Technology and Investment, 2013, 4, 54-66
Published Online February 2013 (http://www.SciRP .org/journal/ti)
Copyright © 2013 SciRes. TI
Cross-Sectional Estimation Biases in Risk Premia and Ze-
ro-Beta Excess Returns
Jianhua Yuan, Robert Savickas
Yuan is at Capital Market Research Division, Fannie Mae, Washington, DC, USA
Savickas is at Department of Finance, George Washington University, Washington, DC, USA
Email: jhyuan@gwu.edu and savickas@gwu.edu
Received 2012
ABSTRACT
This paper shows that the classic cross-sectional asset pricing tests tend to suffer from severe risk-premium estimation
errors because of small variation in betas. We explain how the conventional approach uses low criteria to validate an
asset -pricing model and suffers from the model-misspecification issue because of the complication associated with the
zero-beta excess return. We show that the resulting biases in estimates of risk premia and their standard errors are se-
vere enough to lead researchers into inferring incorrect implications about some asset-pricing theories being tested.
Further, we suggest that one simple method of mitigating these issues is to restrict the zero-beta excess returns to their
theoretical values in the cross-sectional regressions and to conduct the straightforward test of whether the estimated
ex-ante risk premia are consistent with the observed ex-post ones. The empirical testing results not only further affirm
the higher efficiency of the estimates produces by the suggested method, but also show, contrary to some prior evidence,
that the market factor is priced consistently.
Keywords: Cross-Sectional Regression; Consistent Estimator; Efficient Estimator; Risk Premium; Zer o-Beta Return;
Model Misspecification; Beta-Variation
1. Introduction
The two-pass cross-sectional regression (CSR)
methodology, which is used in the two classic studies of
CAPM---one by Black, Jensen, and Scholes [3, 1972]
(BJS) and the other by Fama and MacBeth [12,
1973](FM)---first estimates the factor loadings on the
given risk factors in time series regressions and then uses
the estimated loadings in the second-pass CSR to
estimate the risk premia of the factors. This intuitive
two-pass CSR method is easy to implement and has been
widely used in the empirical studies of linear beta-pricing
models. Despite the wide usage of the two-pass CSR
methodology in asset-pricing tests, quite a few issues
associated with this classic methodology have been
identified in the literature.
First, all the three underlying assumptions- -- normality,
conditional homoskedasticity, and stationarity---of this
two-pass methodology are somewhat disputable. The
distributions of most stock returns exhibit significantly
positive skewness and higher-tha n -normal kurtosis. Fama
[9, 1965] and Blattberg and Gonedes [5, 1974] document
the non-normality shown in security returns. The works
by Barone-Adesi and Talwar [1, 1983], Bollerslev, Engle,
and Wooldridge [6, 1988], and Schwert and Seguin [22,
1990] document the conditional heteroskedasticity of
stock returns. For the least worrisome stationarity
assumption, Blume [4, 1970] shows that it is not totally
inappropriate.
The second issue in the asset-pricing tests is the
cross-sectional dependence among asset returns.
Theoretically, this issue can be handled by a generalized
least-square (GLS) estimation but it would be impractical
to estimate a huge covariance matrix for a large number
of securities. Moreover, such estimated matrix may often
not be positive-definite and hence fail the purpose of the
GLS methodology. As a more practical approach, the
grouping procedure, which was employed in the two
classic studies of CAPM---one by BJS and the other by
FM, is used to form cross-sectional portfolios. Since the
cross-sectional dependence is allowed for grouped data
test, the linear beta-pricing models can be tested on
returns of the cross-sectional portfolios. As pointed out
by BJS, the grouping procedure also allows for any
nonstationarity.
The third issue associated with the two-pass CSR
method is the well-known error-in-variable (EIV)
problem. BJS show that the market risk premium
estimates will be biased because of the measurement
errors in the market beta estimates but argue that the
measurement errors can be ignored with large samples of
many time periods. FM propose to use lagged rolling
beta-estimates to generate a time series of risk premium
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
estimates and then take the mean as the final risk
premium estimate. Shanken [23, 1992] provides an
excellent discussion of this two-pass CSR methodology,
especially the EIV adjustment and the asymptotic
distribution analysis under the conditional homoskedastic
assumption.1
2
R
Kim [15, 1995] provides an EIV correction
met ho dology using maximum likelihood estimation.
The fourth issue of this classic CSR methodology is
related to the common practice of low acceptance criteria
for a beta-pricing model. The literature sometimes
emphasizes the high cross-sectional and the high
t-values of the risk premium estimates (i.e. the estimated
risk premium associated with a particular factor is
significantly different from zero). Kan and Zhang [14,
1999] conduct simulations in which true asset returns are
generated from a one-factor model but the factor in the
two-pass CSR tests is misspecified as a random variable
uncorrelated with the asset returns. They call such a
misspecified factor as a ``useless" factor and show that
the t-value of the ``useless" factor converges to a large
value in the cross-sectional regression and the probability
of a fairly high cross-sectional 2
R
is quite big. Even
though the case they provide is an extreme one and a
useless factor may be relatively easy to detect as
Jaga nnathan and Wang [13, 1998] have shown,
quasi-models whose factors are only weakly correlated
with the true factors will be harder to detect. Lewellen,
Nagel, and Shanken [16, 2008] (LNS) show that a
proposed model with factors only weakly correlated with
the true factors and uncorrelated with the errors is
capable of producing high cross-sectional
2
R
. As part
of their critique, LNS show that the magnitudes of the
zero-beta estimates and the estimated equity risk
premium in many papers are unreasonably high and low,
respectively. In attempting to improve empirical tests,
LNS have offered several prescriptions including GLS
estimation, expansion of the set of testing portfolios
beyond size-B/M portfolios, imposing related constraints
in time-series and/or cross-sectional regressions, and so
on. These suggested approaches may improve empirical
tests but somehow lose the intuition and the simplicity of
the original method.
This paper focuses on the fourth issue described above
and attributes this problem to the complication of the
zero-beta factor. Even though the classic CAPM
developed by Sharpe [24, 1964] and Lintner [17, 1965]
suggests that the excess return (in excess of risk-free rate)
of any asset should be proportional to its market risk
loading by the same multiplier (the market risk premium),
the early CAPM testing results (e.g. BJS, FM, etc.) show
that the estimated CSR intercept tends to be significantly
1Jagannathan and Wang [13, 1998] generalize Shanken's
asymptotic analysis to the case of conditional heteroske-
dastic returns.
positive and the market risk premium seems to be
significantly smaller than the ex post equity premium.
Possibly motivated by such empirical results, Black [2,
1972] extends the CAPM by including borrowing
restriction.2
2
R
Black's extended CAPM postulates that the
excess return of any asset will be linear in terms of its
market beta with the same intercept (the zero-beta excess
return) and the same linear coefficient (the market risk
premium). The extended Black's CAPM not only is a
theoretical extension of the original Sharpe-Lintner
CAPM but also seems to reconcile the early CAPM
testing results. However, the zero-beta excess returns,
which are due to the differences between the costs of
(risk-free) borrowing and the returns of risk-free lending,
are not observable and are somehow ambiguous. With
the complication of the zero-beta excess return, the
observed equity premium is no longer an ex-post
measure of the ex-ante risk premium and this leads to the
common practice of low criteria of model
acceptance---high CSR and high
t
-values of risk
premium estimates. A more serious problem associated
with the conventional cross-sectional regression method
is that the estimation errors will be greatly amplified by
the small variation among the market betas.3 To address
these issues, we advocate focusing on the theoretical
linear beta-pricing model that restricts the zero-beta
expected return to its theoretical value and testing
whether the estimated ex-ante risk premia are consistent
with the observed ex-post ones.4
In the theoretical linear beta-pricing model with a
risk-free rate, the risk premia are assumed to be
observable. In order to find supportive evidence for such
a pricing model, one needs to show that the null
hypothesis that the estimated (ex-ante) risk premia are
equal to the observed (ex-post) ones will fail to be
rejected. For a misspecified model, it is unlikely that the
estimated risk premium of the misspecified factor can
match its observed value.
Next, we discuss the
importance of this simplification.
5
2Brennan [7, 1971] also provides an excellent analysis on
the capital market equilibrium with divergent borrowing
and lending rates.
3Notice that the estimation error of the market risk pre-
miu m is
That is, the testing procedure
)
ˆ
(var),
ˆ
(cov
1
βεβ
.
4In our discussion of the two-pass CSR methodologies,
we focus on the ordinary least-square (OLS)
cross-sectional estimation. But the approach suggested in
this paper can be directly applied to the generalized
least-square (GLS) cross -sectional estimation, as well.
5Kan and Zhang [14, 1999] show that the risk premium
estimate of a useless factor converges to infinity with
probability one as
∞→T
. Suppose the factors are
stationary and ergodic, then the sample means of the
factors will converge to finite numbers and the probabil-
55
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
for the linear beta-pricing model with a risk-free rate will
not suffer from the model-misspecification problem.
It has also been proven that the two-pass CSR
methodology suffers from the EIV problem and the CSR
estimation errors are underestimated. Shanken [23, 1992]
shows that it is very important to make the standard error
correction in order to find supportive evidence for a
linear beta-pricing model with zero-beta excess return,
especially one with multiple factors. The estimated risk
premia that are significantly different from zero with the
uncorrected error estimates may turn into insignificant
with respect to the corrected- --and hence larger---error
estimates. However, the standard error correction will not
be needed if the linear-beta pricing model can be
validated with the uncorrected error estimates. That is, if
the null hypothesis that the estimated risk premia are
equal to the observed ones fails to be rejected with
respect to the uncorrected standard errors, it will also fail
to be rejected with respect to the corrected (larger) ones.
The more important reason that we should focus on the
theoretical linear beta-pricing model with risk-free rate is
that this method can be applied to obtain more efficient
estimates of risk premia. We show that the conventional
two-pass estimates from the unrestricted cross-sectional
regressions suffer from high estimation errors because of
the relatively small variation among betas. Although
there are sophisticated, more efficient econometric
methods available for evaluating linear beta-pricing
models, these methods are generally more complicated
and less robust than the two-pass CSR methodology6
One of the prescriptions suggested by Lewellen, Nagel,
and Shanken [16, 2008] to improve the efficiency of the
WICSR is to impose related constraints in the regressions.
According to Shanken [23, 1992], this means to force the
risk premium of a portfolio-return factor to be the
difference between the mean of the observed factor
returns and the average zero-beta excess returns. Hence,
for a pricing model with only portfolio-return factors, the
cross-sectional estimation reduces to estimating the mean
and in some situations, it is difficult to interpret statistical
inferences obtained from these methods. As an
alternative, the approach of restricted cross-sectional
regressions possesses all the advantages---intuitiveness,
simplicity, and robustness---of the classic two-pass
method while, at the same time, being capable of
producing much more efficient estimates than the
conventio na l method is. It is also straightforward to
interpret the testing results obtained from the method we
advocate.
ity that the risk premium estimate of a useless factor
equals to its ex post risk premium will converge to zero
as
∞→T
.
6Cochrane [8, 2001] makes an excellent discussion about
tradeoffs between the methods for estimating and eva-
luating asset-pricing models.
of the zero-beta excess returns. With this restriction, the
risk premium of any portfolio-return factor is not directly
estimated and the standard error associated with the risk
premium of the portfolio-return factor is not estimated.
Because of this, it becomes difficult to statistically
interpret the estimation results with respect to the
portfolio -return factor, especially when the tested model
has multiple factors. When focusing on the theoretical
linear beta-pricing model with (market equivalent)
risk-free rate, we are able to restrict cross-sectional
regressions by imposing the zero-intercept constraint,
thereby obtaining estimates that are very straightforward
to interpret.
In the rest of the paper, we proceed as follows. In
Section 2, we first describe both the traditional method
and the method we advocate. Then we show that both
methods are
T
-consistent and derive the asymptotic
distributions of the corresponding estimates under the
assumptions of conditional heteroskedasticity and/or
homoskedasticity. In the last subsection of Section 2, we
derive the cross-sectional asymptotic properties and
show that the advocated approach is more efficient than
the traditional method under the assumption of
sufficiently weak cross-sectional dependence. Section 3
presents empirical evidence of the higher efficiency of
the methodology we describe. In Subsection 3.1, we
provide simulation evidence to illustrate that
risk-premium estimates resulting from the restricted
cross-sectional regressions are more efficient than the
conventional estimates. Subsection 3.2 shows that the
warrants for significant zero-beta excess returns are not
as strong as believed. In Subsection 3.3, we reexamine
the Fama-French three-factor model and provide new
cross-sectional supportive evidence. The final section
summarizes our findings.
2. The Cross-Sectional Regression Methods
for Asset Pricing Tests
2.1. The Mathematical Setup
A linear asset pricing model with risk-free rate can be
expressed as follows:
(1)
where
r
is a vector of excess returns (in excess of the
risk--free rate) for
N
assets;
][E f
, the vector of risk
premia, is the mean of the
K
asset-pricing risk factors
f
; and
B
is the
KN ×
matrix of factor loadings of
r
on
f
. Let
Σ
and
denote the covariance
matrix between
r
and
f
and the variance-covariance
matrix of the factors
f
, respectively. Then
.=
1
ΩΣB
(2)
An important case of model (1) is the Sharpe-Lintner's
CAPM, where
f
is the market excess return and
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J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
equation (1) is called the Security Market Line. In the
general case of Merton's ICAPM,
f
is the vector of
excess returns of the market portfolio and the
1K
hedging portfolios and equation (1) is called the Security
Market Hyperplane. In Ross's APT case, equation (1)
will be empirically approximated if returns have a linear
factor structure
f
.
The classic design of testing the model (1) is to test the
following extended linear model:
,=][E
λιγ
Br +
N (3)
where scalar
γ
is interpreted as the zero-beta excess
return; N
ι
is an
N
-dimensional column vector of
ones, and
λ
is the
1×K
column vector of risk
premia. Define
,
1
='
NNNN N
ιι
ID
(4)
where N
I is the
N
-dimensional identity matrix, then
it is easy to see that
.='=
2
NNN
DDD
(5)
Henceforth, we drop the subscript N for
D
and
ι
for simplicity, as the dimensions of
D
and/or
ι
can
be implicitly determined. Multiplying equation (3) by
D
, we have
.=][E
λ
DBrD
(6)
Now assume that
NK <1+
and that the matrix
],[
ι
B
has full rank
1+K
, then
DB
will also have
full rank
K
and
].[E)(=][E)(=
11
rDBDBBrDBDBDB′′′′′′
−−
λ
(7)
Substituting (1) into (7) , we have
].[E=][E)(= 1ffDBBDBB′′
λ
(8)
That is, theoretically the basic model (1) can be tested
by testing the extended model (3) with the conventional
with-intercept cross -sectional regression (WICSR)
estimates (7). On the other hand, suppose that
B
has
full rank
K
, then by (1) the risk premium can be
directly expressed as:
].[E)(=
1
rBBB′′
λ
(9)
And a direct estimation of (9) can be obtained through a
no-intercept cross-sectional regression (NICSR).
Let
R
and
F
be the
TN ×
matrix of the
observed excess returns of the
N
assets and the
TK ×
matrix of the observed values of the
K
factors
for the
T
time periods. Assume that
R
and
F
are
stationary and the sample moments of
R
and
F
converge to the corresponding unconditional population
moments. Then the variance and covariance matrices
and
Σ
can be consistently estimated by
,
1
=
1
=FFDFDFD ′′′
TT
(10)
and
.
1
=
1
=FRDFDRD ′′′
ΣTT
(11)
The consistent estimators for
][ER
and
B
are
,
1
=
ι
RRT
(12)
and
1
1
1
=
= ()
=( ).
ΣΩ
′′ ′′
′′
B
RDD FFDD F
RDF FDF
 
(13)
The classic WICSR estimates for the risk premium
λ
=][E f
and the zero-beta excess return
γ
are
,')'(=
ˆ1RDBBDB

λ
(14)
).
ˆ
('
1
=
ˆ
λιγ
BR
N
(15)
The NICSR estimate for the risk premium is
.)(=
~1R'BB'B

λ
(16)
Denote
],[E= rR
R
ε
(17)
,= BB
B
ε
(18)
,=
λεεε
B
R(19)
then we can obtain estimation errors of the WICSR risk
premium estimate by equations (7) and (14)
.)(=
ˆ
1
ελλ
D'BBD'B

(20)
Similarly, equations (9) and (16) give the errors of the
NICSR risk premium estimate
.)(=
~1
ελλ
'BB'B

(21)
2.2. Asymptotic Distributions of the Estimators
In the previous subsection, we have laid out the
mathematical definitions of both the WICSR and NICSR
estimators. Next we present the asymptotic distribution
properties of the risk premium estimator given by the
57
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
conventional WICSR in (14) and that produced by
NICSR in (16). The assumption of conditional
heteroskedasticity we make here is similar to the one
made by Jagannathan and Wang [13, 1998].
Proposition I: Assume that 1) the time series of
returns
R
and factors
F
are stationary and ergodic,
2) as
∞→T
, the random vector
ε
T
converges
to a zero-mean random vector with covariance matrix
Ψ
, and 3) both
B
and
DB
have full rank of
K
,
then
)
ˆ
(
λλ
T
in (20) converges in distribution to a
zero-mean random vector with covariance matrix
;)()(=
11 −−
Ψ
′′DBBDBDBDBBV
(22)
)
~
(
λλ
T
in (21) converges in distribution to a
zero-mean random vector with covariance matrix
.)()(= 11 −−
Ψ
′′ BBBBBBV
(23)
Proof: Since the time series of returns and factors are
stationary and ergodic,
ΩΣ,
, and
B
will
converge in probability to
ΩΣ,
, and
B
, respectively.
Hence (22) and (23) immediately follow from the
assumption that the random vector
ε
T
converges to
a zero-mean random vector with covariance matrix
Ψ
.7
][E= f
λ
Q.E.D.
Since the true risk premium is not
observable, the CSR risk premium estimate can only be
tested against the sample mean of the observed risk
premium:
.
1
=
ι
FF T
(24)
In order to test the basic model (1), the classic WICSR
methodology tries to test the following null hypotheses8
1
0:H
:
Sufficiently high
2
R
to show that the variation
in cross-sectional average excess returns can be
explained by the variation in the factor loadings
B
;
7These two results can be derived as special cases of Ja-
gannathan and Wang's [13, 1998] Theorem 1.
8To fully validate an asset pricing model, it is very essen-
tial to test the hypothesis that all pricing errors are jointly
zero. Since the focus of this paper is on the risk premium
estimation but not on the validation of any specific model,
we only perform tests associated with risk premium es-
timates.
FH =
ˆ
:
2
0
λ
;
0=
ˆ
:
3
0
γ
H
.
Using the NICSR approach, we would like to
test the next two null hypotheses:
:
1
0
H
Sufficiently high
2
R
to show
that the levels of the cross-sectional average excess
returns can be explained by the levels of the factor
loadings
B
;
FH =
~
:
2
0
λ
.
For these purposes, the empirical time-series
form of (1) is introduced as follows:
,= e+BFR
(25)
where
e
is the
TN ×
matrix of idiosyncratic
errors with
0=][E e
and
0=),(coveF
. And the
empirical cross-sectional form of (1) is
,= e+FBR (26)
where
ι
e
T
e1
=
. By equations (13) and (25), we get
.)(==
1
′′
FFDFDBB
B
e
ε
(27)
After rewriting equation (26) in terms of
B
, we have
.=
εε
+≡−+FBFFBR
B

e(28)
Using equation (28), we obtain the conditional errors of
the CSR risk premium estimators given the factor
realization
F
:
1
ˆ=( ),
λε
FB'DB B'D

(29)
f
1
=( ).
λε
FB'B B'
 
(30)
Similar to the previous results on the asymptotic
distribution of the unconditional error of the risk
premium estimators, the asymptotic distribution
properties of errors of the risk premium estimators
conditioning on the realization of the risk factors
F
in
(29) and (30) can be given as follows.
Proposition II:9
R
Assume that 1) the time series of
returns and factors
F
are stationary and ergodic,
9Shanken [23, 1992] presents a comprehensive asymp-
totic analysis for the conditional error distribution of the
58
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
2) as
∞→T
, the random vector
ε
T
converges
to a zero-mean random vector with covariance matrix
Π
, and 3) both
B
and
DB
have full rank of
K
,
then
)
ˆ
(F
λ
T
in (29) converges in distribution to a
zero-mean random vector with covariance matrix
;)()(=
11 −−
Π
′′DBBDBDBDBBW
(31)
)
~
(F
λ
T
in (30) converges in distribution to a
zero-mean random vector with covariance matrix
.)()(= 11 −−
Π
′′ BBBBBBW
(32)
Proof: Similar to the proof of Proposition I. Q.E.D.
In order to conduct the statistical tests on the
hypotheses
1
0
H
,
2
0
H
,
3
0
H
,
1
0
H
, and
2
0
H
,
researchers typically make the assumption that the
idiosyncratic errors
e
are conditionally homoskedastic
on the realization of the risk factors with constant
covariance matrix
Φ
. Under this conditional
homoskedastic assumption, we obtain the following
result.
Proposition III:10
R
Assume that 1) the time series of
returns and
F
are stationary and ergodic and 2)
the idiosyncratic errors
t
e
are independently,
identically distributed with mean 0 and covariance
matrix
Φ
conditional on the realization of the risk
factor
t
F
, then
.)'(1= 1ΦΩ+Π
λλ
(33)
It is well known that the two-pass cross-sectional method
suffers from the EIV problem. From (28) we see that the
conditional errors
ε
have two components:
e
and
F
B
ˆ
ε
. Proposition III indicates that under the
conditional homoskedasticity, these two error
components will be conditionally independent with
asymptomatic covariance matrices
Φ
and
ΦΩ
λλ
1
', respectively. To account for the
estimation errors in betas, Shanken [23, 1992] suggests
to adjust the standard errors according to (33). That is,
the estimated errors should be inflated by
)
ˆ
ˆ
'
ˆ
(1
1
λλ
Ω+
and by
)
~
ˆ
'
~
(1
1
λλ
Ω+
for the
conventional with-intercept cross-sectional estimates
under the stronger assumption that the idiosyncratic er-
rors
e
are homoskedastic.
10We omit its proof here and readers can refer to Theo-
rem 1 of Shanken [23, 1992], as it directly follows.
conventional WICSR estimates and the NICSR ones,
respectively.
2.3. Cross-Sectional Asymptotic Properties
The asymptotic distribution analyses in Subsection 2.2
show that the estimators of both the classic WICSR and
the NICSR are
T
-consistent. This indicates that the
two-pass estimation leaves little to be desired with regard
to its large-sample properties as
∞→T
. In this
subsection, we examine the cross-sectional asymptotic
properties of these two CSR estimators and show that the
NICSR method is likely more efficient than the
conventional WICSR method, provided that
N
is large
enough and the cross-sectional dependence of
idiosyncratic errors is weak enough.
It is known that the traditional WICSR estimator is not
N
-consistent. That is, the risk premium estimate will
not converge in probability to the average observed
realizations
F
as
∞→N
. One obvious reason for
this is the EIV issue, and the other more fundamental
problem is that the true risk loadings
B
and the
cross-sectional average of the idiosyncratic errors
ε
may be correlated.11
N
The NICSR estimator is certainly
not -consistent, either. But under the general
assumption of sufficiently weak cross-sectional
dependence of idiosyncratic errors, the cross-sectional
average of the idiosyncratic errors
ε
in (28) tends to
cancel away and the advocated NICSR method will be
more efficient than the classic WICSR approach. There
are two empirical evidences for our claim. First, Miller
and Scholes [20, 1972] have documented that high-beta
assets tend to have negative alphas and that low-beta
stocks tend to have positive alphas12
N
. This indicates that
when is large enough, the cross-sectional average of
the estimated alphas will be quite small as negative
alphas of the high-beta stocks and the positive alphas of
the low-beta assets will cancel each other. Notice that the
conditional estimation errors
ε
in equation (28) will
be the estimated alphas in the CAPM case. This implies
that
11Most researchers simply assume that the cross-sectional
error
ε
is uncorrelated with the true risk loading
β
.
Empirically, we see that the correlation between
β
ˆ
and
ε
is quite large compared with the beta-variation.
12Even though the measurement error will contribute to
the negative correlation between
α
and
β
ˆ
, the
measurement error in
β
is minimal for CAPM and
the significant cross-sectional correlation is between al-
phas and true betas.
59
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
BBBBDBDBDBDBΠ≈
′′
Π
'=''=
εεεε
.
Secondly, Black, Jensen, and Scholes [3, 1972] report
that the market betas tend to be concentrated near the
value of one with relatively small variation and one of
the purposes of their grouping procedure is to obtain
maximum possible dispersion among betas. This implies
that the square matrix

B'B
is likely significantly
more positive definite than the square matrix

BD'B
. Combining these two points, we know
that the positive definite matrix
W
in (31) should
have bigger norm than
W
in (32). To formalize this
empirical argument, we produce the following result.
Theorem I: Assume that 1) the idiosyncratic
errors
it
e
have mean 0 and sufficiently weak
cross-sectional dependence; and 2) as
∞→N
,
ι
'
1
B
N
,

B'B
N
1, and
ε
D'B
N
1,
converge to ]
ˆ
[E
β
,
)
ˆ
(var]
ˆ
[E]
ˆ
[E
βββ
+
, and
),
ˆ
(cov
εβ
, respectively, then
as
∞→N
,
0
1→≡
ειε
N
;
as
∞→N
, the estimation error of
the conventional WICSR risk premium estimate
converges to
);,
ˆ
(cov)
ˆ
(var
ˆ
1
εββλ
→− F
(34)
as
∞→N
, the estimation error of
the NICSR risk premium estimate converges to
1
ˆ
(var( )
ˆˆˆ
E[]E[])cov(,).
λβ
β ββε
−→
+
F
(35)
Proof: Suppose that the time-series variances of the
idiosyncratic errors
it
e
are bounded, then by the weak
law of large numbers e
N
ι
1 converges in probability
to vector
0
. Hence by (27) and (28), (i) directly
follows. (ii) is the immediate result of assumption 2) and
equation (29). Notice that
,'='
ειεε

BD'BB +
and from Equation (30), we have that (iii) directly
follows the above results (i) and (ii). Q.E.D.
The two conditions in Theorem I are typically
assumed in the analysis of
N
-consistency of the
two-pass CSR method. If we further assume that the true
risk loading betas are uncorrelated with the idiosyncratic
errors
e
, then (34) becomes
1
1
ˆˆˆ
ˆˆ
var ()cov(,)
=[var()var( )][var( )cov(,)].e
ββ β
β βε
βε εε
++F
(36)
The asymptotic result (36) is basically the one reported
by Black, Jensen, and Scholes [3, 1972]. Under the
further assumption of conditional homoskedasticity, they
argue that errors
β
ε
ˆ
and
e
can be ignored for large
T
and hence the CSR estimator is
N
-consistent.
Shanken [23, 1992] formally proves that the ``OLS
version" of maximum likelihood estimation of the
zero-beta excess return
γ
is
N
-consistent under the
assumptions that the idiosyncratic errors
e
are
homoskedastic and cross-sectionally uncorrelated with
the true betas.
The asymptotic result (34) in Theorem I indicates that
three factors---beta variation, EIV, and the average
idiosyncratic error over time--- will determine the
estimation error of the CSR risk premium estimate. The
existing literature emphasizes EIV correction but pays
little attention to the impact of beta variation. BJS point
out that one of the purposes of the grouping procedure
should be to maximize the variation among betas. But
this seems to be difficult in portfolio grouping. Kim [15,
1995] even suspects that the formation of portfolios for
the CSR estimation might cause a loss of valuable
information about cross-sectional behavior among
individual assets. And one of the prescriptions suggested
by LNS to improve empirical tests is to expand the set of
test portfolios beyond the size-B/M portfolios. The
expansion of the set of test portfolios tends to increase
the variation among betas and hence can improve the
CSR estimations. As the asymptotic property (35)
indicates, our preferred NICSR methodology will barely
suffer from the small beta-variation as the total square
sum of betas is used. Hence the NICSR approach will be
more efficient than the traditional WICSR method.
3. Empirical Results
In the previous section, we have shown that the NICSR
method will be more efficient than the classic WICSR
approach in the tests of the linear beta-pricing models. In
this section, we present our supporting empirical
evidence. First, we employ simulations to show this point.
Then, we test the CAPM with the actual stock returns for
the early subsample. In the last subsection, we reexamine
the Fama-French three-factor model.
3.1. CAPM Simulations
60
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
In this subsection, we employ simulations to demonstrate
that the NICSR approach will be more efficient than the
WICSR method in tests of the linear beta-pricing models.
There are twenty-five series of simulated excess return
data generated according to the CAPM
,= itMtiit e+RR
β
(37)
where the market excess returns
Mt
R
are the actual
Fama-French market factor monthly time series between
July 1926 and June 2007 and the market risk loadings
i
β
are the actual estimated market betas of the 25
Fama-French size-B/M cross-sectional portfolios for the
same period13
it
e
. The simulated idiosyncratic errors
in (37) are cross-sectionally independent and are
independently, identically, normally distributed with
mean zero and variance
2
i
σ
across time, where
2
i
σ
are the variances of the 960-period idiosyncratic
residuals of the 25 value-weighted Fama-French
size-B/M portfolios regressed on the Fama-French
market factor.14 With the 25 simulated cross-sectional
excess returns, we first perform the usual pass-one
time-series regression on the given fixed Fama-French
market factor to estimate betas and then run both the
classic WICSR test and the NICSR one with the
cross-sectional average excess returns. The simulations
and the regressions are repeated 100 times.
Figures 1 and 2 graphically show the actual errors and
Figure 1: Actual Errors in Risk Premium Estimates.
the estimated standard errors of the risk premium
estimates of the 100 simulations, respectively. We clearly
see that the errors in the risk premium estimates for the
WICSRs are in general much larger than those for the
NICSRs. In Table 1, we list some summary
13Twelve data points with missing data between July
1930 and June 1931 are excluded.
14For simplicity, the idiosyncratic errors are assumed to
be cross-sectionally independent. This simplification is
not too restrictive as the NICSR method can be directly
generalized to the GLS case.
Figure 2. Estimated Standard Errors of Risk Pre-
mium Estimates
statistics---means, standard deviations, minima, and
ma xima---of the errors in the risk premium estimates of
the 100 simulations for both the methods. These specific
statistics again show that the NICSR approach is more
efficient than the WICSR method. Table 2 shows the
number of rejections of the true hypotheses or the
Sharp e -Lintner CAPM model for both methods. The
lower frequency of the false rejections for the NICSR
method also indicates that the NICSR method is more
efficient and more robust than the WICSR approach.
Table 1. Statistics of Errors in RP Estimates
Sta-
tistics
Mean
Std.
Dev.
Min.
Max.
Panel A: CSR Tests Using Estimated betas
NICSR
0.000
0.024
0.053
0.070
WICSR
0.026
0.215
0.510
0.667
Panel B: CSR Tests Using Actual betas
NICSR
0.000
0.024
0.052
0.072
WICSR
0.031
0.213
0.461
0.682
In Section 2, we show that the NICSR method tends to
be more efficient because the risk loading betas
cross-sectionally center around one with small variation,
and the time-series average idiosyncratic errors diversify
away considerably when they are averaged in the CSR.
The cross-sectional average and the variance of the
market betas of the Fama-French twenty-five
cross-sectional portfolios are 1.225 and 0.030,
respectively. Hence by Theorem I, the errors in risk
premium estimates generated by the WICSRs can be up
to roughly fifty-one15
15
times as large as those by the
NICSRs. The simulation results reveal that, on average,
this ratio is about ten.
Proposition III specifies the asymptotic standard EIV
adjustment. For our simulations here, the mean and the
variance of excess returns of the Fama-French market
portfolio are 0.688 and 28.714, respectively, and hence
51.300.030)/0.0(1.225
2
≈+
61
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
Table 2: Number of Rejections of the CAPM
Criti-
cal Values
0.01
0.03
0.05
0.1
Panel A: CSR Tests Using Estimated betas
NICSR
0
1
5
12
WICSR
5
10
15
26
Panel B: CSR Tests Using Actual betas
NICSR
0
2
5
9
WICSR
5
9
17
25
the corresponding asymptotic standard error adjustment
will be only 0.016. Panel B of Table 1 and Panel B of
Table 2 show the summary statistics of the errors in the
risk premium estimates and the numbers of false
rejections of CAPM, respectively, among the 100
simulations when the actual betas are used. Obviously
the EIV issue here is statistically insignificant and this is
consistent with the findings of Black, Jensen, and
Scholes [3, 1972] and Shanken [23, 1992]. And this fact
shows that the relatively large errors of the risk premium
estimates in WICSRs (compared with those in NICSRs)
can not be significantly reduced by the maximum
likelihood estimations that try to eliminate the errors in
variables.
Even though a very low CSR
2
R
may indicate that
the hypothetical asset-pricing model should be rejected
by the data, 2
R
by itself can not be the appropriate
measure of the model fit. Recently, Lewellen, Nagel, and
Shanken [16, 2008] have shown that proposed
asset -pricing factors that are even weakly correlated with
the true factors and uncorrelated with the errors are
capable of producing high cross-sectional
2
R
and one
of their recommendations is to take the magnitude of
estimates more seriously. Our simulation results reaffirm
their point. As shown by the scatter-plot in Figure 3, high
2
R
can be associated with estimates with large errors,
and low 2
R
may correspond to relatively good
estimates.
Figure 3. Scatter Plot (P-Value vs
2
R
)
3.2. CAPM Tests with Returns in the Early
Subperiod
In Section 2 we have proven that the NICSR estimation
is more efficient than the classic WICSR estimation. We
have also described some supportive simulation evidence
in the previous subsection. However, we should be aware
that NICSR estimation is not appropriate for the
extend ed model (3). In the empirical asset-pricing
literature, the extended form (3) is typically directly
assumed even though the theoretical models generally
assume the existence of risk-free rate and are all in the
basic form (1). The immediate reason of this practice
may be the extensively common usage of the ordinary
with-intercept linear regressions and the relatively rare
application of the no-intercept linear regressions. But the
main reason of this practice should be the rejection of the
original Sharpe-Lintner CAPM and the endorsement of
the extended Black CAPM by the early classic CAPM
testing works of Black, Jensen, and Scholes [3, 1972]
and Fama and MacBeth [12, 1973]. In this subsection,
we show that the evidence and warrants for rejection of
the basic model (1) are not as strong as believed.
As Lewellen, Nagel, and Shanken [16, 2008] point out,
the magnitudes of the estimated zero-beta excess returns
are unreasonably high in many of the existing empirical
results. According to the theoretical explanation16 of the
zero-beta excess return, it is a weighted average of the
differences between the costs of risk-free borrowing and
the returns of risk-free lending and hence should be
within the approximate range17
between 0 and 0.2% per
month. Black, Jensen, and Scholes [3, 1972] report the
estimates of the zero-beta excess returns to be 0.338%,
0.849%, 0.420%, 0.782%, and 0.997% for the periods
1/31-12/65, 1/31-9/39, 10/39-6/48, 7/48-3/57, and
4/57-12/65. These estimates (4.056%,
10.188%,
5.04%, 9.384%, and 11.964% if annualized) are just too
big to be due to the rate differences between (risk-free)
borrowing and risk-free lending and the huge negative
estimate of
0.849% for period 1/31-9/39 is even more
spurious if it is interpreted by the borrowing restriction
argument.
The analyses in Section 2 and the above simulations in
Subsection 3.1 suggest that at least one cause of the
unreasonable magnitudes of the risk premium estimates
and the zero-beta excess return estimates are the likely
large errors of the WICSR estimates. In the rest of this
subsection, we illustrate this point with our empirical
16Brennan [7, 1971] refers to the zero-beta rate as the
market's equivalent risk-free rate and shows that it is a
weighted average of the risk-free borrowing rate and the
risk-free lending rate.
17Lewellen, Nagel, and Shanken [16, 2008] give their
estimates of zero-beta excess returns to be between 0 and
2% per year.
62
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
CAPM testing results over the early period between July
1931 and December 1965.18
Panel A: Ex Post Market Risk Premia
Our data include both the
value -weighted and equally weighted monthly excess
returns of the widely used twenty-five Fama-French
size-B/M cross-sectional portfolios and the CRSP market
index. The monthly risk-free rates are also obtained from
the Fama-French benchmark factor data.
Table 3. CSR Tests on CAPM with Equally Weighted
Data.
7/31-
7/31-
1/40-
1/48-
1/57-
12/65
12/39
12/47
12/56
12/65
M
R
1.471
2.201
1.577
1.256
1.035
Panel B: NICSR Tests on CAPM
2
R
0.98
0.94
0.95
0.95
0.97
λ
~
1.392
2.099
1.540
1.186
1.029
σ
~
0.043
0.112
0.073
0.055
0.039
)
~
=(
λ
M
pR
0.083
0.369
0.625
0.216
0.881
Panel C: WICSR Tests on CAPM
2
R
0.66
0.63
0.54
0.06
0.49
λ
ˆ
1.235
2.741
1.626
0.416
0.982
)
ˆ
(
ˆ
λσ
0.186
0.439
0.313
0.340
0.208
)
ˆ
=(
λ
M
pR
0.218
0.232
0.877
0.021
0.800
γ
ˆ
0.172
0.713
0.091
0.806
0.047
)
ˆ
(
ˆ
γσ
0.198
0.473
0.325
0.352
0.204
)
ˆ
=(0
γ
p
0.395
0.145
0.781
0.032
0.819
Table 3 presents testing results of the Sharpe-Lintner's
original CAPM with equally weighted data for the early
period between July 1931 and December 1965 and four
subperiods 7/31-12/39, 1/40-12/47, 1/48-12/56, and
1/57-12/65. Panel B gives the testing results by the
NICSR. We see that the risk premium estimates are quite
close to the realized average market excess returns, that
the high
2
R
's strongly suggest that the cross-sectional
average excess returns are proportional to the
cross-sectional betas, and that the Sharpe-Lintner's
CAPM fails to be rejected. Panel C shows testing results
of the conventional WICSR methods. We see that the
estimated standard errors of the risk premium estimates
18The testing period of one of the classic CAPM empiri-
cal works by BJS is January 1931 through December
1965. We try to take the same time period except for the
first six month between January 1931 and June 1931, for
which the 25 Fama-French cross-sectional portfolio re-
turn data are missing.
by the WICSR method are much larger than those by the
NICSR approach. But except for Subperiod 1/48-12/56,
our testing results fail to reject the Sharpe-Lintner's
CAPM. For Subperiod 1/48-12/56, the testing statistics
seem to even reject the extended CAPM. Since the
WICSR estimates have bigger estimation errors and the
NICSR testing results tend to support the
Sharp e -Lintner's CAPM, we think that the evidences
for the rejection of Sharpe-Lintner's CAPM generated by
the early CAPM tests with the WICSR approach are
exaggerated.19
Panel A: Ex Post Market Risk Premia
Table 3: CSR Tests on CAPM with Equally
Weighted Data.
Period
7/31-
7/31-
1/48-
Period
12/65
12/47
12/65
M
R
0.944
0.835
1.044
Panel B: NICSR Tests on CAPM
2
R
0.97
0.95
0.97
λ
~
0.965
1.109
1.038
σ
~
0.034
0.052
0.038
)
~
=(
λ
M
pR
0.540
0.000
0.872
Panel C: WICSR Tests on CAPM
2
R
0.40
0.48
0.01
λ
ˆ
0.722
1.188
0.256
)
ˆ
(
ˆ
λσ
0.184
0.258
0.439
)
ˆ
=(
λ
M
pR
0.239
0.186
0.086
γ
ˆ
0.340
0.115
0.823
)
ˆ
(
ˆ
γσ
0.252
0.372
0.461
)
ˆ
=(0
γ
p
0.190
0.760
0.088
Table 4 shows our testing results of CAPM with the
value -weighted data for the early period 7/31-12/65 and
two half-periods 7/31-12/47 and 1/48-12/65 using both
the NICSR and the WICSR methods. Panel B presents
the NICSR testing results. For the entire early period
7/31-12/65 and the subperiod 1/48-12/65, the NICSR risk
premium estimates
λ
~
are very close to the ex-post
excess market returns M
R (The differences are smaller
than 2 bps per month.) But for the subperiod 7/31-12/47,
19 Notice that the discussion here is based only on
pre-seventies stock returns, thus our findings should be
interpreted as supportive evidence for the basic model (1),
a general linear asset-pricing model with risk-free rate,
rather than for the extended model (3).
63
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
λ
~
is considerably larger than M
R. This evidence
seems to imply the rejection of the Sharpe-Lintner
CAPM. However, we feel that the estimated risk
premium for the subperiod 7/31-12/47 is reasonable
because the subperiod 7/31-12/47 spans over the time of
the Great Depression and the World War II.20
3.3. Multifactor Asset-Pricing Tests
In the above subsection, we have presented empirical
evidence to show that the NICSR method is more
efficient than the WICSR for CAPM testing. In this
subsection, we show the higher efficiency of the NICSR
method when testing the Fama-French three-factor model.
Fama and French ([10, 1993] and [11, 1996]) propose a
three -factor model that explains more than 90% of the
time-series variation in portfolio returns and more than
75% of the cross-sectional variation in their average
returns. But the traditional WICSR estimates of the
market risk premium are not significant at all and the
estimated zero-beta excess returns are unreasonably high.
It is obviously contradicting that the market factor has
strong explanatory power in the time series but little
explanatory capability in the cross-section of stock
returns.
Panel C
shows the corresponding WICSR testing results and
again the estimated standard errors of WICSR are
significantly larger than those of NICSR. The two null
hypotheses fail to be rejected in all the four testing
periods. However, one can hardly be convinced that the
CAPM holds by these results as the errors are so big and
the estimated risk premium is essentially indifferent from
0 for the subperiod 1/48-12 /65.
Kim [15, 1995] shows that the WICSR market risk
premium estimates with size factor present can still be
significant when the errors-in-variables are corrected and
the number of portfolios
N
is large enough (say larger
than 400). As we show in Section 2, small variation
among cross-sectional betas amplifies estimation errors,
and the EIV introduces systematic bias. The empirical
results reported by Kim will be more accurate than the
traditional WICSR testing results since expansion of the
set of test portfolios tends to enlarge the
beta-variatio n-- -this is also one of the prescriptions
suggested by Lewellen, Nagel, and Shanken [16, 2008].
In this subsection, we present our NICSR empirical
testing results on the Fama-French three-factor model,
which not only show that the market risk premium is
cross-sectionally significant but also provide new
20Cochrane [8, 2001] suggests to make the standard error
correction to account for the error of the sample mean of
the market factor. Here the standard error correction will
increase the uncorrected standard error by more than
0.5% per month and the null hypothesis will be fail to be
rejected.
cross-sectional supportive evidence for the Fama-French
three -factor model.
The data consist of monthly value-weighted returns of
the 25 Fama-French size-B/M cross-sectional portfolios
and monthly data of the three Fama-French benchmark
factors, namely the market excess returns and the returns
of the two mimicking portfolios of SMB
(smal l-mi nus-big) and HML (high-minus-low), for the
time period between July 1926 and June 2007. Twelve
data points between July 1930 and June 1931 are
excluded because of missing data in the 25 Fama-French
size-B/M cross-sectional portfolio returns. The CSR
estimation for the Fama-French three-factor model is
performed for the specified whole period and four
subperiods, 7/26-6/47, 7/47-6/67, 7/67-6/87, and
7/87-6/07.
Table 5 shows the CSR testing results on the
Fama-French three-factor model. Panel A lists the
realized average values of the three Fama-French factors.
In Panel B, we see that the market risk premium
estimates are not only significant but also are statistically
indifferent from the realized ex-post market risk
premium. The estimated risk premia of the size factor
and the B/M factor are also found to be statistically
indifferent from the realized average factor values. As
shown in Panel C, the conventional WICSR estimates are
spurious with the negative market risk premium
estimates and the unreasonably high estimated zero-beta
excess returns. Overall, we see that the NICSR method
generates more efficient risk premium estimates (in terms
of smaller estimated standard errors) for the market
factor than the WICSR method and comparable estimates
of the risk premia and the standard errors for the SMB
and the HML factors to those generated the by the
WICSR method.
4. Conclusions
This paper shows that the classic cross-sectional
regression approach to asset pricing tests tends to suffer
from severe estimation errors because of the small
beta-variation. We argue that it uses low criteria to
validate an asset-pricing model and suffers from the
model-misspecification issue because of the complication
of the zero-beta excess return. To address this problem,
we advocate focusing on the theoretic linear beta-pricing
model that assumes a (market equivalent) risk-free rate
and directly testing whether the ex-ante risk premium
estimates are consistent with the observed ex-post risk
premia, which allows one to easily detect misspecified
models. Under the assumption that asset returns follow a
stationary and ergodic process, we derive the asymptotic
distribution of the estimators of the method we advocate.
We also show that this approach will be more efficient
than the conventional method, provided that the
64
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
idiosyncratic errors have sufficiently weak cross-
sectional dependence.21
Panel A: Realized Averages of the FF Factors
Table 5. CSR Tests on FF Three-Factor Model.
Period
7/26-
7/26-
7/47-
7/67-
7/87-
6/07
6/47
6/67
6/87
6/07
MKT
R
0.689
0.851
0.969
0.352
0.584
SMB
R
0.240
0.504
0.072
0.276
0.110
HML
R
0.429
0.468
0.336
0.532
0.381
Panel B: NICSR Tests on FF Three-Factor Model
2
R
0.96
0.95
0.98
0.97
0.94
MKT
λ
~
0.672
0.916
0.995
0.350
0.521
SMB
λ
~
0.115
0.313
0.000
0.253
0.125
HML
λ
~
0.445
0.437
0.282
0.577
0.399
MKT
σ
~
0.068
0.101
0.061
0.041
0.072
SMB
σ
~
0.079
0.107
0.072
0.053
0.088
HML
σ
~
0.107
0.159
0.094
0.066
0.098
)=
~
(R
λ
p
MKT
0.809
0.529
0.676
0.962
0.398
)=
~
(R
λ
p
SMB
0.126
0.088
0.332
0.667
0.860
)=
~
(R
λ
p
HML
0.885
0.845
0.574
0.499
0.856
Panel C: WICSR Tests on FF Three-Factor Model
2
R
0.80
0.55
0.42
0.83
0.63
MKT
λ
ˆ
1.23
2
0.91
2
0.42
3
0.42
9
1.16
7
SMB
λ
ˆ
0.159
0.261
0.020
0.236
0.041
HML
λ
ˆ
0.461
0.547
0.298
0.550
0.414
MKT
σ
ˆ
0.280
0.682
0.443
0.389
0.407
SMB
σ
ˆ
0.045
0.097
0.061
0.050
0.070
21The assumption of weak cross-sectional dependence is
also made by Black, Jensen, and Scholes [3, 1972], and
by Shanken [23, 1992] in their
N
-cons i stency analyses
for the cross-sectional regression methods. Based on our
strong evidence of higher efficiency of the NICSR ap-
proach, this assumption should also be quite reasonable.
HML
σ
ˆ
0.061
0.146
0.079
0.063
0.074
)=
ˆ
(R
λ
p
MKT
0.000
0.017
0.005
0.057
0.000
)=
ˆ
(R
λ
p
SMB
0.087
0.020
0.406
0.431
0.335
)=
ˆ
(R
λ
p
HML
0.607
0.595
0.640
0.783
0.669
γ
ˆ
1.989
1.933
1.408
0.815
1.805
)
ˆ
(
ˆ
γσ
0.290
0.715
0.437
0.405
0.431
0)=
ˆ
(
γ
p
0.000
0.013
0.004
0.057
0.000
Our simulation results provide further empirical
evidence of higher efficiency of the approach we
describe and show that the relatively large estimation
errors of the conventional estimates are not due to the
error-in-variable issues. The simulations also indicate
that the cross-sectional
2
R
alone is an inappropriate
criterion of the model fit even if the correct factors are
used.
We reexamine the original Sharpe-Lintner CAPM with
monthly stock returns for the early period between July
1931 and December 1965. Testing results of CAPM with
the actual stock return data for the early subperiod further
affirm that the estimates of the market risk premium
generated by the approach advocated in this paper are
more efficient than those by the classic method and show
that the warrants of the significant zero-beta excess
return are not as strong as believed.
We also reexamine the Fama-French three-factor
model. It is well known that the market beta loses
explanatory power on the cross-sectional average returns
when the size factor is included as an additional
explanatory variable. With the five conventional
estimations (one for the whole period and four for four
subperiods), the estimated zero-beta excess returns are
unreasonably high and the market risk premium
estimates are all negative. But with all the five
estimations using the suggested approach, not only are all
the market risk premium estimates significant, but also
all the three risk premium estimates are statistically
indifferent from the realized average prices of the three
risk factors. Furthermore, the standard error estimates for
the market factor given by this method are much smaller
than those by the conventional method. These findings
illustrate the spurious nature of the conventional
estimates and show that the market factor is consistently
65
J. YUAN, R. SAVICKAS
Copyright © 2013 SciRes. TI
priced in the Fama-French three-factor model.
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