cai_measurement error in the capm (2)

8/9/2019 Cai_Measurement Error in the CAPM (2)

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University of Constance

Department of Economics

Seminar: Empirical Finance

Lecturer: Prof. Dr. Winfried Pohlmeier

Author: Miaoyang CaiConstance, 12-06-2009

Seminar

Empirical Finance

Measurement error in the CAPM


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Table of contents

1 Introduction......................................................................................................................1

2 Empirical Tests of Sharpe-Lintner and Black CAPM......................................................2

3 Seemingly Unrelated Regression Model for CAPM....................................................7

4 Measurement Error Caused by the Market Proxy Problem..............9

5 Consequences for Performance Measures....................................................................18

6 Conclusion.......................................................................................................................19


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IntroductionThe Capital Asset Pricing Model provides the equilibrium relationship between

risk and expected returns. It was first developed by William Sharpe (1964) and John

Lintner (1965). After that, considerable attention has been focused on its application

efficiency in the field of finance. The early empirical tests during the 1970s conducted

by Black, Jensen and Schloes (1972); Fama and MacBeth (1974) support the CAPM

while recent tests show less favorable evidence against the CAPM by Banz (1981);

French and Fama (1992).

One of the representative studies to support the CAPM was conducted by Black,

Jensen and Scholes (1972). In their paper, they showed that although the data do not

support Sharpe-Lintner CAPM, they do support the Black version which relaxes the

assumption that riskless borrowing and lending is not allowed. Another reason that the

evidence deviated from the Sharpe-Lintner model may be caused by the measurement

error in beta, in this case they tried to use the portfolio grouping procedure to reduce

the problem. The gains are uncertain because the results are still significantly different

from their theoretical value.

However, in recent studies, more evidences show that the CAPM may succeed

only theoretically and not practically. Since French and Fama (1992) claimed in their

studies that beta is insignificant while another factor size is significant with or without

betas. The debate is then about whether beta is dead. Another paper was published by

them in 1996 called Multifactor Explanations of Asset Pricing Abnormalities where

they introduce another two factors to better describe the relationship between the

excess portfolio return and the excess market return.

There are still econometric problems with Black, Jensen, and Scholes (1972)s

empirical work and data choosing as well as sample period effect problems in the

French and Fama (1992) study. For example, in Black, Jensen and Scholes (1972) s

research, they used OLS. But in section , we will see that when the assumption that

the error terms are not heteroscedastic and correlated is violated, the OLS estimator

will not be consistent. Another issue is that they did not consider the market proxy

problem which will also cause beta to be biased. Thus reducing the measurement error


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of beta only by the portfolio grouping procedure may not be enough.

In this seminar paper, I focus on the CAPM and present some additional tests to

avoid the problems as discussed above. In the next section, I introduce the empirical

work published by Black, Jensen and Scholes (1972). Their study focuses on finding

the reasons to support the CAPM. In section , a seemingly unrelated regression

model will be applied which has two aims, one is to avoid the errors in variables

problem in the second-pass regression caused by error terms correlation. The other is

to test if the coefficient alpha jointly equals to zero. We can then conclude if the

CAPM is standard. In section , we consider another source of the measurement error

problem, which is the market proxy problem. We know that the market portfolio is

defined as including all the risky assets, but in reality we could only find the market

proxy for this instead. This will cause the error in variables problem in the first

time-series regression stage and beta will thus be devaluated. Since beta is very

sensitive to the choice of market proxy if the true market portfolio is efficient as

claimed by Roll and Ross (1994), we must be very careful in this case. We also

discuss how the different indices will have different measurement error effects.

Section discusses the consequences of measurement error for performance

measures, Treynor and Jensens index will be incorrect if we do not take the errors in

variables problem into account.

Empirical Tests of Sharpe-Lintner and Black CAPMAfter the CAPM was developed, considerable research has focused on model

testing. Earlier research has found the evidence to support the CAPM conducted by

Black, Jensen, and Scholes (1972). They test the CAPM directly including two stages

called first-pass regression and second-pass regression. In their first-pass regression,

they use the time-series data to estimate the coefficients and . can be used

to test if there are abnormal returns or put in other words, if the market is efficient.

comes in the second stage to be the regressor. In the following part, more details about

these two stages will be introduced and applied to Sharpe-Lintner and Black CAPM


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respectively.

A. Sharpe-Lintner Version: First-pass Regression

In Sharpe-Lintner CAPM, the expected risk premium on each asset is

proportional to its systematic risk. That is:

)()( mii RERE =

where iR is the monthly excess return on each security and mR is the monthly

excess return on the market portfolio. The empirical test for this model conducted by

Black, Jensen and Scholes (1972) used the monthly return data file, which contains all

the stocks listed on the New York Stock Exchange from January, 1926 to March 1966.

In their first-pass regression, the ranking procedure was introduced by regressing

the first five years monthly excess return of each security on the monthly excess

return of the market portfolio to obtain the estimates i . The regression equation is,

mtiiit RR += (1)

Ten portfolios were then grouped based on the i with 10% stocks having the

highest and so on. Then for each portfolio, they calculated the monthly return for

the next 12 months. By doing this process repeatedly, they got 35 years monthly

excess return of monthly excess returns for each portfolio. Regressing them on the

monthly excess market return, they obtained the estimates p and p for each

portfolio by OLS. The regression equation is as follows,

mtRppptR += (2)

p is expected to be zero by the assumption in Sharpe-Lintner CAPM. So, test if the

model is standard CAPM we could test p . The results show that is negative

when 1 > and positive when 1


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nonstationarity and the lack of more complete aggregation. This conclusion was also

confirmed by doing the sub-periods test, in which always changes, making the

estimated value of return different from the predicted value.

Lets look back to the ranking procedure again which was supposed to reduce the

measurement errors of p : p estimated by this procedure will be independent from

the measurement error in those i . Earlier studies grouped the portfolio based on i .

If the portfolios are constructed based on the ranking of each securitys i , then the

portfolio with high tends to have the securities which have positive measurement

errors in their . Once the portfolios were estimated in this way, we find that k is

positively biased and k is negatively biased. Thus they concluded that by using the

ranking procedure, unbiased estimates of k and k will be obtained. And this kind

of beta will eliminate the errors in variables problem in the second-pass regress.

However, they failed to consider the another source may also cause the biased k and

k which is the market proxy problem.

B. Sharpe-Lintner Version: Second-pass Regression

Another important result for CAPM is concerning the security market line which

describes the linear relationship between the expected return of portfolios and the

systemic coefficient beta. It takes the form of

)()( mpp RERE = (3)

So in this second-pass regression stage, they test the linear relationship predicted

by the CAPM by comparing the estimated coefficient of equation 4 with the

theoretically value so as to test the Sharpe-Lintner CAPM.

ppR 10 += (4)

where pR is the average monthly excess return of each portfolio and p is the

result from the first stage. The result is 0.003590 = , 0.01081 = . If CAPM is the


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true model, the theoretical values should be 00 = and 0.01421 = which is the

average market excess return. The results were significantly different from the

theoretical value with 52.6)( 0 =t and 53.6)( 1 =t . However, here the t-statistic

maybe overstates the significance as claimed by Black, Jensen and Scholes.

In this stage, Black, Jensen and Scholes proved that the grouping procedure

could be used to reduce the measurement error of , then 0 and 1 will be the

inconsistent estimates. However, this method can not eliminate the whole problem,

besides, this problem can also be caused by the market proxy problem.

In summary, the results in both parts state that the Sharpe-Lintner model is

violated. But for the second part, we should be cautious about the result that the two

coefficients are significantly different from their theoretical value because they maybe

overstate the difference. However, the CAPM is considered to be too flat in the two

stages. This can be seen in Figure 1 where the line in black indicate the theoretical

SML and the red line is the regression line estimated by Black, Jensen and Scholes

(1972).

C. Black Version: First-pass Regression

The Sharpe-Lintner CAPM allows the borrowing and lending with risk free rate.

Blacks version (1972) relaxes this assumption. The asset pricing model becomes to

))()(()()(zmizirErErErE += (5)

where )( irE is the expected monthly return of each portfolio or individual security

and )( zrE denotes the expected monthly return of the zero-beta portfolio which has

zero covariance with the market portfolio. Subtracting the risk free rate from both

sides and taking the form of a regression equation, we get

mtiiit RR += (6)

where we use a capital letter to indicate the excess return. Notice that equation 6 has

the same form as equation 1, thus the same time-series estimation process can be


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applied and the same results are obtained as in Sharpe-Lintner CAMP. However in

this equation, equals to )1( izR where zR equals to fz rr . The theoretical

value of should no longer be zero. Recall that in part A, the conclusion that the

portfolios with high beta tend to have negative alpha and the portfolios with low beta

tend to have positive alpha makes the data inconsistent with CAPM. But now Blacks

version of the CAPM gives a way to solve this problem that when zR positive, then

high portfolio tends to have negative alpha, and low beta portfolio tends to have

positive alpha. The opposite result comes when zR is negative. The studies show

that in general zR is positive.

D. Black Version: Second-pass Regression

In the second-pass regression stage, the equation they estimated is the same as

equation 4. By applying the exactly same procedure from the Sharpe-Lintner CAPM

to the Black CAPM, the same estimates 0 and 1 can be obtained.

According to the Black version of CAPM, 0 equals to zR and 1 equals

to zm RR . Thus the estimation result in part B can not be explained by the

Sharpe-Lintner CAPM but by the Black CAPM. Here the theoretical values of

0 and 1 are not zero and the average return of market portfolio. When 0 is

positive, 1 will be smaller than mR , and when 0 is negative, 1 will be larger

than mR . Remember that in part B the intercept is 0.00359 which is positive, and the

slope is 0.0108 which is smaller then the average market return is 0.0142 can thus be

explained by Black CAPM. We should also note that, zR is generally positive.

In conclusion, although the zero-beta portfolio returns is hardly to be observed,

empirical results show that it is generally positive, and tends to be larger in recent

periods. In this case, the Sharpe-Lintner CAPMs problem can be solved. The data are

consistent with the Black CAPM.


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Seemingly Unrelated Regression Model for CAPMOne of the econometric problems concerned with the Black, Jensen and Scholes

(1972) empirical test is the error terms are assumed to be uncorrelated. So they

estimate portfolios individually with OLS. Actually in time t, the excess return of

portfolio i must give some information of the excess return of at least some

portfolio j . In this case, OLS will lose some efficiency. That means we should use

other methods to estimate the equations jointly. The variance-covariance matrix is

further evidence to show that the error terms do not satisfy the assumption of

0),cov( =ji and )var( i is constant. In this part we consider the seemingly

unrelated regression model developed by Arnold Zellner (1962) which gains

efficiency in estimation by combining information from different equations.

Srivastava and Giles point out in their book Seemingly Unrelated Regression

Equations Models that the model could generally be used in two ways. First, it can

test whether the data support or contradict CAPM by testing the hypothesis that

equals to zero. Second, correlated error terms will produce parameters with

measurement error of beta and then cause the error in variables problem in the

cross-section regression stage.

However, the CAPM is a special case of the seemingly unrelated regression

model, when each equation in the system has the same regressor. The property is that

applying the Full Information Maximum Likelihood or OLS will obtain the same

parameters. So we can not gain efficiency from the new method of FIML. But that

does not mean there is no need to use this model. The reason we still want to use it is

that it allows us to test the coefficient hypotheses or put more exactly, test the CAPM.

In the following part, we will consider two data files: Fama-French data and Campbell

data.

1. Fama-French data

In French and Famas study, they constructed 25 portfolios by breaking up all

NYSE, AMEX and Nasdaq securities twice into five groups based on size and


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book-to-market equity. Lets use French-Fama 25 portfolios to see whether there is

the evidence to support the Sharpe-Lintner CAPM by using full information

maximum likelihood method. The unconstrained model is as follows:

ptmtRppptR ++= (7)

The results are collected in appendix, table 1.

Now we adopt the Wald statistics to test the null hypothesis: 0H : 0= . and

the alternative hypothesis, 1H : 0 . If we can not reject the null hypothesis, then

we could say that the CAPM is correct in predicting the time-series behavior of

returns.

The Wald test statistic is estimated by using the unconstrained model. The

Chi-squared value is 31.35847 with 25 degrees of freedom. Since the p value equals

0.1774, this indicts that the Wald statistic is insignificant so we cannot reject the

Hypothesis.

The results in appendix, table 2 provide the evidence support the Sharpe-Lintner

CAPM. In summary, alpha is insignificantly different from zero. But we notice that if

we are using the OLS method, some alpha are significantly different from zero, this

means, by using the SUR estimation, we could eliminate the measurement error

caused by the error term correlation.

2. Campbell data

Daniel and Titman (1997) claimed that testing the CAPM by using

characteristics-sorted portfolios could be dangerous, for example, the portfolios based

on size and value like in French-Fama 25 portfolios. The consequence will be that the

model may spuriously explain the average returns to these kinds of portfolios. That

means, it will cause the problem in the second-pass regression stage.

To avoid this problem, Campbell and Vuolteenaho (2004) constructed 20

portfolios based on past risk loadings with VAR state variables.Using the same

procedure as in French-Fama data, they also showed that in Wald test, the results

present evidence also consistent with the Sharpe-Lintner CAPM, with the p value


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equals to 0.273. So we can not reject the hypothesis that alpha are jointly equal to zero.

But when we look at the z statistics for each alpha, we find that four of the total 20

alpha are significantly different from zero at the 5% significance level. The estimation

output is presented in Appendix, table 3 and 4.

Measurement Error Caused by the Market Proxy Problem

Another source of producing betas measurement error in the second-pass

regression is the market proxy problem.

A How Do we Measure the Market Portfolio

The market portfolio is a portfolio including a weighted sum of every asset in the

market. In other words, one should find the market portfolio on the

minimum-variance frontier. The basic idea of CAPM is that an investor should

diversify their investment only in the market portfolio. But the problem in practice is

that researchers can rarely find the true market portfolio. Instead they use a market

proxy, such as indices in testing and applying the CAPM. Roll (1977) claimed that

using an index, especially those major share market indices, as the benchmark will

give rise to the errors in variables problem. So the will largely depend on which

index we chose. He even argues that the market portfolio, which can only be proxied,

will never be tested and analyses will therefore lead to dubious results.

B How can we solve the errors in variables problem

Let us consider again to the Sharpe-Lintner CAPM again. In section , the

results show that CAPM is violated. Figure 1 puts this result more clearly, we see that

in the second-pass regression the predicted returns of the portfolio with high are

too high and the predicted returns of the portfolio with low are too low. This is

consistent with the first-pass regression stage where that the portfolio with 1> has

a lower return than predicted, and the portfolio with 1


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predicted. One way to solve this problem as said above is using the Black CAPM. The

other reason is the market proxy problem, but as I discussed above, Black, Jensen and

Scholes did not take the market proxy problem into consideration which will also

cause the measurement error in .

Figure 1 (here excessr indicates the average excess return of the portfolios, and B

indicates estimated in the first-pass regression stage)

--------------- The line predicted by CAPM (intercept=0, slope=0.0142)

----------------The regression line (intercept=0.00359, slope=0.0108)

----------------The line we hope to be moderated by VE model

take the estimation result directly from Black, Jenson and Scholes (1972) to obtain the

line predicted by CAPM and the regression line.

I now present and formulate the errors in variables mode: Let *m

R represent the

true and unobservable monthly excess return of the market portfolio. In reality we

could only approximate *mR by using mR which equals vRm +* with v

representing the measurement error. So the true model is

.008

.010

.012

.014

.016

.018

.020

.022

0.4 0.6 0.8 1.0 1.2 1.4 1.6

B

EXCESSR

EXCESSR vs. B


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itmtRiiitR ++=* (9)

And the model which was estimated by Black, Jensen, Scholes is

itmtRiiitR ++=

(10)

In equation 10, OLS will be inefficient due to the correlation betweenm

R and mv .

(In the Classic VE model, we only consider the correlation caused bym

R and mv . We

also assume that 0),cov(*

=mmt vR . Because if 0),cov( * mmt vR

and 0),cov( =mmt vR , the estimation of the coefficient is still non-biased and

consistent) So mR and mii v + is correlated. In this case, the coefficient we

estimate will be biased and inconsistent. Substitute vRR mm += * into (9), we get,

mtviitmtRiiitR ++=(11)

As a result of the inconsistency,

10

)var(

),cov(lim

22

2

22

2

*

*

*

>+= mm RRp (12b)

So from the VE Model, we can see that if there are measurement errors in

variables, we estimated is higher than its true value, and the we estimated is

smaller than its true value. If in the first-pass regression is measured with error, in

the second-pass regression the problem of errors in variables will arise which

causes0

and1

to be inconsistent and biased. The estimates will again have the

following properties, |plim 1 |< |1

| and plim 0 >0

. So, we conclude that the

estimation does not support Sharpe-Lintner CAPM probably because of the existence

of an errors in variables problem. This conclusion is based on the 00359.00 =


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which should be smaller and 1 =0.0108 which should be larger. From this point of

view, we can see that in figure 1, the new expected regression line which is in color

green should be closed to the predicted SML by the CAPM.

In the above, we got a general impression concerning how the coefficient will be

affected by this problem. But we do not know the exact form of this measurement

error. Here we will consider different data base.

1) Black, Jenson, Schloes Data:

We take directly the estimate results from Jenson et als first-pass regression

estimation output. Since we dont have the real data of securities and market monthly

return, we estimate by using the following equations

pRp m )1(~ = (13a)

pp =~

(13b)

mR

p

p )1(~

~

=(13c)

Note that22

2

*

*

vR

R

m

m

+

= , so it should be larger than 0 and smaller than one. We get the

average =0.968183. Dividing p~

by , we can reestimate the relation between

average monthly excess return of portfolios and their beta, and we get now the

intercept=0.0036, the slope=0.0104.The slope is not larger as we predict probably

because here is very close to 1. And the market proxy problem could be neglected

here.

2) French-Fama 25 portfolio data againAlthough in section , we found the data are not consistent with

Sharpe-Lintner CAPM, all the statistical tests showed alpha not to be equal to zero.

One reason is that the errors in variables problem or the market proxy problem will

make 0 , resulting in biased estimates of . In this part, we use these data again,


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this time taking the market proxy problem into consideration. Before we start, another

issue arises that in that time using a SUR model to test if the coefficient alpha equals

to zero or not is not supported by Eviews. We ignore of this issue, and put more

emphasis on whether or not there is measurement error in market portfolio.

Now we use the SUR model taking into account the errors in variables problem.

The equation is

ptmtRppmRptR ++= )1(

where pRp m )1(~ = and pp =~

. According to the Sharpe-Lintner

CAPM,we assume here that the true alpha should be zero.

Unlike the above procedure, in which we calculate equation by equation, this

time we will estimate the equations as the whole system with SUR model.

The results are in the appendix. We obtain the constant which is 0.979248

with t-statistic 26.06371 and compare with that of the estimation without considering

errors in variables, and we find that the true beta is higher.

In summary, the French-Fama data shows that the market proxy problem does

exist in the CAPM test. It will make we underestimate beta and overestimate the

intercept by pR m )1( which could explain why our estimate of the intercept

is not equal to 0. So the data in the first stage may support the CAPM assumption-but

to what extent is not clear.

But this result can not explain the CAPM in the second-pass regression stage, see

the Figure below.

Figure 2


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-0.4

0.0

0.4

0.8

1.2

1.6

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

SURECAPM1B

EXRETURE

EXRETURE vs. SURECAPM1B

One might say that the portfolios constructed based on size and Book to market can

not be used in testing CAPM. In addition, French and Fama already proved that only

their three factors model can describe the data well Another case is the empirical test

conducted by Lakonishok, Shleifer and Vishny (1994) which show that if the data are

not grouped based on beta, CAPM loses its value. They generated the portfolios

formed by price rations. In their estimation results, there is no linear relationship

between return and risk as indicated by CAPM.

In Summary, the French-Fama 25 portfolio data is not suitable for testing CAPM

in the second-pass regression stage. However, the French-Fama three factor model

which can better describe the data is considered to be an alternative asset pricing

model applied in practice. But the market proxy problem exists in this context which

will result in a downward biased beta. The market portfolio they use is quite good

which results in a close to one. So the Wald Test show that the hypothesis that

alpha is jointly equal to zero can not be rejected.

3) Campbell data

We use the same procedure but Campbell data to Test the Sharpe-Lintner CAPM


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with market proxy problem. Campbell constructed 20 portfolios based on the risk.

By using Campbell data, =0.696012 with t-value 12.3905. That means, there

exist the market proxy problem. And beta is much larger than that we estimate by

OLS or Full information maximum likelihood.

However in the second stage, 0 is still not smaller and 1 is not larger than

we expected. 1 changes from 0.002549 to 0.00179.

One explanation for that we can not get better result in the second stage although

we consider the market proxy problem is the measurement error in beta itself. Recall

the procedure from Black, Jenson and Scholes portfolio constructing procedure. They

prove that they can virtually eliminate the measurement error problem of beta.

In comparing the in the Fama-French and Campbell data estimation. We find

that the Fama-French is very close to one while in Campbell data is relatively

smaller. One explanation is that the market portfolio is well constructed by

Fama-French. The consequence of measurement error is that it may cause alpha

significantly different from zero. That is what we see in the last section in Campbell

data test, that some statistic value of alpha is larger than the critique value.

C Debate on the best market index

Since the problem is caused by market the portfolio proxy, we could make the

market portfolio includes all risky investments. This seems impossible and trivial, for

example risky investments not only include stocks, bonds but also human capital,

private businesses, if you want to invest abroad, this market portfolio should even

includes the risky asset all over the world. In this part, we discuss the previous

arguments and empirical results concerning of the market proxy problem. And then

we select some of the indices for calculating . The aim is to find the difference effect

level of different indices on beta.

In Black, Jensen and Scholess empirical studies, they constructed the market

portfolio by including all the New York Stock Exchange securities with equal

investment weight. During 35 years of there estimation period, the securities raise


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from 582 to 1094. And now there are more than 2000 stocks listed. Recall the result

from last part=0.968183, which is very close to 1, indicating that this market proxy

performs as market portfolio very well. This way of constructing market portfolio is

also confirmed by Merrill, Lynch, Prerce, Fenner & Smith, they also developed

market return based on the value-weighted transactions of all stocks listed on the New

York and American Stock Exchanges. Most tests use a value-weight return or

equal-weight return of NYSE and AMEX stocks as the market portfolio.

Other empirical results show that although the market portfolio mentioned above

can not include all the risky assets, there is no significant difference when introduce

other assets such as bonds, preferred stocks or international assets.

Now we use two indices to be the market proxy: Dow Jones Industrial Average

and S&P 500 index. If the market proxy problem matters, then they will provide

different results. Before we start, we should notice that S&P Index is compositely

calculated by 500 securities while DJIA is calculated by only 30 securities. We could

expect that estimated based on DJI will be smaller than on S&P 500.

(1) Dow Jones Industrial Average

We construct the market portfolio using Dow Jones Industrial index as the

market proxy. Since this index only includes 30 stocks of the market, we expect that

this is a bad proxy of the true market portfolio. By using the same procedure in

French-Fama data estimation, but replace the market portfolio return by DJIA returns.

We can have the new estimation results.

As we expected before, =0.418725 is small and with z-statistic equal to

10.15403 indicating there is the market measurement error which makes beta much

smaller than the true value. We also test the coefficient of the unconstrained

Sharpe-Lintner model by sue but without market proxy problem. And now we see the

chi-square is 62.40711 with p value equal to 0 that we could reject the hypothesis and

conclude that alpha is significantly different from zero probably because of the market

proxy problem.

Like above, we replace Campbell market portfolio return by Dow Jones

industrial average again. =0.338246, with t-statistics =8.589288 indicating again


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that DJIA is a bad benchmark for market portfolio. And our convulsion that this

problem is large that will cause alpha significantly different from zero when testing

the unconstrained model is confirmed 41.82908 Chi-square 0.0029

(2) S&P 500 Index

Since the estimation procedure is exactly the same as in the last part. Here we

just show the result. Note that the estimation period is smaller which may have

finite-sample problem.

In French-Fama data, is now 0.519696 and significant.

I analyze the results by constructing the following table to put all the outputs

together:

French-Fama Cambell

Wald Test a=0 Wald Test a=0

F-F MP =0.979248 C MP =0.696012

Wald Test 0 Wald Test 0

DJIA =0.418725 DowJ =0.338246

Wald Test 0 Wald Test -

S&P 500 =0.519696 S&P 500 -

It shows that the low follows that are jointly significant different from 0.

This does not mean that we can reject the CAPM because not equal to 0 could be

caused by the market proxy problem. For the Campbell data itself, one explanation

could be that although is smaller than in French-Fama data, but the a is still not

jointly significant different from 0 is the basic finding that the market proxy is good

when it has the correlation with the true market portfolio larger than 70%.

D. The Best Market Index

But the problem still exists. So we try to find the best market index. In some

empirical articles, stock market indices are used as the market proxy such as, DAX,

S&P, and so on. While others may include all available assets, for example, real estate,

preferred stocks, international assets as long as the international market is open.

However, none of them can fully reflect the true market portfolio. Empirical results


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show that test based on the CAPM is largely dependent on which market proxy is

chosen. Stambaugh (1986) states that common stocks are the most important factor,

and thus other assets included in the market proxy will not have more effects.

Consequence for Performance Measures

If the portfolio manager has obtained a high return using his portfolio which also

has high beta, we can not say that he performs well in the market. So we need the

performance measures not only concerning return but also those concerning risk. In

practice, there are three overall indices which are used to rank portfolio performance.

Although they take different forms, the basic idea comes from the CAPM. These three

well-known indices are the Sharpe, the Treynor and the Jensen Index. In the following

part, we discuss them individually. We also want to investigate the consequences of

measurement error in these measures.

Sharpes Index is also called the Sharpe Ratio. According to the Sharpe-Lintner

CAPM, investors should choose their portfolio on the linear relationship between risk

free rate and efficient market portfolio and this line will have the highest Sharpe ratio.

It takes the form that,

p

pp

RES

)(

=

where )( pRE is the excess return of the portfolio and p is the portfolios

variability. Since we can use the historic data, there will be no measurement error

problem here. So Sharpe index will not be influenced.

Treynors index is given by,

p

pp

RET

)(=

where p is defined as systematic risk and estimated by using a time series

regression of pR on mR as we discussed before. In the previous section, we have

proved that p will be devaluated due to the market proxy problem. This factor will

of course then affect the Treynors index. We could say that in general Treynors


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index will be overvalued when the beta is not true.

The Third Index is Jensens Index, which is in the regression form that

mtiiit RJR +=

Notice that it has the same form with equation 1, just use J denoted as Jensens index

to replace . Thus we again use the same method regressing the time-series data to

get estimated J which also means the abnormal return on the portfolio. Since here

exist also market proxy problem, we can just use the conclusion as we got in the

section 3 that we should be very careful to consider the market proxy we use, because

it has two effects. The first one is, it will make high beta portfolios tend to have

negative alpha, and low beta portfolios tend to have positive alpha. The portfolio

manager will gain even it takes the passive strategy which does not mean he is talent

in picking the securities in the portfolio. Another effect is the bad the market proxy,

the large impact will be on alpha. Thus if the manager choose DJIA as the market

portfolio benchmark, he will get more on the J or say, abnormal return even in

practice he can not perform well against the market.

Roll (1977) claimed that if the market portfolio is correctly measured, then there

is a linear relationship between the mean return on portfolio and portfolio beta. But

this can also see in the second-pass regression of French-Fama data and Campbell

data that there is no linear relationship. That means there are also problems with

Treynor and Jensens indices.

ConclusionThe Sharpe-Lintner CAPM is not supported by the data in Black, Jensen and

Schloes empirical work. However this could be solved by relaxing the assumption

that investors could borrow and lend as risk free rate which extended to the Black

version CAPM. They also give another reason that the measurement error will make

beta to be biased. They try to reduce this problem by the portfolio grouping procedure.

But they do not consider other factors may cause this measurement error. First, the

error terms correlation; second, the market proxy problem. We solve the first situation


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by using the seemingly unrelated regression model. For the second problem we

consider the errors in variables problem. The results show that there is the market

proxy problem that will affect beta. And when we want to measure the CAPM to the

portfolio performance, we should consider this market proxy problem into account,

otherwise we will make the mistake.

References

Berndt, E. R., 1996, The Practice of Econometrics: Classic and Contemporary.

Addison-Wesley Publishing Company, Inc.

Black, F., M. C. Jensen and M. Scholes, 1972, The Capital asset pricing model:

Some empirical tests. Studies in the theory of capital markets, ed. Michael Jensen,

pp.79-121. New York: Praeger

Brooks, C., 2002, Introductory Econometrics for Finance. Cambridge University

Press

Campbell, J., A. Lo, and A. C. MacKinlay, 1997, The Econometrics of Financial

Markets (2nd ed.), Princeton University Press, Princeton, New Jersey.

Campbell, J., A. Lo, and T. Vuolteenaho, 2004, Appendix to Bad Beta, Good

Beta Data Construction, Harvard University, Cambrideg, MA.

Cuthbertson, K., 1996, Quantitative Financial Economics, John Wiley & Sons.

1996, Multifactor Explanations of Asset Pricing Anomalities. Journal of Finance,

51, 55-84


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Fama, E. F., and K. R. French, 1926-2006, Data Library

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research

Greene, W.H., 2000,Econometric Analysis (4th ed), Prentice Hall International, Inc.

Jagannathan, R., E. R. McGrattan, 1995, The CAPM Debate.Federal Reserve Bank

of Minneapolis Quarterly Revies(fall), Vol. 19, No. 4, pp.2-17

Perold, A. F., 2004, The Capital Asset Pricing Model. Journal of Economic

Perspecties (summer), V.18, pp. 3-24

Wooldridge, J. M., 2000, Introductory Econometrics: a Modern Approach

South-Western College Publishing.

Yahoo Finance historical prices of Dow Jones Industrial Average, 1926-2006

http://finance.yahoo.com/q/hp?s=%5EDJI&a=09&b=1&c=1928&d=05&e=12&f=200

9&g=m

Yahoo Finance historical prices of S&P 500 Index, 1950-2006

http://finance.yahoo.com/q/hp?s=%5EGSPC&a=00&b=3&c=1950&d=05&e=12&f=

2009&g=m


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Appendix

Table 1: French-Fama data estimated by SUR model

System: SURE1

Estimation Method: Full Information Maximum Likelihood (Marquardt)

Date: 06/06/09 Time: 14:40

Sample: 1926M07 2006M12

Included observations: 966

Total system (balanced) observations 24150

Convergence achieved after 1 iteration

Coefficient Std. Error z-Statistic Prob.

C(1) -0.598880 0.785081 -0.762826 0.4456

C(2) 1.650113 0.155732 10.59584 0.0000

C(3) -0.125455 0.722029 -0.173753 0.8621

C(4) 1.477394 0.141632 10.43121 0.0000

C(5) 0.151754 0.403319 0.376264 0.7067

C(6) 1.393240 0.088737 15.70082 0.0000

C(7) 0.373888 0.412900 0.905516 0.3652

C(8) 1.308024 0.089829 14.56129 0.0000

C(9) 0.536824 0.488839 1.098161 0.2721

C(10) 1.388564 0.103336 13.43741 0.0000

C(11) -0.230520 0.305214 -0.755272 0.4501

C(12) 1.248069 0.072650 17.17929 0.0000

C(13) 0.115845 0.318313 0.363933 0.7159

C(14) 1.275309 0.068711 18.56044 0.0000

C(15) 0.281363 0.286552 0.981891 0.3262

C(16) 1.186671 0.058508 20.28236 0.0000C(17) 0.323633 0.279073 1.159671 0.2462

C(18) 1.228104 0.058068 21.14956 0.0000

C(19) 0.350176 0.363710 0.962791 0.3357

C(20) 1.362205 0.070992 19.18805 0.0000

C(21) -0.159954 0.213415 -0.749498 0.4536

C(22) 1.285962 0.056819 22.63253 0.0000

C(23) 0.168376 0.143466 1.173636 0.2405

C(24) 1.129005 0.031274 36.10047 0.0000

C(25) 0.232158 0.170620 1.360676 0.1736

C(26) 1.146734 0.037418 30.64646 0.0000

C(27) 0.274982 0.219848 1.250781 0.2110


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C(28) 1.126950 0.044000 25.61229 0.0000

C(29) 0.226480 0.295366 0.766776 0.4432

C(30) 1.391099 0.062920 22.10913 0.0000

C(31) -0.012875 0.144963 -0.088819 0.9292

C(32) 1.072316 0.036568 29.32398 0.0000

C(33) 0.028365 0.128066 0.221489 0.8247

C(34) 1.096038 0.024811 44.17563 0.0000

C(35) 0.176738 0.165205 1.069812 0.2847

C(36) 1.086521 0.033602 32.33495 0.0000

C(37) 0.210524 0.217374 0.968489 0.3328

C(38) 1.172552 0.043454 26.98352 0.0000

C(39) 0.165658 0.335181 0.494235 0.6211

C(40) 1.447156 0.062529 23.14374 0.0000

C(41) -0.028141 0.086649 -0.324768 0.7454

C(42) 0.970317 0.025120 38.62678 0.0000

C(43) 0.017660 0.092041 0.191866 0.8478

C(44) 0.923702 0.026996 34.21661 0.0000

C(45) 0.066924 0.152381 0.439190 0.6605

C(46) 0.977763 0.035820 27.29683 0.0000

C(47) 0.010922 0.253775 0.043037 0.9657

C(48) 1.133369 0.047810 23.70576 0.0000

C(49) -1.083148 2.430722 -0.445608 0.6559

C(50) 1.240235 0.494646 2.507321 0.0122

Table 2: Wald Test of French-Fama data

Wald Test:

System: SURE1

Test Statistic Value df Probability

Chi-square 31.35847 25 0.1774

Table 3: Campbell data estimated by SUR model

System: SURE1


Date: 06/11/09 Time: 16:54

Sample: 1929M01 2001M12


Total system (balanced) observations 17520Convergence achieved after 1 iteration


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C(1) 0.002644 0.001261 2.097588 0.0359

C(2) 0.791256 0.025929 30.51663 0.0000

C(3) 0.003218 0.001339 2.402561 0.0163

C(4) 0.988421 0.021426 46.13087 0.0000

C(5) 0.002249 0.001722 1.306098 0.1915

C(6) 1.138937 0.027759 41.02900 0.0000

C(7) 0.001597 0.001871 0.853792 0.3932

C(8) 1.352962 0.029829 45.35744 0.0000

C(9) 0.001120 0.002411 0.464710 0.6421

C(10) 1.655137 0.034409 48.10214 0.0000

C(11) 0.002067 0.000952 2.171490 0.0299

C(12) 0.700533 0.021853 32.05627 0.0000C(13) 0.001804 0.001014 1.779261 0.0752

C(14) 0.864388 0.019374 44.61524 0.0000

C(15) 0.001555 0.000980 1.586815 0.1126

C(16) 1.088686 0.017328 62.82911 0.0000

C(17) 0.000932 0.001392 0.669416 0.5032

C(18) 1.250096 0.022441 55.70665 0.0000

C(19) 0.000729 0.001972 0.369501 0.7118

C(20) 1.511079 0.035995 41.98004 0.0000

C(21) 0.001639 0.001084 1.512035 0.1305

C(22) 0.785063 0.024146 32.51338 0.0000

C(23) 0.002495 0.001103 2.261709 0.0237

C(24) 0.937561 0.017619 53.21455 0.0000

C(25) 0.001830 0.001271 1.440148 0.1498

C(26) 1.129695 0.018869 59.87025 0.0000

C(27) 0.001696 0.001552 1.092639 0.2746

C(28) 1.323008 0.022719 58.23373 0.0000

C(29) 0.000269 0.001967 0.136555 0.8914

C(30) 1.588368 0.034290 46.32164 0.0000

C(31) 0.002082 0.001093 1.905279 0.0567C(32) 0.720417 0.023194 31.06058 0.0000

C(33) 0.002697 0.001141 2.363984 0.0181

C(34) 0.883124 0.019816 44.56623 0.0000

C(35) 0.002128 0.001113 1.912357 0.0558

C(36) 1.028426 0.020964 49.05760 0.0000

C(37) 0.001948 0.001480 1.316288 0.1881

C(38) 1.227303 0.021359 57.46009 0.0000

C(39) 0.001024 0.002006 0.510173 0.6099

C(40) 1.509057 0.034527 43.70714 0.0000


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Table 4: Wald Test of Campbell data

Wald Test:

System: SURE1


Chi-square 23.32695 20 0.2730

Table 5: The processed results of modified

obs p~ p~

pR

p~

1 -0.000829 1.5614 0.0213 1.038842 1.612712

2 -0.001938 1.3838 0.0177 1.109418 1.429275

3 -0.000649 1.2483 0.0171 1.038005 1.289322

4 -0.000167 1.1625 0.0163 1.010220 1.200703

5 -0.000543 1.0572 0.0145 1.037528 1.091942

6 0.000593 0.9229 0.0137 0.956710 0.953229

7 0.000462 0.8531 0.0126 0.963263 0.881135

8 0.000812 0.7534 0.0115 0.929454 0.778159

9 0.001968 0.6291 0.0109 0.819470 0.649774

10 0.002012 0.4992 0.0091 0.778917 0.515605

Market - 1.0000 0.0142 0.968183

Source: Black et.al (1972)

Table 6: The new regression in Sharpe-Lintner model

Dependent Variable: EXCESSRETURN

Method: Least Squares

Date: 06/12/09 Time: 11:53

Sample: 1 10


Variable Coefficient Std. Error t-Statistic Prob.


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MODIFIEDB 0.010439 0.000488 21.41023 0.0000

C 0.003611 0.000532 6.783605 0.0001

R-squared 0.982847 Mean dependent var 0.014470

Adjusted R-squared 0.980703 S.D. dependent var 0.003679

S.E. of regression 0.000511 Akaike info criterion -12.14310

Sum squared resid 2.09E-06 Schwarz criterion -12.08258

Log likelihood 62.71550 F-statistic 458.3981

Durbin-Watson stat 2.529683 Prob(F-statistic) 0.000000

Table 7: in French-Fama data

System: SURECAPM


Date: 06/06/09 Time: 14:37Sample: 1926M07 2006M12





C(1) 0.979248 0.037569 26.06552 0.0000

C(11) 1.675973 0.155246 10.79557 0.0000

C(12) 1.511404 0.148651 10.16744 0.0000C(13) 1.430703 0.097206 14.71827 0.0000

C(14) 1.349117 0.100470 13.42802 0.0000

C(15) 1.435975 0.113207 12.68455 0.0000

C(21) 1.271257 0.083350 15.25211 0.0000

C(22) 1.308047 0.080436 16.26202 0.0000

C(23) 1.221490 0.074036 16.49857 0.0000

C(24) 1.265000 0.073794 17.14220 0.0000

C(25) 1.403532 0.086752 16.17875 0.0000

C(31) 1.311069 0.072985 17.96359 0.0000

C(32) 1.158123 0.054051 21.42648 0.0000

C(33) 1.178102 0.057329 20.54991 0.0000

C(34) 1.159568 0.063590 18.23518 0.0000

C(35) 1.429136 0.080917 17.66178 0.0000

C(41) 1.094232 0.053189 20.57239 0.0000

C(42) 1.120668 0.047862 23.41473 0.0000

C(43) 1.114726 0.054554 20.43340 0.0000

C(44) 1.204074 0.064835 18.57133 0.0000

C(45) 1.484889 0.085307 17.40646 0.0000

C(51) 0.988964 0.042452 23.29606 0.0000C(52) 0.942768 0.045655 20.64996 0.0000


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C(53) 0.999837 0.053444 18.70805 0.0000

C(54) 1.158767 0.064524 17.95860 0.0000

C(55) 1.245579 0.504865 2.467155 0.0136

Table 8: in Campbell data

System: SURECAPM


Date: 06/10/09 Time: 15:17

Sample: 1929M01 2001M12





C(1) 0.696012 0.056173 12.39050 0.0000

C(11) 1.140114 0.096613 11.80078 0.0000

C(12) 1.423963 0.117534 12.11538 0.0000

C(13) 1.636655 0.138390 11.82641 0.0000

C(14) 1.941157 0.164626 11.79133 0.0000

C(15) 2.372352 0.202578 11.71079 0.0000

C(21) 1.008624 0.082458 12.23204 0.0000

C(22) 1.242414 0.103496 12.00446 0.0000

C(23) 1.562774 0.130678 11.95901 0.0000

C(24) 1.792055 0.153688 11.66036 0.0000

C(25) 2.165051 0.190507 11.36467 0.0000

C(31) 1.128376 0.092159 12.24385 0.0000

C(32) 1.349113 0.110705 12.18652 0.0000

C(33) 1.622238 0.133771 12.12694 0.0000

C(34) 1.898589 0.159390 11.91157 0.0000

C(35) 2.274367 0.196467 11.57632 0.0000

C(41) 1.037111 0.087713 11.82394 0.0000

C(42) 1.271768 0.107304 11.85199 0.0000

C(43) 1.478126 0.123714 11.94797 0.0000C(44) 1.762294 0.148906 11.83491 0.0000

C(45) 2.162985 0.188921 11.44918 0.0000

Table 9: Wald Test of French-Fama data (DJIA)

Wald Test:

System: SUREDOW1


Chi-square 62.40711 25 0.0000


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Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

Table 10: in French-Fama data (DJIA)

System: SURECAPMDOW


Date: 06/11/09 Time: 17:41

Sample: 1926M07 2006M12





C(1) 0.418725 0.041237 10.15403 0.0000

C(11) 3.417870 0.486042 7.032046 0.0000

C(12) 3.155389 0.454940 6.935828 0.0000

C(13) 3.090690 0.365263 8.461551 0.0000

C(14) 2.873677 0.360538 7.970534 0.0000

C(15) 3.041613 0.388954 7.819984 0.0000

C(21) 2.653412 0.324214 8.184126 0.0000

C(22) 2.789535 0.321625 8.673251 0.0000

C(23) 2.625461 0.300802 8.728200 0.0000

C(24) 2.746471 0.307641 8.927522 0.0000

C(25) 3.024920 0.348806 8.672208 0.0000

C(31) 2.765235 0.316772 8.729414 0.0000

C(32) 2.554114 0.265540 9.618577 0.0000

C(33) 2.616517 0.272709 9.594525 0.0000

C(34) 2.565757 0.269933 9.505166 0.0000

C(35) 3.161451 0.346561 9.122343 0.0000

C(41) 2.370228 0.258111 9.182979 0.0000C(42) 2.507129 0.253049 9.907682 0.0000

C(43) 2.517515 0.261200 9.638257 0.0000

C(44) 2.690012 0.285109 9.435036 0.0000

C(45) 3.311669 0.356241 9.296145 0.0000

C(51) 2.256325 0.226548 9.959578 0.0000

C(52) 2.174151 0.221174 9.830026 0.0000

C(53) 2.282138 0.235882 9.674933 0.0000

C(54) 2.651440 0.274543 9.657637 0.0000

C(55) 2.834257 1.110563 2.552089 0.0107


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Table 11: Wald Test of Campbell data (DJIA)

Wald Test:

System: SUREDOW1


Chi-square 152.8704 20 0.0000

Null Hypothesis Summary:

Normalized Restriction (= 0) Value Std. Err.

Talbe 12: in Campbell data (DJIA)

System: SURECAPMDOW


Date: 06/11/09 Time: 17:50

Sample: 1929M01 2001M12



Convergence achieved after 4 iterations


C(1) 0.338246 0.039380 8.589288 0.0000

C(11) 2.319598 0.281682 8.234819 0.0000

C(12) 2.902186 0.341023 8.510231 0.0000

C(13) 3.323078 0.388760 8.547889 0.0000

C(14) 3.918287 0.463353 8.456369 0.0000

C(15) 4.726415 0.565432 8.358939 0.0000

C(21) 1.987118 0.237777 8.357050 0.0000

C(22) 2.474510 0.296965 8.332667 0.0000C(23) 3.131232 0.373151 8.391325 0.0000

C(24) 3.562384 0.424458 8.392789 0.0000

C(25) 4.251036 0.516308 8.233531 0.0000

C(31) 2.258266 0.272955 8.273394 0.0000

C(32) 2.727767 0.321212 8.492113 0.0000

C(33) 3.283982 0.387524 8.474260 0.0000

C(34) 3.813288 0.450161 8.470951 0.0000

C(35) 4.493394 0.546318 8.224865 0.0000

C(41) 2.073689 0.250859 8.266353 0.0000

C(42) 2.540617 0.300715 8.448597 0.0000

C(43) 2.954507 0.352111 8.390842 0.0000


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C(44) 3.507770 0.416398 8.424076 0.0000

C(45) 4.251597 0.512108 8.302149 0.0000

Table 13: Wald Test of French-Fama data (S&P 500)

Wald Test:

System: SURESP1


Chi-square 103.3223 25 0.0000

Table 14: in French-Fama data (S&P 500)

System: SURECAPMSP


Date: 06/11/09 Time: 21:07

Sample: 1950M02 2006M12



Convergence achieved after 31 iterations


C(1) 0.519696 0.033164 15.67073 0.0000

C(11) 2.423347 0.245602 9.866969 0.0000

C(12) 2.134250 0.208066 10.25759 0.0000

C(13) 1.867062 0.161675 11.54821 0.0000

C(14) 1.795766 0.152938 11.74178 0.0000

C(15) 1.877158 0.157672 11.90548 0.0000

C(21) 2.433569 0.207711 11.71611 0.0000

C(22) 2.071370 0.155699 13.30367 0.0000

C(23) 1.868942 0.142332 13.13086 0.0000

C(24) 1.836022 0.134636 13.63693 0.0000

C(25) 1.981230 0.153575 12.90072 0.0000

C(31) 2.337964 0.188575 12.39803 0.0000

C(32) 1.990215 0.145282 13.69899 0.0000

C(33) 1.836784 0.135397 13.56589 0.0000

C(34) 1.789406 0.134563 13.29789 0.0000

C(35) 1.927649 0.150896 12.77469 0.0000

C(41) 2.243559 0.169878 13.20691 0.0000

C(42) 1.995388 0.138059 14.45312 0.0000

C(43) 1.890138 0.140073 13.49393 0.0000

C(44) 1.810288 0.136696 13.24314 0.0000

C(45) 1.972137 0.151112 13.05084 0.0000


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C(51) 2.012280 0.130583 15.40992 0.0000

C(52) 1.882653 0.133441 14.10849 0.0000

C(53) 1.727854 0.131406 13.14897 0.0000

C(54) 1.679165 0.135589 12.38420 0.0000

C(55) 1.772848 0.153189 11.57291 0.0000

cai_measurement error in the capm (2)

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