1963) and Fama (1965) initially established

that the distribution of stock returns regression residuals

have leptokurtosis. McDonald and Nelson (1989) and

Harvey and Siddique (1999) found skewness and thick

tails in tests of various stock indices and asset classes.

Harvey and Siddique (2000) found positive skewness and

co-skewness with the stock market for portfolios of

electric and water utility stocks, in addition to other

stock portfolios. Chan and Lakonishok (1992) concluded

that since the distribution of stock returns is non-normal

for so many studies due to kurtosis that OLS estimators

of beta will often be inefficient. They found substantial

efficiency gains using robust methods when returns

contain extreme outliers.

Efficient beta estimation addressing skewness or

kurtosis is also discussed by Fielitz and Smith (1972),Francis (1975), McDonald and Nelson (1989), and Butler

et al. (1990). Theodossiou (1998) rejected the assumption

of normality of returns for multiple stock exchanges

indices, exchange rates, and gold. Akgiray and Booth

(1988, 1991) considered a mixture of normal distributions

and non-normal empirical pdfs in modeling the statistical

property of exchange rates. Bali (2003) fits alternative

pdfs (non-normal) to model the extreme changes in the

US Treasury securities market. Bali and Weinbaum

(2007) also reject the normality of stock market returns

for various indices. The literature concluding that asset

returns pdfs have skewness and kurtosis and therefore are

non-normal pdfs is vast. Generally, stock and other assetreturns pdfs have fat tails due to extreme outliers and

are often asymmetric. OLS estimators are highly sensitive

to extreme values. Electric utility as well as non-utility

stock returns have pdfs that are thick-tailed and skewed.

In these cases, alternatives to OLS can yield more efficient

estimators.

3. Flexible probability distributions

and robust estimation

There are a myriad of robust estimation methods.

Although many are discussed in this paper, some methods

such as those of Yohai and Zamar (1997) and Martin andSimin (2003) are not, as this paper focuses on those

methods that reflect generality in pdf by accommodating

varying levels of skewness and kurtosis and that nest

many pdfs. Martin and Simin (2003), however, do find

some interesting results with the data-dependent weighted

least squares approach that they developed and tested.

Boyer et al. (2003) differentiate robust, or outlier-

resistant estimators, into reweighted least squares (RLS)

or least median squares (LMS) and partially adaptive

estimators. Partially adaptive estimation procedures can

be viewed as being quasi-maximum likelihood estimators

(QMLE) because they maximize a log-likelihood function

corresponding to an approximating error distributionover both regression and distributional parameters. RLS

and LMS address only the explicit choice of regression

parameters. Therefore the pdfs in this investigation are

referred to as flexible pdfs. Boyer et al. (2003) use Monte

Carlo simulations to test the efficiency of flexible pdfs,

RLS and LMS. Using one of the four flexible pdfs and

a more restrictive version of another pdf used in this

paper, they concluded that flexible pdfs were found to

produce more efficient estimators than outlier-resistant

methods that do not accommodate changes in pdf

2 J. B. Mcdonald et al.

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parameters when regression models have skewness or

kurtosis. Therefore, among the myriad of robust estima-

tion methods, this paper focuses on flexible pdfs.

The flexible probability distributions considered in this

investigation can accommodate a wider range of data

characteristics than commonly used distributions such as

the normal, log-normal, Laplace, and T. Although the

Laplace is a pdf and least absolute deviations (LAD) is an

estimation method, they produce the same estimates,analogous to the normal pdf and OLS. The flexible

probability distributions are the SGT from Theodossiou

(1998), the SGED from Theodossiou (2001), the SEGB2

from McDonald and Xu (1995), and the IHS from

Johnson (1949). These distributions have been used by

Hansen et al. (2007) to model various financial time series

with skewed and leptokurtic distributions such as various

stock market index returns, exchange rates, and the price

of gold. The SEGB2 and the more restrictive, non-skewed

version of the SGT, the generalized T (GT), were used by

Boyer et al. (2003).

These distributions nest several well-known distribu-

tions often used in econometric modeling. Some of thedistributions that are nested in the flexible pdfs include:

the normal, T, skewed T (ST) (Hansen 1994), GT

(McDonald and Newey 1988), generalized error (GED)

(Box and Tiao 1962), Laplace, and uniform distributions.

See Appendix A.

4. Estimation of alphas and betas

The CAPM is formulated below in equation (1). This

version assumes that the intercept, , is equal to zero. A

number of empirical tests of the CAPM structure have

tested as evidence against the CAPM structure. Handa

et al. (1993) simultaneously test as a vector of s fora series of stocks within portfolios and find evidence that

s are non-zero with monthly returns data. Other studies

such as Black et al. (1972), Blume and Friend (1973), and

Fama and MacBeth (1974) perform empirical CAPM

tests by estimating the security market line and perform-

ing tests on the intercept and whether the slope is equal to

the market risk premium.

The estimation of the CAPM alpha and beta

parameters for each utility company stock return is

accomplished by estimating the following model via

maximization of the sample log-likelihood function of

equation (2):

Ri,t Rf,t i iRm,t Rf,t "i,t, 1

maxi,i,i,j

li,i, i,jji maxi,i,i,j

XTt1

lnfi,j"i,tji,j

( ), 2

where j SGT, SGED, SEGB2, IHS, Laplace, T, GED,

GT, ST, and Normal, "i,t is the error of the stock return-

generating process for utility stock i (i 1,2, . . . , 36),

t denotes the time period (t 1,2, . . . , T), Ri,t is the stock

return of utility i for period t, Rm,t is the stock market

return, Rf,t is the risk-free rate of return, i and i are the

alpha and beta for utility stock i, i is a vector of

distributional parameters in the pdf j, and i includes the

data for estimation. GMM is also applied as it requires no

prior assumption on the pdf of the error term. A complete

discussion of the special estimation methods and the

flexible pdfs is presented in appendix A.

5. Estimation results

The sample consists of 36 electric and electric and gas

combination companies that were continuously publicly

traded between January 1990 and December 2004. These

include all publicly traded companies with SICs 4911

and 4931. Any stock that stopped trading and did not

have continuous returns during the period was removed

from the sample. This exclusion involved only one utility

stock.

Market and utility stock returns are monthly total

stock returns that are obtained from the University of

Chicagos Center for Research in Security Prices (CRSP)database. The market is defined by the CRSP value-

weighted index that includes all stocks traded on the

NYSE, NASDAQ, and the AMEX. We used monthly

data to be generally consistent with practitioners use of

monthly data for estimation. Monthly data resulted in

180 stock return observations for each utility stock and

the market. The risk-free rate is the one-month return on

the one-month US Treasury Bill. The excess market

return is the same as defined in the FamaFrench

database.

A review of the descriptive statistics of the excess

returns data for the utility stocks (mean, standard

deviation, skewness, excess kurtosis, JarqueBera (JB)

statistic) was performed for the 36 utility stocks for theentire period January 1990 to December 2004. The mean

monthly excess return (in decimal format) is 0.0057 and

its standard deviation of 0.0631 results in an average

return-to-risk, or Sharpe ratio of 0.09. This is a typical

reward-to-risk ratio for stocks. By comparison, the

20-year US Treasury Bond average Sharpe ratio is 0.06

between 1961 and 2002. The mean excess kurtosis and

skewness values are 2.832 and 0.0941, respectively.

The JB statistics show that almost all of the utility stocks

returns distributions are non-normal. The JB statistic

is asymptotically 2 distributed with two degrees of

freedom and has a critical value of 5.99 at the 5% level of

significance. This test shows that the levels of skewnessand excess kurtosis of the returns distributions lead

to the conclusion that the returns are non-normally distri-

buted for 28 of the 36 companies. Rather than testing

the significance of skewness and kurtosis independently,

we reviewed their joint test with the normal pdf and

no excess kurtosis nor skewness as the null hypothesis.

Table 1 displays the beta estimates for each of the

alternative estimators. It also includes the GMM estima-

tor since GMM requires no prior distributional assump-

tion for the error term. GMM parameter estimates are

Robust estimation with flexible parametric distributions 3

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asymptotically consistent despite the non-normality of

the error term.y Generally, the betas are similar in

magnitude across the various pdf estimates. The medians

of the GMM estimates are a little less than obtained using

other methods. The maximum GMM estimate is less than

the maximum of all other methods except for one stock.

However, it is common across the stocks for the OLS

(and GMM) estimate(s) to have the largest difference

from the other estimates, which may agree quite closely.

There does not seem to be any systematic under- or

under-estimation of the beta by OLS compared with the

flexible pdfs. We also did not observe any of the robust

estimators to have a tendency to be more or less similar to

OLS. The case of GMP is an example of a large difference

with the OLS estimate being 0.012 and the flexible pdf

estimates ranging between 0.109 and 0.167. CIN has

an OLS beta of 0.103 and a range of flexible pdf betas

ranging between 0.177 and 0.217. These two stocks betas

are substantially different from the OLS estimates.

The resulting risk premia, i(Rm,t Rft), to estimate

their costs of equity of capital and allowed rates of

return would differ by the same magnitude. Althoughmost of the OLS beta differences are not so dramatic, they

have the largest systematic difference from the other

estimates. The mean of the maximum difference between

the OLS and flexible pdfs betas is 0.0496. Given that

the value-weighted portfolio beta for the utility stocks is

0.21, this is a substantial difference, and would lead to

a large difference in the estimated cost of capital for the

portfolio. It appears that the agreement among the robust

estimators on the estimate of betas indicates that these

methods are controlling for the impact of large unduly

influential data points.

An inspection of electric utility company betas in Value

Line from the MarchMay 2004 issues that include

electric utilities have a mean adjusted beta (Blume 1975)

a 0.33 0.67u of 0.79 and unadjusted Value Line

mean beta of 0.69. Value Line uses OLS to estimate raw

betas then applies the Blume beta adjustment shown

above. To the extent that OLS is used to estimate betas to

compute estimates of the public utilities cost of common

equity capital and allowed (regulated) rates of return on

invested capital, the degree of the difference between OLS

and flexible pdf betas due to possible larger estimation

error should be an important regulatory policy question,

as well as a statistical problem. Note that utility betas are

adjusted by Value Line with the above Blume equation

that assumes that they converge to one, whereas in reality

they do not.

Although not presented, none of the alpha estimates

are statistically significantly different from zero. This is

consistent with the structure of the CAPM when using the

excess-return CAPM equation for empirical testing, given

that, according to theory, alpha should be equal to zero.

A comparison of the log-likelihood values corresponding

to the estimates for 11 regression error distributions

(not presented for brevity), including the four flexible

pdfs and their symmetric counterparts, normal or OLS,

the Laplace or LAD, T, and ST show that the log-

likelihood estimates are generally higher for the more

flexible distributions. The OLS results are associated with

the smallest log-likelihood value, which follows from

the normal being a special or limiting case of many of the

other pdfs being considered and due to its inability to

fit thick tails and skewness. Furthermore, in each case

(SGT, ST, SGED, SEGB2, and IHS) the more general pdf

is seen to provide a statistically significant improvement

relative to the normal for almost every stock using

a likelihood ratio (LR) test.

Table 2 reports values of the LR test statistic

corresponding to testing the hypothesis that the estimateddistributions of the regression errors are observationally

equivalent to the normal. This statistic is asymptotically

distributed as 2 with degrees of freedom equal to the

difference in the number of distributional parameters

when the normal is nested in the estimated pdf as in the

case of the SGED. The 2 is not appropriate for non-

nested pdfs. The LAD does not nest the normal and

therefore the test does not appear for that pdf.

The asymptotic distribution of LR is not 2 distributed

for limiting cases where the parameter is on the

boundary of the parameter space such as when comparing

a T with a normal pdf. While the excess returns are non-

normally distributed as shown from JB tests, we would

not be surprised if the errors behave similarly as found by

Blume (1968).

However, simulations conducted by McDonald and Xu

(1992) suggest that the statistical differences will be at

least as large as those based on the use of a chi-square

distribution.z Most of the reported LR values imply the

rejection of the normality assumption at the 5% level.

The exceptions are ED, DTE, HE, PGN, WPS, and WEC

stock returns. The tests for these stocks indicate that the

alternative pdf regression is not a better fit than the

normal. These stocks also have JB statistics for excess

returns that do not reject the null hypothesis that they are

normally distributed. PSD and PNM have some insignif-

icant chi-square statistics among the alternative pdfs.

yWe used the fully iterative GMM estimator as developed by Newey (1988). We used OLS estimates to provide the starting valuesfor the iterative process, which we allowed to iterate until convergence was achieved. We usedJ 4 moment conditions in definingthe objective function to be optimized, thus the convergence involved simultaneously minimizing the correlation between functionsof the first four moments of the estimated error with the independent variable as outlined by Newey (1988).zBased on simulations of 1000 replications, the size of the LR test associated with estimating various pdfs which nest the normal,exponential, or lognormal distribution was explored. When the nested distribution corresponded to a special case of the estimatedpdf, the size of the LR test was close to that predicted by the asymptotic Chi-square distribution. However, in the case where thenested distribution corresponded to a limiting case of the estimated distribution that violated a regularity condition, the size of theLR test appeared to be less than suggested by the asymptotic Chi-square for large sample sizes. This simulation included the GED,T, GT, and SEGB2, but not the SGT or IHS. This suggests that the statistical significance of the limiting cases in table 2 are evengreater than might be inferred from the Chi-square values.


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Table2.Loglikelihoodratiotestforpdfsv.Normalpdf.

Company

T

GED

EGB

GT

SGED

SEGB

IHS

ST

SGT

GMP

63.86

64.23

59.67

65.24

68.62

62.78

66.00

64.79

68.13

AEP

9.71

7.80

9.54

10.10

7.81

9.57

9.76

9.75

10.15

CMS

67.55

60.24

58.41

68.68

61.03

58.52

67.38

67.57

69.25

PGN

1.50

2.11

1.70

2.11

4.67

2.33

2.16

2.73

4.67

CIN

9.97

12.60

12.07

12.60

14.54

14.15

10.64

10.34

14.54

ED

1.77

2.44

1.77

2.44

2.68

2.09

2.07

1.91

2.68

DPL

14.26

15.45

15.47

15.59

16.04

15.79

14.88

14.33

16.05

DTE

0.00

0.18

0.01

0.18

0.65

0.53

0.54

0.35

0.65

D

12.15

12.77

13.62

13.00

14.24

15.18

14.33

13.96

14.59

DUK

20.91

25.40

25.32

25.40

25.60

25.54

22.63

21.23

25.60

EDE

20.99

17.30

19.86

21.08

18.35

19.87

20.86

21.17

21.20

FPL

24.89

24.06

24.34

25.16

24.15

24.50

25.31

24.98

25.24

HE

1.64

1.47

1.61

1.66

2.07

2.52

2.47

2.07

2.12

IDA

10.86

6.45

9.77

14.67

6.52

9.78

10.49

11.28

14.98

WR

27.16

26.96

27.40

27.31

29.74

29.53

28.11

27.45

29.74

ETR

12.77

9.57

12.15

13.67

10.23

12.89

13.30

13.32

14.23

NI

31.49

26.07

28.67

32.52

26.39

28.92

31.31

32.07

32.97

SRP

72.50

67.57

64.78

72.50

67.38

64.86

73.12

73.03

73.03

NU

11.51

16.18

15.92

16.18

16.60

16.36

12.44

11.57

16.60

OGE

8.24

8.76

8.89

8.85

11.09

10.27

9.89

9.40

11.09

PCG

44.66

45.75

43.98

45.79

45.76

43.98

45.62

44.66

45.81

PPL

36.33

35.63

35.90

36.86

38.12

38.08

38.72

37.63

38.62

PNW

33.41

32.98

33.37

34.17

33.30

33.74

34.11

33.63

34.55

POM

14.51

10.57

13.42

18.23

11.02

13.67

14.34

14.62

18.23

PNM

4.76

6.28

5.92

6.28

9.77

9.82

9.31

8.50

9.77

PEG

36.02

35.59

35.56

36.81

39.59

38.63

39.51

37.92

39.60

PSD

4.80

9.40

8.08

9.40

9.42

8.12

5.65

5.14

9.42

EIX

46.02

46.50

45.10

47.32

46.62

45.14

47.22

46.03

47.42

SCG

11.70

14.83

14.39

14.83

15.22

14.79

12.65

11.77

15.22

SO

6.10

6.68

6.42

6.71

6.74

6.46

6.28

6.10

6.74

TE

8.48

10.25

10.27

10.25

11.48

10.59

9.31

8.84

11.48

TXU

72.25

66.49

65.20

72.25

66.25

66.26

73.43

74.86

75.02

UIL

12.30

12.29

12.81

12.43

14.69

13.65

13.07

13.18

14.69

UTL

12.31

15.82

15.40

15.82

15.96

15.53

12.93

12.33

15.96

WEC

6.36

6.32

6.66

6.55

6.46

6.87

6.75

6.63

6.78

WPS

1.79

2.60

1.93

2.60

4.87

4.17

4.10

3.58

4.87

Minimum

0.00

0.18

0.01

0.18

0.65

0.53

0.54

0.35

0.65

Median

12.31

13.80

14.01

15.21

14.96

14.99

13.19

13.25

15.59

Maximum

72.50

67.57

65.20

72.50

68.62

66.26

73.43

74.86

75.02

Mean

21.54

21.27

21.26

22.92

22.32

22.10

22.52

22.19

23.94

StdDev.

20.95

19.59

18.76

20.69

19.67

18.85

21.03

21.03

20.84

Theratiosapproximatethe

2

distributionwiththedegreesoffreedomequaltothediffe

renceinthenumberofparameterscomparedto

thenormalpdf.Thepdfswiththelargestnumberofparametersis5andthe

normalhas2,sothegreatestdegreeso

ffreedomis3.Thecriticalvalueata5%level

ofsignificanceis7.81.


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These stocks excess returns also have relatively higher

(than the stocks listed above) but insignificant JB

statistics.

The relative impacts of skewness and kurtosis are tested

to determine if departures from normal data and the

resulting inefficiency in beta estimates are generated

more dominantly from skewness or kurtosis. Skewness

is important to understand as negative or positive skew-

ness may be an indicator of adverse or favorable (fromthe shareholders perspective) regulation. See Brigham

and Crum (1978).y The SGT, ST, SGED, and SEGB2

regressions were estimated along with their symmetric

counterparts (GT, T, GED, EGB2) of these pdfs by

constraining the skewness parameters to the values that

represent no skewness. We have performed LR tests

(not shown) comparing their fits. All but two stocks 2

statistics are insignificant. None of the SGT-GT and ST-T

LR tests are significant and only 2.8% (one stock) of the

36 stocks LRs for the SEGB2-EGB2 and the SGED-GED

are significant. Therefore, kurtosis appears to be the

dominant non-normality parameter affecting the utility

stock regression fits.The empirical testing leads to the conclusion that

the SGT, IHS, and SEGB2 yield similar results in the

presence of kurtosis and skewness, which can differ

significantly from OLS. Additionally, since partially

adaptive estimation based on SGT includes the LAD,

OLS, and Lk (minimizes (sum of estimated errors)k; see

appendix A) estimators as special cases, the user may

want to consider SGT estimation. It performs better than

the SGED, T, LAD, and the normal based on LR tests.

It performs similarly to the SEGB2 and the IHS. One

must be careful to consider the loss of efficiency from

over-parameterization in using pdfs such as the SGT as it

is defined by five parameters. Note that likelihood ratio

tests between the SGT, SEGB2, and IHS cannot beperformed as they do not nest one another.

One issue that the empirical estimations do not address

is the performance of the flexible pdfs vis-a-vis OLS when

the CAPM regression errors are normally distributed.

When the errors are independently, identically, distri-

buted (i.i.d.) as normal, then OLS is the minimum

variance of any unbiased estimator. However, if the

errors are i.i.d. as non-normal, OLS is still the minimum

variance linear unbiased estimator, but nonlinear robust

estimators may be unbiased and have a smaller variance

than OLS. The next section involves a series of

simulations that compare the efficiency of betas estimated

with OLS and the flexible pdfs when the error term is

normally distributed, is mixed normally distributed

(varying variances and therefore has thick tails), and is

asymmetric (has skewness).

6. Simulations and estimator performance

McDonald and White (1993) and Boyer et al. (2003) used

Monte Carlo methods to compare the relative efficiency

of several regression estimators. Some of the estimators

considered included OLS, LAD, a normal kernel esti-

mator (Manski 1984), GMM (Newey 1988), and partially

adaptive maximum likelihood estimators based on the

assumption of the error terms being independently

and identically distributed as GT, GED, or SEGB2.

The actual error distributions considered included the

(1) normal, (2) a thick-tailed variance-contaminated

normal (normal mixture), and (3) a skewed log-normal.z

Major findings included (1) little efficiency loss for

partially adaptive estimation based on an over-parame-

terization of the distribution of normally distributed

errors; (2) very similar performance of fully iterative

adaptive and partially adaptive estimators in the case of

symmetric thick-tailed distributions;x and (3) clear dom-

inance of the SEGB2 partially adaptive estimator over all

other estimators considered in the case of a skewed error

distribution. Ramirez et al. (2003) used the same sample

design and found that partially adaptive estimation based

on the IHS distribution yields nearly identical results

to the SEGB2 for normal, symmetric thick-tailed, and

skewed error distributions.

The simulations reported in this section are based

on the same error distributions considered in thepapers by Johnson (1949), McDonald and Xu (1995),

and Theodossiou (1998), and use similar data-generating

processes as used by Manski (1984), Newey (1988),

McDonald and White (1993), and Ramirez et al. (2003).

The model simulated in this paper is

yt xt ut 0 0:21xt ut, 3

where the xts correspond to the 180 observed monthly

excess returns on the market, the error terms ut have zero

mean and unitary variance, and the scale parameter is

selected to generate an R2 similar to those obtained from

yThe importance of determining whether non-normality is due to asymmetry or thick tails is important since skewness can result inbiased intercepts (Jensens alphas) under specific conditions, as shown by McDonaldet al. (2009). This is also an importantestimation issue if using the single index model (RiiiRm "i) to predict stock returns. Note that the T-values of the interceptfor the log-normal pdf in the simulations shown in table 3 are substantially higher for the non-normal symmetric pdfs (LAD, GED,T, and GT). The LAD and T intercepts are statistically significant, which is an indicator of bias, as the expected value of the estimateof this parameter is 0. Secondly, negative or positive stock returns skewness can be generated by adverse or favorable regulatorytreatment of utility profits, respectively. Although beyond the scope of this paper, skewness caused by regulatory treatment ofutilities can effect the efficient and unbiased estimation of the models used to estimate the cost of capital.zAll three error distributions are standardized to have a zero mean and unitary variance. The first error distribution is merely a unitnormal generated by u1 z, where z$N(0,1). The thick-tailed variance contaminated error distribution is generated byu2 wz1 (1 w)z2, where z1$N(0, 1/9), z2$N(0,9), and w is 1 with probability 0.9 and 0, otherwise. This distribution is symmetricand has a standardized kurtosis of 24.33. The log-normal distribution is generated byu3 (e

ze0.5)/(e2e)0.5 where z$N(0, 1). Thisdistribution has standardized skewness and kurtosis of 6.185 and 113.94, respectively.xThe fully iterative adaptive kernel and GMM estimators performed much better than the corresponding two-step estimators.


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the regressions discussed in section 4. Appendix B

derives the standard error for scaling the error term in

equation (3) for the simulations. The selected value for

beta is 0.21, which is the market-value weighted OLS

beta for the portfolio of utility stocks. The selected value

of alpha is 0. This is based on the insignificance of the

estimates for the stocks. Ten thousand replications

were performed using the methods of OLS, LAD,

skewed Laplace (SL), GED, SGED, T, ST, GT, SGT,

IHS, SEGB2 and GMM estimators. These results extend

those reported by McDonald and White (1993) to include

the SGT, SGED, and IHS partially adaptive regression

estimators of the slope.

Table 3, Panel 1, presents the mean of the intercept,

slope and expected portfolio return and their T-values

based on 10,000 simulations. T-values for the intercept,

slope and expected portfolio return test for bias; the

expected value for the estimated parameter values of

the intercept and slope are 0 and 0.21, respectively.

The expected value for the T-value is 0. The error

distributions considered are the normal, thick-tailed

contaminated normal, and the skewed log-normal.Panel 2 contains the slope and expected portfolio returns

relative (compared with OLS) root mean square errors

(RRMSEs). Intercept RRMSEs are not presented for

brevity (they are available upon request). The expected

portfolio returns are simulated and also reported in

table 3 to test the relative pricing performance of the

CAPM with the various estimators. We predicted the

excess portfolio returns to assess those pdfs that generate

the least forecast error. The RRMSEs of the returns are

indicators of forecast error.

The results for the intercept show that it is not

significantly different from 0 and therefore is unbiased

except when the error pdf is skewed and a symmetricdensity was applied. Note that the LAD and T pdfs

estimate biased intercepts when the error term pdf is

skewed since their T-values are significantly different

from zero. The T-value for the GT is somewhat high.

Skewness generates intercept bias with the opposite

algebraic sign as all of the symmetric densities yield

intercepts with negative signs yet the log-normal is

positively skewed (see footnote 3). The RRMSEs show

that the efficiency of the intercept estimators of the

flexible pdfs is generally better than the remaining

estimators with skewed errors. The results for the slope

indicate no bias. Their values are close to 0.21 and all of

the T-values are insignificant.y All models, symmetric and

non-symmetric, appear to correspond to unbiased esti-mators of the slope coefficient. Unbiased estimates of the

slope do not mean that there will not be cost of capital

estimation errors since, on average, the slope will tend

toward its true value. Inefficiency leads to larger errors in

slope estimation since the distribution of the slope will

have greater dispersion with higher mean squared errors.

The LAD and SL are the only slope estimators that result

in substantial efficiency losses when the error term is

normally distributed. However, this is not unexpected

since the LAD distribution is the only pdf considered that

does not nest the normal pdf. For the thick-tailed and

symmetric errors distribution, OLS provides the largest

slope RMSE of any of the estimators considered. All of

the estimators except for LAD and SL yield similar

RMSEs with a thick-tailed symmetric error term. In the

case of skewed and thick-tailed error distributions, OLS is

again the worst performing slope estimator. It is in this

case that the partially adaptive estimators (ST, SGT, IHS,

SGED, and SEGB2) give evidence of the potential of

significant increased efficiency for the slope relative to

OLS. GMM also performs well given that it is an

estimator that is pdf independent.

The simulations of the portfolio of utility stocks

expected returns were performed to determine which of

the estimators produces the lowest asset pricing errors.

Simulations of the expected value of the portfolio returnwere performed with an assumed zero intercept and the

simulated slopes multiplied by 0.6, which is the expected

value of the market excess return for the sample period.

As expected, the predicted portfolio returns are unbiased

as shown by their values and their T-values. The efficiency

of the expected portfolio return is of interest since higher

efficiency is an indicator of lower returns prediction and

pricing errors. The portfolio returns simulations reflect

estimation risk, since the variance in the simulated error

term reflects the estimation risk associated with estimated

slopes to fit thick-tailed or skewed errors. The results

clearly show that when the error term is thick-tailed or

skewed, the flexible pdfs and GMM have lower RRMSEsand therefore lower pricing errors. This result essentially

shows that the flexible estimators and GMM are the most

reliable estimators of beta and produce the least pricing

errors.

The inefficiency of the OLS beta estimators can be

visualized as a probability distribution of OLS betas with

greater dispersion than a distribution of betas estimated

by robust methods. This will be reflected in higher

errors in beta estimation, and resulting predictions of

the cost of capital. Envision two beta distributions, one

that is generated from OLS and the other from the IHS

pdf estimates of beta. Although not shown (available

upon request), the beta RMSE for OLS is 0.07645 and

for the IHS the value is 0.04745. All of the four flexiblepdfs (SGT, EBG2, IHS, SGED) have lower RMSEs than

the others and OLS, with an average of about 0.05, but

the IHS is slightly lower than the others. If a beta estimate

from each CAPM beta distribution is chosen at the

yA T-test was used to compare the sample mean of beta from the 10,000 simulation estimates with the hypothesized value of 0.21.The RMSE was used as the standard error of the sample mean for the T-test. If the sample mean was significantly different than0.21, there was a bias in the sample mean estimate. A beta equal to 0.21, alpha equal to 0, a desired R2 of 0.04, 180 randomlygenerated observations, and a standard deviation of 2.1468 for x are the parametric assumptions based on averages of empiricalresults. The corresponding value of can be calculated, which is multiplied times the standardized error terms (u has a mean of 0and variance of 1 using the standard normal, scaled mixture of normals, and scaled and shifted log-normal variable).


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same point, such as one RMSE below the expected value

of 0.21, the OLS value is 0.134 and for IHS the value is

0.163. This would lead to 22% greater under-estimation

of the cost of capital risk premium from OLS due to lower

efficiency. The economic impact of such an allowed rate

of return on the public utility required by regulators

would be a windfall in consumer surplus to customers, an

adverse impact on the financial viability of the public

utilities, resulting in a reduction in economic welfare.Statistically, this may generate negative skewness in the

public utilities stock returns as discussed by Brigham and

Crum (1978).

We recommend that the final choice of slope and

intercept estimator is one of the flexible pdfs if thick tails

and/or asymmetry is present. If there are concerns about

over-parameterization (reduction in efficiency from esti-

mating unnecessary pdf parameters), we recommend the

use of the SGT, IHS, or SEGB2 with testing for statistical

improvements relative to special case pdfs, and then use

the simplest model that is observationally equivalent to

the most general formulation which allows for possible

skewness and/kurtosis. GMM is almost as efficient anestimator of the beta as the flexible pdfs if the flexible pdf

estimators are not readily computable.

7. Summary and conclusions

We apply several flexible pdfs for estimating the betas of

public utility company stocks. Estimation methods based

on the flexible pdfs for the error distributions in regression

models, or partially adaptive estimators, were tested

against OLS, other estimation methods involving the

choice of pdf, and GMM. Partially adaptive estimators

were found to be the most efficient of those considered in

the presence of skewed and fat-tailed distributions.The recommended partially adaptive estimator is based

on the SGT, SEGB2, or IHS if skewness/kurtosis are

present. GMM, which does not assume a particular pdf,

also performed well with non-normal errors.

OLS estimation is efficient when neither skewness nor

kurtosis exist in the pdf of the data, although the partially

adaptive methods that include the normal as a special case

yield similar results. Similar to other stocks, public utility

stock returns and their CAPM regression errors typically

have kurtosis and many have skewness. Therefore, OLS is

an inefficient estimator of beta. This leads to an inefficient

estimate of the cost of common equity capital and

produces the most prediction and pricing error whenthick tails are present. The magnitude of the error in

estimating the cost of common equity capital and allowed

(regulated) rate of return on utility investment is

proportional to the estimation error in beta. Errors in

setting the utilitys allowed rate of return create dis-

balances in consumer and producer surplus and therefore

losses in social economic welfare, result in inefficient

pricing of utility stocks, and general disequilibrium in

energy and related markets as other fuels and conserva-

tion inefficiently substitute mis-priced electric power.

More generally, our results here also suggest that further

research may demonstrate that the substitution of

statistical methods for OLS may result in more efficient

equity markets. Mean and variance estimation risk

affects the choice of optimal portfolios and portfolio

performance.

In appendix A, we review the main distributional

characteristics associated with the flexible pdfs, the

normal, and many other pdfs that they nest. Partially

adaptive estimation based on these distributions was used

to estimate the CAPM for 36 electric utility stocks.

The motivation for selecting electric utility companies

is due to the unique characteristic of regulated rates of

return that appear to be manifest in their skewed and

leptokurtic stock returns and the role estimated betas

have in calculating the cost of capital and in setting

electric utility rates. Based on LR tests and simulations,

the partially adaptive estimators provided significant

improvements relative to OLS in applications in which

the error distribution is skewed and/or thick-tailed.

The statistical performance of the partially adaptive

estimators was also explored using a Monte Carlo study.

The Monte Carlo design was adapted from one used ina number of other papers which explores the impact of

thick tails and asymmetry as well as the efficiency loss of

over-fitting in the presence of normally distributed errors.

Our results show that potential exists for significant

improvements in estimation efficiency in the presence of

leptokurtosis or skewness in the data, and the efficiency

loss from over-fitting the error distribution in the

standard normal linear model was modest.

Lastly, portfolio prediction simulations show that the

SGT, SEGB2, and IHS produce the lowest pricing

prediction errors, and the next best performer is GMM.

Future research should consider estimating time-varying

conditional betas using the conditional or intertemporal

CAPM with robust estimators as discussed by Bali (2008).

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Appendix A: Special estimation methods and

flexible density functions

As indicated previously, many well-known distributions

are nested within the four flexible distributions. Thus,

maximization of the above likelihood function gives, in

the special cases of OLS,

i,iOLS arg mini,i

XTt1

"2i,t

!,

Laplace, the LAD estimator (Koenker and Bassett 1978),

i,iLAD arg mini,i

XTt1

j"i,tj

!,

and the GED (for fixed k), the Lk estimator,

i,iGED arg mini,i

XTt1

j"i,tjk

!,

for an arbitrary but fixed value of k. The skewed

Laplace (SL), the trimmed regression quantile (Chan

and Lakonishok 1992), and the estimators of Koenker

and Bassett (1982) are given by

mini,i

XTt1

"i,t or mini,i

XTt1

1

1 sign"i,tj"i,tj,

where, if"i,t j"i,tj and "i,t 0,

"i,t 1 j"i,tj if"i,t50for0551 or 1551,

and are therefore nested within the flexible pdfs.

Therefore, by definition, the flexible pdfs are increasingly

accommodating of the characteristics of the empirical

distributions of stock returns than typically used pdfs.

Characteristics of the flexible probability

density functions

All of the following flexible pdfs nest the normal among

many other symmetric and asymmetric distributions.


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They have either four or five parameters that describe

their shapes in contrast to the normal that has two

parameters (mean and variance distributions) and there-

fore is less flexible. As shown below, they exhibit similar

performance in fitting an approximating pdf to data that

has skewness and kurtosis. The choice of one of the

following four is recommended below.

Skewed generalized T

The SGT is a five-parameter pdf (mean, standard

deviation, two kurtosis, and skewness parameters).

The and parameters generate the conditional means.

The SGT pdf is

fSGT";,, ,, k, n

C

j"= jk

n 2=k1 sign" kk

n1=k,

where

C k=2n 2=k1=kB1=k, n=k,

S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 32 4A22p

,

k=n 21=kB1=k, n=k0:5B3=k, n 2=k0:5S1,

A B2=k, n 1=kB1=k, n=k0:5B3=k, n 2=k0:5,

2AS1, B() is the beta function, and are the

mean and standard deviation of", k controls the peakness

of the distribution, n controls the thickness of the tails,

50 generate negative skewness, and 40 generates

positive skewness.

The major nested distributions, or special cases of the

SGT, are as follows: 0 gives the GT of McDonald and Newey (1988) and

Boyer et al. (2003),

k 2 gives the ST of Hansen (1994),

k 2 and 0 give the Students T,

n ! 1 gives the SGED of Theodossiou (2001),

n ! 1 and 0 give the GED of Box and Tiao (1962),

n ! 1 and k 1 give the skewed Laplace (SLAD),

n ! 1, k 1 and 0 give the Laplace,

n ! 1, k 2 and 0 give the normal, and

n ! 1, k!1 and 0 give the uniform.

Standardized values for regression error skewness

and kurtosis in the ranges (1, 1) and (1.8, 1) can be

modeled with the SGT. However, not all combinations ofskewness and kurtosis are defined by the SGT.

Skewed generalized error

The SGED is a four-parameter distribution (mean,

standard deviation, kurtosis, and skewness). The SGED

pdf is

fSGED";, , ,, k C

exp

j"= jk

1 sign" kk

:

where

C k=21=k,

2AS1,

1=k0:53=k0:5S1,

S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 32 4A22

p,

A 2=k1=k0:5

3=k,

and k controls the peakness of the distribution, 50

generates negative skewness, and 40 generates positive

skewness. The nested distributions, or special cases of the

SGED, are

0 gives the GED,

k 1 gives the SL,

k 1 and 0 give the Laplace,

k 2 and 0 give the normal, and

k ! 1 and 0 give the uniform.

Skewed exponential generalized beta of the second kind

The SEGB2 is a four-parameter pdf (mean, standarddeviation, and two joint skewness/kurtosis parameters).

The SEGB2 pdf is

fSEGB2";, , ,p, qCep="=

1 ep="=

pq,

C 1=Bp, q,

p q,

1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0p 0q

p,

where p and q are positive scaling constants, B(p, q) is the

beta function, and (z) d ln (z)/dz is the psi function

and psi-prime; the first derivative of the psi function is

known as the digamma function. A smaller value ofp results in a more leptokurtic pdf, q4p reflects negative

skewness, and q5p reflects positive skewness. When

p q, the symmetric EGB2 obtains, and when p q ! 1

the normal distribution obtains.

Inverse hyperbolic sine

The IHS is a four-parameter pdf (mean, standard

deviation, skewness, kurtosis). The pdf is

fIHS";,,,, k kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

22 "2=2 2p

exp k2

2ln = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 "2=2 p ln 2 ,

w sinh z=k,

1=w,

w 1

21 e4jj 2 e2jjk

2

1=21 ek2

1=2ejjk2

,

w=w,

w sign1

21 e2jj ejj1=2k

2

,

where a smaller k results in a more leptokurtic pdf,

50 indicates negative skewness and 40 indicates


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positive skewness. 0 gives the symmetric IHS. 0

and k ! 1 give the normal distribution.

Appendix B: Scaling the standardized error

term for simulations (r ut)

Given that

R2 1 SSE=n

SST=n,

for the simple bivariate regression model,

SST

n

1

n

Xnt1

yt "y2

1

n

Xnt1

xt "x ut "u2:

For OLS estimation and the related orthogonality

conditions, SST/n can be rewritten as

SST

n

1

n 2 Xn

t1

xt "x2X

n

t1

u2t" #:

Thus, for large n and stationary x, R2 approaches

R2 1 2

2varx 2

2varx

2varx 2and 1 R2

2

2varx 2:

Therefore,

2 1 R2

R2

2varx:

Given the variance of the xs used in the simulation,

a beta equal to 0.21, alpha equal to 0, a desired R2 of 0.04,

180 randomly generated observations similar to the 180

observations in the empirical estimations, and a standard

deviation of 2.1468 of x are the parametric assumptions

based on the means of the empirical results. The

corresponding value of can be calculated which is

multiplied by the standardized error terms (uihas a mean

of 0 and variance of 1 using the standard normal, scaled

mixture of normals, and scaled and shifted log-normal

variables).


DownloadedBy:

[informa


2010

rls_beta

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