Specification error, random coefficient and the risk-return relationship test in capital asset pricing

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Faculty Working Papers

G>llege of Commerce and Business Administration

Univvrsity of Illinois at Urbana - Cho mpoig n

FACULTY WORKING PAPERS
College of Connnerce and Business Adniinistration
University of Illinois at Urbana-Champaign
May 9, 1980

SPECIFICATION ERROR, RANDOM COEFFICIENT AND THE
RISK-RETURN RELATIONSHIP TEST IN CAPITAL ASSET
PRICING

Cheng F. Lee, Professor, Department of Finance

Frank J. Fabozzi, Hofstra University

Jack Clark Francis, II, Baruch College, CUNY

#676

Summary

Both specification error and random coefficient concepts are used to show
why the traditional method of testing the risk-return relationship in capital
asset pricing may not be appropriate. Empirical studies of capital asset pricing
are reviewed in detail to show how the random coefficient behavior can occur in
the estimated parameters of the equation used to test the positive theory of
capital asset pricing.

SPECIFICATION ERROR, RANDOM COEFFICIENT AND THE RISK-RETURN
RELATIONSHIP TEST IN CAPITAL ASSET PRICING

I. INTRODUCTION

The theory of equilibrium in the capital markets developed inde-
pendently by Sharpe (1964), Lintner (1965) and Treynor (1961) provides
a risk-return relationship for assets and portfolios. This relation-
ship, shown as equation (1), is called the security market line (SML)
by Sharpe and the capital asset pricing model (CAPM) by others.

(1) E(R^) = E(R^) + B^[E(R^)-E(R^)]

where E(R, ) denotes the expected rate of return from the jth market

asset, E(R.) represents the expected value of the risk-free interest

rate, B. is called the beta systematic risk coefficient, E(R ) is the
2 ' m

expected return from the market, and

[E(R ) - E(R-)] is the theoretical market risk premium.
m f

The risk-retvim relationship has been used to evaluate portfolio
performance, test for market efficiency, determine the required equity
return for regulated industries and approximate the hurdle rate for
capital investment projects. Because of the pivotal role of the
relationship in financial theory, researchers have empirically tested
the model to determine if the estimated parameters are equal to the
theoretical values. Although the results of researchers have consis-
tently rejected the hypothesis that the estimated and theoretical
values are equal, the findings do suggest that systematic risk is a
meaningful measure of risk and stocks with high systematic risk yield

-2-

correspondingly higji rates of return.

This paper presents the empirical results of a different test of
the validity of the risk return relationship. Rather than estimate
the model using a fixed regression coefficient model (as is suggested
by theory since the market risk premivm is theoretically constant for
all securities) , a stochastic parameter regression model is tested.
With this latter model the market risk premium can be tested to
determine if it changes randomly from security to security.

The next section provides theoretical reasons to justify the use
of a stochastic parameter regression model to describe the risk-return
relationship. Section III formulates the specific stochastic parameter
model employed. The data and resvilts are described in the fourth section,
followed by the conclusions in section five.

II. JUSTIFICATION FOR EMPLOYING THE STOCHASTIC PARAMETER REGRESSION MODEL
The empirical analogue of equation (1) is shown as equation (2) .

(2) Rj = ^0 "^ ^l^j ^ ®j

where

R. = the arithmetic mean return for security j

3

B. = the estimated systematic risk for security j, and
e. = the stochastic error for security j

Equation (2) is estimated using cross-sectional data with the beta
coefficient estimated from a first-pass regression based on a time

-3-

series of historical returns. If equation (1) is a true description
of the risk-return relationshp, then X-. and X, should equal the arith-
metic mean return for the risk-free asset (R-) and arithmetic average

excess of the market return (R ) over the risk- free rate (that is, the

m

market risk premium) . Moreover, the slope should be constant for all

securities.

In a survey article about stochastic parameter regressions,

3arr Rosenberg (1973, p. 381) states:

The stochastic parameter problem arises when para-
meter variation includes a component which is a
realization of some stochastic process in addition
to whatever component is related to observable
variables. Thus, stochastic parameter regression
is a generalization of ordinary regression. Ideally,
a model would be so well defined that no stochastic
parameter variation would be present, and no general-
ization would be needed J, but the world is less than
ideal.

If the risk-return relationship is well specified, we would not observe

that its slope or market risk premium would vary stochastically.

However, there exist both theoretical economic and econometric reasons

to suspect the risk-return relationship is, in fact, misspecified.

First, Arditti (1967), Kraus and Litzenberger (1976) and others

have published theoretical and empirical work showing that the equil-

ibrixmi return of an asset is influenced by both the second and third

statistical moments of its return distribution. These findings extend

the two parameter model to a third parameter, namely, skewness. Since

the risk-return relationship ignores the impact of skewness, or skew-

2
ness related factors, it suffers from omitted variables.

Second, other studies have suggested the possibility of additional

emitted variables. Sharpe (1977), for example, has given the risk-

-4-

retum relationship a "multi-beta" interpretation. Similarly, Ross
(1976, 1977) uses an arbitrage approach to derive a multi-factor
risk-return relationship. Brennan (1970) has analyzed the impact of
the tax effect due to the different treatment of dividend income and
capital gains. He derived a multi- index model including average excess
dividend yield as an additional explanatory variable in the risk-return
relationship. Bachrach and Calai (1979) have s'hovm that the price of
the stock should be included in the risk-return relationship while
Lanstein and Sharpe (1978) and Joehnk and Petty (1980) have shown that
duration or interest rate risk should also be considered.

Statistically, the multi- index risk-return model can be specified
by eqtiation (3) .

(3) Rj = ^0 + hh ■" ^2^^2j + • • • + Vnj ■" 'i

A, A

where X_ , . . ., X are estimates of omitted factors discussed above,
X_, X , X„ , ... X are cross-section regression parameters, and t. is
the stochastic error for security j. It should be noted that equation (3)
is a generalized case of equation (2) .

If we use the specification method specified by Theil (1971, pp.
548-549) it can be shown that

A A I A A

(4) X- = X, + b„X„ + . . . + b X

1 1 z z n n

3
where b„, b_, . . . b are so-called auxiliary regression coefficients.

In addition we also know that

5A) ^0 " ^j " ^l^j

-5-

and

A| _^ A| AAA A ~

5B) ^0 = Rj - ^i3j - X2X2 - . . . - W

n _ — n ^

where R. = E R./n, e. = E B./n
J j=l ^ ^ j=l ^

If all auxilliary regression coefficients are zero (i.e., all the

A A

omitted variables are independent of 3.), then X. is an unbiased estimate

A f A A f

for X- . However, X^ is no longer an unbiased estimate for X^.

A

Therefore, X^ cannot be used to test the null hypothesis that Xq is
equal to R^ if the multi-index model is appropriate. If equation (3)
does hold and all auxilliary regression coefficients are approximately

A A

equal to zero, X- may well still be an unbiased estimator of X- . However,

A

X^ becomes a random instead of a fixed variable.

Third, Roll (1977) and others have suggested beta estimates obtained
by regressing returns from common stocks on stock market average returns
are a form of partial equilibrium analysis which ignores investment in
other capital assets. They suggest a general equilibrium analysis
which includes other assets (such as, investments in human capital,
commodities, real estate, etc.) should be used to obtain a risk-return
tradeoff. If the more general equilibrium analysis suggested by Roll
produces a risk-return relationship which departs significantly from
the usual partial equilibrium analysis, then all previous empirical
estimates of the market risk premium are conceptuallly flawed and may
explain why the model may exhibit the characteristics of a stochastic
parameter regression model.

Fourth, Levy (1978) and Hessel (1978) have demonstrated that
imperfect capital markets will modify the risk-return relationship

-6-

which is predicated on the assiimption of perfectly competitive markets.
Levy (1978), for example, has developed a generalized risk-return
relationship when (i) market participants differ in their investment
strategies and do not adhere to the same risky portfolio given by
their market portfolio and (ii) do not hold many risky assets in their
portfolio. Levy concludes that the true risk index is somewhere between
the total variance of the security and the systematic risk implied by
capital market theory.

Finally, numerous studies have documented that the explanatory
variable in the risk-return relationship, the beta coefficient, is subject
to estimation error. The beta coefficient is estimated in the first-
pass regression. However, in the first pass regression, the true market
model may be a multi-index model rather than a single index model. As
indicated in the discussion of the second pass regression above, the

estimated beta of the market model will then exhibit the characteristics

4
of a stochastic parameter regression model.

III. TEST FORMULATION

Previous research employed the classical OLS fixed-coefficient
approach to estimate equation C2) . The purpose here is to determine
if a random coefficient relationship between return and systematic risk
exists. That is, does the proportionality constant, X , which represents
the market risk premium fluctuate randomly from one security to the next?

There are several stochastic parameter regression models suggested in the
literature. The random coefficient model formulated by Thiel (1971) is
used in this investigation. The fixed coefficient model given by equation
(2) can be converted to the random coefficient model (RCM) shown by
equation (6) .

-7-

(6) R. = X_ +X,B. + w.
J 0 1 J J

where

and X, is the mean of X, . . Moreover, it is assumed that the distribution
1 Ij

of X, . is homoscedastistic and the e. values are uncorrelated with the
Ij J

X, . values.

To test whether the RCM is a description of the risk-return relation-
ship, two statistics must be estimated. First X of equation (6), and
second, the variance of the distribution of X , around its mean X ,
var (X ) , must be estimated. If no statistically significant variance

for X, . around X, is found, then the RCM can be rejected and the

Ij 1

traditional fixed-coefficient model accepted.

Theil (1971, p. 623) has shown that the OLS estimator of X
in equation (6) is unbiased but will result in an inefficient estimator
for the variance of the estimate of X , var C'*^-.^)* The procedure suggested
by Theil to estimate X and var(X^ .) is described briefly below.

First, the ordinary least squares residuals, denoted by e., must
be calculated from equation (2). Second, equation (7) must be estimated
using OLS.

(7) e? = m_P. + m,Q. + f .

where

J
P. = 1 -',

* 2

and

f. = stochastic error term.
3

The coefficients m and dl. are to be estimated. They represent
the variance of the error term in eqiiation (6) and var (^]^^)»
respectively. The statistical significance of vl. (as measured by its
t-statistic) then determines whether the RCM is appropriate. However,
because of the heteroscedasticity in equation (7), Theil suggests that

Q

eqiiation (6) be estimated using generalized least squares (GLS) . The

GLS estimate for X in equation (Jo) is defined in equation (8).

(8) X^ =

No

te that if m^ is not statistically different from zero, equation C8)

reduces to the OLS estimate for X .

IV. EMPIRICAL RESULTS

The securities used to estimate the risk-return relationship are
the common stock of 694 New York Stock Exchange companies. For each
stock, B. was estimated from the single- index market model, equation
(9), using monthly non-compounded price change plus dividend returns
for the 72 month period from Janiiary, 1966 to December, 1971.

C9) R. = a. + g.R + u.^

jt J ""j mt jt

-9-

where

R, = return on stock i in month t
Jt

R ^ = market return in month t
mt

u.^ = stochastic error term in month t for stock i, and,
jt j» »

a. and 3. are the parameters to be estimated.

The S&P 500 index with dividends included was used for the market index.

The time period was also partitioned into two non-overlapping
36 month periods — January, 1966 to December, 1968 and January, 1969
to December, 1971. Equation (9) was estimated for both time periods.
The market risk premivim was positive for the first sample period and
negative for the second sample time period.

Estimates of the fixed coefficient OLS model equation C2) for each
of the three time periods are presented in Table 1. The theoretical
values for X and X are also shown in Table 1. For each time period,
the signs of the estimated parameters were the same as theory suggests.
And, each parameter was significantly different from zero at the 1% level
of significance. Other researchers \Ao estimated the risk-return
relationship found that the estimated values for the parameters were
significantly different from the theoretical values. Table 1 suggests
that for each of the three time periods, X^ was significantly different
from the theoretical value of the market risk premium. For the two 36

month time periods, X was not statistically different from the theoretical

9
value, R-.

-10-

The results for X and m. [= var (A ,)] for the RCM are summarized
in Table 1. For the 72 month period, the variance of X was positive
and significantly different from zero at the 5% level of significance.
This was also found for the 36 month period January, 1966 to December,
1968 in which the market risk premium was positive. Hence, for the two
periods in which the market risk premiian was positive, the SML was
found to exhibit the property of a RCM. Howevier, when the market risk
premium was negative, namely, from January, 1968 to December, 1971,
the variance of X . was not statistically significant.

It is also interesting to note the degree of randomness of the
market risk premiimi for the two cases in which m was statistically

/a a

significant. The coefficient of variation,^m /X , for the 72 month
and 36 month time periods were 1.57 and .97, respectively. This indicates
considerable random movement in relation to X , If a 95% confidence
interval was constructed for the movements around X based onx/m^ , the
interval would include the theoretical value for the market risk premium.

V. CONCLUSIONS

The risk-return relationship of capital market theory is not simply
a model accepted by some academicians. Regulators have used the model
to estimate the appropriate return on equity for regulated firms.
Corporate management has been encouraged to use the model to evaluate
the performance of in-house or independent pension portfolio managers.
The performance of an entire industry has been questioned based on
empirical results which have used the theoretical model. We must,
therefore, continue to evaluate the model both theoretically and
empirically.

-11-

In this paper, we disclose a disturbing empirical result of
the risk- return relationship. Employing a stochastic parameter
regression model, we find that the market risk premium varies randomly
from one security to the next. Moreover, the observed randomness
was substantial. General plausible explanations for such results
were suggested. Even if the reader rejects any or all of these
argunfints, it is difficult to refute the empirical findings. It is
worthwhile to note that the empirical results of this paper have
indirectly supported Roll and Ross's (1979) empirical results of
testing the Arbirage pricing theory. The direct relationship
between the results of this study and Roll and Ross's results will
be developed in the future research.

-12-

FOOTNOTES

This is essentially the conclusion reached by Modigliani and
Pogue (1974) in their review of the empirical tests of the risk-return
relationship. Research subsequent to that reviewed by Modigliani and
Pogue has not altered that conclusion.

rrevious research on skewness and the risk-return relationship
is summarized in footnote 2 of Kraus and Litzenberger (1976).

A more precise definition can be found in Theil (1971, p. 549).

This was found true for a substantial number of stocks
by Fabozzi and Francis (1978) using the stochastic parameter regression
model described in the next section. In such cases, the total risk
can be partitioned as follows:

X i

where

2
0. = variance for the returns for stock i

2

a = variance for the market return
m

2
a = unsystematic risk for stock i

^i
and

2
Ot, = variance for the systematic risk of stock i
o.

1

In such a case, equation CI) is then

R. = Xg + X^dl + aln/2 .

"2
Hence, equation (2) is misspecified in that a is not considered.

Note also that if the traditional procedure for computing unsystematic

risk is employed but the market model is a RCM, then the unsystematic

2 2
risk would improperly include a a . This might explain why some

i ™
researchers have found a positive relationship between average returns

and unsystematic risk.

-13-

See. Rosenberg (1973) for a description of various stochastic
parameter variation models.

This inefficiency results from the fact that w. in equation
(6) is heteroscedastic [see Theil (1971, pp. 623)]. -^This may help explain
why Mller and Scholes (1972) found heteroscedasticity when they
estimated equation (2) .

7 ^ —

In the equations below B. and R, represent deviations of

each variable from their respective means. The summation is over all

observations.

g
Theil (1971) has shown that to estimate the variance-covariance

matrix for generalized least squares in this case, the following weights

should be used:

Z^ = 1/2 (^j + ^Q.)~^

where Z. = the weight for the jth observation and m- and m^ are the
ordinaiT' least sqxiares estimates of nu and m^ for eqxiation (7),

9
This result was scirfiwhat surprising in light of the analytical

results derived in Section II. It was shown there that A-, will be a

biased estimate of R^ if the multi-index model is appropriate.

A one-tail test is used since the alternative hypothesis is
that the variance of X^ . is positive.

Roll and Ross have found that there exist at least three and
probably four "priced" factors in addition to market factor in the
generating process of return.

-14-

REFERENCES

Arditti, F. D. (1967), "Risk and the Required Rate of Return,"
Journal of Finance (March), pp. 19-36.

Bachrach B. and Galai, D. (1969), "The Risk-Return Relationship and
Stock Prices," Joxirnal of Financial and Qtiantitative Analysis
(June), pp. 421-442.

Brennan, M. J. (1970), "Taxes, Market Valuation and Corporate Financial
Policy," National Tax Journal, pp. 417-427.

Fabozzi, F. J. and Francis, J. G. (1978), "Beta as a Random Coefficient,"
Journal of Financial and Quantitative Analysis (March) , pp. 101-115.

Hessel, C, A. (1978), "The Effects of Taxes and Imperfect Competition on
an Investor's Optimum Portfolio," unpublished doctoral dissertation.
New York University.

Joehnk, M. D. and Petty, J. W. (1980), "The Interest Sensitivity

of Common Stock Prices," Journal of Portfolio Management (Winter),
pp. 19-25.

Kraus, A. and Litzenberger, R. H. (1976), "Skewness Preference and the
Valuation of Risk Assets," Journal of Finance (September), pp.
1084-1100.

Lanstein, R. and Sharpe, W. F. (1978), "Duration and Security Risk,"
Journal of Financial and Quantitative Analysis (November), pp.
653-668.

Levy, H. (1978), "Equilibrium in an Imperfect Market: A Constraint on
the Number of Securities in the Portfolio," American Economic
Review (September), pp. 643-658.

Lintner, J. (1965), "The Valuation of Risk Assets and the Selection
of Risky Investments in Stock Portfolios and Capital Budgets,"
The Review of Economics and Statistics 47 (February), pp. 13-37.

Miller, M. M. and Scholes, M. (1972), "Rates of Returns in Relation
to Risk: A Re- examination of Recent Findings," published in
Studies in the Theory of Capital Markets, edited by Michael
Jensen Praeger, pp. 47-78.

Modigliani, F. and Rogue, G. A. (1974), "An Introduction to Risk
and Return," Financial Analysts Journal (March-April) , pp.
68-80; (May-June), pp. 69-86.

Roll, R. (1977), "A Critique of the Asset Pricing Theory's Tests;
Part I: On Past and Potential Testability of the Theory,"
Journal of Financial Economics, (March), pp. 129-176.

-15-

Roll, R. and S. A. Ross (1979), "An Empirical Investigation of The
Arbitrage Pricing Theory," Working paper 15-79, Gradiiate School
of Management, UCLA.

Rosenberg, B. (1973), "A Survey of Stochastic Regression Parameters,"
Annals of Economic and Social Measurement, Vol. 2, pp. 381-397.

Ross, S. A. (1976), "Arbitrage Theory of Capital Asset Pricing,"
Journal of Economic Theory, 8, pp. 341-360.

Ross, S. A. (1977), "Return, Risk and Arbitrage," In I. Friend and
J. L. Bicksler, Risk and Return in Finance, Vol. 1 (edited),
Ballinger Publishing Company.

Sharpe, W. F. (1964), "Capital Asset Prices: A Theory of Market
Equilibrium Under Conditions of Risk," Journal of Finance, •
(September), pp. 425-442.

Sharpe, W. F. (1977), "The Capital Asset Pricing Model: A Multi-Beta
Interpretation," in Levy, H. and M. Samat, eds., Financial
Decision Making Under Uncertainty, (New York: Academic Press,
1977).

Theil, H. (1971), Principles of Econometrics, John Wiley & Sons.

Treynor, J. L. (1961), "Toward A Theory of Market Value of Risky
Assets," unpublished paper.

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