b. 6. Nonnormal and Non-IID Returns

we have

1 i=i

(5.6.2)

The GMM estimator в is chosen to minimize the quadratic form

Q7.(0) = gr(0) Wgr(e), (5-6.3)

where W is a positive definite (2Nx2/V) weighting matrix. Since in this case we have 2N moment condition equations and 2N unknown parameters, the system is exactly identified and в can be chosen to set the average of the sample moments gr(0) equal to zero. The GMM estimator will not depend on W since Qr(6) will attain its minimum of zero for any weighting matrix. The estimators from this GMM procedure are equivalent to the maximum likelihood estimators in (5.3.13) and (5.3.14). The estimators are

a =

(3 =

A -Pfrm

El=l №mt ~ Am)

(5.6.4) (5.6.5)

The importance of the GMM approach for this application is that a robust covariance matrix of the estimators can be formed. The variances of d and (3 will differ from the variances in the maximum likelihood approach. The covariance matrix of the GMM estimator в follows from equation (A.2.8) in the Appendix. It is

where

V = [D0S-D0] 3gr(0)

Do = E

gr(fl)] 90 J

(5.616)

(5.617)

(5.68)

So = £ E[f,(0)f,-i(0)].

;=-oo

The asymptotic distribution of 0 is normal. Thus we have

0 ~ Jve.ifXSoDor1).

The application of the distributional result in (5.6.9) requires consistent estimators of D0 and S0 since ihey are unknown. In this case, for D0 we haye

D = - . 7 (5-6.10)

(5.6j9)

д. К + Ю

In this sec lion we arc concerned with inferences when there are deviations from ihe assumption that returns are jointly normal and I ID through time. We consider tests which accommodate non-normality, heteroskedasticity, and temporal dependence of returns. Such tests are of interest for two reasons. First, while the normality assumption is sufficient, it is not necessary lo derive the CAlM as a theoretical model. Rather, the normality assumption is adopted for statistical purposes. Without this assumption, linile-sample properties of asset pricing model tests are difficult to derive. Second, departures of monthly security returns from normality have been documented.1 There is also abundant evidence ol heteroskedasticity and temporal dependence in slock returns.1 F.ven though temporal dependence makes the CAlM unlikely to hold as an exact theoretical model, it is still of inteiesi 10 examine the empirical performance of the model. It is therefore desirable to consider the elfecls of relaxing these statistical assumptions.

Robust tests of tin CAlM can be constructed using a Generalized Method of Moments (GMM) framework. We focus on lests of the Sharpc-I.inlner version; however, robust leslsof the Шаек version can be constructed in the same manner. Within the GMM framework, the distribution of returns conditional on the markei return can be both serially dependent and conditionally heteroskedaslic. We need only assume that excess asset returns ate stationary and ergodic with Funic fourth moments. The subsequent analysis draws on Section A.2 of the Appendix which contains a general development of the GMM methodology. We continue with a sample of T lime-series observations and N assets. Following the Appendix, wc need to set up the vector of moment conditions with zero expectation. The required moment conditions follow from die excess-return markei model. The residual vector provides N moment conditions, and the product of the excess return of the market and the residual vector provides another N moment conditions. Using the notation of the Appendix, for f,(6>) we have

f,(0) = h, ®e (5.0.1)

where h, = [ I / , e, = Z, - a - ft / , and 0 = [a (3\.

Ihe specification of the excess-return market model implies the moment condition F.f,(0 )] = 0, where 0 is the true parameter vector. This moment condition forms the basis for estimation and testing using a GMM approach. GMM chooses the estimator so that linear combinations of the sample average of this moment condition are zero. For the sample average,

V.- I.iim.i (l!Hi.\ I!l7li, IH.inUiu, iniiI Connie-. (l! 7-l), Alllcck-Cravrs and McDonald (IlK.lhand Tal.lr II m (Ik,I.

Sec C.lwjiUis 1 and СЛ and die u-leieiu es Wen in iliosc chapters.

A consistent estimator D-r can easily be constructed using the maximum likelihood estimators of д, and ofn. To compute a consistent estimator of S0, an assumption is necessary to reduce the summation in (5.6.8) to a finite number of terms. Section A.3 in the Appendix discusses possible assumptions. Defining St as a consistent estimator of So, (l/VfDySDr]-1 is a consistent estimator of the covariance matrix of 0. Noting that d = R0 where R = (1 0) ® lN, a robust estimator оГ Var(d) is (\/ПЩЪТ), ]~lR. Using this we can construct a chi-square test of the SharpeT.intncr model as in (5.3.22). The test statistic is

/, = 7aJR[DrSrDг]-1 R] d. (5.6.11)

Under the null hypothesis a = 0,

h ~ Xl- (5-6.12)

{MacKinlay and Richardson (1991) illustrate the bias in standard CAlM statistics that can result from violations of the standard distributional assumptions. Specifically, they consider the case of contemporaneous conditional heteroskedasticity. With contemporaneous conditional hetcroskedas-liciiy, the variance of the market-model residuals of equation (5.3.3) depends on the contemporaneous market return. In their example, the assumption that excess returns arc IID andjointly multivariate Student t leads to conditional heteroskedasticity. The multivariate Student / assumption for < excess returns can be motivated both empirically and theoretically. One empirical stylized fact from the distribution of returns literature is that rent rr s have fatter tails and arc more peaked than one would expect from a nornal distribution. This is consistent with returns coming from a mulli-vari He Student t. Further, the multivariate Student / is a return distribution for ivhich mean-variance analysis is consistent with expected utility maximization, making the choice theoretically appealing/

The bias in the size of the standard CAPM test for the Student / case depends on the Sharpe ratio of the market portfolio and the degrees of freedom of the Student I. MacKinlay and Richardson (1991) present some estimates of the potential bias for various Sharpe ratios and for Student I degrees of freedom equal to 5 and 10. They find that in general the bias is small, but if the Sharpe ratio is high and the degrees of freedom small, the bias can be substantial and lead to incorrect inferences. Calculation of the lest statistic Ji based on the GMM framework provides a simple check for the possibility thai the rejection of the model is the result of heteroskedasticity in the data.

See iivjicvsoii (1У87), p. KM.

5.7 Implementation of Tests

In this set lion we consider issues relating lo empirical implementation of the test methodology. A summary of empirical results, an illustrative implementation, and discussion of the observability of the market portfolio arc included.

5.7.1 Summary of Empirical Evidence

An enormous amount of literature presenting empirical evidence on the CAlM has evolved since the development of the model in the 1960s. The early evidence was largely positive, with Black, Jensen, and Scholes (1972), Fama and MacBcth (1973), and Blumc and Friend (1973) all reporting evidence consistent with the mean-variance efficiency of the market portfolio. There was some evidence against the Sharpc-I.inlner version of the CAPM as the estimated mean return on the zero-beta portfolio was higher than the riskfree return, but this could be accounted for by the Black version of the model.

In the late 1970s less favorable evidence for the CAlM began lo appear in the so-called anomalies literature. In the context of ihe tests discussed in this chapter, the anomalies can be thought of as firm characteristics which can be used to group assets together so thai the langcncy portfolio of the included portfolios has a high ex post Sharpe ratio relative to the Sharpe ratio of the market proxy. Alternatively, contrary to the prediction of the CAPM, the firm characteristics provide explanatory power lor die cross section of sample mean returns beyond the beta of the ( АРМ.

Early anomalies included the price-earnings-ratio effect and the size effect. Basu (1977) first reported the pricc-eai nings-ralio clfcct. Basils finding is thai the market portfolio appears not to be mean-variance efficient relative to portfolios formed on the basis of the price-earnings ratios оГ firms. Firms with low price-earnings ratios have higher sample returns, and linns with high price-earnings ratios have lower mean returns than would be the case if the market portfolio was mean-variance efficient. The size effect, which was first documented by Ban/. \1981), is the result that low markei capitalization firms have higher sample mean returns than would be expected if the market portfolio was mean-variance efficient. These two anomalies arc al least partially related, as the low price-earnings-ratio firms lend to be small.

A number of other anomalies have been discovered more recently. Fama and French (1992, 1993) find that beta cannot explain the difference in return between portfolios formed on the basis of the ratio of book value of equity to market value of equity. Finns with high book-market ratios have higher average returns than is predicted by the (АРМ. Similarly,

J. I lie (.a/iilal Л sset 1iiriiif! Mattel

DcHondt and Tlialn (1985) andcgadecsh and Tiinun (1995) find thai a portfolio lormcd In buying stocks whose value Iras declined in die past (/mm) and selling slocks whose value has risen in ihe past {winners) has a higher average relurn than ihe CAPM predicts. Fama (1991) provides a good discussion of these and oilier anomalies.

Although die results in the anomalies literature may signal economically important deviations from the CAPM, there is little theoretical motivation lor the linn characteristics studied in this literature. This opens up the possibility dial the evidence against the CAPM is overstated because ol data-snooping and sample selection biases. We briefly discuss these possibilities.

Data-snooping biases refer to the biases in statistical inference that result Irom using information from data lo guide subsequent research with the same or related data. These biases are almost impossible to avoid due lo the iionexperimeiilal nature of economics. We do not have the luxury of running another experiment lo create a new data set. l.o and MacKinlay (1990b) illustrate the potential magnitude of data-snooping biases in a lest of the Sharpe-Lintner version of the CAPM. They consider the case where the characteristic used to group stocks into portfolios (e.g. si/c or price-earnings ratio) is selected not from theory but from previous observations of mean stock returns using related data. (lomparisoiis of the null distribution of the lesl statistic with and without data-snooping suggests that the magnitude of the biases can be immense. I lowever, in practice, il is difficult lo specify the adjustment that should be made for data-snooping. Thus, the main message is a warning that the biases should al least be considered as a potential explanation for model deviations.

Sample selection biases can arise when data availability leads to certain subsets of stocks being excluded from the analysis. For example, Kothari. Shanken, and Sloan (I 995) argue thai data requirements for studies looking at book-market ratios lead to failing slocks being excluded and a resulting survivorship bias. Since the failing stocks would be expected lo have low reliirnsand high book-market ratios, the average return of the included high book-markcl-ralio storks would have an upward bias. Kothari, Shanken, aiid Sloan (I r ) argue that ibis bias is largely responsible for the previouslvcilcd result of Fama and French (1992, 1993). I lowever, the importance of this particular survivorship bias is not lully resolved as Fama and French (1996b) dispute Ihe conclusions of Kothari, Shanken, and Sloan. In anv event, il is clear thai researchers should be aware ol the potential problems that can arise from sample selection biases.

7.2 Illustrative liii/ilemenlalioit

We present IcstsolihcSharped ininei model lo illustrate die testing methodology. We consider lour lest statistics: /1 from (5.3.23), /- from (5.3.33), /,

5.7. Implementation of Tests

213 ~

from (5.3.41), and /7 from (5.6.11). The tests are conducted using a thirty-year sample of monthly returns on ten portfolios. Stocks listed on theJNew York Slock F.xchange and on the American Stock Exchange are allocated to the portfolios based on the market value of equity and are value-weifited within the portfolios. The CRSP value-weighted index is used as a proxy for the market portfolio, and the one-month US Treasury bill relurn is used for the riskfree return. The sample extends from January 1965 through December 1994.

Tests are conducted for the overall period, three ten-year subperiods, and six five-year subperiods. The subperiods are also used to form overall aggregate test statistics by assuming that the subperiod statistics are independent. The aggregate statistics for J2, J3, and /7 are the sum of the individual statistics. The distribution of the sum under the null hypothesis will be chi-squarc with degrees of freedom equal to the number of subperiods times the degrees of freedom for each subperiod. The aggregate statistic for J\ is calculated byscaling and summing the F statistics. The scale factor is calculated by approximating the F distribution with a scaled chi-square distribution. The approximation matches the first two moments. The degrees of freedom of the null distribution of the scaled sum of the subperiod 7is is the number of subperiods times the degrees of freedom of the chi-square approximation.

The empirical results are reported in Table 5.3. The results present evidence against the Sharpe-Lintner CAPM. Using /1, the -value for the overall thirty-year period is 0.020, indicating that the null hypothesis is rejected at the 5% significance level. The five- and ten-year subperiod results suggest that the strongest evidence against the restrictions imposed by the model is in the first ten years of the sample from January 1965 to December 1974.

Comparisons of the results across test statistics reveal that in finite samples inferences can differ. A comparison of the results for J\ versus }i illustrates the previously discussed fact that the asymptotic likelihood ratio test tends to reject too often. The finite-sample adjustment to }i works well as inferences with J% are almost identical to those with J\. j

5.7.3 Unobservability of the Market Portfolio

In the preceding analysis, we have not addressed the problem that the rtjturn on the market portfolio is unobserved and a proxy is used in the tests. Most tests use a value- or equal-weighted basket of NYSE and AMEX stocks ajs the market proxy, whereas theoretically the market portfolio contains all assets. Roll (1977) emphasizes that tests of the CAPM really only reject the rrean-variance efficiency of the proxy and that the model might not berejected if the return on the true market portfolio were used. Several approaches have