testing the capm

8/2/2019 Testing the CAPM

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Testing the CAPM:

A Simple Alternative to Fama and MacBeth

(1973)

Paper Number: 06/04

Cherif Guermat, George Bulkley*, Mark C. Freeman* and RichardD.F. Harris*

Department of Economics and *Xfi Centre for Finance and Investment

University of Exeter

Author for CorrespondenceXfi Centre for Finance and Investment

University of ExeterExeter EX4 4STEngland

Tel: +44-(0) 1392-263155

Fax: +44-(0) 1392-262525

Email: [email protected]@exeter.ac.uk

R.D.F. [email protected]

I.G. [email protected]


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Testing the CAPM:A Simple Alternative to Fama and MacBeth

(1973)

Abstract

Most studies that use the method of Fama and MacBeth (1973) to test

the Capital Asset Pricing Model (CAPM) are unable to reject the null hy-

pothesis that beta and expected returns are uncorrelated. This paper in-

troduces a simple new test which has the CAPM as its null hypothesis. We

show theoretically, by simulation and by using market data that our pro-

posed test has greater power than that of Fama and MacBeth. The new test

helps to establish whether the findings of previous studies result from the

low power of the Fama and MacBeth test or from a failure of the CAPM.

JEL classifi

cation: G10; G12; C12; C15Keywords: Capital Asset Pricing, Fama and MacBeth Test.

1. Introduction

Empirical tests of the Capital Asset Pricing Model (CAPM) have been

unable to find a statistically significant correlation between stock returns and

beta (see, for example, Reinganum, 1982; Lakonishok and Shapiro, 1986;

Ritter and Chopra, 1989; Fama and French, 1992). These tests typically

employ the method of Fama and MacBeth (1973, hereafter FM), which

involves estimating a series of monthly cross-section regressions of individual

stock returns on beta, and then testing whether the average slope coefficient

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in these regressions is statistically different from zero using the time series

variation of the estimated slope coefficient in order to calculate the standard

error of the average slope coefficient.

Typically these empirical studies do not reject the null hypothesis of

the FM test that beta risk is not priced. A question raised by Chan and

Lakonishok (1993) is whether the null hypothesis is true or whether the

tests just lack the power to reject it in finite samples. They show that,

for realistic levels of market return volatility, the FM test is likely to have

low power in samples of the size typically employed in practice. The test

cannot reject the null that beta is not priced but it also cannot reject the

null of the CAPM assuming any plausible value for the expected return on

the market. A researcher who has strong prior beliefs in the CAPM would

not be compelled to infer from the point estimate and associated standard

error that the CAPM is false.

The low power of the FM test arises from the fact that, under both the

null and alternate hypotheses, a component of the slope coefficient in each

monthly cross-section regression is the realized excess market return . This

is very volatile, which yields a highly noisy estimated series of monthly slope

coefficients. It is this noise, which is common to both null and alternative

hypotheses, that is responsible for the low power of the FM test.

In this paper, we present a simple test in the spirit of Fama and Mac-

Beth (1973) that addresses this problem of low power. Our modified FM

test (hereafter Modified FM) involves subtracting the realized excess market

return each month from the estimated FM slope coefficient. The variance

of the test statistic is then significantly reduced and this results in a sub-

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stantial increase in power. This is demonstrated analytically, by simulation

and by using real market data.

A natural consequence of subtracting the observed excess market return

from the estimated FM slope coefficient is that the CAPM becomes the null

hypothesis of the Modified FM test. This has a major advantage. Hav-

ing the null as the hypothesis that beta is unpriced means that rejecting

the null using the FM test is sufficient to establish that beta helps explain

cross-sectional differences in expected return. It is, though, not directly

informative about whether the price of beta risk is equal to the equity pre-

mium. In contrast, by having the CAPM as its null, finding a Modified FM

test statistic that is significantly different from zero is sufficient to reject the

model. The Modified FM test is therefore a more specific test of the CAPM

than FM.

There are other recent alternative methods to FM for testing the CAPM,

such as Hansens (1982) generalized method of moments (GMM) and the

semi-parametric method of Hodgson et al. (2002). GMM based models

are theoretically superior to FM as they relax both the normality and the

conditional homoscedasticity assumptions (Jagannathan and Wang, 2002).

However, GMM based methods do not generally lead to fully efficient es-

timates (Vorkink, 2003). In addition, Harvey and Zhou (1993) find little

difference between OLS and GMM based tests, while Ferson and Foerster

(1994) provide evidence suggesting that GMM methods lead to asset pricing

tests with aberrant properties. The semi-parametric method of Hodgson et

al. (2002) seems promising, but the evidence is scant and is based only on

a small-scale simulation study (Vorkink, 2003).

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Thus, although these alternative methods may have more attractive the-

oretical properties, there is still a lack of empirical evidence in their favor.

From a practical point of view, both methods suffer from the fact that they

are difficult to implement, and that no standard software is available for car-

rying out estimation and testing using these methods. In contrast, both the

FM method and our modification to it combine simplicity with robustness

to cross-sectional correlation. They are also easily implemented using stan-

dard statistical and econometric software, which makes them potentially

attractive methods for practitioners. The simplicity of the FM method-

ology has resulted in its continued popularity. Among the recent studies

that have employed the FM methodology are Coval and Moskowitz (2001),

Lettau and Ludvingson (2001), Gomes, Kogan and Zhang (2003), Menzly,

Santos and Veronesi (2004), and Gompers and Metrick (2001). Variants of

the FM method are also used by Gompers, Ishii and Metrick (2003) and

Baker, Stein and Wurgler (2003).

The outline of the paper is as follows. In the following section, we present

the FM test in detail and discuss its small sample properties. In Section

3, we introduce the Modified FM test and compare it with the FM test.

Section 4 reports the results of simulation experiments to ascertain the size

and power of both the FM test and the Modified FM test. In section 5,

we run the FM and Modified FM tests on US stock market data. We test

the CAPM using the FM and Modified FM tests over several subperiods

of US market data since 1950. These confirm that the Modified FM test

statistic has lower standard error than the FM test statistic. Even with

this increased power, though, it is in many cases not possible to reject the

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CAPM under the Modified FM test for beta ranked portfolios. Section 6

offers some concluding remarks.

2. The Fama-MacBeth Test

Suppose that there are N securities, with the return to asset i {1,...,N}between times t 1 and t being denoted by rit. The expected excess returnto any asset is given by

E[Rit] = i + i (1)

where Rit = rit rf,t1 is the excess return between period t1 and periodt for security i, rf,t1 is the one-period risk free rate at time t 1. i =Cov(Rit, Rmt)/V ar(Rmt) is the CAPM beta of security i which is assumed to

be constant across time and Rmt is the excess return on the market portfolio.

is the price of beta risk. Under the CAPM, = E[Rmt] while, if beta is

not priced, = 0. i incorporates the elements of the expected return that

are not captured by beta. This may include size effects, book-to-market

effects or additional Arbitrage Pricing Theory effects. It is assumed that

i does not vary across time. Again, if the CAPM is true, i = 0 for all i.

Now consider the realized excess return to each asset minus its expecta-

tion; it = Rit E[Rit]. This can, without loss of generality, be linearlyprojected against mt, the realized excess return to the market portfolio

minus its expectation:

it = proj(it|mt) + vit (2)

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where the projection term is defined by

proj(it|mt) =E[itmt]

E[2mt]mt = imt (3)

i, defined above, is the true, or population, CAPM beta. For ease of

exposition, it will be assumed that this is always estimated without error.

It is easily established that E[vit] = E[vitmt] = 0.1 We will further impose

the restriction that E[vitvj] = 0 for all i, j including i = j and 6= t. This,

essentially, is restricting asset excess returns to display zero autocorrelation.

There is no requirement here that the disturbance terms are cross-sectionally

independent as they may, for example, within an Arbitrage Pricing Theory

context, include co-movement with other macroeconomic factors. This

general framework emphasizes that betas are informative about the ex-post

relationship between individual asset returns and the realized return to the

market portfolio. This, though, is not the central element of the CAPM,

which makes predictions about the ex-ante, pricing, effects of beta captured

by = E[Rmt] and i = 0. Equation 2 can be re-expressed as:

Rit = E[Rit] + i(Rmt E[Rmt]) + vit (4)

Consider the following cross-section regression between excess returns

and beta, which is commonly used in empirical studies of the CAPM:

Rit = at + ti + it (5)1 To establish these results (i) take expectations of equation 2 and (ii) multiply both

sides of this equation by mt and then take expectations. Here we are invoking a standard

asset pricing technique that is often used within a stochastic discount factor setting. See,

for example, Cochrane (2001, p.18).

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The OLS estimate oft is given by

bt = Covi(Rit,i)V ari(i)

=Covi(E[Rit],i)

V ari(i)+ Rmt E[Rmt] + t

(6)

where t = S1

Pi(i )vit, S =

Pi(i )2 and = (1/N)

Pi i.

The notations Covi and V ari denote the sample covariance and variance,

respectively, evaluated over the cross-section dimension. By substituting in

for equation 1, it is clear that:

bt = + + Rmt E[Rmt] + t (7)where = S1

Pi(i)(i) and = (1/N)

Pi i. Empirical tests

of the CAPM can therefore be formulated as tests of hypotheses about E[t]

in regression (5). In particular, most tests of the CAPM involve testing the

null hypothesis that E[t] = 0, which is consistent with = 0, against the

alternative that E[t] 6= 0.2

If the CAPM holds, then E[t] = E[Rmt] and

the null hypothesis (E[t] = 0) should be rejected. The slope coefficient t

can be estimated straightforwardly by the usual OLS estimator, t, defined

above. However, the hypothesis that E[t] = 0 cannot be tested using a

conventional OLS t-statistic because the error term is cross-sectionally cor-

related under the null hypothesis. Consequently, the usual OLS estimate

of the standard error of t will be both biased and inconsistent.3 Indeed,

2 Notice that E(t) = Pt S1 Pi(i

)E[vit] = 0. In Fama-MacBeth type tests it is

also usually assumed that = 0.3 Both the pooled OLS regression and the Black, Jensen and Scholes (1972) pure cross

section regression suffer from this shortcoming. See Cochrane (2001) for a discussion.

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Cochrane (2001) reports that the errors are so highly cross-sectionally cor-

related that standard errors can often be underestimated by a factor of ten

or more.

In a seminal paper, Fama and MacBeth (1973) propose a solution to this

problem that exploits the fact that, while the estimated standard error in

the regression will be biased, the estimated slope coefficient will be unbi-

ased. Fama and MacBeth propose repeatedly estimating the cross section

regression equation 5. The point estimate is the time-series mean of the

estimated slope coefficient in each of the cross-section regressions:

FM =1

T

TXt=1

t (8)

Under the null hypothesis that E[t] = 0, E[FM] = 0, while under the

alternative hypothesis that E[t] = E[Rmt], E[bFM] = E[Rmt]. In order totest the null hypothesis that E[t] = 0, Fama and MacBeth propose the test

statistic

tFM =FM

SE(FM)(9)

where SE(FM) = SD(t)/

T 1 and SD(t) is the sample standarddeviation of the T cross-section regressions estimates t. Under the null

hypothesis that E[t] = 0, the FM test statistic has a t-distribution with

T1 degrees of freedom. The FM approach has been used by many empiricalstudies, including Reinganum (1982), Lakonishok and Shapiro (1986), Ritter

and Chopra (1989), Fama and French (1992). These studies have invariably

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found that the null hypothesis that E[t] = 0 cannot be rejected.4

Since the null hypothesis is that expected returns and beta are uncor-

related, the ability of the FM test to reject non-pricing depends upon the

power of the test to detect the correlation between beta and expected re-

turns. This in turn depends upon the standard error of FM, and hence,

from equation 8, on the variance of t. In the Appendix, we show that the

variance of t is given by

V ar(FM) =2mT

+2

T S+

T(10)

where 2m is the variance of Rmt and

= 2S2

Xi

Xj6=i

(i )(j )Cov(vit, vjt) (11)

. The power of the FM test therefore depends on the variance of the realized

market return, 2m. The higher the variance of2m, the higher the standard

deviation oft, the higher the standard error ofFM and the lower the power

of the FM test. Chan and Lakonishok (1993) show that, for realistic levels

of market return volatility, the FM test is likely to have low power in data

sets of the lengths typically employed in practice, and is therefore unlikely to

reject the null hypothesis that the average slope coefficient between returns

and beta is zero, even if the CAPM is true. We provide more evidence on

the small sample properties of the FM test in Section 4.

Note that since the OLS estimator t is unbiased, from equation 7:

4 Fama and MacBeth (1973) is one of the very few studies that found support for the

CAPM.

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E[t] = E[t] = + (12)

The null hypothesis in the FM test, therefore, is whether + = 0,

whereas the alternative is that + 6= 0. Thus, the FM test does not

specifically test the CAPM model which makes three specific predictions;

= E[Rmt], = 0 and that there are no sources of expected return that

are orthogonal to beta. captures the effects of sources of risk that affect

expected returns and are correlated with beta. This might include, for

example, size effects. Usually the rejection of the Fama-MacBeth null is

seen to support the view that beta risk is priced. Again, though, this need

not be the case if 6= 0. The proposed Modified FM test focuses specifically

on testing the CAPM hypothesis.

3. The Modified Fama-MacBeth Test

The low power of the FM test comes from the fact that the slope coeffi-

cient in each monthly regression contains the realized excess market return.

The high volatility of the realized market return translates into a high volatil-

ity of the estimated monthly slope coefficient, and consequently into a high

standard error of the average estimated slope coefficient. In this section, we

propose a modification of the FM test that leads to a substantial increase

in power to discriminate between the CAPM and non-CAPM hypotheses.

To motivate the new test, note that from equation 7, the slope coefficient

in the FM test has a common component, Rmt, under any hypothesis on

the relationship between expected returns and beta (i.e. for any value of

). Our proposal, therefore, is to modify the FM test by first subtracting

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the common component, Rmt, from the estimated slope coefficient, t, each

month. If the CAPM holds, = E[Rmt] and hence

t = t Rmt = + E[Rmt] + t (13)

Now consider the null hypothesis that the CAPM is true; = E[Rmt]

and = 0. In this case E[bt] = 0. Rejection of this null is sufficient toreject the CAPM. It will be rejected either because 6= E[Rmt] or because

6= 0.

The Modified FM test is given by

tMFM =MFM

SE(MFM)(14)

where MFM =1

T

Pt t, SE(MFM) = SD(bt)/T 1 and SD(bt) is the

sample standard deviation of

bt over the T cross-section regressions esti-

mates. Under the null hypothesis that the CAPM holds, the Modified FM

test statistic has a t-distribution with T 1 degrees of freedom. By sub-tracting the realized market return each month, the adjusted slope coefficient

estimate has lower variance but the same signal, and hence the Modified FM

statistic should have higher power to discriminate between the null and al-

ternative hypotheses. In the Appendix, we show that the variance of the

modified statistic, MFM, is equal to

V ar(MFM) =2

T S+

T(15)

Thus the variance of MFM is lower than the variance ofFM by 2m/T.

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Equivalently, we can note that under their respective nulls:

SE2(bFM) SE2(bMFM) = 2mT (16)In the next section we evaluate the power of the modified FM test to

distinguish between CAPM and the alternative hypothesis that returns and

beta are uncorrelated using simulation experiments.

4. Simulation Experiments

In this section, we evaluate the small sample properties of both the

FM test and the Modified FM test using simulated data. In particular, we

generate data under two distinct hypotheses. Under the first hypothesis, the

CAPM holds, while under the second hypothesis, expected returns and beta

are uncorrelated. The two models used to generate the simulated data are

calibrated using realized monthly returns on 1000 stocks randomly drawn

from stocks in the CRSP database that had as least 24 observations over

the period January 1968 to December 2000.

For each randomly drawn stock i, we use the available time series data

from January 1968 to December 2000 to estimate i as the slope of the time

series regression of the monthly excess return of each stock on the stock

market excess return. The variance of the residuals from this regression 2i

is saved. The excess market return is the monthly CRSP market return

minus the US Treasury bill rate (IMF series USI60C).

For the first model, under which the CAPM holds, returns, Rit, are

generated as follows. At each date t, excess market returns, Rmt, are drawn

from a normal distribution with mean E(Rmt) and variance 2m. We use

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E(Rmt) = 0.007 and m = 0.055. These values closely match the sample

moments of the CRSP data and are similar to those used in Jagannathan

and Wang (2002). In order to examine the impact of the volatility of the

market return on the small sample properties of the two tests, we also use

half the standard deviation (m = 0.0275). Four time-series sample sizes,

T, are considered (60, 120, 240, and 360 months). These time horizons are

commonly used in testing the CAPM (see, for example, Fama and French,

1992, and Jagannathan and Wang, 1996). We also consider some cases with

a large sample size (T = 1000).

Individual excess returns under the null hypothesis of the CAPM are

generated by

R1,it = iRmt + vit vit N(0, 2i ) (17)

where i is randomly drawn, with replacement, from the vector of 1000

estimates ofi, and it is randomly drawn from a normal distribution with

variance 2i .

For the second model, where beta risk is not priced, individual excess

returns are generated by

R2,it = E(Rmt) + i(Rmt E(Rmt)) + vit vit N(0, 2i ) (18)

The simulation exercise is based on 1000 replications. In each replica-

tion, equations 17 and 18 are used to generate individual stock return data.

In each replication, and for each of the return-generating models, j = 1, 2,

we estimate a series of cross-section regressions Rj,it = j,t + j,ti + ej,it,

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t = 1, , T, to yield two series of estimated slope coefficients, 1,t, and

2,t, t = 1, , T. For each of the two models, j = 1, 2, the FM statistic

is the t-statistic of the sample mean of j,t, given by equation 9, while the

Modified FM test the t-statistic of the sample mean of j,t = j,t Rmt,given by equation 14. The FM statistic is then used to test the null hypoth-

esis that expected returns and beta are uncorrelated against the alternative

hypothesis that beta is priced, and the Modified FM test is used to test

the null hypothesis that the CAPM holds against the alternative hypothe-

sis that 6= E[Rmt]. In order to compute the size of the FM and Modified

FM tests, we calculate the proportion of times each test rejects its respective

null hypothesis, when the respective null hypothesis is true. To compute the

power of the two tests, we calculate the proportions of times each test rejects

its respective null hypothesis, when the respective alternative hypothesis is

true. The tests are conducted at the 5% and 10% two-sided significance

levels.

The simulation results for the FM and Modified FM tests are reported in

Table 1. For all three significance levels, the empirical size of both tests un-

der their respective null hypotheses is close to the nominal significance level.

However, the Modified FM test clearly has greater power to discriminate be-

tween the two hypotheses. Even at the smallest sample size, the power of

the Modified FM test is very close to unity regardless of the volatility of the

market return. For the FM test, however, the power is around 20 percent

for the smallest sample size when m = 0.055. Even when the variance of

the market return is halved, the power is still low, at most 56 percent at the

10% significance level. To achieve reasonable power, the FM test requires

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at least 1000 months of data, and even then, the power is still marginally

lower than that of the Modified FM test.

[Insert Table 1 around here]

Some interesting properties of the two tests are shown in Table 2, which

reports the mean, standard deviation, and skewness and excess kurtosis

coefficients of the simulated FM and Modified FM statistics. The two

tests show little diff

erence under their respective null hypotheses. But thedifference is striking under their respective alternatives. The means of the

two statistics have the expected opposite sign, but the difference in their

variance is substantial. Owing to its reduced variance, the Modified FM

statistic is around four and a half times greater at all sample sizes. The

distribution of the Modified FM statistic under the alternative is to the left

of zero as suggested by its 5th and 95th percentiles. The FM statistic is

far closer to its value under the null hypothesis, with the 5th percentile

becoming negative at lower sample sizes.


4.1. Robustness Check

In this subsection, in order to establish the robustness of our simula-

tion experiments, we consider how the FM and Modified FM tests perform

against some alternative data generating processes for returns. As an alter-

native to the CAPM, Jagannathan and Wang (2002) suggest a model of the

form

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Rit = i + iRmt + vit (19)

where i = E(Rit) iE(Rmt). In relation to our general specification,this is the case where i +i is unconstrained. This model allows for other

firm-specific determinants of expected returns, such as market capitalization

and the ratio of book to market value of equity. In the simulation, i is

calibrated using the 1000 randomly drawn stocks from the CRSP database,

and is calculated as

i = Ri iRm (20)

Kan and Zhou (1999) suggest an alternative model that adds noise to

the market return. In particular, one of their suggestions is to use

Rit = ift + vit vit

N(0,2i ) (21)

where ft = (Rmt + nt).p

1 + 2n , nt is a zero mean measurement error

with finite variance 2n, and which is uncorrelated with vit. Since, E(ft) =

E(Rmt).p

1 + 2n , this is consistent with the CAPM. In the simulation we

use equation 21 to generate the data, using nt N(0, 0.01).Table 3 presents the rejection rates when the data are generated by the

Jagannathan and Wang (2002) and the Kan and Zhou (1999) models. For

a well specified test, the CAPM should be accepted for the Kan and Zhou

model and rejected for the Jagannathan and Wang model. Therefore the

FM (Modified FM) test should reject (fail to reject) the null for the Kan and

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Zhou process. For the Jagannathan and Wang process the FM (Modified

FM) test should fail to reject (reject) the null.


Panels A and B of the table show that the noisy CAPM of Kan and

Zhou is not identified by the FM test. The Modified FM rejects it with the

correct size for both levels of market volatility and both significance levels.

Panels C and D of the table show the rejection rate of the two tests whenthe data are generated by the Jagannathan and Wang noisy Non-CAPM

model. While the additional noise does affect the Modified FM test for very

small sample sizes, this effect virtually disappears once T = 120. On the

other hand, the FM test yields a surprising result. Contrary to expectation,

the rejection rate increases with the sample size. Typically, rejection rates

of the FM test should remain close to the nominal sizes of 5% and 10%.

Indeed, as can be seen from Table 1, the rejection rates for the Non-CAPM

data generated by equation 18 are all close to their respective nominal sizes,

irrespective of the sample size. What is different in the Jagannathan and

Wang model is that the Non-CAPM model contains additional noise. This

appears to distort the size of the FM test, especially for large values of T.

Thus, noise not only seems to worsen the power of the FM test when the

null hypothesis is false, but also seems to distort the size of the FM test

when the null is true. This suggests that in the presence of additional noise,

the FM test may not be able to detect the presence (or lack) of correlation

between beta and average returns. Thus, the FM test can potentially fail

completely under certain alternative models. In contrast, the Modified FM

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test performs reasonably well in both cases examined here.

5. Testing the CAPM

In this section, we test the CAPM on US data using the FM and Modi-

fied FM tests. We randomly selected 9,000 stocks from the CRSP dataset

of 22,716 companies. Stocks with less than 60 consecutive monthly observa-

tions throughout the period January 1968 to December 2000 were removed

from the sample. This left slightly more thanfi

ve thousand stocks available,of which the first five thousand were selected for the test.

In the first stage, the estimated beta of each stock, bi, i = 1,..., 5000,was calculated. Following standard procedures, these were calculated using

a time series regression over all the available observations for each stock

Rit = i + iRmt + eit (22)

On the basis of these estimated individual stock betas, 100 beta-ranked

portfolios were formed with 50 stocks in each. Again, following standard

Fama-MacBeth procedure, the beta of each portfolio was re-estimated using

the 60 months of data prior to the test period in order to remove potential

measurement error bias. This provided estimates of portfolio betas bp,p = 1,..., 100. Then t was estimated from the cross section regression

Rpt = at + tp + vpt (23)

The FM and Modified FM tests are then simply t-tests on the sample

mean of the time series t and (t Rmt), respectively. The results, shown

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in Table 4, contrast the conclusions of the two tests for a selection of starting

dates and sample sizes, T.


As expected, the Modified FM test statistic has a substantially lower

standard error than the FM test, potentially giving it much greater power.

According to equation 16:

2m = ThSE2(bFM) SE2(bMFM)i (24)Using this equation to estimate 2m from the standard errors in Table 4,

the average estimated annualized standard deviation of the market is 14.6%

with a minimum and maximum estimate of 11.6% and 16.9% respectively,

which is broadly consistent with observed returns.

Despite the increase in power, the Modified FM test statistic is statis-

tically significant at the 10% level only twice. From Table 1, it has been

established that the null hypothesis that this test statistic is equal to zero is

rejected almost 100% of the time if beta is unpriced and if market returns

in the test period are normally distributed with a mean of 0.7% per month.

These results do, then, give some support to the view that beta can help

explain cross-sectional variations in average returns. This is backed by the

FM test statistics, which are always positive, of similar value to the esti-

mated equity premium and where, despite the tests low power, the p-value

is less than 10% in three out of the nine cases.

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6. Conclusion

It is known from a large number of studies in empirical asset pricing that

it is very difficult to reject the null hypothesis that the correlation between

beta and expected returns is zero. This has led some to infer that the

CAPM beta is not priced. However, others have noted that, given the low

power of the Fama and MacBeth test and the short test periods employed,

the inability to reject the null would be unsurprising even if the CAPM held.

In this paper, we note that the key factor that lies behind the low power

of the standard FM test is the volatility of realized market returns. The

ex-post excess return to the market is a common component of the test sta-

tistic under both the null and alternate hypotheses. We therefore construct

an alternate test, the Modified FM, where the realized return to the market

in each period is removed from the FM statistic. It is shown theoretically,

by simulation and through an empirical study using market data that the

Modified FM has greater power than the FM statistic. In principle, there-

fore, the Modified FM is better able to distinguish between the CAPM and

alternative hypotheses.

In the final section of the paper, we have conducted FM and Modified

FM tests on various samples of US data. Using beta ranked portfolios, the

decreased standard errors of the Modified FM test is clearly demonstrated

and is of the expected order of magnitude. It is interesting to see in theseresults that, when the modified test with substantially enhanced power is

applied, in most cases it is not possible to reject the null hypothesis that

the CAPM is true.

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Appendix

As in equation 4, the data generating process is given by:

Rit = E[Rit] + i (Rmt E[Rmt]) + vit (25)

where Rmt = E[Rmt] + ut, ut iid(0,2m), E(vit) = 0, E(vitvjs) = 0 fort 6= s, i,j. We also assume V ar(vit) = 2v, and E(utvis) = 0, i,t,s.

The OLS estimator oft, as in equation 7, is given by

t = + + Rmt E[Rmt] + t (26)

The variance of t is given by

V ar(t) = V ar(Rmt) + V ar(t) + 2Cov(Rmt, t) (27)

Consider the covariance term. Since E[t] = 0 this can be re-expressedas E[Rmtt]. Substituting in for t this is equal to E[RmtS

1

Pi(i)vit].

Rearranging gives S1P

i(i )E[Rmtvit] = 0 from the properties of vitgiven above. It follows that

V ar(t) = V ar(Rmt) + V ar(t) (28)

As asset excess returns are assumed to be zero autocorrelated, Cov(t, s) =

0, t 6= s, the variance of FM is given by

V ar(FM) =1

TV ar(t) (29)

We have V ar(Rmt) = 2m and, assuming i fixed,

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V ar(t) = V ar

1

S

Xi

(i )vit!

=

=1

S2V ar

Xi

(i )vit!

=2vS

+1

S2

Xi

Xj6=i

(i )(j ) Cov(vit, vjt)

=2vS

+

=2vP

i(i )2+

where = S2P

i

Pj6=i(i )(j ) Cov(vit, vjt). Hence

V ar(FM) =2mT

+2v

T S+

T(30)

The estimated variance of FM is defined as

Var(FM) = 1T2FM (31)

where 2FM =1

T1

Pt(t FM)2.

To compute E[Var(FM)] rewrite t and FM using (26)

t FM = Rmt Rm + t (32)

= ut u + 1S

Xi

(i )(vit vi). (33)

Note that E[2FM] = 1T1Pt E[(t FM)2] = 1T1Pt V ar(t FM),

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since E[t

FM] = 0. Hence,

V ar(t FM) = E

(ut u)2

+1

S2

Xi

(i )2E

(vit vi)2

+1

S2

Xi

Xi6=j

(i )(j ) E[(vit vi)(vjt vj)]

=

1 1

T

2m +

1 1

T

1

S2

Xi

(i )22v

+

1 1

T

1

S2Xi Xi6=j(i )(j ) Cov(vit, vjt)

=T 1

T

2m +

1

S2v + .

Therefore,

E[2FM] =1

T 1T 1

T

2m +

1

S2v + T =

2

m +1

S2v +

and

E[Var(FM)] =2mT

+2v

T S+

T. (34)

Thus, the sample variance of the FM estimator is an unbiased estimator of

its true variance.

Now turn to the variance of the modified FM statistic, which is given by

MFM =1

T

Xt

t =1

T

Xt

(t Rmt) (35)

Using (26), t is given by

t = + E[Rmt] + t (36)

Repeating the same steps used above, it follows trivially that the true

variance of MFM is given by

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V ar(MFM) =2v

T S+

T

and that the sample variance is unbiased, E[Var(MFM)] = V ar(MFM).

25


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29

Table 1. Simulation Results for the Fama-MacBeth and Modified FM tests.

Fama-MacBeth Modified FM Test

T 5% 10% 5% 10%

Panel A: Size of tests (m=0.0275)60 0.043 0.090 0.057 0.111

120 0.053 0.106 0.042 0.086

240 0.044 0.086 0.054 0.102

360 0.049 0.112 0.053 0.098

Panel B: Size of tests (m=0.055)

60 0.047 0.100 0.045 0.100

120 0.049 0.097 0.049 0.093

240 0.055 0.093 0.049 0.092

360 0.036 0.082 0.058 0.105

1000 0.051 0.091 0.044 0.097

Panel C: Power of tests (

m=0.0275)60 0.404 0.556 0.988 0.998

120 0.711 0.806 1.000 1.000

240 0.958 0.981 1.000 1.000

360 0.991 0.996 1.000 1.000

Panel D: Power of tests (m=0.055)

60 0.162 0.244 0.987 0.996

120 0.266 0.380 1.000 1.000

240 0.498 0.611 1.000 1.000

360 0.651 0.762 1.000 1.000

1000 0.971 0.987 1.000 1.000


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30

Table 2. Sample Properties of Simulated Statistics (m=0.055).SampleMean

SampleVariance

5thPercentile

95thPercentile Skewness

ExcessKurtosis

JB*Statistic

p-value(JB)

T=60

Fama-MacBeth (H0) 0.020 1.031 -1.647 1.681 -0.095 0.071 1.706 0.426

Modified FM (H0) -0.031 1.017 -1.718 1.644 -0.040 -0.109 0.763 0.683

Fama-MacBeth (HA) 0.995 1.031 -0.666 2.696 -0.053 0.021 0.492 0.782

Modified FM (HA) -4.377 1.216 -6.227 -2.630 -0.200 0.111 7.192 0.027

T=120

Fama-MacBeth (H0) -0.012 1.014 -1.661 1.625 0.015 0.084 0.328 0.849

Modified FM (H0) -0.040 1.010 -1.614 1.650 0.109 -0.050 2.067 0.356

Fama-MacBeth (HA) 1.356 1.028 -0.276 2.987 0.052 0.100 0.865 0.649

Modified FM (HA) -6.155 1.195 -7.925 -4.357 -0.099 -0.003 1.625 0.444

T=240

Fama-MacBeth (H0) 0.016 0.984 -1.669 1.527 -0.163 -0.067 4.621 0.099

Modified FM (H0) -0.055 0.973 -1.644 1.566 -0.033 0.084 0.474 0.789

Fama-MacBeth (HA) 1.938 0.990 0.224 3.472 -0.144 -0.086 3.741 0.154

Modified FM (HA) -8.679 1.135 -10.519 -6.998 -0.163 0.185 5.859 0.053

T=360

Fama-MacBeth (H0) -0.027 0.938 -1.554 1.583 0.146 0.412 10.639 0.005

Modified FM (H0) -0.013 1.032 -1.686 1.640 -0.005 0.357 5.305 0.070

Fama-MacBeth (HA) 2.334 0.938 0.821 3.916 0.159 0.437 12.169 0.002

Modified FM (HA) -10.557 1.166 -12.383 -8.781 -0.031 0.175 1.437 0.487

T=1000

Fama-MacBeth (H0) -0.070 0.968 -1.607 1.513 0.040 0.048 0.361 0.835Modified FM (H0) 0.018 0.993 -1.595 1.684 0.026 -0.116 0.673 0.714

Fama-MacBeth (HA) 3.859 0.973 2.339 5.478 0.043 0.047 0.407 0.816

Modified FM (HA) -17.534 1.190 -19.338 -15.776 -0.076 -0.131 1.670 0.434


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31

Table 3. Simulation Results for the FM and Modified FM tests.

Fama-MacBeth Modified FM Test

T 5% 10% 5% 10%

Panel A: Rejection rate (Kan&Zhou, CAPM) (

m=0.0275)60 0.072 0.140 0.042 0.090

120 0.093 0.173 0.041 0.087

240 0.164 0.256 0.048 0.096

360 0.243 0.365 0.050 0.096

Panel B: Rejection rate (Kan&Zhou, CAPM) (m=0.055)

60 0.073 0.137 0.058 0.105

120 0.108 0.166 0.049 0.096

240 0.167 0.256 0.051 0.109

360 0.183 0.287 0.038 0.092

1000 0.497 0.616 0.056 0.103

Panel C: Rejection rate (Jag&Wang, Non-CAPM) (m=0.0275)

60 0.134 0.220 0.593 0.726

120 0.223 0.335 0.871 0.925

240 0.409 0.552 0.992 0.998

360 0.566 0.681 1.000 1.000

Panel D: : Rejection rate (Jag&Wang, Non-CAPM) (m=0.055)

60 0.094 0.154 0.580 0.700

120 0.102 0.165 0.876 0.933

240 0.141 0.221 0.995 0.998

360 0.202 0.306 1.000 1.000

1000 0.459 0.610 1.000 1.000


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Table 4. Estimation Results for the FM and Modified FM using US market data

FM test Modified FM test

T Sample St. Sample St.

(months) Mean Error Mean Error

Start (%) (%) t-statistic p-value (%) (%) t-statistic p-value

Jan-73 120 0.553 0.535 1.035 0.303 0.527 0.298 1.769 0.08

240 0.497 0.351 1.415 0.158 0.13 0.193 0.671 0.503

336 0.583 0.305 1.915 0.056 0.058 0.183 0.319 0.75

Jan-78 120 0.577 0.451 1.278 0.204 0.026 0.235 0.112 0.911

240 0.512 0.297 1.728 0.085 -0.234 0.168 -1.392 0.165

Jan-83 120 0.440 0.457 0.964 0.337 -0.267 0.243 -1.099 0.274

216 0.600 0.370 1.621 0.107 -0.202 0.230 -0.875 0.383

Jan-88 120 0.448 0.387 1.158 0.249 -0.494 0.239 -2.072 0.04

156 0.771 0.449 1.718 0.088 -0.09 0.294 -0.306 0.76

testing the capm

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