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ROBUST ESTIMATION WITH FLEXIBLE PARAMETRIC DISTRIBUTIONS: ESTIMATION OF UTILITY STOCK BETAS

James B. McDonald Brigham Young University, USA

Richard A. Michelfelder Rutgers University, USA

Panayiotis Theodossiou Rutgers University, USA Quantitative Finance, Forthcoming

January 2009

Keywords:

Robust estimation, beta, skewness, kurtosis, flexible distributions, electric utilities, stock returns, CAPM JEL Classification: G12, C13, C15, C22

ROBUST ESTIMATION WITH FLEXIBLE PARAMETRIC DISTRIBUTIONS: ESTIMATION OF UTILITY STOCK BETAS

ABSTRACT

The distribution of stock returns and capital asset pricing model (CAPM) regression residuals are typically characterized by skewness and kurtosis. We apply four flexible probability density functions (pdf’s) to model possible skewness and kurtosis in estimating the parameters of the CAPM and compare the corresponding estimates with OLS and other symmetric distribution estimates. Estimation using the flexible pdf’s provides more efficient results than OLS when the errors are non-normal and similar results when the errors are normal. Large estimation differences correspond with clear departures from normality. Our results show that OLS is not the best estimator of betas using this type of data. Our results suggest that the use of OLS CAPM betas may lead to erroneous estimates of the cost of capital for public utility stocks.

Keywords:

Robust estimation, beta, skewness, kurtosis, flexible distributions, public utilities stock returns, cost of capital, CAPM

JEL Classification: G12, C13, C15, C22

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 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 are more 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) conducts two surveys on the use of the CAPM for utility regulation and finds that the model has either been considered or being used by 38 utility commissions. Cooley (1981) reviews the use of the CAPM in estimating the cost of equity capital for public utility companies and concludes that its use “has not been merely nominal.” In a review of surveys, Cooley (1981) finds that the Federal Communications Commission and a minimum of 20 state utility commissions have 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) finds that the outcomes of public utility rate cases have a tremendous impact on financial health of both the consumers and the utility companies. He concludes 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 for setting the regulated utility’s 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 the investor’s 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 pdf’s in regression estimation 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 pdf’s include the skewed generalized T (SGT), the skewed generalized error distribution (SGED), the skewed exponential generalized beta of the second kind (SEGB2) and the inverse hyperbolic since distribution (IHS). The first three pdf’s have been recently developed in the last ten years. The IHS was first introduced in 1949 but has remained obscure until the recent interest in robust estimation and addressing non-normality in regression. The flexible pdf’s accommodate wide ranges of skewness and kurtosis and therefore may result in more efficient estimated betas when the data are non-normally distributed. Although these applications herein involve electric utility stocks, the estimation methods universally apply to all types of company stock CAPM parameters as their stock returns pdf’s 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.

I. 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 in Butler, McDonald, Nelson, White (1990), McDonald and Nelson (1989), Francis (1975), and Fielitz and Smith (1972). 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 pdf’s in modeling the statistical property of exchange rates. Bali (2003) fits alternative pdf’s (non-normal) to model the extreme changes in the US Treasury securities market. Bali and Weinbaum (2007) also rejects the normality of stock market returns for various indices. The literature

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concluding that asset returns pdf’s have skewness and kurtosis and therefore are non-normal pdf’s is vast. Generally, stock and other asset returns pdf’s 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 pdf’s that are thick-tailed and skewed. In these cases, alternatives to OLS can yield more efficient estimators.

II. 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 in Yohai and Zamar (1997) and Martin and Simin (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 pdf’s. Martin and Simin (2003), however, do find some interesting results with the data-dependent weighted least squares approach that they developed and tested. Boyer, McDonald, and Newey (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 distribution over both regression and distributional parameters. RLS and LMS address only the explicit choice of regression parameters. Therefore the pdf’s in this investigation are referred to as flexible pdf’s. Boyer, McDonald, and Newey (2003) use Monte Carlo simulations to test the efficiency of flexible pdf’s, RLS and LMS. Using one of the four flexible pdf’s and a more restrictive version of another pdf used in this paper, they concluded that flexible pdf’s were found to produce more efficient estimators than outlierresistant methods that do not accommodate changes in pdf parameters when regression models have skewness or kurtosis. Therefore, among the myriad of robust estimation methods, this paper focuses on flexible pdf’s. 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 LAD (least absolute deviations) 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 in Hansen, McDonald, and Theodossiou (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

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SEGB2 and the more restrictive, non-skewed version of the SGT, the generalized T (GT), were used in Boyer, McDonald, and Newey (2003). These distributions nest several well-known distributions often used in econometric modeling. The distributions that are nested in the flexible pdf’s 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.

III. Estimation of Alphas and Betas The CAPM model is formulated below in equation I. This version assumes that the intercept, α, is equal to zero. A number of empirical tests of the CAPM structure have tested α as evidence against the CAPM structure. Handa, Kothari, and Wasley (1993) simultaneously test α as a vector of α’s for a series of stocks within portfolios and find evidence that α’s are non-zero with monthly returns data. Other studies such as Black, Jensen, and Scholes (1972), Blume and Friend (1973), and Fama and MacBeth (1974) perform empirical CAPM tests by estimating the security market line and performing 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 II:

(Ri ,t − R f ,t ) = α i + β i (Rm,t − R f ,t ) + ε i ,t

(

(I)

)

⎧T ⎫ max l (α i , β i , θ i , j φi ) = max ⎨∑ ln f i , j ε i ,t θ i , j ⎬ , α i , β i ,θ i , j α i , β i ,θ i , j ⎩ t =1 ⎭

(II)

j = SGT, SGED, SEGB2, IHS, Laplace, T, GED, GT, ST, and Normal, where εi,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, αi and βi are the alpha and beta for utility stock i, θi 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 pdf’s are presented in Appendix A.

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IV. 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 publiclytraded companies with SIC’s 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 Chicago’s 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 Fama-French database. A review of the descriptive statistics of the excess returns data for the utility stocks (mean, standard deviation, skewness, excess kurtosis, Jarque-Bera (JB) statistic) was performed for the 36 utility stocks for the entire 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 χ2 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 skewness and excess kurtosis of the returns distributions lead to the conclusion that the returns are non-normally distributed 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 estimator since GMM requires no prior distributional assumption for the error term. GMM 1

parameter estimates are asymptotically consistent despite the non-normality of the error term. Generally, the betas are similar in magnitude across the various pdf estimates. The median 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

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does not seem to be any systematic under- or under-estimation of the beta by OLS compared with the flexible pdf’s. 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, βi(Rm,t - Rft), to estimate their costs of equity of capital and allowed rates of return would differ by the same magnitude. Although most 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 pdf’s’ betas is 0.081. 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 March - May 2004 issues that include electric utilities have a mean adjusted beta [Blume (1975)] βa =.33+.67 βu 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. Please 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 eleven regression error distributions (not presented for brevity), including the four flexible pdf’s 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 pdf’s 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.

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Table 2 reports values of the LR test statistic corresponding to testing the hypothesis that the estimated distributions of the regression errors are observationally equivalent to the normal. This statistic is asymptotically distributed as χ2 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 χ2 is not appropriate for non-nested pdf’s. The LAD does not nest the normal and therefore the test does not appear for that pdf. The asymptotic distribution of LR is not χ2 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 2

differences will be at least as large as those based on the use of a chi-square distribution. 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 insignificant chi-square statistics among the alternative pdf’s. 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 skewness may be indicators of 3

adverse or favorable (from the shareholder’s perspective) regulation. See Brigham and Crum (1978).

The SGT, ST, SGED, and SEGB2 regressions were estimated along with their symmetric counterparts (GT, T, GED, EGB2) of these pdf’s 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 χ2 statistics are insignificant. None of the SGT-GT and ST-T LR tests are significant and only 2.8% (1 stock) of the 36 stocks LR’s 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 as the SEGB2 and the IHS. One must be careful to consider the loss of efficiency from over-parameterization in

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using pdf’s such as the SGT as it is defined by five parameters. Note that likelihood ratio tests between the SGT, SEGB2, and IHS cannot be performed as they do not nest one another. One issue that the empirical estimations do not address is the performance of the flexible pdf’s vis-a-vis OLS when the CAPM regression errors are normally distributed. When the errors are independently, identically, distributed (i.i.d.) as normal, then OLS is 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 pdf’s when the error term is normally distributed, is mixed normally distributed (varying variances and therefore has thick tails), and is asymmetric (has skewness).

V. Simulations and Estimator Performance McDonald and White (1993) and Boyer, McDonald, and Newey (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 estimator [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), 4

and (3) a skewed log-normal. Major findings included (1) little efficiency loss for partially adaptive estimation based on an over-parameterization of the distribution of normally distributed errors; (2) very similar performance of fully-iterative adaptive and partially adaptive estimators in the case of symmetric 5

thick-tailed distributions ; and (3) clear dominance of the SEGB2 partially adaptive estimator over all other estimators considered in the case of a skewed error distribution. Ramirez, Misra, and Nelson (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 the papers by Theodossiou (1998), McDonald and Xu (1995), Johnson (1949), and use similar data generating processes used by Manski (1984), Newey (1988), McDonald and White (1993), and Ramirez, Misra, and Nelson (2003). The model simulated in this paper is:

y t = α + βxt + σut = 0 + 0.21xt + σut ,

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(III)

where the xt’s 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 σ is selected to generate an R2 similar to those obtained from the regressions discussed in section IV. Appendix B derives the standard error for scaling the error term in equation (III) 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 in 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 to OLS) root mean square errors (RRMSE’s). Intercept RRMSE’s are not presented for brevity (they are available upon request). The expected portfolio returns are simulated and also reported on Table 3 to test the relative pricing performance of the CAPM with the various estimators. We predicted the excess portfolio returns to assess those pdf’s that generate the least forecast error. The RRMSE’s 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 symmetric density was applied. Note that the LAD and T pdf’s estimated 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 endnote 3. The RRMSE’s show that the efficiency of the intercept estimators of the flexible pdf’s are 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 6

insignificant. All models, symmetric and non-symmetric, appear to correspond to unbiased estimators of the slope coefficient. Unbiased estimates of the slope does not mean that there will not be cost of capital estimation errors since, on average, the slope will tend toward it’s 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

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the only pdf considered which 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 RMSE’s 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 return were 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 pdf’s and GMM have lower RRMSE’s and 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 flexible pdf’s (SGT, EBG2, IHS, SGED) have lower RMSE’s than the others and OLS, with an average of about 0.05, but the IHS is the slightly lower than the others. If a beta estimate from each CAPM beta distribution is chosen at the 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 to 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 in Brigham and Crum (1978).

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We recommend that the final choice of slope and intercept estimator is one of the flexible pdf’s if thick tails and / or asymmetry is present. If there are concerns about over-parameterization (reduction in efficiency from estimating un-necessary pdf parameters), we recommend the use of the SGT, IHS, or SEGB2 with testing for statistical improvements relative to special case pdf’s, and then use the simplest model which is observationally equivalent to the most general formulation which allows for possible skewness and / kurtosis. GMM is almost as efficient an estimator of the beta as the flexible pdf’s if the flexible pdf estimators are not readily computable.

VI. Summary and Conclusions We apply several flexible pdf’s for estimating the betas of public utility company stocks. Estimation methods based on the flexible pdf’s 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 which 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 when thick 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 utility’s allowed rate of return create dis-balances in consumer and producer surplus and therefore losses in social economic welfare, results in inefficient pricing of utility stocks, and general disequilibrium in energy and related markets as other fuels and conservation 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 reviewed the main distributional characteristics associated with the flexible pdf’s, the normal, and many other pdf’s 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

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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 in a number of other papers which explores the impact of thick-tails and asymmetry as well as efficiency loss of over-fitting in the presence of normally distributed errors. Our results showed 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 in 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:

(α , β ) i

i

OLS

T ⎛ ⎞ = arg ⎜ min ∑ εi , t 2 ⎟ , ⎝ αi , βi t =1 ⎠

Laplace, the LAD estimator [see Koenker and Bassett (1978)]:

(α , β ) i

i

LAD

T ⎛ ⎞ = arg ⎜ min ∑ εi ,t ⎟ , ⎝ αi , βi t =1 ⎠

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

(α , β ) i

i

GED

k T ⎛ ⎞ ⎜ = arg min ∑ εi , t ⎟ , ⎝ αi , βi t =1 ⎠

for an arbitrary but fixed value for k. The skewed Laplace (SL), the trimmed regression quantile [see Chan and Lakonishok (1992)], and estimators of Koenker and Bassett (1982) are given by: T

T

min ∑ ρ (ε i ,t ) or min ∑ αi , βi

t =1

αi , βi

t =1

1 ε i ,t , where (1 + sign(ε i ,t )λ )

If ρ (ε i ,t ) = φ ε i ,t and ε i ,t ≤ 0 ,

ρ (ε i ,t ) = (1 − φ ) ε i ,t if ε i ,t < 0 for 0 < φ < 1 or − 1 < λ < 1 ,

14

therefore are nested within the flexible pdf’s. Therefore, by definition, the flexible pdf’s are increasingly accommodative of the characteristics of the empirical distributions of stock returns than typically used pdf’s. Characteristics of the Flexible Probability Density Functions All of the following flexible pdf’s nest the normal among many other symmetric and asymmetric distributions. They have either four or five parameters that describe their shapes contrasted to the normal that has two parameters (mean and variance distributions) and therefore 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 α and β parameters generate the conditional means. The SGT pdf is:

k ⎛ ε ⎜ +δ C⎜ σ f SGT (ε ; α , β , σ , λ , k , n ) = ⎜ σ ((n − 2 ) / k )(1 + sign (ε + δσ ))k θ k ⎜⎜ ⎝

(

⎞ ⎟ ⎟ ⎟ ⎟⎟ ⎠

− ( n +1) k

, where,

)

C = k / 2 ((n − 2 ) / k ) θB (1 / k , n / k ) , 1/ k

S (λ ) = 1 + 3 λ2 − 4 A 2 λ2 ,

θ = (k / (n − 2 )) B (1 / k , n / k ) B (3 / k , (n − 2 ) / k ) 1/ k

−0 .5

0 .5

A = B (2 / k , (n − 1) / k )B (1 / k , n / k )

−0 .5

S (λ ) , −1

B (3 / k , (n − 2 ) / k )

−0 .5

,

δ = 2λAS (λ )−1 , B( ) is the beta function, μ and σ are the mean and standard deviation of ε, k controls the peakness of the distribution, n controls the thickness of the tails, λ < 0 generate negative skewness, and λ > 0 generates positive skewness. The major nested distributions, or special cases of the SGT are as follows:

15

λ = 0 gives the GT of McDonald and Newey (1988) and Boyer, McDonald, and, Newey (2003), k = 2 gives the ST of Hansen (1994), k = 2 and λ = 0 gives the student’s T, n → ∞ gives the SGED of Theodossiou (2001), n → ∞ and λ = 0 gives the GED of Box and Tiao (1962), n → ∞ and k = 1 gives the skewed Laplace (SLAD), n → ∞ , k = 1 and λ = 0 gives the Laplace, n → ∞ , k = 2 and λ = 0 gives the normal, and n → ∞ , k → ∞ and λ = 0 gives the uniform. Standardized values for regression error skewness and kurtosis in the ranges (- ∞, ∞ ) and ( 1.8, ∞) can be modeled with the SGT. However, not all combinations of skewness 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: k ⎛ ε ⎜ C σ +δ ⎜ f SGED (ε ; α , β , σ , λ , k ) = exp⎜ k σ ( 1 + sign (ε + δσ )λ ) θ k ⎜⎜ ⎝

⎞ ⎟ ⎟ ⎟ , where, ⎟⎟ ⎠

C = k / (2θΓ(1 / k )),

δ = 2λAS (λ )−1 , θ = Γ(1 / k )0.5 Γ(3 / k )−0.5 S (λ )−1 , S (λ ) = 1 + 3λ2 − 4 A 2 λ2 ,

A = Γ(2 / k )Γ(1 / k )

−0.5

Γ(3 / k ) ,

and k controls the peakness of the distribution, λ < 0 generates negative skewness, and λ > 0 generates positive skewness. The nested distributions, or special cases of the SGED are: λ = 0 gives the GED,

16

k = 1 gives the SL, k = 1 and λ = 0 gives the Laplace, k = 2 and λ = 0 gives the normal, and, k → ∞ and λ = 0 gives the uniform.

Skewed Exponential Generalized Beta of the Second Kind The SEGB2 is a four parameter pdf (mean, standard deviation, and two joint skewness / kurtosis parameters. The SEGB2 pdf is:

f SEGB 2 (ε ; α , β , σ , p, q )C

e ( p / θ )((ε / σ )+δ )

(1 + e (

)

p / θ )( (ε / σ )+δ ) p + q

,

C = 1 / (B( p, q )θσ ) ,

δ = (ψ ( p ) − ψ (q ))θ , θ = 1 / ψ ' ( p ) + ψ ' (q ) , Where p and q are positive scaling constants, B(p,q) is the beta function, and ψ

( z ) = d ln Γ ( z ) / dz is

the psi function and and psi-prime, the first derivative of the psi function is known as the digamma function. A smaller value of p results in a more leptokurtic pdf, q > p reflects negative skewness, and q < p reflects positive skewness. When p = q, the symmetric EGB2 obtains, and when p = q → ∞ the normal distribution obtains. Inverse Hyperbolic Sine The IHS is a four parameter pdf (mean, standard deviation, skewness, kurtosis). The pdf is:

f IHS (ε ; α , β , σ , λ , k ) =

((

k

( ( ) ) ⎞ + (ε / σ ) + δ ) − (λ + ln (θ ))) ⎟⎟

2π θ + ε / σ 2 + δ σ 2

⎛ k2 exp⎜⎜ − ln μ / σ + θ 2 2 ⎝

2

2

2

2

2

⎠

17

(

w = sinh λ + z

k

),

θ = 1σ , w σw =

(

1 −4 λ −2 λ −k −2 1+ e + 2e 2

) (1 − e ) 1/ 2

−k −2

1/ 2

e

λ +k −2

,

δ = μw σ , w

μ w = sign (λ )

(

)

1

λ + k −2 1 −2 λ 1− e e 2 2

where a smaller k results in a more leptokurtic pdf, λ < 0 indicates negative skewness and λ > 0 indicates positive skewness. λ = 0 gives the symmetric IHS. λ = 0 and k → ∞ gives the normal distribution.

Appendix B Scaling the Standardized Error Term for Simulations (σ ut) Given that

R2 = 1−

SSE / n , SST / n

For the simple bivariate regression model,

(

SST ⎛ 1 ⎞ n = ⎜ ⎟∑ y t − y n ⎝ n ⎠ t =1

)

2

((

)

⎛1⎞ n = ⎜ ⎟∑ βˆ xt − x + u t − u ⎝ n ⎠ t =1

)

2

For OLS estimation and the related orthogonality conditions sst/n can be rewritten as:

(

)

n 2 SST ⎛ 1 ⎞ ⎡ ˆ 2 n ⎤ = ⎜ ⎟ ⎢ β ∑ xt − x + ∑ u t2 ⎥ . n ⎝ n ⎠ ⎣ t =1 t =1 ⎦

Thus, for large n and stationary x, R2 approaches

R2 = 1 −

σ2

β 2 var ( x ) + σ 2

=

β 2 var ( x ) σ2 2 1 − R = and β 2 var ( x ) + σ 2 β 2 var ( x ) + σ 2

18

⎛1 − R2 ⎞ 2 ⎟⎟ β var ( x ) . ⎠

Therefore σ 2 = ⎜⎜ 2 ⎝ R

Given the variance of the x’s 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 means of empirical results. The corresponding value of σ can be calculated which is multiplied times the “standardized” error terms (ui has a mean of 0 and variance of 1 using the standard normal, scaled mixture of normals, and scaled and shifted log-normal variable).

19

References Akgiray, V. and G.G. Booth, 1988, “Mixed Diffusion-Jump Process Modeling of Exchange Rate Movements,” The Review of Economics and Statistics, 70, 631-637. Akgiray, V., and G.G. Booth, 1991, “Modeling the Stochastic Behavior of Canadian Foreign Exchange Rates, Journal of Multinational Financial Management, 1, 43-72. Bali. T.G., 2008, “The Intertemporal Relation between Expected Returns and Risk,” Journal of Financial Economics, 87, 101-131. Bali, T.G., and D. Weinbaum, 2007, “A Conditional Extreme Value Volatility Estimator Based on HighFrequency Returns,” Journal of Economic Dynamics and Control, 31, 361-397. Bali, T.G., 2003, “An Extreme Value Approach to Estimating Volatility and Value at Risk,” The Journal of Business, 76, 83-108. Bey, R., 1983, “Market Model Stationarity of Individual Public Utilities,” Journal of Financial and Quantitative Analysis, 18, 67-88. Black, F., M. Jensen, and M. Scholes, 1972, “ The Capital Asset Pricing Model: Some Empirical Tests,” In Studies in the Theory of Capital Markets, Michael Jensen, ed., New York: Praeger. Blume, M, 1968, “The Assessment of Portfolio Performance,” Ph.D. dissertation, University of Chicago. Blume, M., 1975, “Betas and Their Regression Tendencies,” Journal of Finance, 30, 785-795. Blume, M., and I. Friend, 1973, “ A New Look at the Capital Asset Pricing Model,” Journal of Finance, 28, 19-33. Box, G. E. P., and G . C. Tiao, 1962, “A Further Look at Robustness Via Bayes Theorem,” Biometrika, 49, 419-432. Boyer, B.H., J.B. McDonald, and W. K. Newey, 2003, “A Comparison of Partially Adaptive and Reweighted Least Squares Estimation,” Econometric Reviews, 22, 115-134. Brigham, E., and R. Crum, 1978, “Reply to Comments on the Use of the CAPM in Public Utility Rate Cases, Financial Management, 7, 72-76. Butler, R. J., J.B. McDonald, R. D. Nelson, and S. White, 1990, “Robust and Partially Adaptive Estimation of Regression Models,” Review of Economics and Statistics, 72, 321-327. Chan, L.K.C., and J. Lakonishok, 1992, “Robust Measurement of Beta, Journal of Financial and Quantitative Analysis, 27, 265-282. Cooley, P., 1981, “A Review of the Use of Beta in Regulatory Proceedings,” Financial Management, 10, 75. Fama, E., 1965, “The Behavior of Stock Market Movements,” Journal of Business, 38, 1749-1778.

20

Fama, E., and J. D. MacBeth, 1974, “Tests of the Multiperiod Two-Parameter Model,” Journal of Financial Economics, 1, 43-66. Fielitz, B.D., and E.W. Smith, 1972, “Asymmetric Distributions of Stock Price Changes,” Journal of the American Statistical Association, 67, 813-814. Francis, J.C., 1975, “Skewness and Investor’s Decisions,” Journal of Financial and Quantitative Analysis, 10, 163-172. Handa, P., S.P. Kothari, and C. Wasley, 1993, “Sensitivity of Multivariate Tests of the Capital Asset Pricing Model to the Return Measurement Interval,” Journal of Finance, 48, 1543-1551. Hansen, B.E., 1994, “Autoregressive Conditional Density Estimation,” International Economic Review, 35, 705-730. Hansen, C.J., J.B. McDonald, and P. Theodossiou, 2007. “Some Flexible Models for Partially Adaptive Estimators of Econometric Models,” Economics E-Journal. Harrington, D.R., 1980, “The Changing Use of the Capital Asset Pricing Model in Utility Regulation,” Public Utilities Fortnightly, 105, 28. Harvey, C.R., and A. Siddique, 1999, “Autogressive Conditional Skewness,” Journal of Financial and Quantitative Analysis, 34, 465-487. Harvey, C.R., and A. Siddique, 2000, “Conditional Skewness in Asset Pricing Tests,” Journal of Finance, 55, 1263-1295. Johnson, N.L., 1949, ?Systems of Frequency Curves Generated by Methods of Translation,? Biometrika, 36, 149-176. Klein, R.W. and V.S. Bawa, 1976, “The Effect of Estimation Risk on Optimal Portfolio Choice,” Journal of Financial Economics, 3, 215-231. Koenker, R., and G. Bassett, 1978, “Asymptotic Theory of Least Absolute Regression Error,” Journal of the American Statistical Association, 73, 618-622. Koenker, R., and G. Bassett, 1982, “Robust Methods in Econometrics,” (with discussion), Econometric Reviews, 1, 213-255. Mandelbrot, B., 1963, “The Variation of Certain Speculative Prices,” Journal of Business, 36, 394-419. Manski, C. F., 1984, “Adaptive Estimation of Non-Linear Regression Models,” Econometric Reviews, 3, 145-194. Martin, R.G., and T.T. Simin, 2003, “Outlier-Resistant Estimates of Beta ,” Financial Analysts Journal, 59, 56-69. McDonald, J.B. and R.T. Nelson, 1989, “Alternative Beta Estimation for the Market Model using Partially Adaptive Estimation,” Communications in Statistics: Theory and Methods, 18, 4039-4058.

21

McDonald, J.B., and W.K. Newey, 1988, “Partially Adaptive Estimation of Regression Models Via the Generalized T Distribution,” Econometric Theory , 4, 428-457. McDonald, J.B. and S.B. White,1993, “A Comparison of Some Robust, Adaptive, and Partially Adaptive Estimators of Regression Models,” Econometric Reviews, 12, 103-124. McDonald, J.B. and Y.J. Xu, 1995, “A Generalization of the Beta Distribution with Applications,” Journal of Econometrics, 66, 133-152. Errata 69 (1995), 427-428. McDonald, J.B. and Y.J. Xu, 1992, “An Empirical Investigation of the Likelihood Ratio Test when the Boundary Condition is Violated,” Communications in Statistics: Simulations, 21, 879-892. McDonald. J.B., R.A. Michelfelder, and P. Theodossiou, 2009, “Robust Regression Methods and Intercept Bias: A Capital Asset Pricing Model Application,” working paper. Newey, W.K., 1988, “Adaptive Estimation of Regression Models via Moment Restrictions,” Journal of Econometrics, 38, 301-339. Ramirez, O.A., S.K. Misra, and J. Nelson, 2003. “Efficient Estimation of Agricultural Times Series Models with Non-Normal Dependent Variables,” American Journal of Agricultural Economics, 85, 1029-1040. Siegel, A.F., and A. Woodgate, 2007, “Performance of Portfolios Optimized with Estimation Error,” Management Science, 53, 1005-1015. Theodossiou. P, 1998, “Financial Data and the Skewed Generalized T Distribution,” Management Science, 44, 1650-1661. Theodossiou, P. 2001, “Skewness and Kurtosis in Financial Data and the Pricing of Options,” mimeo, Rutgers University. The Value Line Investment Survey, “Ratings and Reports,” March, April, and May 2004. Yohai, V.J. and R.H. Zamar, 1997. “Optimal Locally Robust M-estimates of Regression,” Journal of Statistical Planning and Inference, 64, 309-324.

22

Table 1: Beta Estimates from Alternative PDF’s Company

OLS

GMM

LAD

T

GED

GT

SGED

SEGB2

IHS

ST

SGT

Median

Max Δ

GMP

-0.0119

0.1091

0.1445

0.1558

0.1689

0.1457

0.1087

0.1298

0.1624

0.1657

0.1661

0.1457

-0.1779

AEP

0.2270

0.1943

0.3439

0.3387

0.3142

0.3417

0.3154

0.3298

0.3345

0.3350

0.3337

0.3337

-0.1074

CMS

0.7755

0.6240

0.6454

0.5691

0.6781

0.5220

0.6767

0.6282

0.5771

0.5635

0.4492

0.6240

0.0988

PGN

0.2416

0.1941

0.2684

0.2570

0.2544

0.2545

0.3092

0.2887

0.2831

0.2945

0.3092

0.2684

-0.0676

CIN

0.1028

0.1095

0.2035

0.2043

0.2062

0.2063

0.1776

0.1765

0.2167

0.2153

0.1775

0.2035

-0.1139

ED

0.0881

0.0896

0.1113

0.1306

0.1156

0.1156

0.1157

0.1354

0.1354

0.1336

0.1157

0.1157

-0.0473

DPL

0.3864

0.3488

0.3746

0.3861

0.3714

0.3721

0.3812

0.3741

0.3840

0.3835

0.3789

0.3789

0.0024

DTE

0.1139

0.0805

0.1751

0.1139

0.1059

0.1059

0.0959

0.1073

0.1068

0.1091

0.0959

0.1068

0.0067

D

0.2024

0.1417

0.1936

0.1810

0.1859

0.1810

0.1407

0.1626

0.1607

0.1605

0.1547

0.1626

0.0398

DUK

0.2917

0.1950

0.2657

0.2472

0.2657

0.2657

0.2542

0.2535

0.2542

0.2439

0.2542

0.2542

0.0375

EDE

0.1588

0.1733

0.1084

0.1577

0.1307

0.1579

0.1613

0.1526

0.1552

0.1569

0.1578

0.1577

-0.0025

FPL

0.1847

0.2261

0.3439

0.3482

0.3439

0.3470

0.3470

0.3255

0.3448

0.3469

0.3446

0.3446

-0.1623

HE

0.2112

0.2124

0.2435

0.2240

0.2202

0.2237

0.2267

0.2202

0.2206

0.2272

0.2276

0.2237

-0.0164

IDA

0.3587

0.3201

0.3647

0.4203

0.4023

0.3705

0.4002

0.4149

0.4184

0.4175

0.3779

0.4002

-0.0597

WR

0.4956

0.4741

0.4879

0.5341

0.4869

0.5370

0.5083

0.5074

0.5329

0.5257

0.5083

0.5083

-0.0372

ETR

0.1453

0.1457

0.3943

0.2713

0.2919

0.2316

0.3325

0.2822

0.2851

0.2815

0.2434

0.2815

-0.1872

NI

0.2644

0.2319

0.2232

0.2755

0.2293

0.2809

0.2598

0.2604

0.2590

0.2414

0.2503

0.2590

0.0040

SRP

0.5977

0.4847

0.5795

0.4747

0.7370

0.4773

0.6847

0.5474

0.4998

0.4825

0.4851

0.4998

-0.0870

NU

0.3946

0.3531

0.3177

0.3596

0.3183

0.3183

0.3283

0.3273

0.3554

0.3607

0.3283

0.3283

0.0392

OGE

0.1877

0.2392

0.1772

0.2118

0.2145

0.2150

0.2128

0.2169

0.2103

0.2068

0.2128

0.2128

-0.0292

PCG

0.3023

0.2967

0.2890

0.3173

0.2901

0.2890

0.2901

0.2872

0.3097

0.3184

0.2901

0.2901

-0.0074

PPL

0.4221

0.4097

0.4696

0.4024

0.4722

0.4207

0.4377

0.4413

0.4190

0.4117

0.4292

0.4221

-0.0191

PNW

0.3913

0.3526

0.3062

0.3623

0.3062

0.3318

0.4121

0.3493

0.3739

0.3797

0.3598

0.3598

-0.0208

POM

0.2614

0.3155

0.3381

0.2377

0.2636

0.2605

0.2768

0.2393

0.2377

0.2380

0.2605

0.2605

-0.0154

PNM

0.5966

0.6286

0.5914

0.6428

0.6175

0.6175

0.6358

0.6473

0.6478

0.6483

0.6358

0.6358

-0.0513

PEG

0.2758

0.4898

0.4596

0.5190

0.5395

0.5254

0.5035

0.4967

0.5282

0.5247

0.5035

0.5035

-0.2524

PSD

0.2174

0.2443

0.2483

0.2566

0.2433

0.2433

0.2418

0.2424

0.2568

0.2538

0.2418

0.2433

-0.0394

EIX

0.2423

0.3246

0.3778

0.4118

0.4087

0.4107

0.4068

0.3737

0.4112

0.4143

0.4087

0.4087

-0.1689

SCG

0.2254

0.2013

0.2883

0.2949

0.2787

0.2787

0.2329

0.2705

0.2971

0.2931

0.2329

0.2787

-0.0717

SO

0.0005

-0.0235

0.0379

-0.0057

0.0152

0.0110

0.0139

0.0014

-0.0020

-0.0057

0.0116

0.0014

-0.0134

TE

0.2465

0.2073

0.3159

0.2607

0.2991

0.2991

0.2727

0.2817

0.2618

0.2522

0.2727

0.2727

-0.0352

TXU

-0.0517

0.0607

0.1917

0.1194

0.1950

0.1209

0.0285

0.1650

0.1433

0.1376

0.1455

0.1376

-0.2167

UIL

0.3749

0.3350

0.2761

0.3355

0.2868

0.3080

0.3142

0.3124

0.3311

0.3414

0.3142

0.3142

0.0438

UTL

0.4000

0.3318

0.2983

0.3185

0.2990

0.2990

0.3081

0.3040

0.3134

0.3194

0.3081

0.3081

0.0866

WEC

0.1097

0.0538

0.2805

0.2128

0.2004

0.2127

0.2144

0.2264

0.2235

0.2207

0.2226

0.2144

-0.1167

WPS

0.0992

0.0968

0.1369

0.1111

0.1163

0.1163

0.1211

0.1101

0.1094

0.1077

0.1211

0.1111

-0.0219

Minimum

-0.0517

-0.0235

0.0379

-0.0057

0.0152

0.0110

0.0139

0.0014

-0.0020

-0.0057

0.0116

-0.0517

-0.2524

Median

0.2420

0.2290

0.2886

0.2734

0.2828

0.2798

0.2834

0.2820

0.2841

0.2873

0.2666

0.2820

-0.0322

Maximum

0.7755

0.6286

0.6454

0.6428

0.7370

0.6175

0.6847

0.6473

0.6478

0.6483

0.6358

0.6473

0.0988

Mean

0.2647

0.2576

0.3020

0.2961

0.3020

0.2893

0.2985

0.2941

0.2988

0.2974

0.2869

0.2961

-0.0496

Std Dev.

0.1755

0.1551

0.1386

0.1419

0.1594

0.1362

0.1642

0.1457

0.1433

0.1423

0.1334

0.1433

0.0825

Max Δ is the maximum difference between the OLS and the flexible pdf betas (SGT, SEGB2, IHS, SGED).

23

Table 2: Log Likelihood Ratio Test for PDF’s V. Normal PDF Company GMP AEP CMS PGN CIN ED DPL DTE D DUK EDE FPL HE IDA WR ETR NI SRP NU OGE PCG PPL PNW POM PNM PEG PSD EIX SCG SO TE TXU UIL UTL WEC WPS Minimum Median Maximum Mean Std Dev.

T 63.86 9.71 67.55 1.50 9.97 1.77 14.26 0.00 12.15 20.91 20.99 24.89 1.64 10.86 27.16 12.77 31.49 72.50 11.51 8.24 44.66 36.33 33.41 14.51 4.76 36.02 4.80 46.02 11.70 6.10 8.48 72.25 12.30 12.31 6.36 1.79 0.00 12.31 72.50 21.54 20.95

GED 64.23 7.80 60.24 2.11 12.60 2.44 15.45 0.18 12.77 25.40 17.30 24.06 1.47 6.45 26.96 9.57 26.07 67.57 16.18 8.76 45.75 35.63 32.98 10.57 6.28 35.59 9.40 46.50 14.83 6.68 10.25 66.49 12.29 15.82 6.32 2.60 0.18 13.80 67.57 21.27 19.59

EGB2 59.67 9.54 58.41 1.70 12.07 1.77 15.47 -0.01 13.62 25.32 19.86 24.34 1.61 9.77 27.40 12.15 28.67 64.78 15.92 8.89 43.98 35.90 33.37 13.42 5.92 35.56 8.08 45.10 14.39 6.42 10.27 65.20 12.81 15.40 6.66 1.93 -0.01 14.01 65.20 21.26 18.76

GT 65.24 10.10 68.68 2.11 12.60 2.44 15.59 0.18 13.00 25.40 21.08 25.16 1.66 14.67 27.31 13.67 32.52 72.50 16.18 8.85 45.79 36.86 34.17 18.23 6.28 36.81 9.40 47.32 14.83 6.71 10.25 72.25 12.43 15.82 6.55 2.60 0.18 15.21 72.50 22.92 20.69

SGED 68.62 7.81 61.03 4.67 14.54 2.68 16.04 0.65 14.24 25.60 18.35 24.15 2.07 6.52 29.74 10.23 26.39 67.38 16.60 11.09 45.76 38.12 33.30 11.02 9.77 39.59 9.42 46.62 15.22 6.74 11.48 66.25 14.69 15.96 6.46 4.87 0.65 14.96 68.62 22.32 19.67

SEGB2 62.78 9.57 58.52 2.33 14.15 2.09 15.79 0.53 15.18 25.54 19.87 24.50 2.52 9.78 29.53 12.89 28.92 64.86 16.36 10.27 43.98 38.08 33.74 13.67 9.82 38.63 8.12 45.14 14.79 6.46 10.59 66.26 13.65 15.53 6.87 4.17 0.53 14.99 66.26 22.10 18.85

IHS 66.00 9.76 67.38 2.16 10.64 2.07 14.88 0.54 14.33 22.63 20.86 25.31 2.47 10.49 28.11 13.30 31.31 73.12 12.44 9.89 45.62 38.72 34.11 14.34 9.31 39.51 5.65 47.22 12.65 6.28 9.31 73.43 13.07 12.93 6.75 4.10 0.54 13.19 73.43 22.52 21.03

ST 64.79 9.75 67.57 2.73 10.34 1.91 14.33 0.35 13.96 21.23 21.17 24.98 2.07 11.28 27.45 13.32 32.07 73.03 11.57 9.40 44.66 37.63 33.63 14.62 8.50 37.92 5.14 46.03 11.77 6.10 8.84 74.86 13.18 12.33 6.63 3.58 0.35 13.25 74.86 22.19 21.03

SGT 68.13 10.15 69.25 4.67 14.54 2.68 16.05 0.65 14.59 25.60 21.20 25.24 2.12 14.98 29.74 14.23 32.97 73.03 16.60 11.09 45.81 38.62 34.55 18.23 9.77 39.60 9.42 47.42 15.22 6.74 11.48 75.02 14.69 15.96 6.78 4.87 0.65 15.59 75.02 23.94 20.84

The ratios approximate the χ 2 distribution with the degrees of freedom equal to the difference in the number of parameters the normal pdf. The pdf's with the largest number of parameters is 5 and the normal has 2, so the greatest degrees of freedom is 3. The critical value at a 5% level of significance is 7.81.

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Table 3: Simulations of Betas, Alphas, and Predictions of Utility Stock Returns (Page 1 of 2) Panel 1: Mean of Intercept, Slope and Expected Portfolio Return Based on 10,000 Simulations A. Intercept Means OLS LAD SL GED SGED T ST Normal 0.00209 -0.00085 0.00292 0.00123 0.00199 0.00169 0.00223 Mix-Normal -0.00923 -0.00476 -0.01081 -0.00368 -0.00653 -0.00272 -0.00724 0.01266 -0.35473 0.13823 -0.74909 -0.01791 Log-Normal -0.00031 -0.90233

GT 0.00276 -0.00338 -0.52268

SGT 0.00270 -0.01428 -0.01709

IHS 0.00000 -0.00004 0.00010

SEGB2 0.00692 -0.00005 0.00001

GMM 0.00976 0.00415 0.00682

T-values Normal Mix-Normal Log-Normal

OLS 0.00627 -0.02776 -0.00094

LAD -0.00200 -0.02741 -2.60688

SL 0.00850 -0.02674 0.03768

GED 0.00363 -0.02376 -0.76211

SGED 0.00594 -0.02597 0.36736

T 0.00500 -0.02007 -2.03484

ST 0.00669 -0.02026 -0.05310

GT 0.00807 -0.02490 -1.10216

SGT 0.00799 -0.01821 -0.05088

IHS 0.00037 -0.01229 0.02817

SEGB2 0.14521 -0.01423 0.00166

GMM 0.29347 0.12437 0.20631

B. Slope Means Normal Mix-Normal Log-Normal

OLS 0.20989 0.21012 0.20989

LAD 0.20955 0.21020 0.20856

SL 0.20978 0.21027 0.20783

GED 0.20969 0.21033 0.20920

SGED 0.20968 0.21035 0.20813

T 0.21001 0.21033 0.20932

ST 0.21000 0.21039 0.20791

GT 0.20960 0.21054 0.20865

SGT 0.20966 0.21067 0.20775

IHS 0.20946 0.21004 0.20776

SEGB2 0.20821 0.20988 0.20784

GMM 0.20912 0.20995 0.20770

T-values Normal Mix-Normal Log-Normal

OLS -0.00149 0.00154 -0.00149

LAD -0.00470 0.00471 -0.01833

SL -0.00225 0.00657 -0.03585

GED -0.00402 0.00917 -0.01074

SGED -0.00414 0.00965 -0.03598

T 0.00007 0.01062 -0.01032

ST 0.00004 0.01223 -0.04064

GT -0.00505 0.01741 -0.02130

SGT -0.00433 0.02122 -0.04274

IHS -0.00687 0.00120 -0.04736

SEGB2 -0.02223 -0.00344 -0.04507

GMM -0.01053 -0.00152 -0.04197

C. Expected Portfolio Return { E(R i )= â i *0.6 } Means OLS LAD SL Normal 0.12593 0.12573 0.12587 Mix-Normal 0.12607 0.12612 0.12616 Log-Normal 0.12593 0.12513 0.12470

GED 0.12581 0.12620 0.12552

SGED 0.12581 0.12621 0.12488

T 0.12600 0.12620 0.12559

ST 0.12600 0.12623 0.12475

GT 0.12576 0.12633 0.12519

SGT 0.12579 0.12640 0.12465

IHS 0.12568 0.12602 0.12466

SEGB2 0.12493 0.12593 0.12470

GMM 0.12547 0.12597 0.12462

GED -0.00402 0.00917 -0.01074

SGED -0.00414 0.00965 -0.03598

T 0.00007 0.01062 25 -0.01032

ST 0.00004 0.01223 -0.04064

GT -0.00505 0.01741 -0.02130

SGT -0.00433 0.02122 -0.04274

IHS -0.00687 0.00117 -0.04736

SEGB2 -0.02223 -0.00344 -0.04507

GMM -0.01053 -0.00152 -0.04197

T-values Normal Mix-Normal Log-Normal

OLS -0.00149 0.00154 -0.00149

LAD -0.00470 0.00471 -0.01833

SL -0.00225 0.00657 -0.03585

Table 3: Simulations of Betas, Alphas, and Predictions of Utility Stock Returns (Page 2 of 2) Panel 2: Relative RMSE of Intercept, Slope and Expected Return Based on 10,000 Simulations A. Intercept Rel. RMSE Normal Mix-Normal Log-Normal

OLS 1 1 1

LAD 1.27002 0.52175 2.89146

SL 1.03041 1.21603 1.00557

GED 1.01245 0.46599 1.75086

SGED 1.00283 0.75553 1.19931

T 1.01277 0.40750 2.49718

ST 1.00183 1.07375 1.01038

GT 1.02708 0.40792 2.11148

SGT 1.01434 2.35791 1.00585

IHS 1.00060 0.85925 0.99914

SEGB2 13.75912 0.99242 0.99542

GMM 10.24300 9.98450 9.97450

1 1 1

LAD 1.24902 0.52606 1.03000

SL 1.24482 0.51844 0.79201

GED 1.00895 0.46097 0.97401

SGED 1.01498 0.46008 0.67903

T 1.01022 0.39866 0.86121

ST 1.01892 0.40137 0.67182

GT 1.02316 0.39696 0.82807

SGT 1.03356 0.39912 0.69019

IHS 1.01190 0.42594 0.61389

SEGB2 1.04990 0.45086 0.62184

GMM 1.08530 0.42132 0.70882

LAD 1.24902 0.52606 1.03000

SL 1.24482 0.51844 0.79201

GED 1.00895 0.46097 0.97401

SGED 1.01498 0.46008 0.67903

T 1.01022 0.39866 0.86121

ST 1.01892 0.40137 0.67182

GT 1.02316 0.39696 0.82807

SGT 1.03356 0.39912 0.69019

IHS 1.01990 0.42594 0.61389

SEGB2 1.04990 0.45086 0.62184

GMM 1.08530 0.42132 0.70882

B. Slope Rel. RMSE Normal Mix-Normal Log-Normal

OLS

C. Expected Portfolio Return Rel. RMSE Normal Mix-Normal Log-Normal

OLS 1 1 1

RRMSE is the relative RMSE = RMSE (pdf) / RMSE (normal). RMSE is calculated as follows:

RMSE (β ) =

(

N 1 ∑ βi − β 10,000 i =1

) + (β − 0.21) , β = 10,1000 ∑ β 2

10 , 000

2

i =1

The simulated CAPM equations are based on:

(Ri ,t − R f ,t ) = 0 + 0.21(Rm,t − R f ,t ) + ε i , where 0.21 is the market-value weighted portfolio beta for the utility stocks.

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i

.

ENDNOTES 1. We used the fully iterative GMM estimator as developed by Newey (1988). We used OLS estimates to provide the starting values for the iterative process which we allowed to iterate until "convergence" was achieved. We used J=4 moment conditions in defining the objective function to be optimized, thus the convergence involved simultaneously minimizing correlation between functions of the first four moments of the estimated error with the independent variable as outlined in the Newey (1988). 2. Based on simulations 1000 replications, the size of the LR test associated with estimating various pdf’s which nest the normal, exponential, or Lognormal was explored. When the nested distribution corresponded to a special case of the estimated pdf, the size of the LR test was close to that predicted by the “asymptotic Chi-square distribution.” However, in the case in which the nested distribution corresponded to a limiting case of the estimated distribution which violated a regularity condition, the size of the LR 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 even greater than might be inferred from the “Chi-square” values.

3.The importance of determining whether non-normality is due to asymmetry or thick tails is important since skewness can result in biased intercepts (Jensen’s alphas) under specific conditions, as shown in McDonald, Michelfelder, and Theodossiou (2009). This is also an important estimation issue if using the single index model (Ri=αi+βiRm+εi) to predict stock returns. Note that the T-values of the intercept for the log-normal pdf in the simulations shown on Table 3 are substantially higher for the non-normal symmetric pdf’s (LAD, GED, T, and GT). The LAD and T intercepts are statistically significant, which is a indicator of bias as the expected value of the estimate of this parameter is 0. Secondly, negative or positive stock returns skewness can be generated by adverse or favorable regulatory treatment of utility profits, respectively. Although beyond the scope of this paper, skewness caused by regulatory treatment of utilities can effect the efficient and unbiased estimation of the models used to estimate the cost of capital. 4. All three error distributions are standardized to have a zero mean and unitary variance. The first error distribution is merely a unit normal generated by u1 = z, where z ~ N(0,1). The thick-tailed variance contaminated error distribution is generated by u2 = w z1 +(1 – w) z2, where z1 ~ N(0,1/9), z2 ~ N(0,9), and w is 1 with probability .9 and 0, otherwise. This distribution is symmetric and has a standardized kurtosis of 24.33. The log-normal distribution is generated by u3 = (ez - e0.5) / (e2 - e)0.5 where z ~ N(0,1). This distribution has standardized skewness and kurtosis of 6.185 and 113.94, respectively.

5. The fully iterative adaptive kernel and GMM estimators performed much better than the corresponding two-step estimators. 6. A T-test was used to compare the sample mean of beta from the 10,000 simulation estimates to 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 than 0.21, 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 randomly generated observations, and a standard deviation of 2.1468 for x are the parametric assumptions based on averages of empirical results. The corresponding value of σ can be calculated which is multiplied times the “standardized” error terms (u has a mean of 0 and variance of 1 using the standard normal, scaled mixture of normals, and scaled and shifted log-normal variable).

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