Jouxnal of Fhmtial
Economics ‘?I (3958) 213-254. North-Holland
THE EMPIRICAL FOUNDATIONSOF THE
ARBITRAGE PRICING
THEORY*
David ha. MODEST
Received 0ctober 1985, &nal version received January 1988
The arbitrage pricing theory (APT) developed by Ross (1976,1977) is a
major attempt to overcome the problems with testzbility’ and anomalous
euqkical evidence that have plagued the static and iktertemporalcapital asset
pricing models (CAPMs). The main assumption of the theory is that the
returns of a large (in the limit infinite) number of assets can be broken into
two components: nondiver&able, systematic rislc,which can be measured as
exposure to a small number of common factors, and idiosyncratic risk, which
*We are grateful to the Faculty Research Fund of the Columbia Business School and i&z
Institute for Quantitative Research iz Finance for their support and to Wayne Ferson and Allan
Kleidon for comments on earlier drafts. We owe a special debt of gratitude to Jay Shauken (the
refer) for signikantly improving the paper through his incisive comments and for pointing out
severai errors in earlier drafts. The usual disclaimer applies.
‘The model does not resolve all empirical ambiguities of the type discuss4 in Roll (1977). Jtn
particular, !Shanken(1982,198Sa) has emphasized that the absence of riskless arbitrage opportunities coupled with the linear factor model for security returns does not place sufficiently precise
restrictions on expected returns. In section 2 we address this i$nt further and sqecify a set of
additional assumptions, that is sufficient to perform preci§e statistical tests.
0304405X/88,/,93.50 0 1,988,Ekevier Science Publishers B.V. (North-
214
B.N Lehmann and D.M. Modest, Empirical basis of the a&rage pricing theory
em be eliminated in large weW.iversified portfolios. This assumption, comb&d tith the presumption that investors prefer more to less, leads to an
approtiate
theory
of expected returns through the preclusion of riskless
arbitrage opportunity.
Unfortunately, the apparent simplicity of the AFT conceals serious dishcultis associated with its implementation. In particular, the theory cannot be
tested without a strategy for measuring thecommon factors. Most investigators have turned to factor analysis to measure these common factors implicitly.
This approach exchanges the problem of identif@g tie factors o priori for the
computational problems of performing maximum4ikelihood factor analysis on
large cross-sections using conventional software pa&ages.
As a consequence, most previous researchers have performed factor analysis
on relatively small cross-sections. This resolution of the problem of common
factor measurement can adversely aff&cttests of the APT’in two ways. First,
the use of small cross-sections can yield imprecise estimates of the common
factors because the reliability of these estimates is low with small cross-sections. Second the reliance on a small number of securities in the analysis
nHkes it dif&u’lt to confront the theory with the anomalies that have proven
puzzling in the CAPM context. Both problems can lead to tests of the APT
that reject the theory when it is true or fail to reject rt when it is false.’
In this paper we remove some of the empirical
.*. ambiguity s~urrounding the
APT by performing more comprehensive tests than have previously been
feasible. In the next section, we review the basic theory of the APT and its
different formulations. The third section contains a brief literature review and
a detailed description of our tests. in the fourth section, w’c de&be our
procedure for forming basis portfolios to mimic the common factors and the
impact of measurement error in the basis portfolios on our tests. The fifth
section presents our empirical results. Among the most striking results are: (i)
rejection of the hyp&esis that our basis portfolios span the mean-variance
frontier of listed equities on the New York and American Stock Exchanges,
(ii) evidence that the APT can explain the dividendyield and own-variance
anomalies where the usual CAPM market proxies fail, and (iii) a distinct
inability of the APT to explain the relation between firm size and average
returns- The final section is devoted to concluding remarks.
2. T&e APT
ROSS(1976,1977) argued that the key intuition underleg the CAPM was
the distinction between systematic and unsystematic riik. Ross noted that
systematic risk need no: be adequately represented by a single common factor
such as the return on the market and instead assumed that asset returns tie
5 N Lehmann and D.M. Modest,Empiricalbasisoj th? at&rage pricing theory
215
generated by a hear K-factor model:
In (l), R, is the return on asset i between dates t - 1 and t, Ei is the asset’s
expected return, & is the realization of the kth common factor (normalized
to have a zero population mean), bik is the sentitivity of the return of asset i
to the kth common factor (called the factor loading), and Eitis tLe idiosyncratic return on the ith asset, which is assumed to have zero mean and finite
variance, and to be suf&iently independent across securities so that idiosyncratic risk can be eliminated in large, well-diversified portfolios.
Ross and many subsequent authors have proven that the absence of riskless
arbitrage opporuunities implies that expected returns must satisfy (approximately):
(2)
as the number of assets satisfying the factor model (1) tends toward infinity
where A, is the intercept of the pricing relation and X, is the risk premimn cm
the kth common factor, k= l,..., K.
The approxir&e pricing relation given by (2) should price most assets with
negligible error but, unfortunately, need not price all assets arbitrarily well. If
the pricing errors for most assets were not trifling, one could construct zero net
investment arbitrage portfolios that are riskless and earn nonzero profits.
Unfortunately, the same argument cannot be used to guarantee that all assets
will be priced correctly, since arbitrage portfolios must place appreciable
nf assets to exploit a few signScant pricing
weight on a small INdEi
u
deviations- These portfolios, in general, wii not be well-divers&d and need
not have negligible total risk.2 u
As a consequence, the central risk-expected return relation of the APT
given by (2) is not testable without further assumptions, since a small number
of assets could be priced arbitrarily badly.3 Not surprisingly, many investigators have examined the circumstances in which the pricing errors for all assets
under consideration are negligible. Chamberlain (1983, corollary I, pa 1315)
provided the conditions required to transform the approximate pricing relation
(2) into an exact pricing relation - the e&tence of a riskv well-diversified
portfolio on the mean-variance efficient frontier of the assets under considera28ti1arly,
assets.
these heuristic arguments can fail when applied to a large but finite number of
‘This point is made by Shanken (1982) and is a focus of the exchange between 5 -;bvig and Ross
(1985) and Shanken (1985a).
tionis both
and sufikient for exact factor pricing.4 Note that the
requirement for exact factor pricing on a large subset of traded assets whose
returns follow a factor structure(i.e., that there exist a well-div~ed portfolio
of these assets on their mean=varianceefkient frontier)does not preclude the
existence of nontraded assets, such as human capital, or traded assets whose
returns do not satisfy a linear factor model.
Since Ross% approximate pricing relation is not testable, our tests must be
considered joint tests of Ross’s basic theory plus the additional assumptions
requked to turn (2) into an exact factor pricing relation. In addition, as
discussed in section 3, the empirical formulation of our tests requires us to
construct po&*oKosto mimic the factors that span the factor space and do not
contain any idiosyncratic risk, As a conseclue
any rejection of the APT
could reflect a fail= of the exact factor pricing version of the theory or of our
inability to construct r&able estimates of the common factors.
mceswy
The .APT has several formulations that follow from the diversikation
possibilities arising when portfolios are formed from large(in the limit infinite)
cross-sections of securities that satisfy a linear factor structure. As noted by
Chamberlain (1983), the principal distinction among exact factor pricing
models is whether the entire mean-variance frontier is well-diversifiedor
whether only one potiolio on the frontier is welldiversified. This in turn
depends ;~lhwhether the limiting minimum variance potiolio of riw assets
contains (i) no risk (i.e., no factor 0~idiosyncratic risk), (ii) both factor and
idiosyncratic risk, or (iii) only .U factor.Each case yields a difkrent formulation of the 4UT.
These three possibilitiesregardingthe risk of the limiting minimum vatiance
portfolio lead to threepossible exact factor pricing versions of the APT’.If it is
possible to construct a limi*Gngportfolio that costs a dollar and whose returns
do nat vary, then the pricing intercept A, in (2) corresponds to the riskless
rate (IQ). M**reover,in this case, there exists a well-diversifiedmean-variance
efficient portfolio of *he K basis portfolios that with the risklessasset (i.e.,the
limiting riskless portfolio of these risky mts) spans the mean-vtince
efficient fkontierof the individual assets - although the K basis portfolios by
themselves do not spzm the frontier.
‘Using Chamberlain’sterminology,a sequenceof portfolios is omsiderd well-diver&xl if in
the limit (as the number of assets in the portfolio goes to infmity) the idiosyncratic variance of the
sequence of portfolios tends to zero. ‘3
k, en and Ingersoll (1983), COMOr (1984), Dybvig (1983),
Grinblatt and Titmm (1983), and !&a&en (198%) provideequilibriumconditions under which
the pricing deviations for all assets will be small.
‘Huberman aud Kandel(1987) refer to the formercircumstanceas spanning of the efficient
frontierand the latter as intersectionof the eikient frontier.
It is not possible to form a limiting portfolio [from an infinite subset of
securities whose retsatisfy (1)) whose returns do not vary if, under an
appropriate normalization of the factor spaeV the factor loadings on one of
the factors are identical for all but a tite number of securities.Put diff~&rent,ly~
*&isinability to form a limiting riskless portfolio arises when a vector of one
lies approximat&y in the cohunn space of the factor loading matrk This
would occur, for instance, if most security returns are equally a@&ctedby
unexpected &anges in some mal3xBonomk variable
asGlWorinfla=
tion.
Inthiscase,~~aretwoversionsoftheApT,dependi~onwhetherthe
limiting minimum variance potiolio is welbdivaed
and contains ody
factor risk or whether it is not w&divers&d and, neaCe,also contains some
idiosyncratic risk, In the former c8se, the entire &an-&ce
eflkient set
will be welldiversi6ed and the K basis portfolios will span the frontier, given
exact factor pricing. Under this formulation, the pricing intercept A, will be
zero.
However, if the &3iting minimum variance portfolio is not ~e&diversiEed
and, hens con’aias some idiosyncratic risk, the K basis
span the frontier. In these circumstan~althoughalinear
K basis portfolios will be mean-variance e&ient and well-diversikd, its
orthogonal partner (Le.,its a!Bociatedminimum varian4x&zero beta portfolio)
will contain idiosyncratic risk and hence the entire mean-variance frontier will
not be welldivers&d. Under this formulation,there is no restrictionon the
pricing interpret A,.
Since A, could equal the riskless rate, these intercepts cannot be used to
distinguish this model from the limiting riskless asset case dkussed above.
Moreover, it is not poss&le to form a riskless portfolio with finitely many
linearlyindependent securitieseven if the limitingminimumvarianceportfolio
is r&less. Ekize, the etiteme of a limiting riskkss minimum variance
po?tfolio is without empirical content in finite cross-sections. As a consequence, the two empirically distinct models are the spanuing formulation in
which A, = 0 and the K basis portfolios span the frontier, and the alternative
formulation in which X0 is unrestricted and a portfolio of the K basis
portfolios is mean-variance efficient.
2.3. Empirical formulations
Hence we have two exact factor prickg models. If the K basis portfolios
span the frontier, then security returns ,s._kF~
3
218
B.N LAmann
and D.M.
Modest, Empitical
basis
of tk arbitrage pricing theory
where R,, is the return on the basis portfolio that has unit sensitivity to the
kth factor, zero sensitivity to aII other factors, and no idiosyncratic risk. In
this case, tests of the APT mean restriction can be performed by regressing the
raw secutity returns on a constant and the returns Gf the basis portfoIios and
examining whether the constants are significantly different from zero. We
sometimes refer to this as the raw-returnversion of the APT’.In addition, the
K basis porafolios span the mean-variance efficient frontier in these circumstances and it follows from RoII(1977) and Huberman and Kandel(l987)
that the sum of the factor loadings must equal one. In section 3.3, we formahy
discuss our procedures for testing these restrictions.
If the minimum variance portfoIio is not we&diversified, the K basis
portfoIio wiII not span the mean-variance frontier of the individual assets and
hence the sum of the factor loadings need not equaI one. In these circumstances, individual security returns are described by
where fizt is the return on the minimum variance portfolio orthogonal td the
returns of the K basis portfoIios? Thus, tests of the APT mean restriction can
be performed by regre&q excess returns (over &) on a constant and the
excess returns of the basis portfolios and examining whether the constants are
fiigdicantly difkent from zero. We sometimes refer to this as the excess-retum
v&sion of the APT.’
3. Hypothesis-testing3.1. Previous tests of the APT
Previous tests of the restriction given by (2) have typically taken three
forms:* (i) tests for the equ&y of intercepts across smah subgroups of
securities, (ii) tests for the joint significanceof the factor risk premiums in each
%ee
also Gibbons, Ross, and Shanken(1987).
‘Pantematively,we could perform such tests using returns in excess of X0 (i.e., E&J). Of course,
both X0 and i?,, are unobservable in practice. Our reasons for performing the tests based on (4)
are discussed below.
*Among the papers that have examined the implications of the factor-pricing relation are Roll
and Ross (3980), Brown and Weinstein (1983), Chen (1983), Dhrymes, Friend, and Gultekin
(1984), Dhrymes, Friend, Gultekin, and Gultekin (1985), and Connor and Korajczyk (‘1986).
Papers that have examined the K-factor assumption underlying the theory include Gibbons (1986)
and Shanken (1987a).
B.N Lehmann and D.M. Modest,Empiricalbasisof the arbitrage pricing theory
219
subgroup,g and (iii) tests for the si@can~
of nonfactor risk measures in
explaining expected returns. Although most of the existing empi&al literature
has failed to provide sharp evidence against the theory, this body of work
suffers from a serious problem: the tests often lack the power to reject the
theory when it is false. Some of the problems with earlier tests stem from the
division of the universe of securities into small groups to perform maximumlikelihood factor analysis with Wnventional software packages.
This forced dependence on small cross-sections has two deleterious consequences. First, it results in imprecise estimates of the pricing intercept and the
factor risk premiums, which make statistid tests particularly susceptible to
Type II errors. Second, this practi~ prevents the implementation of tests that
have proven useful in the CAPM context, such as the examination of the
risk~adj~ustedreturns on portfolios sorted on the basis of some characteristic
such as dividend yield or firm size. Our maximum4ikelihood factor analysis
procedure permits us to use many securities in our examination of the APT.
3.2. Testing exact fucttw pricing
We implement +be tests uy estimating the factor loadings and idiosyncratic
variances for a large cross-section of securities and using these estimate to
construct basis portfolios to mimic realizations of the common factors. Second, we form portfolios of securities ranked on characteristics such as firm
size, dividend yield, and own variance. We then regress the returns of the
sorted portfolios on the corresponding basis portfolio returns and a constant.
The usual F test for the hypothesis that the intercepts for each portfolio are
jointly insigniucantly Werent from zero provides a test of exact factor
pricing.
To guard against potential power dithculties caused by possible nonlinearities of the dividend yield, own variance, and size effects, we consistently
perform this test on five, ten, and twenty sorted portfolios. In addition, we use
similar procedures to test the mean- var&na &&iCY & -&e~aJly-ly_wei$#@d
and valueweighted CRSP indices. The failure to reject the APT and simulta~
neous rejection of the mean-variance efficiency of the usual market proxies
suggest that our tests have power against reasonable alternatives.
3.2.1. The test statistic
Fomdy, our tests are as follows. Let a,, be the vector of excess returns on
the sorted characteristics portfolios when the K basis portfolios do not&sFan
the mean-variance frontier and be the corresponding raw returns when there
‘This involves the additional assumption that investors are risk-averse. Unfortunately, the
sample rotation of the factors may got be the same across different factor analysis runs.
Consequently, there is no prediction that the factor risk pretiums should be equal across groups
w&e kior ioadings may correspond to different rotations of the factor space.
220
B.N L.ehmmn and B. &L.X&s~~ &qvirica! busis of the arbitrage pricing theoy
is spanning. Similarly, let a,, be the corresponding vectors of excess or raw
returns on the basis portfolios (where appropriate), which are assumed to be
perfectly correlated with the factors. Consider the fitted multivariate regression
of &,, on 8,1 and a constant:
where ap is tie estimated constant term vector, 3 is the estimated factor
loading matrix, and gPt is the fitted residual vector. If there is exact factor
p&in% the basis portfolios are measured without error, and we observe fizt
when appropriate, kP should be statistically insignificantly different from zero.
On the assumption that &,, md 1, are jointly norm&y distributed random
vectors, &e usual F statistic for testing this hypothesis is
(6)
where fiP is the sample residual covariance matrix of &, h, is the vector of
sample mean returns on the basis portfolios, and e,,, is the sample covariance
matrix of their returns.
3.2.2. On the cmsttuction of excess tetwns
To construct our orthogonal portfolio we choose the N portfolio weights Use
so that they
mi.nwjDwrj
@%f
s.t.
w$bk = 0, Qk,
and
w$,, = 1,
where bk is the kth cohunn of the factor loading matrix B and D is the
diagonal matrix consisting of the estimated variances of the idiosyncratic
disturbances.‘*
The assumption that our orthogonal portfolio perfectly mimics the returns
on the true minimum-variance orthogonal portfolio can fail to hold for several
rasons. If the conjiructed orthogonal portfolio were free of ‘excess’ idiosyncratic risk but not factor risk, then the use of its ~-eturn1?,*1in computing
excess returns [i.e., replacing R,, with 82 in (4)] would still lead to valid tests
‘*This is ptisely
the portfolio for the intercept that is produced by the Fama-MacBeth style
cros~sectional regression on a constant and the factor loadings. Note that it will not be possible
to solve this programming problem in the population if Bak= dpl,that is when the ~&niting
minimum-variance portfolio contains no idiosyncratic risk and the basis portfolios spankthe
mean-variance &Sent frontier.
B. N Lehmann and D. M. Modest,Empirical basis of the arbitragepricing theory
221
of the APT mean restriction while using its mean return E&] would not? If,
on the other hand, the orthogod portfolio were free of factor risk but some
‘excess’ idiosyncratic risk remained, it would be appropriate to construct the
excess returns in relation to R,*,] rather than the actual retwns, since the
mean return would quai h, in this case.
In practice, it seems likely that the orthogonal portfolios will contain
negligible ‘excess’ idiosyncratic risk when they are well=diversSed portfolios
constructed from large cross+ections. However, since the orthogonal po~-Vinc
are constructed to have weights orthogonal to the estimated factor loadings
(but not necessarily orthog~=zl to the true loadings), some factor risk is likely
10 remain - biasing these ~&folios’ mean returns even as the number c;’
secur5ti.e-sin the cross=section grows large - leading to inference problems if
the mean return on the orthogonal portfolio is used to construct excess
returns?* Co nsequently, the results presented below for exact factor pricing
without spanning are based on excess returns computed in relation to the
actual returns on the orthogonal portfolio.13 In additioq we report summary
s’tatistics for the orthogonal portfolios and their relationship to one-month
Treasury bill returns.
3.3.
Testsfor q?anrai?lg
Pn sections 2.2 and 2.3, we noted that distinctions among exact factor
pricing versions of the APT hinge on whether the K basis portfolios span the
mean-variance frontier of the individual assets. As previously noted, it follows
‘ITo see thig asspe that actual purity returns are generated by a one-factor model:
Rit = R:c + BicR,* - R,,) + i&r where R,, is the return on the @ctor@isider the mulatjon
regression coefficients (at and bf ) from ~UIU@ the regression: R, - R, = a,? + b,+( R,, - Rz)
+ Zit, where i?;Y,is the ret= on a proxy ‘orthogonal’ pcrtfolio wKch contains some factor risk
but no idiosyn&atic risk, i.e., Rz = (1 - v)R,, + &,,,. Since Bit - 83 = (fli - r)(R,,,, - R,,) + Zfr
a@d _Rmr--_jQ = (1 - y)(i?,, - R,,). it is my to ShOW tbt b: =COV(Ri, - 82, R,,-- k,*,)/
- y) (which Is a b&ed estimate of pi U&SS y = 0), but that 4r =
“ar(Rm, - Rz) = (pi - y)\(l
E[Rit - R$] - b,*Qfimt - &I = 0. Note that subtracting the mean retum on the orthogonal
portfolio would lead to biased tests since b? would be an unbiased estimate of & but El&] + A0
?ice the portfolio contains some factor r&k.
12The i&ition behind th& result is clearest in the one-factor case. Consider Fama-B&Beth
style cross-sectional regressions of inditiduaI securitjr returns on a constant and the estimated
factor loadings. As is well knom measurement error in the independent variable (the estimated
factor loadings) will unambiguously bias tie slope term (the estiated return on the factor)
toward zero and the constant term (the estimated return on the orthogonal portfolio) away from
zero. However, as the c:oss-section grows large, idiosyncratic risk will be eliminated in the
p~flf~lio mimicking the retum on the factor. Hace, wMe &&isbasis portfoliu return will II& have
a sensitivity of one to the factor because of the bias, it ti in the limit be perfectly correlated with
the factor* which is all that is needed to test the mean restriction. Unfortunately, since the bias
in the regressioncoefficientsdoes not disappearin the limit, inferencesregarding the mean return
on the orthogonal portfolio remainproblematic.
?paSa c&k on this assumption,selectedresultsarepresew.3 for the excess-returntests, w&&e
th sample mean retwn on the or%ogonalportfoliois used to constructexcess retums.
229
B.N Lehmann and D.M. Mudest,Empiricalbasiscfthe arbitragepricing theory
from Roll (1977) and Huberman and Kandel(l987) that spanning implies that
the sum of the factor loadings must equal one, and hence that Br, = c,.
As a consequence, the spanning versus no-spanning dichotomy can be
examined by testing whether the-sum of t@ecoefficients in either (3) or (4) is
unity. On the assumption that R,, and R, are jointly normally d&?buted
random vectors, the usual F statistic for testing this hypothesis is
where again fiP is the matrix of estimated portfolio factor loadings and fiP is
the sample residual covariance matrix of the regression residuals. We perform
this test on both the excess-return and raw-return regressions to guard against
possible digerences in the two test formulations -both specifications are
appropriate under the null hypothesis that BI, = B,, and there is exact factor
pricing.
The aualysis in section 3 presupposes the existence of basis portfolios that
are perfectly correlated tith the common factors underlying security returns.
In this section, we discuss the construction of basis portfolios designed to
mimic the realizations of the common factors and the impact of measurement
erro: h the estimated factor loatigs and in the basis portfolios on our tests.
The construction of basis portfolios is a two-step procedure. In step one, the
sensitivities to the common factors are estimated for a collection of individual
securities. In the second step, these estimated factor loadings are used to form
the basis portfolios.
The primary assumption of the APT is that security returns are generated
by a K factor linear structure. Given the structure in (1) combined with the
assumptions E[$#,] = 0 and I@$#“] = 0 (a a positive definite symmetric
matrix) and the normalization of the factors so that E!!8t]= 0 and E&8;] = 1,
the covariance matrix of security returns, 2, can be written as 27= B.?? -+$2.
Theoretically, the APT places no restrictions on Q other than the requirement
that the off-diagonal elements be sufficiently sparse so that the residual risks
are diversifiable (in the limit) and, hence, security returns satisfy au approximate factor structure.i”
14The formal requirement is t&t the e’~gecv&es cf D remain bounded as N + 00.
B. N Lehmann and D.M. Modest,Empirical basis
ofthe arbitrage pricing theory
223
Chamberlain and Rothschild (1983)have shown that consistent estimates of
the factor loa&ngs can be obtained from the eigenvectorsassociated with the
K largest ezgenvaluesof the matrix T-2, where T is any arbitrary positivedeiknitemat& with eigenvalues bounded away from zero and i&&y. Standard maximum likelihoodfector zplar*;m
(m
Jyw \mder
the normahty assumption) is
numericakI +tivalent
qa
to calc$aring the largest K eigenvectorsof the matrix
T-!S, where T is set equal to D, which is a diagonal matrix consisting of the
estimated residual variances and S is the sample covariance matrix.r5Although this is a conceptuallysimple exercise,it is computationallyinfeasible to
obtain these estimates by iteratively solving the tit-order conditions when the
number of securities being analyzed is substantial. We employ a signifkantly
cheaper alternative: the EM algorithm of Dempster, Laird, and Rubin (1977)
applied to factor analysis in Rubin and Thayer (1982).
A variety of methods can be used to construct portfolios to mimic the
factors. The niost commonly used procedure involves treating the factor
sensitivities &) in (1) as explanatoq variables, gi, - Ei as the dependent
variables, and *thefactor realizations (I&J as parameters to be estima%d, and
running cross-sectional regressions perid by period along the lines of Fama
and IMacBeth(1973).Given the true factor loa&ngs (B) and the true idiosyncratic covariance matrix (a), the generalized least squares version of this
estimator [i.e., portfolio weights given by (B12-1B)-1M2-1] provides the
minimum-variance linear unbiased estimate of the factors. In practice, all APT
applications that we are aware of replace Q with a diagonai matrix consisting
of estimates of the idiosyncratic variances, thereby ignoring the .ofMiagonal
elements. Hence, the procedure is more accurately referred to as a weigh&d
least squares (WLS) procedure.
Unfortunately, the true factor loadings and the *&ueidiosyncraticcovariance
matrix are not known and estimates must be used in constructing these
portfolios. The presence of this measurement error reduces the population
correlation between these WLS basis portfolios and the common factors and,
in fact, alternative basis portfolio formation techniques may have higher
likelihood function: S(ZiS) = (-- NT/2)ln(2rrr) “Formally, this involves maxim@@e
(T/2)lx@( - $Xr_,(& - @‘P(R,
- R). Principal components involves calculating the corresponding eigenvectors of the matrix T-IS with T set equal to a~ identity matrix. Asymptotically,
both methods provide the same estimated factor loading up to an arbitrary rotation. Unfortunately, the relative small-sample properties of the two estimation procedures are unknowrz,
assuming only an approximate factor structure holds. Connor and Korajczyk (1986) prove the
consistency of the factor estimates (as N + GU)of the principal components estimator. A similar
proof can be used to show the consistency of maximum likelihood factor analysis with T set equal
to a diagonal matrix of estimated idiosyncratic variances with elements bounded away from zero
and intinity.
224
B.N Lehmann
D.&i. Mdest,
basis of
arbitragepricing
population correlation with the common factors? This is possible because the
WLS procedure tends to give greater weight to security returns associated with
large estimated factor loadings and typically downweights those with small
loading estimates. This is appropriate in the absence of measurement error,
since the returns of securities with large factor loadings are more informative
the WLS ;++hg
prw
about fhictuations in the common factor
dure is less approtxiate when the sample loadings reflect measurement error in
addition to the t&e loadings.
We employ a method we refer to as the minimumidbsyncraticrisk procedure
as an aluznative to the WLS potiolio formation procedure. In particular, our
procedure involves choosing the portfolio weights 7 which solve
minw;Dwi
wJ
s.t.
w;bk =O,Vj#k,
and
+=l,
where 1c(i’bi
is unrestricted.17 These portfolios are similar to the WLS ones in
that they minim&e the sample idiosycratic variance of the basis portfolios
subject to the constraint that the weights be orthogonal to the sample loadings
of the factors not being mimicked [i.e., w;bk = 0, Vj i k].‘8 ‘I%~d8brence
between the two procedures lie in the requirement that the WLS portfolios
have a sample loading of unity on the factor being mimicked (before normal=
ization to unit net-investment), whereas the minimum idiosyncratic risk
portfolios must simply cost a dollar. As a consequence, the minimum idiosyncratic risk procedure largely ignores the information in the factor loadings: a
16’Ihisis formally shown fcs the case of a single common factor in Lehmann and Modest
(1985). One other problem with the WLS procedure is that in factor model estimation it is
conventional to normalixe the factors so that they are uncorrelatedad have unit variances and to
normal& the factor loadings so that B’D’IB 2s &god.
lfhis practice yields typical factor
loading estknates that are much less than one - on the order of 0.001 to 0 9901 in daily data A.: a
co~uence,
the WLS procedure must place large positive and negative weights on at least some
see&ties to ensure both that 9% = 1 and $bk = 0, Vj # k. For instance, we have found that the
WLS procedure coupkd with the conventional normalization of the factor model typically
produces portfolio weights in excess of 100% in absolute vale, yielding poorlydiversified
reference portfolios. The evidence presented in Lehmann and Modest (1985) suggests that the
minimum idiosyncratic risk proceduredkussed below performed at least as well as (and usually
better than) its competitors.
“This minimum idiosyncratic risk estimator for the jth factor is D-LB*[B4’D-‘B*J-iej, where
B* =(b,q...r
. . . bk), 1 is a vector of ones in the jthcohrmnand ei is a vector of zeros except for
a one in the jth position fn large crossIsections, this procedure can be shown to produce valid
mimicking portfolios in the sense of proposition 1 of Huberman, Kandel, and Stambaugh (1987).
A proof is available upon request.
‘*As pointed out by Litzenbergerand Ramaswamy (1979) and Rosenberg and Marathe (1979),
the WLs estimator is equivalent to choosing the N portfolio weights q (to mimic the jth factor)
so that they min w/D9 subject to qbk - 0,Vj + k, and BJ%~
- 1, Vj - k, and then normalizing
the weights ?5“&c they sum to one,
B.N Lehmann ad D.M. Mod&t, Empirical b&s of the erbitmge prizing theory
225
bad decision in the absence of meassurement error and a potentially
choice in its presence.19
We emphasize that the distinction between the minimum idiosynaatic risk
procedure and the WLS method affects only the results obtained for the
raw-fetum verson of the model *muse of our use of constructed orthogonal
portfolios to create excess returns. Wt mte the excess returns on the
mimicking portfolios by subtracting the retbm on the orthogonal portfolio
from the returns on the minimum idiosyncratic tisk potiolios. These excess
returns are identical (up to a factor of proporti,?nality)to the coefficients
obtained from the WLS cross-sectional regression 3 individual security returns on the factor loadings and a ve$.?;ccG<
: : : a-- T’
-.s e&e rs&s
&&&d
for the excess+etm version of the
*,~2ldon the Merences in
these procedureP
4.3. On the impact oj?n4wsure~dnten-or
A critical assumption in our tests of exact factor pricing is that the bak
portfolios span the factor space and contain no idiosyncratic risk, an assumption that is literally correct only as the number of securities in the cross-section
tends toward infinity. To the extent that this assumption is violated, the
intercepts in (5) will be biased away from zero when the APT is true, yielding
F statistics that are biased toward rejection. Fortunately, although measurement error in the estimated factor loadings will tend to lead to biased
estimates of the individual factors, linear combinations of the K bask pw+folios will still tend to span the factor space. Furthermore,as long as the basis
portfolio weights are we&diversified (i.e., of order l/N), the basis portfolios
will contain minimal idiosyncratic risk.21Hence9in the limit, the use of the K
basis nortfolios leads to valid tests of the APT mean restriction even in the
presence of measurement error in the factor loadings.
We eschew an alternative strategy that would partially mitigate the effecd of
measurement error in the estimated factor loadings. In particular, we-could
have estimated the basis portfolio returns by straightfonvard application of
the Fama-MacBeth style cross-sectional regresfiions(at each date t) of the
19The two strategies yield basis portfolios with very different diversification properties in actual
practice. The average sum of squared basis portfolio weig#.s (per factor) using the minirn~um.
idiosyncratic risk procclure is 0.016 for the five-factormodd, 0.022 for the ten-factor rn~&A,and
0.025 for b&efiftce&~_ctor model, GE c&c to -the minimum attainable sum of 0.00213. By
contrast, the corresponding averages for the WLS procedure coupled with the conventional
normalization of the factor model are 0.659, 72.689, and 2SS.234pclearly far from the minimum
attainable. Footnote 16 discusses why this occurs.
2oSee Lehmann and Modest (1985) for a detailed analysjs of this equivalence in the one-factor
case. This relation between the two procedures, however, dims not obtain when measured riskless
rates are used instead of the constructed orthogonal portfoE returns.
“a two-factor example is available on request.
226
B.N Lehmannand D.M. Modest,Empiricalbasisof the arbitragepricingtheoly
sorted portfolio returns on a constant md their estimated factor loadings. The
analog to hp in (5) could then be computed from the time series mean of the
portfolio residuals. This approach would tend to alleviate the measurement
ahave. since the measurement error in the factor
error problem discuss,,@A-u..+
loadings of the sorted portfolios should be much smaller than the typical error
in the loadings of the individual securities as long as the portfolios are
well-diversified and not formed on the basis of their sample loadings.
Vv’eforego this seemingly superior statistical procedure because it involves
estimating the factor risk premiums using the portfolios formed from wellknown emtGrical anomalies. Suppose that the APT is false and we construct
the test stitistics in this revised fashion. The cross-sectional regressions will
choose estimates of the factor risk premiums that minim& the weighted sum
of squared residuals (i.e., tend to fit the anomalies). This, in turn, will tend to
make the F statistic small, and hence can cause a failure to reject the null
hypothesis when it is false. In our procedure, we estimate the factor risk
premiums from the whole sample of securities underlying the factor analysis, a
sample that is not biased with regard to firm size, dividend yield, or own
variance. These premiums are then used to estimtite “iP and to test its
significance.
5.I. Data considerations
The CRSP files provide both daily and monthly equity returns. The potential benefit associated with the use of daily data in the estimation of variances
a9.d covariances is enormous, since the precision of these parameter estimates
hinges on the frquency of observation. Of course, daily data has the wellknown probiems of asynchronolus trading, which bias the estimates of second
moments, and the bid-ask spread, which bias the estimates of first moments.
Moreover, daily data provide no advantage when estimatiag mean returns
whose precision depends on the length of tl& estimation interval.
In choosing an observation frequency, we opt for a comprotise solution.
Following Roll and Ross (1980) and most subsequent empirical investigators,
we estimate out factor models fd>rsecurity returns with daily data, since we
surmise that the ga’%nin precisioii offsets the thin trading biases in the
estimation of covariance matrices .** We test the theory and its various aspects,
however, using weekly returns data, formed by compounding daily returns
%sa check,we also pfesd t resultsbasedon factormodels estimatedwithweekly arndmonthly
retUfUS.
B.N Luhmannand D. M. Modest, Empirical basis of the arbitragepricing theory
227
from Wednesday k~ Tuesday. 23 Consequently, basis portfolio returns are
computed by multiplying the portfolio weights by the corresponding weekly
returns on individual securities. The weekly returns on the CRSP qua,@weighted and value-weighted indices are compttted by compound&g th&
daily returns in the same fashion. Thus our market proxies contain both New
York Stock Exchange (NYSE) and American Stock Exchange (AMEX) securities. Excess returns, when needed, are computed in relation to the orthogonal
portfolio returns. All of the relevant test statistics are constructed with these
weekIy returns with one exception.” The tests comparing the returns on the
orthogonaI portfolios and Treasury bibs are performed with monthly data,
since ‘Zeasury bills with one week to maturity are not actively traded and
hence reliable weekly interest rates cannot be obtained.
Two other important choices involve the length of the estimation interval
and which firms to include in our ample. We assume stationarity over
five-year subperiods and divide the time interval covered by the CRSP daily
returns fi3e into four periods: 1963-1967, 1968-1972, 1973-1977, and
1978-1982. Within each period, we exclude securities that are not continuously listed or which have missing returns and ignore the possible selection
bias inherent in this strategy. The remaining securities number 1,001, 1,359,
1,346, and 1,281 in the four periods. The number of daily observations in these
samples totals 2,259, 1,234, 1,263, and 1,264, respectively, and there are 260
weekly observations in each five-year period. The CRSP daily file (with few
exceptions) lists securities in alphabetical order. We randomljf reorder the
securities in each subperiod to guard against any biases induced by the natural
progression of letters (IBM, International Paper, etc.). The usual sample
~variance mmix of these security returns provides the basic input to our
subsequent anr~ysis. Bach period we estimate five-,, ten-, and fifteen-factor
models using the first 750 securities in 3ur randomly reordered data file.
5.2. Tests of t!k APT mean restriction
Our strategy for testing the AFT involves examining the ability to the theory
to account for well-documented empirical anomalies that provide the basis for
rejecting the mean-variance efficiency of the usual market proxies. ‘rabies 1
through 6 provide tests based on three such anomalies: firm size, dividend
23T~ be more precise, our weeks begin on the fitst trading day after Tuei ‘J (usually
Wednesday)and end on the last tradingday priorto Wednesday(usuallyTuesday).We madethis
choice because &there
are fewer tradingholidayson Tuesdaysand Wednesdaysand to mitigate
biasescausedby the day-of-the-week
eflrect..
Also, it was sometimesnecessaryto dropobservations
at the beginning and end of our fiveaye= subperiodsto insure that within each subperiodour
weeks began on Wednesdaysand ended on Tuesdays.
“We i&Fated rmostof our tests SBzxx&E~+I
,a ta and verifiedthat the conclusionisreportedhc1=e
are robustwith respectto this choice. We reportthe weeklyresultsbecauseof the potentialgain
associatedwith more pwdbclestimationof the residualcovariancematrix.
228
1%N khtnann ad D.iU. Mode@, Empirica! basis of the arbitragepri&g theory
I
I
,
I
,
r
r
c
5
Y
I
B.N khmmn lad D.M. Mdest, Empiricalbasisof the arbitragepric@g thewy
229
3.18
(0.01)
value
Wt?igh!d
0.62
(0.63)
3.18
(O+Ol)
1.25
(0.29)
9.17
’
9.17
(0.62E-O.6) (0.62E-06)
proxies
(K-= 1)
W&ly
(0.63)
0.62
1.25
(0.29)
Monthly
1968-1972
Moz~tbly
We&y
J%dY
W4figbd
.
marLa
CRSP
Pricing relation
1963-l&7
(0.02)
3.12
3.28
(0.01)
Weekly
(O&2)
3.12
3.28
(0.01)
Monthly
1973-1977
F statistics for five-yearsubperiods
(p-values are given in parenth~55s)~
Ma&&
(0.02)
2.96
(0.02)
2.96
6.50
6.50
(O.S4E-04) (OS4E-04)
Weekly
1978-1982
y#&..y
39.60
(0.89E-03)
r.r - - .._.MOKlUii~
(0.89E-03)
39.60
80.77
(O.l2E-09)
196=82
80.77
(O.l2E-09)
-I.
Aggregate &i-squared
statistic (p-value)”
Tests are performed oti tie quail y-we&k&d portfotios of NOSE and AMEYCsecurities group&
_- on the basis of the market value of equity. F statistics
for four subpetiods and aggregate cl&squared statistic for the ehe period for the joint hypothesis that ui = 0 (i = 1,. . . ,5) in’ the pricing relation
Ei - X0 = ai + Z&lbjh X,; where I$ is the expected return OIP&e portfallio
i,6ik is the seusitivity of portfolio i to factor k, X, is the risk premitim of
thz k*& factor, and a0 is the pricing inter&@. The factor pricing models allow for five and ten factors (K = $10) estimated from a preliminary
analysis of the weekly and monthly returns of 750 randomly selected securities.~ The mean-variance efficiency tests use the CRSP equally-weighted
and vahe~weighted indices (K = 1). F tests use 260 weekly observations and ci&squa~& tests vm 1,040 weekly ob3?:rvatidns.
the interval 1963-1982.
Size-based tests of exact factor pricing models (APT) using different factor estimates and of the mean-variance efficiency of CUSP market proxies in
Table 2
Excessreturns
(A, # 0)
Raw returns
(A, = 0)
(K-5)
Tenfactor
APT
(K=lO)
4.58
(0.52E-03)
3.10
(0.99E82)
3.30
(0.67E-02)
3.07
(0.01)
2.89
(0.01)
1.77
(0.12)
2.80
(0.02)
2.00
(0.08)
4.20
(O.llE-02)
3.45
(0.49E-02)
3.61
(0.36E-02)
2.26
(0.05)
2.47
(0.03)
2.83
(0.02)
2.32
(0.04)
69.95
57.10
(0.19E-06) (0.2OE-04)
3.38
(0.57E-02)
2.99
(0.01)
73.85
57.50
(0.42E-07) (0.17E-04)
74.25
3.52
3.49
52.60
(0.44E-02) (0.47E-02) (0.34E-07) (0.93E-04)
4.21
(O.llE-02)
81.25
63.20
4.82
4.04
4.32
3.31
3; (W6E-02) (0.31E-03) (O.lSE-03) (0.24E-08) (0.23E-05)
(0.86E-O
‘The estimated factor model is used to compute portfolio weights which are combined with the weekly returns of the 750 securities to produce
weekly estimates of the facto=
bEach F statisticis for the test of the joint hypothesis that ui = 0 in the appropriatepricing relation. For the API’ tests, the ~~~ahteis the probability
of obtaining a realization greater than the test statistic from an F distribution with N numerator degrees of freedom and 235 - K - N denominator
degrees of freedom. For the CRSP mean-variance efficiencytests, the p-value is the probability of obtaining a realizationgreater than the test statistic
from an F distribution with N - 1 numerator degrees of freedom and 235 - N denominator degrees of freedom.
‘Each x2 statistic for the ste
time period is N times the sum of the four subperiod F statistics.The pvalue umber ee& x2 %idic is the
probability of obWung a realization greater than the test statistic from a x2 distribution with degrees of freedom equal to 20 when N = 5 and 80
when N = 20 for the APT tests and with degreesof freedom equal to 16 when N = 5 and 76 when N = 20 for the CRSP mean-variance efficiencytests.
3.36
3.72
(0.59E-02) (0.29E-02)
280
(0.02)
3.47
(0.48E-02)
3.32
3.29
3.79
(0.64E-02) (0.68E-02) (0.2SE-02)
Excegsreturns
.t 37
(0.4&2)
(A, + 0)
Raw returns
(%I = 0)
Five
factor
Raw returns
I;?‘,*
= 0)
Excess returns
& # 0)
W= 5)
Weighted
Value
WiSightd
W&Y
Five-
proxies
W= 1)
-ZR$P
Pricingrelation
1.76
(0.12)
(O*
W
OH
0387
(0.6q
I.08
(0.37)
1.98
(0.10)
1.37
(0.24)
1.94
(0.01)
Iv=20
6.15
(O.lOEAI3)
N==5
196301%7
1.22
(0.30)
2.01
@08)
0.20
(0.94)
0.26
(0.91)
N=5
1.12
(0.33)
1.54
(0.07)
1.51
(0.09)
1.40
(0.13)
N=2O
1968-1972
2.12
(O-06)
1.47
(0.20)
1.56
(0.19)
0.87
(0.48)
N=S
1.12
(0.33)
0.93
(0.55)
1.01
(0.45)
0.76
(0.75)
N=20
1973-1977
F statistics fez five-yearsubperiodsb
(~Va&E!sare givez k pSeA!ses)
3.45
(0.005)
3.56
(O*OW
2.38
(0.05)
4.44
(0.002)
N=j
1.07
(0.39)
I.13
(0.32)
0.79
(0.72)
1.36
(0.15)
N=20
1978-1982
42.76
(0.002)
42.08
(0.003)
30.57
(0.06)
46.83
(0.72E-04)
N=S
h%3-1982
84.89
(0.33)
89.42
(0.22)
83.35
(0.26)
103.77
(0.02)
N--20
Aggregate&i-squared
statistic (p-value)=
Size-based tests of exact factor pricing models (APT) and of the mean-variance efficiency of CRSP market proties in the interval L963A982,
excludkg January retums.
portfolios of WSE and AMEX secmitks grouped on the basis of the
Tests are pe&med 0~ tie (N = 5) and twenty (N = 39) qy_TMeighted
market value of equity. F statisticsfor four subperiods and aggry te &i-squared statisticfor the entire period for the joint hypothesis that ai = 0
(i=l,..., Sori==l,..., 20) in the p&&g relatisn Ei - A, = Ui+ xk_lbikXk, where Ei is the expected return 011 S& portfolio i, bik is the seasititity of
potiolio i to factor k, A, is the risk premium of the kth factor, and X, is the pricing intercept. me factor pricing models allow for five, ten, and
fifteen factors (K = $10,
15)estimated from a prelimimuy factor analysis of the daily ret~tns of 750 randomly selected securitka The mean-variance
efficiency tests use the CRSP equally-weightedand valueweighted indices (K = 1). F testsuse 260 weekly observations and &i-squared tests use 1,040
weekly 0bservatkMUL
Table 3
Excessreturus
(X0# 0)
(K=lS)
1.89
(0.10)
206
(0907)
1.81
(0.11)
206
(O-07)
0.81
(0.71)
0.85
(0.65)
0.88
(0.61)
0.95
(0.53)
1.23
(0.29)
1.01
(0.41)
1.80
(0.11)
1.68
(0.14)
1.04
(0.41)
1.17
(0.28)
1.26
(0.12)
1.37
(0.14)
1.08
(0.37)
0.87
(0.50)
1.12
(0.35)
0.89
(0.49)
0.74
(0.78)
0.70
(0.82)
0.87
(0.63)
0.78
(0.73)
2.21
(0.W
2.52
(0.03)
2.29
(0.05)
2.22
(0.05)
0.78
(0.73)
0.92
(0.57)
0.78
(0.74)
0.n
(0.74)
32.96
(0*04)
32.26
(0*04)
35.14
(0.02)
34.24
(0.03)
67.47
(0.84)
72.79
(0.70)
75.66
(0.62)
77.55
(0.56)
‘The estimated factormodel is used to computeporGotioweightswhich are combinedwith the weeklyretums of the 750 securitiesto product
weeklyestimatesof the factors.
bEach F statisticis for the test of thejoiut hypothesisthat a, = 0 in the appropriatepricingrelatiort.For the APT tests,the p-valueis the probabiity
of obtai&g a C
grcatcrthau the test statisticfrom au F &&ibuhn
with N numeratordegreesof freedomaud 260 - X - iVdeuominaiwr
degreesof freedom.For the CIUP mean-variauceefficiencytests,the p-valueis the probabilityof obtainiuga reahzatiortgteaterthau the test statistic
from au F &tribution with N - 1 uumeratordegreesof freedomaud 260- N deuomiuatordegreesof freedom.
%a& x2 statistic for the aggregatetime periodis N times the sum of the four subperiodF statistics.The p-value under each x2 statisticis the
probabilityof obtaiuiuga realizatiougreaterthau the test statisticfrom a x2 distributionwith degreesof freedomequal to 20 when N = 5 and 80
when N = 20 for the APT testsaud with degreesof freedomequalto 16 when N = 5 and 76 when N = 20 for the CR!Wuxau-variauccetRcieucytests.
Raw Mums
(ho = 0)
Excessreturus
(ho # 0)
=(K= 10)
Fifteeufactor
Raw returus
(A0= 0)
T-0
factor
1.47
(s.20)
1.44
(0.21)
Excess returns
(h.0f 0)
W= 5)
0.92
(0.56)
0,91
(0.57)
1.13
(0.32)
2.18
(MU)
Raw re&cns
(X0 e 0)
=&*@i
FivefW
2.40
(0.001)
9.54
(0.33M)
V&R
Equally
Weighted
N=20
N=5
1%3-1967
CRSP
market
proxies
&x= l>-
Pricing relation
,
0.80
(0.55)
1.70
(0.14)
0.26
(0.90)
0.96
(0.43)
N=5
0.62
(0.89)
x.11
(0.35)
0.44
(0.98)
0.55
(0.94)
N=20
1968-1972
1.76
(0.12)
1.56
(0.17)
2.64
(0.04)
3.43
(0.009)
N=5
0.73
(0.79)
0.80
(0.71)
1.03
(0#43)
x.21
(0.25)
N=20
1973-1977
N=20
N=s
Iv=20
1963-1982
1.20
(0.31)
0.77
(0.57)
2.46
(0.05)
27.48
(0.12)
25.99
(0*17)
0.77
(0.75)
30.15
(0.02)
0.69
(O-84)
1.02
(O-44)
61.01
(0.94)
70.11
(0.78)
68.72
(0.71)
127.78
2.56
86.08
7.58
(0.88E=O5) (0.5lE=O3) (O.lSE~lO) (O.l9E-03)
N=5
1978-1982
Tests are performed ORfive (N = 5) and twenty (N = 20) equally~weightedportf&os of NYSE and AMEX sec~nitiesgrouped on the basis of
divide~~&y.i&!.~F statistics for four subperiods ad aggregate&i-squared statistic for the entire period for the joint hypothesis that ai = 0 (i = 1,. . . ,5
ori= ,...,20) ia thepzicing relation Ei - X0 = Ui+ Zf-lb,X, 9where Ei is the expected netwn OII dividend-yield pottfolio i, bik is the sensitivityOf
portfolio i ta factor k, A, is the riskpremium of the kth factor, and X0 is the pricing intercept. The factor pricing models allow for five, ten, and
fifteen factors (# = 5, IO,15) estimated from a preliminary factor analysis of the daily returns of 750 randomly selected securities! The mean-variance
eficiency tests use the CRSP eq~~IIy~w@hted
and value-weightedindim (K= 1). F testsuse 260 weekly observations and &i-squared tests use 1,040
weekly observations.
-r
Aggregatechi~squared
F statisticsfor five-yearsubperiods
statistic ( pqvalue)d
(~~a&s are given in parentheseQc
tests of exact factor pricing models (APT) and of the mean-variance efficiency of CRSP market proxies in the interval
1963-1982.
Table 4
Excess returns
(K= 10)
1.12
(0.33j
1.15
(0.30)
1.12
(0.33)
1.19
(0.26)
0.48
(0=79)
0.64
(0.67)
0.58
(0.72)
1.10
(0.36)
0.52
(O*%j
0.75
(0.77)
0.54
(0.95j
0.96
(0.52)
1.15
(0.33)
1.07
(0.38)
0.97
(0~4)
0.85
(0.52)
0.51
(00%)
0.54
(0.95)
0.55
(0.94)
0.55
(094)
1.05
(0.39)
1.56
(0.17)
I.12
(0.35)
1.58
(0.17)
0.64
(0.88)
0.68
(O-85)
0.67
(0.85)
0.70
(0.83)
25.96
(0.17)
29.62
(0‘08)
25.54
(0.18)
31.01
(0%)
55.59
(0.98)
62.32
(0.93)
57.69
(0997)
67.87
(0.83)
“The Grstportfolio is an equally-weighted portfolio of those stocks that paid no dividends k the preceding period. The remaining four or nineteen
portfolios are equally-weighted potiolios of the remaining stocks ranked by their ditidend yieid.
bThe estimated factor model is used to compute portfolio weights which are combined with the weekly returns of the 750 securities to produce
weekly estimates of the factors.
‘Each F statistic is for the test of the joint hypothesis that q = 0 in the appropriatepricing re&&on.For the AH’ tests, the p_Vatueis the probabihty
of obtaining a realization greater than the test statistic from an F distribution with N numerator degrees of fr&om and 260 - K - N denominator
degrees of freedom*For the CRSP mean-variance efficienqf tests, the pevabte is the probability of obtaining a realization greater than the test statistic
from an F distribution with _WY 1 numerator degrees of freedom and 260 - N denominator degrees of freedom.
dEach x2 statistic for the aggregate time period is N times the sum of the four subperiod F statistics. The p-value under each x2 statistic i;i;the
probability of obtaining a r&Jization greater than the test statistic from a x2 distribution with degrees of freedom equal to 20 when N = 5 and 80
when N = 20 for the APT’tests and with degrees of freedom equal to 16 when N = 5 and 76 when N = 20 for the CRSP mean-variance efficiency tests.
(&I + Oj
(0.03)
Eluxssretums 2.51
(K=15)
2.65
(0.02)
Raw returns
(k, = 0)
2.44
(0.04)
2.67
(0.02)
Fifteenfactor
(&I if 0)
Raw returns
(A, = 0)
Tenfactor
tests of exact factor pricing models (APT) and of the mean-variance efficiencyof CmP market proxies in the interval 1963-1982.
Excessretums
(ho 90)
(K=5)
1.70
(0.14)
1.70
(0.13)
1.55
(0.07)
1.55
(0.07)
1.82
(0.02)
3.27
(0.01)
Weighted
xaw returns
(A, = 0)
N-5
0.35
0.53
(0.75)
1.38
(0.23)
(O&Q
0.47
(0.98)
0.77
(0.75)
(0.73)
0.78
0.93
(0.55)
@‘=a
1968-1972
2.75
1.14
(O.l9E-Q3) (0.34)
Value
9.05
N=2!!
(0.7FM6)
Q&Y
N=5
Wt!igh&!d
Five
factor
(K=l)
market
proxies
CRSP
Pricing relation
1%3-185%
1.72
(0.13)
1.55
(0.18)
(0.19)
1.56
2.39
(0.05)
N=5
0.76
(0.76)
0.75
(0.77)
(0.49)
0.98
1.10
(0.35)
N=2(1,
1973-1977
F statistics for five-yearsubpetiods
(Pvapues are given ti ytientk~~$~
0.65
(0.66)
1.15
(0.34;
(0.05)
2.36
6.86
(0.3OE-04)
--i-=5
0.62
(0.90)
0.70
(0.82)
(0.43)
1.03
1.95
(0.01)
N,20
1978-1982
22.%
(0.29)
28.88
(0909)
30.17
(0.02)
81.71
(0.82E-10)
N35
67.98
(0.83)
75.50
(0.62)
(0.18)
87.38
127.94
(0.18Eo3)
N=20
1%3-1.982
Aggregatechi-squared
statistic (p-value)=
Tats are performed on five (N= 5) and twenty (N= 20) equaUy=weighted
portfolios of NYSE and AMEX securities grouped on the basis of
owxwarhce.
F statisticsfor four wbpekds and qgreg ate chi~squaredstatistic for the entire p&od for the joint hypothesis that ai = 0 (i = 1,. . . ,5
or i--l ,...,2O)inthepricingrelati~
Ei-$j=$d_Zkp1
Qikx k, where Ei is the exgeC&dMUrn OnOW!kV&B
POdOfiO
is bik & &e S#Zd~V@ Of
partfolio
i t0 factor k, xk is the risk prermutu of the kth factor, and ho is the pricing intercept. The factor piking models aNow for five, ten, and
fifteen factors (K = S,lO,15) estimatt from a prelimimuy factor ana@is of the daily retums of 750 randomly selected securitk~~ The mean-variance
efficiency tests use the CRSP equaUy=weighted
and value-weightedindices (K= 1). F testsuse 260 weekly observations and chi=sqnaredtests use 1,040
weekly observations.
0wnwuianF’-based
Table 5
.
1.90
@.iO)
1.96
Excessreturns
(WO)
(K= 10)
Excessreturns
(A, # 0)
factor
(K==
1.81
(0.11)
(O-09)
1.50
1.28
(0.19)
1.35
(0.15)
1.39
(0.13)
(0.08)
0.74
0.67
(0.65)
0.73
(0.60)
0.48
(0.79)
(0.60)
0.60
0.54
(0.95)
0.62
(0.90)
0.48
(0.97)
(0.91)
0.91
(0.47)
0.63
(0.68)
0.89
(0.49)
0.51
(0.77)
0.50
(O.%)
0.50
(0.97)
0.52
(O.%)
0.48
(0.97)
0.32
(0.90)
0*25
(0994)
0.39
(0.85)
0.36
(0.88)
0.48
(0.97)
0.56
(0.93)
0.49
(0.97)
0.49
(097)
18.54
(0.55)
17.84
(0.60)
18.30
(0.57)
18.80
(0.53)
56.18
(0.98)
60.42
(0.95)
57.75
(0.97.)
61.20
(094)
VIuz estimated factor mode1is used to computeportfolioweights which are combinedwith the w&y returnsof the 750 securitiesto produce
weekly estimatesof the factors.
%a& F statistic is for the teat of thejoint hypothesisthat a,- = 0 in the appropriate
pricingrelation.For the APT tests, the p-valueis the probability
of oWaining a realization greaterthan the test statisticfrom an F distibution with N numeratordegreesof freedomand 260 - K - M denominator
degreesof freedom.For the CRSPmean-varianceefkkncy tests,the p-vatueis the p_&&GIityof obtaininga realizationgreaterthsn the tet statistic
fkxu an F distributionwith N - 1 numeratordegreesof freedomand.260 - N denominatordegreesof freedom.
‘Each x2 statistic for the aggregatetime periodis N times the sum of t?e four subperiodF statistics. The p-value under each x2 statisticis the
probabilityof obtaining a realizationgreaterthan the test statisticfrom a xt distributionwith degreesof freedomequal to 20 when N = S and 80
when N = 20 for the APT tests and with degreesof freedomequalto 16 when N = 5 and 76 when N = 20 for the CRSPmean-varianceefkiency tests.
15)
Raw returns
(A, = 0)
Fifteen-
(0%)
(&) - 0)
2.15
Raw r,4%urnn
svaw=
factar
clr-
238
BAAILehmann and D.M. Modest, Empirical twis of the arbitrage pricing thmy
Table 6
Tests of exact factor pricing models (APT) and of the mean-vtiance efficiencyof CRSP market
proxies in the interval 1963-1982, omitting securitks used to estimate the preliminq
factor
Tests are performed on five equally-weighted portfoxos & NOSE 2nd AMEX securities grouped
on the basis of the market v&&~~f eq&y (Size), dividend yield (Div. yi&I), and own variance
(Own var.). F statistics for four subperiods and aggregate c&qua&
statistic for the entire
period for the joint hy@ds
that g-m0 (i-1,...,5)
in the pti~@ &ti~n Ei-X,mai+
zf- 1bikXk9whm IZ is the errpectedanUn 011sorted PortfoliOs’,bik iS aheS&!SJI OfpOdoK i
to factor k, A, is the risk pre&m of the kth factor, ad X0 is the pricing intercept. The factor
pricing models allow for five, tea. and Mteen factors (K = 5,10,15) estimated from a pAimimuy
factor analysis of the daily returns of 750 rant amly selected securitka The mean-variance
efficiency tests use the CRSP equally-weighted and value-wei@ed indices (K= 1). F tests use 260
weekIy *ations
and chi-4uared &s& alce1,040 weekly observations
F statistics for fin+year subperk&
(JMN@esaregi!&qMuentbeses)b
1%3-1%7
Pricingrelation
t”Rsp
market
Epuauy
1973-1977
1%~1972
size
Div.
yield
own
var.
Size
Div.
yield
own
var.
!Sze
Div.
yield
own
var.
4.69
4.67
5.47
1.24
0.24
0.67
2.80
2.81
1.87
wreighted
(0.001) (0,001) (0.31E-03) (0.29) (0.91) (0.61) (0.03) (0.03) (0.12)
value
weightted
279
(0.03)
1.18
(0.32)
Fivefactor
Rawretums
@PO)
1.40
(0.23)
1.67
(0.14)
1.32
(0.25)
1.23 1.54 0.63
2.56 0.73
1.28
(0.03) (0.60) (0.30) (0.18) (0.68) (0.27)
(K==5)
Excess
1.37
(0.24)
1.59
(0.16)
1.35
(0.25)
0.47
1.80 0.29
1.83 0.73
1.31
(0.11) (0.92) (0.80) (0.11) (0.61) (0.26)
Tenfactor
Raw returns 1.49
(0.19)
(A, = 0)
1.61
(0.16)
1.49
(0.19)
1.97 0.38
0.84 0.77 0.36
0.36
(0.08) (0.86) (0.52) (0.57) (0.88) (0.88)
(K-10)
Excess
1.43
(0.22)
1.40
(0.22)
1.49
(0.19)
0.39
1.03 0.32
0.49
1.92 0.18
(0.09) (0.97) (0.86) (0.40) (0.90) (0.78)
Fifteenfactor
Raw returns 1.26
(0.28)
(&I = 0)
1.39
(0.23)
1.23
‘(0.30)
1.41 0.24
0.46 0.78 0.40
0.45
(0.22) (0.94) (0.80) (0.56) (0.85) (0.81)
(K=15)
Excess
returns
& + 0)
1.25
(0.28)
1.23
(0.29)
0.34
I.46
0.20
1.08 0.33
0.53
(0.20) (O.%) (0.89) (0.37) (0.89) (0.76)
pties
(K=l)
return!!!
(A, * 0)
I.25
(0.29)
0. ,;
257
2.51
1.46
4.05
1.08 0.37
(0.33E-02) (0.37) (0.83) (0.58) (0.04) (0.04) (0.21)
Table 6 (contiaued)
F statistics
for five-yearsubpeziods
(pvaluesare~veninparentheses)b
Aggre&ate**qu
statistic (JMhi&e)=
Pricingrdation
CRSP
market
proxies
W= 1)
Fivefactar
A?T
<X=5)
Tenfactor
(K=lO)
EBualtY
5.06
(0.6lE-03)
4.73
(0.001)
5.47
(0.3lE-03)
Valut
*ted
1.$5
(0.12)
1.98
(0.10)
2.30
(0.W
33.19
(O.oar)
24.11
(0.09)
34.12
(0.52E-u2)
1.41
(0.22)
0.68
(0.64)
0.79
(0.57)
34.35
(0*02)
18.54
(0.55)
23.13
(0.28)
0.35
(0.58)
0.64
(0.63
0.53
$H2)
28.33
(0*09)
16.20
(0.V
(0.55)
1.03
WO)
1.34
(OJ%
0.63
(0.67)
z&34
(0.15)
to-=)
16.58
(0.68)
0.76
(O.f8)
0.92
(0.43
_
0.42
(0.83)
25.72
(0.18)
IBlO
(U.83)
13.95
(0.53)
1.13
(0.34)
1.28
(0.27)
0.48
(0.79)
22.92
(0.29)
16.56
(0.68)
13.13
@=87)
0.81
(0.W
0.85
(0.52)
0.34
(0.89)
23.02
(0.29)
13.17
$kq
12.21
(Q.91)
wei@ed
RaWrcturns
&-0)
EXoesS
RaWrrtunrs
(X0 e 0)
Excess
fetums
(h#o)
Fifkelb
fix&M
@,=O)
(K==lS)
Euzess
RaWmWllS
55.19
49.86
53.92
(0.33E-M) (OJ4E-04) (0.53E-UQ
18.47
bEa& Fsoalisticisfat~teStofthejohrth~thatuii~Oi;rtise~riatepricing
relation. For the APT tests, the ~~alue is the probability of obtaining a reabatiori greater than
the test statistic fkom an F distn’butiaawith N mameram ikgpiESOf~~and26U-K-N
denominator degrees of freedom, For the CRSP mean-varb~~ e&c&q tests, the p-value is the
probability of obtaG.ng 4 &ization greater than the test e&istic from an F dis&ii~tron
with
N-laumeratordegreeso~~mand260-hldeMwninatordegreesofEreedom.
‘Eadr~~statisticfortheaggregaletimeperiadisNtimesthesumofthefowsubperiod
F
sMstics.
The p-value
under each x2 statistic is the probability of obtaining a realization grater
than the test statistic from a x2 distr&ution wit& degreesof freedom equal to 20 when N = 5 and
80~~N=U)forthe~tests~d~~d~offreedom~~to16wi?ennr=Sand76
when Iv = 20 for tpleCRSP mean-varbce eIkietlcy tests.
240
B. N
Lehmann and 2 M. Modest, Empirical basis of the arbitragepricing theory
yield, and ow~1 vaniafl~e.Each table reports the F statistics for both the
rm-teturn and excess-return formulations of the AFT and for five-, ten-, ad
tables present the large sample F
em-factor models. In
of the CRSP equally-weighted and
value-Wtighredindices 0
relevant F statistic for
the four subperiod F statistics.26 The mumber report& under each test
basisoffkmsize.The
spWi.ng the ranked securities into either five or twenty groups consisting of
equal numbers, and then co~tructing equally-weighted portstocks in each group.27 Tables 2 and 3 provide the same
information as table 3, but serve as decks that the results in table 1 do not
hinge on peculiar&ies jll,oivvingthin trading or January. The!sole &fkrence
between tables 1 and ri! that table 2 presents resuh,swheL the factor mod&
are estimated using week@ and monthly data rather than daily data. In a__
similar spirit, the tests in table 3 are based on returns that exclude those
occurring in January to ensure that we are not confusing the turn-of-the-year
and size effects. Tabk 4 is similar to the Grsttable except that portfoliosare
formed GAthe basis of dividend yield in the year preceding the test period.
The e
procedure is somewhat different as well, since we form an
equally-weighted portfolio of the kms that paid zero dividends and then form
the remaining four or nineteen portfolios by ranking the remaining dividend-
=The F statistics for the CRSP indices, derived in Shanken (1985b), diifer slightly from those
givea in (6) for the APT tests. First, &* is the vector of residuals from the cross-sectional
et model ktercepts on one minus th_L estimated betas. The other
is replaced with N - 1. Note that these statistics are formed from
the sorted portfolios and are subject to the potentid power diEulties discussed in the text.
However, this does not sze,.n to be a problem in thi: application.
._
26The distrihiun or’ hi times an F statistic with Ad numeqtor degrees of freedom and N
&&i&GGfi
Rib
om is tiudy
~&stingui&&~e
from Q
denominator degrees of
M degrees of freedom for
ranging between 5 and 20 and deoq+ajis
of freedom of at
least 215.
27
also performed tests based (?LIten such portfolios but the results were similar to those
obtained with either the five or twenty portfolios and so we omit them to conserve sbace.
paying
stocks
inthe
the fmgind si&nificancelevels
subperiod results reveals consid
spending subperiod F statistics that in&de &wary retums or correct for
thin trading are large enough to rejed’tthe APT at cmventional signifi@ance
or of the version of the
are large enough to r@xt in the subperiods as w&L29
trament
‘*We also carried out testsby raking on dividend yield without
nitsaltswerevery
dividend group.
alsodonotseemtohgeonwbe
excess-m versionof the
29The
test!3 of
ft!tunrs~OO~tKUCted~thetzturnsO~&~
ortihogonal portfolio. w* excessre~~~~45
orthogonal portfolio, the correspond F statis
factor x-e&s reported in table 1 for the four s
and 3.3P(O.Oi) - alI not very Merent from
F tests do, however, ignore the estimation error in the sample man
portfolios, imparting a MS of Untnomm direction and nqpimdc
return
of the zero
2452
B.lV Lehmann and D.iU. Modest, Empirical basis ofthe czbitragqwicing theory
hese results are not just reflections of unusually large intercepts for the
est firm (i.e., fifth quintile) portfolio. To be sure, this portfolio has large
teo
as in all four periods. owever, the size e!Tectis not
e fourth quintile appears to plot above t
the largest-firm portfolio consistently plots below the
1
r-r p firm effect receives st
rane. This aaagwg conformaeighted index on the basis portfolios,
t iwtercepts on the order of - 2.5% to -4.5% per year
year period. Although some of this effect may be
to nonstationanty associated v ’ th the changing weights of the
effect remains (albeit with lesser magnitude) when
firm portfolio.
-variance efficiency of the equally-weighted and
ed on five firm&,e portfolios provide similarly
umentation of the magnitude qf the size efkct. The aggregate x2
statistic rejects the mean-variance efficiency of the equally-weighted index at
marginal significance levels below 10a9, and the same statistic constructed
excluding January returns rejects at marginal signiknce levels below 10e4.
The antiogous
egate marginal signifkance levels for the tests of the
mean-variance e
ency of the value-weighted index are below 10m3 for the
whole sample and at the 6% level when January returns are excluded. Again,
le uniformity in the subperiod results, since the mean-variciency of both indices is rejected in all but the second period when
returns are included and in the first and fourth periods when they are
on twenty portfolios reported under M = 20 in
e size-related results b
1 tell a somewhat di
nt story. The mean-variance efficiency of the
usual market proxies is rejected in aggregate at marginal significance levels
for the equally-weighted index and at the 5% level for the
dex &I the whole sample, while only the equally-weighted
level) when January returns are excluded. In
-return version of the five-factor model is
is no accident, since in
.
case were we
tabk 5, whkh exclude the 39 securities that were used to const
at an aggregate marginal significance level near 2% and the five-factor e~cezz~retui-fi W&III was marginally rejected at just over the 9% Ilevel,
probably &kct the size of these subsamples - there are fewer
very small firms with which to reject the theory.
The tests for the two
report&! in table 4, reject euTmean-varian
from the earlier work of Litzenberger
and Elton, Goober, and
equally-weighted index at a mar
statistics reject the null hypothesis for both indices at conventional sign.%cance
e mean-variance e
the equally-weighted index is also
and in the subs
examination of
244
B.N &hmann ad D.M. A&&w, Empirical basis of the arbitrage pi&~
#wury
related portfolios. The only evidence against the theory in the individual
subperiods occurs in the first sttbperiod: both versions of the ten- and
fifteen-factor models are rejected at marginal significance levels between 2%
%md4% with five dividend-sorted portfolios. There is no evidence against the
APT in the remaining five ,portfolio. results and no evidence at all in the
twenty-i>ortfolio test statistics or those in table 6, which exclude the 750
securities used to create the basis portfolios.3o
The results for portfolios based on own variance mirror those obtained in
the dividend-yield case. The mean-variance efficiency of the CRSP indices is
rejected at almost identical marginal significance levels in most cases and more
sharply for some of the remaining statistics, particularly those relating to the
value-weighted index. The AIT basis portfolios do not yield intercepts that are
significant over the whole sample with the exception of the five-factor r;;c1wreturn model which receives a marginal rejection at the 9% level. In addition,
manv of the F statistics are marginally sig,nXcant (between the 6% and 10%
level) in the first five-year period, although the remaining test statistics are
typically grossly ins&&ant (many at the 90% level and larger). Again, the
information in the test statistics is a reliable guide to the behavior of the
individual portfolio intercepts, except perhaps for the highest own-variance
portfolio, which has moderately large intercepts across three of the four
subperiods = although they are not very precisely estimated and usually are
statistically insignificant. The basic message is similar: the rejections of the
mean-variance efficiency of both CIUB indices suggest the ability to reject
some asset pricing models, and the failure to reject the APT pricing restriction
suggests that the theory provides an adequate account of the risk and return of
the own-variance portfolios.
Finally, in table 7 we present summary x2 statistics for the joint significance
of the mean returns of the basis portfolios constructed from the five-, ten-, and
fifteen-factor models. These x2 statistics are of interest, since the AIT implies
that at least one of the factor risk premiums should be significantly different
from zero under the assumption that investors are risk-averse. As is readily
apparent, our large cross-sections yielded basis portfolios with highly significant mean returns in aggregate and in most of the individual subperiods as
well. This is in sharp contrast to the frequency insignificant mean returns of
basis portfolios constructed from smaller cross-sections.
3oExamination
of the individualportfolio interceptsprovidessome evidenceof positive (but
usually insignificant)interceptsfor the zero-dividendand high-dividendportfolios.In the tests
based on twenty dividend-yielaportfolios,the interceptsfor the remainiugportfoliosoften have
mixed signs and are typicallyeconomicallyand statisticallyinsiguificzu&Howevcr9in the tests
based ou five sorted portfolios:the lowest dividend-yieldportfolio (excludingthe zero-dividend
portfolio) has on average the smallest intercept and the remaininginterceptsincrease in a
monotonic fashion - producinga dividend~yieldelTectthat is qualitativelysimilar to the one
found using the traditionalmarketproxies,but not nearlyso pronounced.
EWSsfetums
(A, # 09
(X0 - 09
Raw-
(X0f;lo9
(X0- 01
(X0+ 09
35.52
(0.21E-02)
35.48
(0.2lE=O2)
21.25
(0.13)
49.21
(0.16E-04)
12.74
(0.62)
27.43
{0#025)
24.58
(0.056)
37.48
(O.llE-02)
94.10
(0.32E-02)
149.60
(0.13E-08)
73.83
(0.9CE-03j
23.35
(0.95E-02)
10*36
(0.41)
12.85
(0.23)
27.28
(0.233-02)
Elsessreturns
121.29
(0.4iE-Wj
35.23
(O.llE-03)
23.25
(0.99E-02)
36.83
(0.61E-04)
25.98
(0.38E-02)
Raw returns
42.20
(0.26E-02)
7.56
(0.18)
1363-1982
77.69
(O.%E-O8$
3.27
(0.66)
(&I = a)
1978-1982
12.96
(0.024)
15.88
(0.82EAI2)
1973-1977
20.23
(O.llE-02)
_-
Excessreturns
28.62
(0.28E-04)
1968-1972
20.53
(0.99E=O3)
1963-1967
Rawretums
.
Aggregatechb4uared
statistic (~value)=
K) in the appropriate prWg relation. The pvatUe is the
“~~%*s~~~i8for~~tofthejointh~~thatX,-O(k~l,...,
pr&bWy of obtJlininga realizationgreaterthan the test statistic from a x2 distribution with K degrees of Worn.
b&ch
x2 statktk ftr the qgrega& time period is *&e sum of the four subperiod x2 statistks. The p-value is the probability of &&i&g
a
.
.
rcdatm
gceater than the test stMistic from a x2 distributbn with 4K degtees of Morn.
(Km 15)
FifWXl-
Fivs?factor
Mcing reuion
Chi~4uared statistics for five-yearsubperiods
(~Values ate gkn in paren&Ses)b
chi-squared statistics for four subperiods and for the entire period for the joint hypothesis that A, = 0 (k = 1,. . . , K) in the,:pricing relation
Ei - A@= Ui+ CRKllb,khk,where Ei is the wtd
ROn securify i, bik b tk SeDsittitiQ~
of secufity i to faCtOrk, A, is the&k P&W
Of the
kth factor, and A, is the prkibg intercept.The fabctorp&kg m& -Is Z&W for five, te3, and fifteen factors(K = S,lO, 15) estimated from a preGminary
factor anal@ of the daily returns of 750 randomly selected securities! The snbp&ods co&& 260 weekly observations a&the whole wk.
contains 1,040 weekly observations.
Tats of the joint tignikance of the factor Gsk prembms Ak (k = 1,. . . , K) b the interval 1663-1982.
Table 7
246
B.N Lehmann~and D.M. Modes, ~mpirdca~basis of the arbitrage pricing theory
2.57
(0.027)
Excess ;:tums
(&I f 0)
2165
(0.023)
2.68
(0.22E-01)
10.10
(0.83E-08)
3.20
(O.SlE-02)
2.18
(0.058)
0.52
(0.76)
13.33
(O.l7E-06)
1.68
(0.14)
268.89
(O=OQ
7.36
(O.l9E-OS)
own var.
Size
192.22
(0.39E-29)
27.37
(0.13)
276.43
(0.W
28.02
(0.11)
685.98
WOO)
127.37
(O.l2E-16)
199.12
(0.17E30)
38.72
(0.72E02)
256.54
(O=OQ
59.07
(0.99E-05)
945.24
(O=OO)
109.40
(0.25E-13)
Div. yield
AX12
(0.92Eo27)
27.93
(Urn)
233.36
(0.23E-37)
41.50
(0.32E-12)
2195.00
(0.00)
102.48
(0.45&12)
own var.
-
‘The estimated factor model is used to compute portfolio weights which are combined with the weekly returns of the 750 securities to produce
weekly estima+~sof the factors.
bEacb F ~tatktk is for the test of the joint hypothesis that ~~_lbik = 1 in the appropriate pricing relation. The p-value is the probability of
obtain@ %Irealization @eaterthan the test statistic from an F distribution with 5 numerator degreesof freedom and 260 - K - 5 denominator degrees
of freedom
‘Each x2 statistic for the aggregatetime period is N times the sum of the four subperiod F statistics.The p-value under each xz statistic is the
pmbbility
of obtaining a realization greater than the test statistic from a x2 distribution with degrees of freedom equal to 20 when N = 5 and 80
when N - 20 for the AP’p tests and with degreesof freedom equal to 16 when N = 5 and 76 when N = 20 for the CRSP mean-variance efficiencytests.
1.45
(0.21)
14.17
(0.35E-11)
Excess returns
(A, f: 0)
Raw returns
(&I = 0)
0.95
(0.45)
Raw returns
(&I = 0)
Tenfactor
APT
(K= 10)
Fifteenfactor
APT
(K= 15)
59.78
(O-00)
Excess returns
(&I + 0)
(K= 5)
142.04
(O=W
5.93
(0.34E-04)
Raw returns
(A, = 0)
Fivefactor
7.41
(O.l7E-05)
Div. yield
1963-1982
1978-1982
Pricing relation
Aggregatechi-squared
statistic (JWhle)’
F statistics for five-yearsubperiods
(p-values are given in paicntheseQb
Table 8 (continued)
s;t.
Mean
std
dev:
1968-1972
t
stat.
Mean
std.
dcv.
1973-1977
t
stat.
-
&cPa
returns
over
T-bills
(K=5)
-0.0039
Raw returns -0.00055 0.018
1.59 0.0039
(0.11)
-
0.22 0.0084 0.022
(0.82)
1.36
(0.17)
0.0028
-
3.00
0.0081 0.022
(0.0027)
1.01
(0.31)
2.91
(0.0036j
0.0033 0.00056 45.95 0.0046 0.0011 33.65
0.0053 0.0011 36.25
(O*O@
(OJJ@
(0.W
Fivefactor
Treasurybill returns
MeaIl
Std.
dev.
1963-1967
Summaq statistics for five-yearsubperiods
(p-values are given in parentheses)=
-0.0024
t
stat.
Mean ( p-value)d
sib.
Y2
1963-1982
-
0.82 O.OoQl 6.08
(0.42)
(0.19)
2.211 QOO56 22.41
(0.027)
(0.00017)
0.0023 29.36 0.0055 5420.17
(O=W
(O=OQ
0.0064 0.022
OAKI
Mesa
std.
dev.
1978-1982
ior the aggregate
time period
summary
Statistic
‘he orthogonaJ portfolios are constructed to have minimum variance and returns uncorrelated with the realizations of the common factors in the
eturn-generatingprW Ri, - Ei =Zfml&ikBkr+ eit, whm Ri, is the return of securify i at time t, Ei is the expected return OII~W$ty i, 6ikis the
en&My of security i to factor k, &, is the realization of the kth factor, and ei, is the idiosyncratic disturbance of security i. The factor pricing
nodels allow for five, ten, and fifteen factors (K= 5,10,15) estimatqi from a preliminq factor analysis of the dGly returns of 750 rand&nIy sekcted
securities.b
Table 9
Means, standard deviations, and t statistics for monthly returns on Treasury bills and on orthogonal portfolios in the interval 1%3-1982.a
Raw retlu!N
-
-0.QO2i
-
0.0012 0.014
-0.0023
0.00110 0.014
-
1.78
0.0029
(0.075)
-
3.70
ON82 0.020
(0.ooo21)
1.19 0.045
(OW
-
2.21
0.0028
(0.027)
-
0.7l 0.0091 0.016 4.45
0.0081 0.020
(0.84E-05)
(0.48)
131 0.0043
(0.19)
0.59 O.W88 0.018
(0.W
1.08
(0.28)
3.11
(iMol9)
1.12
(0.26)
3.13
(O.aw)
-
-0.oo20
.
0.0068 0.021
-0.0013
Mw?s 0.022
6.36
(0.17)
(0.47)
0.73
0.0008
(O.o!n)
7.99
247
0.0063 36.12
(0.013)
(0.27E-06)
0.47 o.an!J
(O*MI
269
J !.+I64 31.13
(0.29E-05)
(o.oon:
aThe Tmury bill returns are derived from the discount yields for Treasury bills with one month to maturity presented in Salomon Brothers’
Ana@id Reud qf Yiiel& ad YM Spmds.
bathe estimated fwtor model is used to compute portfolio weights which are combined with the monthly returns of the 750 securities to produce
monthly estimates of the f&ctors
‘The p-values under the t statistics are the probability of obtaining realizationgreater than the test statistic from a t distribution with 59 degrees of
freedom.
dThe x2 statistics for the aggregatetime peziod are obtained by squaring and summing the individual subperiod t statistics.‘I%epvalws under each
x2 statistic is the probabiity of obt&iog a realizationgreater than the test statistic from a x2 distribution with four degrees of freedom.
tYFii!s
(K= 15) Excess
returns
Fifteen- Raw atwns
factor
EZLS
(K=lO) Excess
returns
Terrfactor
250
B.N Lehmann
ad D..M Modest, Empirical basis of the arbittage pricing theory
5.3. Tests for spanning
Table 8 provides evidence on whether the K basis portfolios span the
mean-variance effkient frontier of the individual assets. In particular, it
reports the relevant test statistics for the hypothesis that the portfolios formed
on the basis of firm size, dividend yield, and own variance have loazlings that
sum to unity - an implication of mean-variance spanning. Attention is refive such portfolios in order to conserve space,
stricted to the results bas4
which involves little sacrifke, since the results for ten and twenty portfolios
were much less favorable to this formulation. The results in table 8 suggest
overwhelming rejection of the joint hypothesis that the raw-return formulation
of the APT is correct and that the basis portfolios span the mean-variance
efficient frontier and are measured without error. The aggregate test statistics
based on excess return regressions reject the null hypothesis at marginal
significance levels below 1O-27,and the least significant subperiod F statistic
has a marginal significance level of 2.3%. The hypothesis fares only slightly
better in the raw-return regressions - all but three of the eighteen aggregate
test statistics have marginal significance levels below 0.002, as do many of the
subperiod statistics. Moreover, these results are most favorable to this version
of the APT - we would have made no such caveats had we chosen to report
the twenty portfolio results.
5.4. I%e k&adot of the orthogonalpott&olios
Table 9 reports summary statistics on the sample behavior of the monthly
returns on the orthogonal portfolios constructed from fivemiten-, and fifteenfactor models anA
d one-month Treasury bills. For each orthogonal portfolio
a&&o for one-month Treasury bills, we report the mean return, the sampie
standard deviation of its return, and the t statistic for the hypothesis that the
mean return is significantly diEerent from zero. We also present the mean
return md associated t statistic for the difference in returns between each
orthogonal portfolio and onemonth Treasury bilk. ‘Z”nesestatistics are presented for each of the four five-year periods. In addition, we present approximate x2 statistics31 for the hypothesis that the mean returns are. jointly
significantly different from zero across the four periods. In all cases we also
report the marginal significance level of the test statistics.
The results in table 9 suggest considerable uniformity in the behavior of the
orthogonal portfolios from the five-, ten-, az% fifteen-factor models The t
siatls~ics
for a-x Y”it%rn retarm
ruAaa~ cm the orthogonal portfolioo of all three factor
311t proved to be convenient with our software to produce these approximate x2 statistics
instead of the usual F statistics. Fortunately, with these sanqk sizes the difference in marginal
significance is very small.
-_
B.N Lehmann and D.M. Modest, Enapiricd baris of the arbitragepricing theory
251
models are highly significant in the final three subperiods, although t&y ate
insignificant in the first period. The x2 statistics for the joint significance of
the mean returns for each orthogonal portfolio across the four sample periods
have marginal significance levels below lOa for each factor model. In con_
tradistinction, the corresponding t statistics for the differences in mean returns
between these portfolios and the one-month Treasury bill are ins&n&ant in
each subperiod with two exceptions: in the second subperiod, the mean return
difference fcr the orthogonal portfolio from the ten-factor model is marginally
sign&ant at the 7.5% level, whereas that from the fifteen-factor model is
significant at the 2.7% level. In addition, these mean-return differences have
mixed signs, negative in the Crst and fourth periods and positive in +&emiddle
two periods. Moreover, the aggregate x2 statistics for the joint signifkance of
the mean-return differences have marginal significance levels of 0.19 for the
five-factor model, 0.17 for the ten-factor znod4,s and 0.892 for the fifteen-factor
model. Only the fifteen-factor results reflect a marginal rejection of tlze
hypothesis that this mean-return difference is zero, a rejection attributable
solely to the results from the second subperiod.
The sun~~~ry statistics rcport& in table 9 indicate orthogonal portfolio
returns with means that are signifkantly different from zero and insignificantly
different from the mean return on one-month Treasury bills. This is additional
evidence against the spanning model that is far more consistent with either of
the excess-retm models rather than the mixed results obtain& by the
previous authors, since most previous studies have been unable to fine
statip);flQtl=r
&rr&fi;r.
OaV.,ant pricing intercepts. -Moreover, these results stand in
UUWUUJ
sharp contrast to those obtain& in studies ~4 the CAPM, where the estimated
zero-beta rates are typically significantly greater than Treasury bill rates.
Finally, the sample standard deviations of these orthogonal portfolios are
far from zero: typi&lly ten to thirty times those on one-month Treasury bills.
This could arise for several reasons, as discus& in section 3.2.2. For example,
the limiting minimum~variance portfolio could be riskless but our procedures
are incapable of eliminating idiosyncratic risk ln Unite cross-sections. This
interpretation is consistent with the observation that the mean returns on the
orthogonal portfolios are typically insignificantly diflerent from those of
one-month Treasury bills. Alternatively, the limiting minkmum-variance portfolio could be risky and, hence, it is shear happenstance that the mean returns
on these portfolios are insignificantly different from one-month Treastu-yMl
rates. It is certainly worth noting that these mean returns are insignificantly
different from many positive n~~mbers!
6. Conclusion
This paper is devoted to the accumulation of facts a_n,A,
the sifting of
evidence regarding +he validity of the APT in its various formulations. By far
252
RN Lehmannand D.M. Modest,Empiricalbasinof the arbitrage pricing theory
the most interesting results concern the validity of the API’ itself. The APT
fares well when confronted with the strong relationship between average
returns and either dividend yield or own variance. The APT provides an
adequate account of the relation between risk and return of the dividend-yield
and ow~ariance
portfolios where risk adjustment with the usual CAPM
market proxies fails. It is noteworthy that the APT provides a risk-based
explanation of these phenornexa in contrast to the usual tax-related explanation of the dividend effect and the transitions-cost account of the relationship
b&we3 own variance and average retums.32 In contradistinction, the tests
hsed
on firm size provide sharp evidence against the APT, although the form
of size effect appears Merent from that documented in CAPM studies.
One interpretation of this failure to account for the size effect centers on
sample size, asynchronous trading, or any of the 0th~ potential problems
discussed earlier. We are persuaded, though, the large cross-sections that XT
employ largely mitigate the effect of measurement error. Similarly, the thin.
trading corrections yield no suggestion 4&atthe size-related results are attr%W
able to this problem. Moreover, the sharpness of the rejections reported in
tables 1-3 suggests that they cannot be attributed to peculiar small-sample
properties of the test statistics such as those that might result, for example,
from nonnormality. These considerations suggest the failure of the APT to
account for the size effect is credible.
The obvious interpretation is that we have rejected the exact factor pricing
versions of the APT or9 to be more precise, our implementation of the theory.
‘I’he abilitv of a m~~~~re of wt=*ra+~~~:yIpuf3~~~a~~~
risk
to
expiain risk-adjusted
returns vi&a&s the theory. The concentration of the size effect in the very
smallest and largest tims, however, suggests at least one potential alternat&
explanation of these results. Suppose there is a small-firm factor, in that the
business-cycle risk of small firms, especially *thoseprimarily traded over the
counter or closely held, is much greater than that of larger firms. In addition,
suppose that the exposure to this source of risk is small for most listed quities
but that the risk premium for this factor is positive and large.33 In these
circumstances, our procedures for estimating the common factors would fail to
measure this factor well, since few &IKE in our cross-s&on would be
materially affected by it. Hence, this account of the size effect involves
measurement error in the factors, measurement error that follows from the
assets selected for the analysis rather than arising from our statistical proa-
321t
is possible *hat the absence of a measured dividend effect in our APT results is consistent
with the tax story. This could occur, for example, if one of the risk factors reflected random
marginal tax rates impinging on asset pricing and the corresponding factor loadings arc the
dividend yields of the individual securities.
33“f’hisscenario is5 in principle, consonant with our rqjectionsof the APT where the small&
firmshme small pHive loadings and the large firms have small negative loadings.
.
B. N khmann and D.M. Modest, Empiricalbasis of the arbitragepricing theory
253
dures.34 This is a manifestation of Shanken’s (1982, 198Sa) observation that
tests of the APT involve joint hypotheses aboat the relations between factors
extracted from a subset of assets and the relevant equilibrium pricing aggregates.
In short, the size and the turn-of-the-year effect have thus far evaded a
satisfactory risk-based explanation. It is worth emphasiig, however, that our
size effect is largely concentrated in the largest and smahest fums. This
observation, in conjunction with the model’s success when confronted with the
dividend-yield and own-variance anomalies, suggests that the APT is priang
most listed equities with little error.
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atjba,