Technology and Investment, 2013, 4, 5466 Published Online February 2013 (http://www.SciRP .org/journal/ti) Copyright © 2013 SciRes. TI CrossSectional Estimation Biases in Risk Premia and Ze roBeta Excess Returns Jianhua Yuan, Robert Savickas Yuan is at Capital Market Research Division, Fannie Mae, Washington, DC, USA Savickas is at Department of Finance, George Washington University, Washington, DC, USA Email: jhyuan@gwu.edu and savickas@gwu.edu Received 2012 ABSTRACT This paper shows that the classic crosssectional asset pricing tests tend to suffer from severe riskpremium estimation errors because of small variation in betas. We explain how the conventional approach uses low criteria to validate an asset pricing model and suffers from the modelmisspecification issue because of the complication associated with the zerobeta excess return. We show that the resulting biases in estimates of risk premia and their standard errors are se vere enough to lead researchers into inferring incorrect implications about some assetpricing theories being tested. Further, we suggest that one simple method of mitigating these issues is to restrict the zerobeta excess returns to their theoretical values in the crosssectional regressions and to conduct the straightforward test of whether the estimated exante risk premia are consistent with the observed expost ones. The empirical testing results not only further affirm the higher efficiency of the estimates produces by the suggested method, but also show, contrary to some prior evidence, that the market factor is priced consistently. Keywords: CrossSectional Regression; Consistent Estimator; Efficient Estimator; Risk Premium; Zer oBeta Return; Model Misspecification; BetaVariation 1. Introduction The twopass crosssectional regression (CSR) methodology, which is used in the two classic studies of CAPMone by Black, Jensen, and Scholes [3, 1972] (BJS) and the other by Fama and MacBeth [12, 1973](FM)first estimates the factor loadings on the given risk factors in time series regressions and then uses the estimated loadings in the secondpass CSR to estimate the risk premia of the factors. This intuitive twopass CSR method is easy to implement and has been widely used in the empirical studies of linear betapricing models. Despite the wide usage of the twopass CSR methodology in assetpricing tests, quite a few issues associated with this classic methodology have been identified in the literature. First, all the three underlying assumptions  normality, conditional homoskedasticity, and stationarityof this twopass methodology are somewhat disputable. The distributions of most stock returns exhibit significantly positive skewness and highertha n normal kurtosis. Fama [9, 1965] and Blattberg and Gonedes [5, 1974] document the nonnormality shown in security returns. The works by BaroneAdesi and Talwar [1, 1983], Bollerslev, Engle, and Wooldridge [6, 1988], and Schwert and Seguin [22, 1990] document the conditional heteroskedasticity of stock returns. For the least worrisome stationarity assumption, Blume [4, 1970] shows that it is not totally inappropriate. The second issue in the assetpricing tests is the crosssectional dependence among asset returns. Theoretically, this issue can be handled by a generalized leastsquare (GLS) estimation but it would be impractical to estimate a huge covariance matrix for a large number of securities. Moreover, such estimated matrix may often not be positivedefinite and hence fail the purpose of the GLS methodology. As a more practical approach, the grouping procedure, which was employed in the two classic studies of CAPMone by BJS and the other by FM, is used to form crosssectional portfolios. Since the crosssectional dependence is allowed for grouped data test, the linear betapricing models can be tested on returns of the crosssectional portfolios. As pointed out by BJS, the grouping procedure also allows for any nonstationarity. The third issue associated with the twopass CSR method is the wellknown errorinvariable (EIV) problem. BJS show that the market risk premium estimates will be biased because of the measurement errors in the market beta estimates but argue that the measurement errors can be ignored with large samples of many time periods. FM propose to use lagged rolling betaestimates to generate a time series of risk premium
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI estimates and then take the mean as the final risk premium estimate. Shanken [23, 1992] provides an excellent discussion of this twopass CSR methodology, especially the EIV adjustment and the asymptotic distribution analysis under the conditional homoskedastic assumption.1 Kim [15, 1995] provides an EIV correction met ho dology using maximum likelihood estimation. The fourth issue of this classic CSR methodology is related to the common practice of low acceptance criteria for a betapricing model. The literature sometimes emphasizes the high crosssectional and the high tvalues of the risk premium estimates (i.e. the estimated risk premium associated with a particular factor is significantly different from zero). Kan and Zhang [14, 1999] conduct simulations in which true asset returns are generated from a onefactor model but the factor in the twopass CSR tests is misspecified as a random variable uncorrelated with the asset returns. They call such a misspecified factor as a ``useless" factor and show that the tvalue of the ``useless" factor converges to a large value in the crosssectional regression and the probability of a fairly high crosssectional 2 is quite big. Even though the case they provide is an extreme one and a useless factor may be relatively easy to detect as Jaga nnathan and Wang [13, 1998] have shown, quasimodels whose factors are only weakly correlated with the true factors will be harder to detect. Lewellen, Nagel, and Shanken [16, 2008] (LNS) show that a proposed model with factors only weakly correlated with the true factors and uncorrelated with the errors is capable of producing high crosssectional . As part of their critique, LNS show that the magnitudes of the zerobeta estimates and the estimated equity risk premium in many papers are unreasonably high and low, respectively. In attempting to improve empirical tests, LNS have offered several prescriptions including GLS estimation, expansion of the set of testing portfolios beyond sizeB/M portfolios, imposing related constraints in timeseries and/or crosssectional regressions, and so on. These suggested approaches may improve empirical tests but somehow lose the intuition and the simplicity of the original method. This paper focuses on the fourth issue described above and attributes this problem to the complication of the zerobeta factor. Even though the classic CAPM developed by Sharpe [24, 1964] and Lintner [17, 1965] suggests that the excess return (in excess of riskfree rate) of any asset should be proportional to its market risk loading by the same multiplier (the market risk premium), the early CAPM testing results (e.g. BJS, FM, etc.) show that the estimated CSR intercept tends to be significantly 1Jagannathan and Wang [13, 1998] generalize Shanken's asymptotic analysis to the case of conditional heteroske dastic returns. positive and the market risk premium seems to be significantly smaller than the ex post equity premium. Possibly motivated by such empirical results, Black [2, 1972] extends the CAPM by including borrowing restriction.2 Black's extended CAPM postulates that the excess return of any asset will be linear in terms of its market beta with the same intercept (the zerobeta excess return) and the same linear coefficient (the market risk premium). The extended Black's CAPM not only is a theoretical extension of the original SharpeLintner CAPM but also seems to reconcile the early CAPM testing results. However, the zerobeta excess returns, which are due to the differences between the costs of (riskfree) borrowing and the returns of riskfree lending, are not observable and are somehow ambiguous. With the complication of the zerobeta excess return, the observed equity premium is no longer an expost measure of the exante risk premium and this leads to the common practice of low criteria of model acceptancehigh CSR and high values of risk premium estimates. A more serious problem associated with the conventional crosssectional regression method is that the estimation errors will be greatly amplified by the small variation among the market betas.3 To address these issues, we advocate focusing on the theoretical linear betapricing model that restricts the zerobeta expected return to its theoretical value and testing whether the estimated exante risk premia are consistent with the observed expost ones.4 In the theoretical linear betapricing model with a riskfree rate, the risk premia are assumed to be observable. In order to find supportive evidence for such a pricing model, one needs to show that the null hypothesis that the estimated (exante) risk premia are equal to the observed (expost) ones will fail to be rejected. For a misspecified model, it is unlikely that the estimated risk premium of the misspecified factor can match its observed value. Next, we discuss the importance of this simplification. 5 2Brennan [7, 1971] also provides an excellent analysis on the capital market equilibrium with divergent borrowing and lending rates. 3Notice that the estimation error of the market risk pre miu m is That is, the testing procedure . 4In our discussion of the twopass CSR methodologies, we focus on the ordinary leastsquare (OLS) crosssectional estimation. But the approach suggested in this paper can be directly applied to the generalized leastsquare (GLS) cross sectional estimation, as well. 5Kan and Zhang [14, 1999] show that the risk premium estimate of a useless factor converges to infinity with probability one as . Suppose the factors are stationary and ergodic, then the sample means of the factors will converge to finite numbers and the probabil
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI for the linear betapricing model with a riskfree rate will not suffer from the modelmisspecification problem. It has also been proven that the twopass CSR methodology suffers from the EIV problem and the CSR estimation errors are underestimated. Shanken [23, 1992] shows that it is very important to make the standard error correction in order to find supportive evidence for a linear betapricing model with zerobeta excess return, especially one with multiple factors. The estimated risk premia that are significantly different from zero with the uncorrected error estimates may turn into insignificant with respect to the corrected and hence largererror estimates. However, the standard error correction will not be needed if the linearbeta pricing model can be validated with the uncorrected error estimates. That is, if the null hypothesis that the estimated risk premia are equal to the observed ones fails to be rejected with respect to the uncorrected standard errors, it will also fail to be rejected with respect to the corrected (larger) ones. The more important reason that we should focus on the theoretical linear betapricing model with riskfree rate is that this method can be applied to obtain more efficient estimates of risk premia. We show that the conventional twopass estimates from the unrestricted crosssectional regressions suffer from high estimation errors because of the relatively small variation among betas. Although there are sophisticated, more efficient econometric methods available for evaluating linear betapricing models, these methods are generally more complicated and less robust than the twopass CSR methodology6 One of the prescriptions suggested by Lewellen, Nagel, and Shanken [16, 2008] to improve the efficiency of the WICSR is to impose related constraints in the regressions. According to Shanken [23, 1992], this means to force the risk premium of a portfolioreturn factor to be the difference between the mean of the observed factor returns and the average zerobeta excess returns. Hence, for a pricing model with only portfolioreturn factors, the crosssectional estimation reduces to estimating the mean and in some situations, it is difficult to interpret statistical inferences obtained from these methods. As an alternative, the approach of restricted crosssectional regressions possesses all the advantagesintuitiveness, simplicity, and robustnessof the classic twopass method while, at the same time, being capable of producing much more efficient estimates than the conventio na l method is. It is also straightforward to interpret the testing results obtained from the method we advocate. ity that the risk premium estimate of a useless factor equals to its ex post risk premium will converge to zero as . 6Cochrane [8, 2001] makes an excellent discussion about tradeoffs between the methods for estimating and eva luating assetpricing models. of the zerobeta excess returns. With this restriction, the risk premium of any portfolioreturn factor is not directly estimated and the standard error associated with the risk premium of the portfolioreturn factor is not estimated. Because of this, it becomes difficult to statistically interpret the estimation results with respect to the portfolio return factor, especially when the tested model has multiple factors. When focusing on the theoretical linear betapricing model with (market equivalent) riskfree rate, we are able to restrict crosssectional regressions by imposing the zerointercept constraint, thereby obtaining estimates that are very straightforward to interpret. In the rest of the paper, we proceed as follows. In Section 2, we first describe both the traditional method and the method we advocate. Then we show that both methods are consistent and derive the asymptotic distributions of the corresponding estimates under the assumptions of conditional heteroskedasticity and/or homoskedasticity. In the last subsection of Section 2, we derive the crosssectional asymptotic properties and show that the advocated approach is more efficient than the traditional method under the assumption of sufficiently weak crosssectional dependence. Section 3 presents empirical evidence of the higher efficiency of the methodology we describe. In Subsection 3.1, we provide simulation evidence to illustrate that riskpremium estimates resulting from the restricted crosssectional regressions are more efficient than the conventional estimates. Subsection 3.2 shows that the warrants for significant zerobeta excess returns are not as strong as believed. In Subsection 3.3, we reexamine the FamaFrench threefactor model and provide new crosssectional supportive evidence. The final section summarizes our findings. 2. The CrossSectional Regression Methods for Asset Pricing Tests 2.1. The Mathematical Setup A linear asset pricing model with riskfree rate can be expressed as follows: (1) where is a vector of excess returns (in excess of the riskfree rate) for assets; , the vector of risk premia, is the mean of the assetpricing risk factors ; and is the matrix of factor loadings of on . Let and denote the covariance matrix between and and the variancecovariance matrix of the factors , respectively. Then (2) An important case of model (1) is the SharpeLintner's CAPM, where is the market excess return and
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI equation (1) is called the Security Market Line. In the general case of Merton's ICAPM, is the vector of excess returns of the market portfolio and the hedging portfolios and equation (1) is called the Security Market Hyperplane. In Ross's APT case, equation (1) will be empirically approximated if returns have a linear factor structure . The classic design of testing the model (1) is to test the following extended linear model: ,=][E λιγ Br + N (3) where scalar is interpreted as the zerobeta excess return; N ι is an dimensional column vector of ones, and is the column vector of risk premia. Define (4) where N I is the dimensional identity matrix, then it is easy to see that (5) Henceforth, we drop the subscript N for and for simplicity, as the dimensions of and/or can be implicitly determined. Multiplying equation (3) by , we have (6) Now assume that and that the matrix has full rank , then will also have full rank and ].[E)(=][E)(= 11 rDBDBBrDBDBDB′′′′′′ −− λ (7) Substituting (1) into (7) , we have ].[E=][E)(= 1ffDBBDBB′′ − λ (8) That is, theoretically the basic model (1) can be tested by testing the extended model (3) with the conventional withintercept cross sectional regression (WICSR) estimates (7). On the other hand, suppose that has full rank , then by (1) the risk premium can be directly expressed as: (9) And a direct estimation of (9) can be obtained through a nointercept crosssectional regression (NICSR). Let and be the matrix of the observed excess returns of the assets and the matrix of the observed values of the factors for the time periods. Assume that and are stationary and the sample moments of and converge to the corresponding unconditional population moments. Then the variance and covariance matrices and can be consistently estimated by , 1 = 1 =FFDFDFD ′′′ Ω (10) and (11) The consistent estimators for and are (12) and 1 1 1 = = () =( ). − − − ΣΩ ′′ ′′ ′′ B RDD FFDD F RDF FDF (13) The classic WICSR estimates for the risk premium and the zerobeta excess return are (14) ). ˆ (' 1 = ˆ λιγ BR − (15) The NICSR estimate for the risk premium is (16) Denote (17) (18) ,= λεεε B R−(19) then we can obtain estimation errors of the WICSR risk premium estimate by equations (7) and (14) (20) Similarly, equations (9) and (16) give the errors of the NICSR risk premium estimate .)(= ~1 ελλ 'BB'B − −(21) 2.2. Asymptotic Distributions of the Estimators In the previous subsection, we have laid out the mathematical definitions of both the WICSR and NICSR estimators. Next we present the asymptotic distribution properties of the risk premium estimator given by the
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI conventional WICSR in (14) and that produced by NICSR in (16). The assumption of conditional heteroskedasticity we make here is similar to the one made by Jagannathan and Wang [13, 1998]. Proposition I: Assume that 1) the time series of returns and factors are stationary and ergodic, 2) as , the random vector converges to a zeromean random vector with covariance matrix , and 3) both and have full rank of , then in (20) converges in distribution to a zeromean random vector with covariance matrix ;)()(= 11 −− ′ Ψ ′′DBBDBDBDBBV (22) in (21) converges in distribution to a zeromean random vector with covariance matrix .)()(= 11 −−′ Ψ ′′ BBBBBBV (23) Proof: Since the time series of returns and factors are stationary and ergodic, , and will converge in probability to , and , respectively. Hence (22) and (23) immediately follow from the assumption that the random vector converges to a zeromean random vector with covariance matrix .7 Q.E.D. Since the true risk premium is not observable, the CSR risk premium estimate can only be tested against the sample mean of the observed risk premium: (24) In order to test the basic model (1), the classic WICSR methodology tries to test the following null hypotheses8 1 0:H : Sufficiently high to show that the variation in crosssectional average excess returns can be explained by the variation in the factor loadings ; 7These two results can be derived as special cases of Ja gannathan and Wang's [13, 1998] Theorem 1. 8To fully validate an asset pricing model, it is very essen tial to test the hypothesis that all pricing errors are jointly zero. Since the focus of this paper is on the risk premium estimation but not on the validation of any specific model, we only perform tests associated with risk premium es timates. • ; • . Using the NICSR approach, we would like to test the next two null hypotheses: • Sufficiently high to show that the levels of the crosssectional average excess returns can be explained by the levels of the factor loadings ; • . For these purposes, the empirical timeseries form of (1) is introduced as follows: (25) where is the matrix of idiosyncratic errors with and . And the empirical crosssectional form of (1) is ,= e+FBR (26) where . By equations (13) and (25), we get (27) After rewriting equation (26) in terms of , we have .= εε +≡−+FBFFBR B e(28) Using equation (28), we obtain the conditional errors of the CSR risk premium estimators given the factor realization : (29) f (30) Similar to the previous results on the asymptotic distribution of the unconditional error of the risk premium estimators, the asymptotic distribution properties of errors of the risk premium estimators conditioning on the realization of the risk factors in (29) and (30) can be given as follows. Proposition II:9 Assume that 1) the time series of returns and factors are stationary and ergodic, 9Shanken [23, 1992] presents a comprehensive asymp totic analysis for the conditional error distribution of the
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI 2) as , the random vector converges to a zeromean random vector with covariance matrix , and 3) both and have full rank of , then in (29) converges in distribution to a zeromean random vector with covariance matrix ;)()(= 11 −− ′ Π ′′DBBDBDBDBBW (31) in (30) converges in distribution to a zeromean random vector with covariance matrix .)()(= 11 −− ′ Π ′′ BBBBBBW (32) Proof: Similar to the proof of Proposition I. Q.E.D. In order to conduct the statistical tests on the hypotheses , , , 1 0 H , and 2 0 H , researchers typically make the assumption that the idiosyncratic errors are conditionally homoskedastic on the realization of the risk factors with constant covariance matrix . Under this conditional homoskedastic assumption, we obtain the following result. Proposition III:10 Assume that 1) the time series of returns and are stationary and ergodic and 2) the idiosyncratic errors are independently, identically distributed with mean 0 and covariance matrix conditional on the realization of the risk factor , then .)'(1= 1ΦΩ+Π − λλ (33) It is well known that the twopass crosssectional method suffers from the EIV problem. From (28) we see that the conditional errors have two components: and . Proposition III indicates that under the conditional homoskedasticity, these two error components will be conditionally independent with asymptomatic covariance matrices and ΦΩ− λλ 1 ', respectively. To account for the estimation errors in betas, Shanken [23, 1992] suggests to adjust the standard errors according to (33). That is, the estimated errors should be inflated by and by for the conventional withintercept crosssectional estimates under the stronger assumption that the idiosyncratic er rors are homoskedastic. 10We omit its proof here and readers can refer to Theo rem 1 of Shanken [23, 1992], as it directly follows. conventional WICSR estimates and the NICSR ones, respectively. 2.3. CrossSectional Asymptotic Properties The asymptotic distribution analyses in Subsection 2.2 show that the estimators of both the classic WICSR and the NICSR are consistent. This indicates that the twopass estimation leaves little to be desired with regard to its largesample properties as . In this subsection, we examine the crosssectional asymptotic properties of these two CSR estimators and show that the NICSR method is likely more efficient than the conventional WICSR method, provided that is large enough and the crosssectional dependence of idiosyncratic errors is weak enough. It is known that the traditional WICSR estimator is not consistent. That is, the risk premium estimate will not converge in probability to the average observed realizations as . One obvious reason for this is the EIV issue, and the other more fundamental problem is that the true risk loadings and the crosssectional average of the idiosyncratic errors may be correlated.11 The NICSR estimator is certainly not consistent, either. But under the general assumption of sufficiently weak crosssectional dependence of idiosyncratic errors, the crosssectional average of the idiosyncratic errors in (28) tends to cancel away and the advocated NICSR method will be more efficient than the classic WICSR approach. There are two empirical evidences for our claim. First, Miller and Scholes [20, 1972] have documented that highbeta assets tend to have negative alphas and that lowbeta stocks tend to have positive alphas12 . This indicates that when is large enough, the crosssectional average of the estimated alphas will be quite small as negative alphas of the highbeta stocks and the positive alphas of the lowbeta assets will cancel each other. Notice that the conditional estimation errors in equation (28) will be the estimated alphas in the CAPM case. This implies that 11Most researchers simply assume that the crosssectional error is uncorrelated with the true risk loading . Empirically, we see that the correlation between β and is quite large compared with the betavariation. 12Even though the measurement error will contribute to the negative correlation between and , the measurement error in is minimal for CAPM and the significant crosssectional correlation is between al phas and true betas.
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI BBBBDBDBDBDBΠ≈ ′′ Π ′'=''= εεεε . Secondly, Black, Jensen, and Scholes [3, 1972] report that the market betas tend to be concentrated near the value of one with relatively small variation and one of the purposes of their grouping procedure is to obtain maximum possible dispersion among betas. This implies that the square matrix is likely significantly more positive definite than the square matrix . Combining these two points, we know that the positive definite matrix in (31) should have bigger norm than in (32). To formalize this empirical argument, we produce the following result. Theorem I: Assume that 1) the idiosyncratic errors have mean 0 and sufficiently weak crosssectional dependence; and 2) as , , B'B 1, and ε D'B 1, converge to ] [E β , , and , respectively, then • as , ; • as , the estimation error of the conventional WICSR risk premium estimate converges to );, ˆ (cov) ˆ (var ˆ 1 εββλ − →− F (34) • as , the estimation error of the NICSR risk premium estimate converges to 1 ˆ (var( ) ˆˆˆ E[]E[])cov(,). λβ β ββε − −→ ′ + F (35) Proof: Suppose that the timeseries variances of the idiosyncratic errors are bounded, then by the weak law of large numbers e ι ′ 1 converges in probability to vector . Hence by (27) and (28), (i) directly follows. (ii) is the immediate result of assumption 2) and equation (29). Notice that and from Equation (30), we have that (iii) directly follows the above results (i) and (ii). Q.E.D. The two conditions in Theorem I are typically assumed in the analysis of consistency of the twopass CSR method. If we further assume that the true risk loading betas are uncorrelated with the idiosyncratic errors , then (34) becomes 1 1 ˆˆˆ ˆˆ var ()cov(,) =[var()var( )][var( )cov(,)].e ββ β β βε βε εε − − ++F (36) The asymptotic result (36) is basically the one reported by Black, Jensen, and Scholes [3, 1972]. Under the further assumption of conditional homoskedasticity, they argue that errors and can be ignored for large and hence the CSR estimator is consistent. Shanken [23, 1992] formally proves that the ``OLS version" of maximum likelihood estimation of the zerobeta excess return is consistent under the assumptions that the idiosyncratic errors are homoskedastic and crosssectionally uncorrelated with the true betas. The asymptotic result (34) in Theorem I indicates that three factorsbeta variation, EIV, and the average idiosyncratic error over time will determine the estimation error of the CSR risk premium estimate. The existing literature emphasizes EIV correction but pays little attention to the impact of beta variation. BJS point out that one of the purposes of the grouping procedure should be to maximize the variation among betas. But this seems to be difficult in portfolio grouping. Kim [15, 1995] even suspects that the formation of portfolios for the CSR estimation might cause a loss of valuable information about crosssectional behavior among individual assets. And one of the prescriptions suggested by LNS to improve empirical tests is to expand the set of test portfolios beyond the sizeB/M portfolios. The expansion of the set of test portfolios tends to increase the variation among betas and hence can improve the CSR estimations. As the asymptotic property (35) indicates, our preferred NICSR methodology will barely suffer from the small betavariation as the total square sum of betas is used. Hence the NICSR approach will be more efficient than the traditional WICSR method. 3. Empirical Results In the previous section, we have shown that the NICSR method will be more efficient than the classic WICSR approach in the tests of the linear betapricing models. In this section, we present our supporting empirical evidence. First, we employ simulations to show this point. Then, we test the CAPM with the actual stock returns for the early subsample. In the last subsection, we reexamine the FamaFrench threefactor model. 3.1. CAPM Simulations
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI In this subsection, we employ simulations to demonstrate that the NICSR approach will be more efficient than the WICSR method in tests of the linear betapricing models. There are twentyfive series of simulated excess return data generated according to the CAPM ,= itMtiit e+RR β (37) where the market excess returns are the actual FamaFrench market factor monthly time series between July 1926 and June 2007 and the market risk loadings are the actual estimated market betas of the 25 FamaFrench sizeB/M crosssectional portfolios for the same period13 . The simulated idiosyncratic errors in (37) are crosssectionally independent and are independently, identically, normally distributed with mean zero and variance across time, where are the variances of the 960period idiosyncratic residuals of the 25 valueweighted FamaFrench sizeB/M portfolios regressed on the FamaFrench market factor.14 With the 25 simulated crosssectional excess returns, we first perform the usual passone timeseries regression on the given fixed FamaFrench market factor to estimate betas and then run both the classic WICSR test and the NICSR one with the crosssectional average excess returns. The simulations and the regressions are repeated 100 times. Figures 1 and 2 graphically show the actual errors and Figure 1: Actual Errors in Risk Premium Estimates. the estimated standard errors of the risk premium estimates of the 100 simulations, respectively. We clearly see that the errors in the risk premium estimates for the WICSRs are in general much larger than those for the NICSRs. In Table 1, we list some summary 13Twelve data points with missing data between July 1930 and June 1931 are excluded. 14For simplicity, the idiosyncratic errors are assumed to be crosssectionally independent. This simplification is not too restrictive as the NICSR method can be directly generalized to the GLS case. Figure 2. Estimated Standard Errors of Risk Pre mium Estimates statisticsmeans, standard deviations, minima, and ma ximaof the errors in the risk premium estimates of the 100 simulations for both the methods. These specific statistics again show that the NICSR approach is more efficient than the WICSR method. Table 2 shows the number of rejections of the true hypotheses or the Sharp e Lintner CAPM model for both methods. The lower frequency of the false rejections for the NICSR method also indicates that the NICSR method is more efficient and more robust than the WICSR approach. Table 1. Statistics of Errors in RP Estimates Dev. Panel A: CSR Tests Using Estimated betas Panel B: CSR Tests Using Actual betas In Section 2, we show that the NICSR method tends to be more efficient because the risk loading betas crosssectionally center around one with small variation, and the timeseries average idiosyncratic errors diversify away considerably when they are averaged in the CSR. The crosssectional average and the variance of the market betas of the FamaFrench twentyfive crosssectional portfolios are 1.225 and 0.030, respectively. Hence by Theorem I, the errors in risk premium estimates generated by the WICSRs can be up to roughly fiftyone15 15 times as large as those by the NICSRs. The simulation results reveal that, on average, this ratio is about ten. Proposition III specifies the asymptotic standard EIV adjustment. For our simulations here, the mean and the variance of excess returns of the FamaFrench market portfolio are 0.688 and 28.714, respectively, and hence
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI Table 2: Number of Rejections of the CAPM cal Values Panel A: CSR Tests Using Estimated betas Panel B: CSR Tests Using Actual betas the corresponding asymptotic standard error adjustment will be only 0.016. Panel B of Table 1 and Panel B of Table 2 show the summary statistics of the errors in the risk premium estimates and the numbers of false rejections of CAPM, respectively, among the 100 simulations when the actual betas are used. Obviously the EIV issue here is statistically insignificant and this is consistent with the findings of Black, Jensen, and Scholes [3, 1972] and Shanken [23, 1992]. And this fact shows that the relatively large errors of the risk premium estimates in WICSRs (compared with those in NICSRs) can not be significantly reduced by the maximum likelihood estimations that try to eliminate the errors in variables. Even though a very low CSR may indicate that the hypothetical assetpricing model should be rejected by the data, 2 by itself can not be the appropriate measure of the model fit. Recently, Lewellen, Nagel, and Shanken [16, 2008] have shown that proposed asset pricing factors that are even weakly correlated with the true factors and uncorrelated with the errors are capable of producing high crosssectional and one of their recommendations is to take the magnitude of estimates more seriously. Our simulation results reaffirm their point. As shown by the scatterplot in Figure 3, high can be associated with estimates with large errors, and low 2 may correspond to relatively good estimates. Figure 3. Scatter Plot (PValue vs ) 3.2. CAPM Tests with Returns in the Early Subperiod In Section 2 we have proven that the NICSR estimation is more efficient than the classic WICSR estimation. We have also described some supportive simulation evidence in the previous subsection. However, we should be aware that NICSR estimation is not appropriate for the extend ed model (3). In the empirical assetpricing literature, the extended form (3) is typically directly assumed even though the theoretical models generally assume the existence of riskfree rate and are all in the basic form (1). The immediate reason of this practice may be the extensively common usage of the ordinary withintercept linear regressions and the relatively rare application of the nointercept linear regressions. But the main reason of this practice should be the rejection of the original SharpeLintner CAPM and the endorsement of the extended Black CAPM by the early classic CAPM testing works of Black, Jensen, and Scholes [3, 1972] and Fama and MacBeth [12, 1973]. In this subsection, we show that the evidence and warrants for rejection of the basic model (1) are not as strong as believed. As Lewellen, Nagel, and Shanken [16, 2008] point out, the magnitudes of the estimated zerobeta excess returns are unreasonably high in many of the existing empirical results. According to the theoretical explanation16 of the zerobeta excess return, it is a weighted average of the differences between the costs of riskfree borrowing and the returns of riskfree lending and hence should be within the approximate range17 between 0 and 0.2% per month. Black, Jensen, and Scholes [3, 1972] report the estimates of the zerobeta excess returns to be 0.338%, 0.849%, 0.420%, 0.782%, and 0.997% for the periods 1/3112/65, 1/319/39, 10/396/48, 7/483/57, and 4/5712/65. These estimates (4.056%, 10.188%, 5.04%, 9.384%, and 11.964% if annualized) are just too big to be due to the rate differences between (riskfree) borrowing and riskfree lending and the huge negative estimate of 0.849% for period 1/319/39 is even more spurious if it is interpreted by the borrowing restriction argument. The analyses in Section 2 and the above simulations in Subsection 3.1 suggest that at least one cause of the unreasonable magnitudes of the risk premium estimates and the zerobeta excess return estimates are the likely large errors of the WICSR estimates. In the rest of this subsection, we illustrate this point with our empirical 16Brennan [7, 1971] refers to the zerobeta rate as the market's equivalent riskfree rate and shows that it is a weighted average of the riskfree borrowing rate and the riskfree lending rate. 17Lewellen, Nagel, and Shanken [16, 2008] give their estimates of zerobeta excess returns to be between 0 and 2% per year.
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI CAPM testing results over the early period between July 1931 and December 1965.18 Panel A: Ex Post Market Risk Premia Our data include both the value weighted and equally weighted monthly excess returns of the widely used twentyfive FamaFrench sizeB/M crosssectional portfolios and the CRSP market index. The monthly riskfree rates are also obtained from the FamaFrench benchmark factor data. Table 3. CSR Tests on CAPM with Equally Weighted Data. M R Panel B: NICSR Tests on CAPM Panel C: WICSR Tests on CAPM λ ) =( λ M pR ) ( γσ Table 3 presents testing results of the SharpeLintner's original CAPM with equally weighted data for the early period between July 1931 and December 1965 and four subperiods 7/3112/39, 1/4012/47, 1/4812/56, and 1/5712/65. Panel B gives the testing results by the NICSR. We see that the risk premium estimates are quite close to the realized average market excess returns, that the high 's strongly suggest that the crosssectional average excess returns are proportional to the crosssectional betas, and that the SharpeLintner's CAPM fails to be rejected. Panel C shows testing results of the conventional WICSR methods. We see that the estimated standard errors of the risk premium estimates 18The testing period of one of the classic CAPM empiri cal works by BJS is January 1931 through December 1965. We try to take the same time period except for the first six month between January 1931 and June 1931, for which the 25 FamaFrench crosssectional portfolio re turn data are missing. by the WICSR method are much larger than those by the NICSR approach. But except for Subperiod 1/4812/56, our testing results fail to reject the SharpeLintner's CAPM. For Subperiod 1/4812/56, the testing statistics seem to even reject the extended CAPM. Since the WICSR estimates have bigger estimation errors and the NICSR testing results tend to support the Sharp e Lintner's CAPM, we think that the evidences for the rejection of SharpeLintner's CAPM generated by the early CAPM tests with the WICSR approach are exaggerated.19 Panel A: Ex Post Market Risk Premia Table 3: CSR Tests on CAPM with Equally Weighted Data. M R Panel B: NICSR Tests on CAPM Panel C: WICSR Tests on CAPM Table 4 shows our testing results of CAPM with the value weighted data for the early period 7/3112/65 and two halfperiods 7/3112/47 and 1/4812/65 using both the NICSR and the WICSR methods. Panel B presents the NICSR testing results. For the entire early period 7/3112/65 and the subperiod 1/4812/65, the NICSR risk premium estimates are very close to the expost excess market returns M R (The differences are smaller than 2 bps per month.) But for the subperiod 7/3112/47, 19 Notice that the discussion here is based only on preseventies stock returns, thus our findings should be interpreted as supportive evidence for the basic model (1), a general linear assetpricing model with riskfree rate, rather than for the extended model (3).
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI is considerably larger than M R. This evidence seems to imply the rejection of the SharpeLintner CAPM. However, we feel that the estimated risk premium for the subperiod 7/3112/47 is reasonable because the subperiod 7/3112/47 spans over the time of the Great Depression and the World War II.20 3.3. Multifactor AssetPricing Tests In the above subsection, we have presented empirical evidence to show that the NICSR method is more efficient than the WICSR for CAPM testing. In this subsection, we show the higher efficiency of the NICSR method when testing the FamaFrench threefactor model. Fama and French ([10, 1993] and [11, 1996]) propose a three factor model that explains more than 90% of the timeseries variation in portfolio returns and more than 75% of the crosssectional variation in their average returns. But the traditional WICSR estimates of the market risk premium are not significant at all and the estimated zerobeta excess returns are unreasonably high. It is obviously contradicting that the market factor has strong explanatory power in the time series but little explanatory capability in the crosssection of stock returns. Panel C shows the corresponding WICSR testing results and again the estimated standard errors of WICSR are significantly larger than those of NICSR. The two null hypotheses fail to be rejected in all the four testing periods. However, one can hardly be convinced that the CAPM holds by these results as the errors are so big and the estimated risk premium is essentially indifferent from 0 for the subperiod 1/4812 /65. Kim [15, 1995] shows that the WICSR market risk premium estimates with size factor present can still be significant when the errorsinvariables are corrected and the number of portfolios is large enough (say larger than 400). As we show in Section 2, small variation among crosssectional betas amplifies estimation errors, and the EIV introduces systematic bias. The empirical results reported by Kim will be more accurate than the traditional WICSR testing results since expansion of the set of test portfolios tends to enlarge the betavariatio n this is also one of the prescriptions suggested by Lewellen, Nagel, and Shanken [16, 2008]. In this subsection, we present our NICSR empirical testing results on the FamaFrench threefactor model, which not only show that the market risk premium is crosssectionally significant but also provide new 20Cochrane [8, 2001] suggests to make the standard error correction to account for the error of the sample mean of the market factor. Here the standard error correction will increase the uncorrected standard error by more than 0.5% per month and the null hypothesis will be fail to be rejected. crosssectional supportive evidence for the FamaFrench three factor model. The data consist of monthly valueweighted returns of the 25 FamaFrench sizeB/M crosssectional portfolios and monthly data of the three FamaFrench benchmark factors, namely the market excess returns and the returns of the two mimicking portfolios of SMB (smal lmi nusbig) and HML (highminuslow), for the time period between July 1926 and June 2007. Twelve data points between July 1930 and June 1931 are excluded because of missing data in the 25 FamaFrench sizeB/M crosssectional portfolio returns. The CSR estimation for the FamaFrench threefactor model is performed for the specified whole period and four subperiods, 7/266/47, 7/476/67, 7/676/87, and 7/876/07. Table 5 shows the CSR testing results on the FamaFrench threefactor model. Panel A lists the realized average values of the three FamaFrench factors. In Panel B, we see that the market risk premium estimates are not only significant but also are statistically indifferent from the realized expost market risk premium. The estimated risk premia of the size factor and the B/M factor are also found to be statistically indifferent from the realized average factor values. As shown in Panel C, the conventional WICSR estimates are spurious with the negative market risk premium estimates and the unreasonably high estimated zerobeta excess returns. Overall, we see that the NICSR method generates more efficient risk premium estimates (in terms of smaller estimated standard errors) for the market factor than the WICSR method and comparable estimates of the risk premia and the standard errors for the SMB and the HML factors to those generated the by the WICSR method. 4. Conclusions This paper shows that the classic crosssectional regression approach to asset pricing tests tends to suffer from severe estimation errors because of the small betavariation. We argue that it uses low criteria to validate an assetpricing model and suffers from the modelmisspecification issue because of the complication of the zerobeta excess return. To address this problem, we advocate focusing on the theoretic linear betapricing model that assumes a (market equivalent) riskfree rate and directly testing whether the exante risk premium estimates are consistent with the observed expost risk premia, which allows one to easily detect misspecified models. Under the assumption that asset returns follow a stationary and ergodic process, we derive the asymptotic distribution of the estimators of the method we advocate. We also show that this approach will be more efficient than the conventional method, provided that the
J. YUAN, R. SAVICKAS Copyright © 2013 SciRes. TI idiosyncratic errors have sufficiently weak cross sectional dependence.21 Panel A: Realized Averages of the FF Factors Table 5. CSR Tests on FF ThreeFactor Model. SMB R Panel B: NICSR Tests on FF ThreeFactor Model SMB Panel C: WICSR Tests on FF ThreeFactor Model 2 2 3 9 7 21The assumption of weak crosssectional dependence is also made by Black, Jensen, and Scholes [3, 1972], and by Shanken [23, 1992] in their cons i stency analyses for the crosssectional regression methods. Based on our strong evidence of higher efficiency of the NICSR ap proach, this assumption should also be quite reasonable. SMB Our simulation results provide further empirical evidence of higher efficiency of the approach we describe and show that the relatively large estimation errors of the conventional estimates are not due to the errorinvariable issues. The simulations also indicate that the crosssectional alone is an inappropriate criterion of the model fit even if the correct factors are used. We reexamine the original SharpeLintner CAPM with monthly stock returns for the early period between July 1931 and December 1965. Testing results of CAPM with the actual stock return data for the early subperiod further affirm that the estimates of the market risk premium generated by the approach advocated in this paper are more efficient than those by the classic method and show that the warrants of the significant zerobeta excess return are not as strong as believed. We also reexamine the FamaFrench threefactor model. It is well known that the market beta loses explanatory power on the crosssectional average returns when the size factor is included as an additional explanatory variable. With the five conventional estimations (one for the whole period and four for four subperiods), the estimated zerobeta excess returns are unreasonably high and the market risk premium estimates are all negative. But with all the five estimations using the suggested approach, not only are all the market risk premium estimates significant, but also all the three risk premium estimates are statistically indifferent from the realized average prices of the three risk factors. Furthermore, the standard error estimates for the market factor given by this method are much smaller than those by the conventional method. These findings illustrate the spurious nature of the conventional estimates and show that the market factor is consistently
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