HD28
.MA14
M5.l37f
Devwey
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
NEW EVIDENCE ON THE NATURE OF SIZE
RELATED ANOMALIES IN STOCK PRICES
by
Terry A. Marsh***
Philip -^rown*
Allan W. Kleidon**
//1379-82
November 1982
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
NEW EVIDENCE ON THE NATURE OF SIZE
RELATED ANOMALIES IN STOCK PRICES
by
Terry A. Marsh***
Philip fir own*
Allan W. Kleidon**
#1379-82
November 1982
NEW EVIDENCE ON THE NATURE OF SIZE RELATED ANOMALIES IN STOCK PRICES
by
Philip Brown*
Allan W. Kleldon»»
Terry A. Marsh***
*Universlty of Western Australia
•*Univer3lty of Chicago
•**Massachusetts Institute of Technology
Last Revised:
Forthoomlng:
March 1982
Journal of Financial Economics
We are 'indebted to Craig Ansley for assistance with stochastic
Eugene Fame, Michael Gibbons, John
estimation procedures; to Petor Dodd
Gould, Drucc Grundy, Jon Ina'^rsoll, Bob Korajczyk, Richard Leftwlch, Merton
Miller, Richard Rubaok, Myron Scholes and Arnold Zellner for helpful
comments; to Marc Relnganum for furnishing data used in his doctoral
thesis; and to the Center for Research in Security Prices at the University
of Chicaeo for access to their data files and for financial assistance.
,
0745337
I.
ABSTRACT
Thl8
paper
Is
concerned
returns reported by Banz
small
the
with
(1981)
and
the
Reinganum
(1981).
CAPM.
We
find
that
the
size
stock
in
They showed
that
than those predicted by
firms have tended to yield returns greater
traditional
anomalies
size-related
effect
is
linear
In
the
logarithm of size, but reject the hypothesis that the ex ante excess return
attributable
to
size
is
stable
through
time.
Seemingly Unrelated Regression Model (SURM) and
alternative estimators of the size effect.
effect,
studied.
we
find
that
the
estimates
are
Due
We
a
briefly
analyze
the
two-step procedure as two
to
the Instability of the
sensitive
to
the
time
period
-1-
DUCTION
INTRO
.1^
^B _.
1 .
-
There
.
.
evidence
i3
that
excess returns
excess
that
returns
common
on
earned
be
time
over
by
Basu (1977) and Relnganum (1981)
ranking securities on certain variables.
report
can
stocks
are
function of the ranks of their earnings-price
(E/P)
decreasing
monotone
a
Banz (1981)
ratios.
shows that rankings based on firm size can be used to earn excess returns,
and
Relngonum
(1981,
returns
'abnormal'
20)
shows
further
that
market
value
effects,
one
p.
for
controlling
"after
could
detect
not
an
independent E/P effect."
The
size effect has manifested
example,
Beta
In
the
Dead?"
Institutional Investor
well-publicized
article,
Michaud
Richard
"Instead of using the beta model,
such
factor
as
Itself beyond academic" circles.
capitalization.
I
would
It's
Bache
of
is
(1980,
p.
quoted
as
"passive management
related
return."
to
class of "small
Expansion Fund" of small
.Market
..
firm growth
considered
stocks,"
(1980) program, provides another illustration.
follows:
The
in
American
It set up
firm stocks
.
firm stocks
typical
growth
The
Street Week
the Wall
Indeed, the implications of
the anomaly for "small firm growth stocks" are especially interesting.
small
"Is
probably like to use one other
National Bank and Trust Company of Chicago went a step further.
a
29)
For
As
they allegedly earn positive excess returns; while the
stock
has
a
low.
not
ratio,
high,
E/P
that
their
and
(presumably)
negative, not positive oxooaa roturnn.
Banz and
Relnganum
with misspeciflcation
used
assess excess
to
also Ball
(1978)].
in
conclude
both
the capital
returns
However,
in
rather
evidence
is
asset pricing model. (CAPH)
than
with
market
consistent
benchmark
inefficiency [see
common with the examples Just cited, they
-2-
distribution of the
effect, or the
size encv.
» ^^^af
exp ..Pd Size
that the eXDCCteu
assume
Implicitly
examined.
over the periods
t.nt °''
•
Bl« «^^-^MQftl.
""'^"'
^^
.
long
iontj
oe
to be
.
f«H
constructed
mean
nrp^ents
presenva
1c^
15)
p.
^t^^u
monthly
small
in
r*»turns
returna
firms andH
on
,hnrt
short
„3,Mtrage"
"aroitragc
in
Banz
example.
For
portfolios
»~
firms over
large
iarg
the
4„m-
^
He
of 2.99 over
.vv, a
« r..norted
reported t-statistic
with
time
over
average excess returns
have t-statistics
the nine subpcriods
of
two
only
But
, ,
the total period.
strategy
po
..3,,,,,,,e" portfolio
subperiods the arbitrage
two
in
and
greater than 2.0.
attributes the
16)
p.
(1981.
Banz
ess returns.
yields negative excess
in
nortfollo concentration
oH by
hv porttoix
caused
diversification
1, to a lack of
..... ..
„e
„„,,„,
-„
..p."-
...
...
^^^^^
.t >3t
acoo...
...t ...
or
other
the
years
effect
....
..U.
o.
„.3
,„. ...,.
^^^ ^^^ ^^^^^^^„^,„„
in
while
to
U U,eU
„U..c.ne.t.„.
sta.UU.
..e3tuat. a. ...ct t.
.
value,
"Unou
dUtrlDutlon.
stationary
.U.P,0
..turns o.tain.
in
... .^r.
na, a n..at..
The
reversed.
is
.^^
...ct.
Issue
of
the
o. tn. 3U. .no„aU
pot.ntu. .xp.3n3t.on3
.portant.,.
a3
oiw i<i itself
itseU described properly,
the anomaly is
If
evaluated more readily
insofar
Regression
nrocedure
as
M H 1
Model
to
is
measurement
(SURM)
measure.
and
,ize
size
a
n.d
concerned.
two-step
effects
clieot
and
we
use
a
generalized
find
that
Seemingly
least
they
are
Unrelated
squares
(GLS)
,,„.i,tically
statistically
-3-
l„,igniflc.„t
.ubperlcd
m
over
January
maignlfioant
size
perlc.
to
,967
effect,
flnC
Ov,r
the
-poaltlve-
but
1979.
J.n.
-967, to
January
a
December
,975.
We
Is.
small
flms have negative excess
that
positive excess returns.
returns and large firms have
However. Banx (,98,)
t-statistlc
period ,966-,975 with a
negative slxe effect over the
subperiod. we find a
In addition, over the ,973-1979
of ,.55.
results
These seemingly contradictory
Significantly negative size effect.
of th. size
of our finding of instability
.r. not surprising in light
reports
a
effect.
existence
some explanations of their
If size effects are not constant,
can
be-ruled out.
If
themselves.
others
small
firm
need
stocks
modifying,
are
and
expected
to
still
earn
others
positive
suggest
excess
stocks,
commission costs in trading those
returns because of differential
large
less diversification service than
or because they tend to provide
every period.
premium should be positive in
firms, then the expected return
terms of non-trading induced
explanation of size effects in
results.
not consistent with our
estimated beta coefficients is
Roll's
(1981)
biases
m
of the
unclear whether a modified version
AS we discuss in Section 5. it is
non-trading across
in the pattern of
explanation which allowed instability
results.
firms would be consistent with the
There appears to be
•
a
of both small and
seasonal affecting the returns
by Kelm
with the "January effect" reported
large firms, which is consistent
effect, the
taking account of a January
after
even"
However,
(1981).
Although excess returns are
remains.
instability in the size effect
unstable
1950's.
since
1926.
the
phenomenon
is
most
pronounced
since
the
late
.
-u-
the
estimation procedures and
Alternative
alternative
vh.
the
dlsouaa
^^^^ T,:.o:->if,slen
In Section 2. we
estimation procedures
.
nt., the alternative .,^,.
evaluates
^,,
Section
unstable slope.
those
over
anomaly
..„H« and size of the
magnitude
the
on
^^'^^^^ 4^^and presents evidence
Some
assumption.
reasonable
.
KP
A
to be a
appears
Which statlonarlty
periods
^
Sectlo
.
ar,,.<,nslde^
anomaly
size
the
of
^^^^^^^
potential explanations
^
m
^
given in Section 6.
brief summary is
.
M rrlfoVolVY' AND SAM PLE
2.
2
-'
-'"""••
„i sni^!
,ii
'fscJcnoo
:fo;,
.
:.i
o«-
e,-
»ir
.
Size Anomaly
The CAPM and the
^uo b^lL'T'sd nao
Is:
premla
written in terms of risk
model
market
^^^^^^^^^^
The well-known
1
^ht'
''FtV=
.«
Whore
R
n^^
^M^V
«i/
it
- "Ft^
^
V,
•
-
-.axi
R
"Mt
on
the rate of return
c
-
t;
f
<
«n
tl
nnrlod t.
perloa
on asset 1 in
the rate of return
'.v;.j.P.«oq od bluoris mulmsiQ
muJs-;
,
a
s
.-.
-.M,,.,.
f>,*
the
ao;i
ui
fiRUP
h-i.-.
,
.«h
period
portfolio of assets in
market port
2jni:ioXi;yor. aJ£>d
disturbance term;
security specific
'''"/^''^^Kskless
Y^^J
'^f'"'^
rate in period t;,.,,,,,^
b6J\
^^
,^^^^^.,.
^^-
,
,
espsld
nx
^^^^^^^
^^.
"Ft
a
N
.
E
constant;
an arbitrary
«
total number of assets;
operators,
covariance and variance
e.ecVation.
are
and Var
and Where E. Cov
Of
^
linearity properties
justified by the
.espectiv;..
.uatlon
multivariate normal
,
cms
multlvaT^^ate studcnt-t)
ror
(or multivaria
Chapter
returns [Fama (1976.
0.
E
(f,,. c^,)
=
0. s / t.
oM
2)]
of asset
Joint distribution
assumption E (e
nr
bv the direct
or by
r
K^^>^
-
-5-
The Sharpe-Llntner2 CAPM implies that
a
a
is an "excess return", i.e.,
*<
If there is a size anomaly relative to the Sharpe-Llntner CAPM,
0.
will be related
to
for
size
least
at
asset 1.
some
hypothesis obviously provides no clue
as
CAPM as
The
null
a
excess
the relation between
to
<»j
returns and size.
Benz's work (1979,
on
Merton's
Klein
Bawa
and
negative
(1977)
services model
consumer
return
large
very
for
positive excess returns for very small
the
that
appropriate
(market) value.
form
a
is
Relnganum (1981)
the
former
whereas
firms
model
declining
reports
similar result.
a
implies
work implies
function
nonlinear
a
based
implies
latter
the
His empirical
firms.
3
information model of
the
end
that
argues
Banz
(1977).
excess
implies two possible functional forms
1981)
equity
of
However, the
dlBtribution of market value of equity across the securities they studied
which may at least in part account for the
is extremely positively skewed
reported
non-linearities.
Further,
for
Relnganum does not always adjust
risk differences across size-ranked portfolios.
Rather thnn form
the size of
.estimate
firm 1,
the
prior speoifiontion for the relation between
a
we
use
the
cross-sectional
CRSP daily
pattern
of
(beta)
average
excess return file
excess
returns
and
*.
5
for
to
ten
portfolios ranked on size.
2.2
Data
Our
(1981)
in
primary
which
sample
a
consists of the
size-related
566
anomaly is
firms
studied
reported.
The
by
Relnganum
sample
turn, a subset of 577 companies analyzed by Latane and Jones (1977).
reason
size
for using
anomaly,
and
this
sample
Relnganum
is
has
is,
in
The
that it has proven informative about the
generously
supplied
us
with
his
data.
Since all 566 firms were required to have complete quarterly data from June
-6-
1967 to December
1975,
Relnganum extended
survivorship bias is possible.
a
the series for eight quarters from the fourth quarter of 1975, and we have
further extended it to the fourth quarter of 1979.
o.-»"
Of the 566 firms in exlstenoe at December
Dooember
(Relnganum'
1977
196
and
sample)
3
through
December
our
primary
focus
Is
on
size,
firm
incomplete data as at December 31.
Table
1979
tabulates
1
through
Since
1979.
5'Sis
•
r,-
':9--^'-;'-vi
535 survived
1975.
ST.
for
bnfi
reasons
the
Tor each size decile where size Is
defined as market value of equity at December 1975.
Assuming
a
combination
of two iistedf firms is more likely to result in the exchange's delisting
the stock of the smaller than of the larger,
surprising that 45
is not
it
of.
of the 62 mergers and acquisitions resulted in the disappearance of firms
When we consider firms below the median
smaller than the median firm size.
there is no obvious relation between size and the Incompleteness of
size,
'''•"
'.
j£-:>
data (except possibly that due to bankruptcy).
""^"'A
of
feature
including
all
of
Reinganura'
the
a.,
de.ta, 3,et
is
that
3rV,. are
not
In
smallest
98
of
CRSP
the
relative file because they were not traded on the NYSE.
the
98
had
returns
on
th^. CR3P
construct monthly returns.
NYSE
listed
companies.
informative about
a
It
Da^lly Jleturns
file
the
stocks,
566
monthly
price
All but three of
which
were
used
to
Because the latter file contains both AMEX and
la
that
likely
Relnganum's
size-related anomaly than
firm stociks traded on the NYSE.
,•
^^^j
a
data
set
more
Is
set confined to the larger
y
Table 2 reports Spearman's rank order correlation coefficients between
size (as measured by market value of equity) at June 1967, size at December
1975,
leverage
at
December
1975
(with
equity
market values) and total assets in December
assets
and
book
value
of
debt
and
equity
measured
at
both
1966 and December
are
taken
from
book
1975.
the
and
Total
Compustat
-7-
annual
tape,
and
market
value
of
equity
from
monthly
CRSP
price
files
(checked against the Compustat tape).'
Table
shows
2
high degree of stability in market
a
value of equity
rankings over the period June 1957 to December 1975, the strong correlation
between total assets and market value of equity, and the stability in asset
ranking from
1966 to
1975.
The high degree of stability and correlation
among the alternative size measures suggest that our results are not likely
size variable used, nor to the use of a
to be sensitive to the particular
constant
size
measure
discussed in Section
2.3
through
time.
The
results
other
In
Table
2
are
below,
5
Linearity of the Size Effect
We turn now to
size.
To
avoid
the
the
form of the
possible
relation
between
survivorship
bias
securities on market value of equity
at
average daily excess return
for
security
s econd
to
trading day in
1976)
daily excess returns tape.
magnitude,
relative
to
December 29,
Table
standard
3
of the
1976,
and
1975
1978,
using
the
1976
CRSP
summary of the
securities'
rank
we
estimate
from January 6,
contains a
error,
until
the
(the
(beta)
sign
and
excess returns
While it is not at all clear how t-statistics are to
.across size deciles.
be interpreted in the
estimation procedure,
each
December
excess returns and
context of CRSP's population and
Q
its excess return
one conclusion is reasonably certain:
does not appear to be powered by small stocks alone.
the anomaly
All of the largest 56
firms studied had negative excess returns, on average, from January 1976 to
December 1978.
To test whether these results are sample specific, the experiment was
repea.tcd for the population of all stocks for which market values of equity
were available as at the first day of the CRSP daily returns tape (July 2,
-8-
which ony exoeaa returns were available.
for
and
1962)
Although it appears that the monotone effect in
also presented In Table 3-
Reinganum
the
example,
In
is
the
longer
covered
period
larger
this
by.
effect
the
specific,
sample
not
sample
over
attenuated
results are
The
clearly
is
For
sample.
although 10. 8t of the t-statistics in Reinganum's sample exceed 2
absolute value,
5.2% in this category for the expanded
are only
there
sample.
Both
deciles
size
across
effect
monotonic
apparently
the
and
the
of
attenuation over the longer sample period are also shown by the values
the average dally excess return
Table
In
Again,
3.
the
main
difference
average
between the results appears to be that the absolute values of the
largest firms are greater in the
excess returns for both the smallest and
These results foreshadow Section
sub-period.
which shows that although
3
the excess returns are linear in log size, the slope of the linear relation
Consequently,
changes' sign through time.
the effect measured over
a
long
period during which the slope changes sign will be attenuated relative to a
sub-period In which the sign is constant.
Finally,
linearity
the
using
here
graphically,
most
perhaps
is
formed
portfolios
ten
when
apparent
from
a
illustrated
cross-sectional
ranking of the sample securities in ascending order of size which we will
use in most of the later results.
dully
oxcosa
calculated
as
returns
the
moan
for
Figure
Reinganum's
size
of
all
1
although
the
(here .portfolio)
relation
between
and
snmp]rt
firms
natural logarithm of portfolio size, and (c)
that
shows scatter plots of average
excess
In
(•)
size
each
size decile.
return
and
portfolio
decile,
log
size or
size decile
Is
used.
the
The plots imply
untransformed
size may be non-linear, an approximately linear
exists when either
(b)
size
firm
relation
Further the results
-9-
are very aimilar
for
size
log
Indicating -that mere size
size decile.
and
the anomaly as size per se.
ranking may contain as much information about
the
with our analysis of monthly data on
In either case, we proceed
return and log size.
assumption of a linear relation between excess
2.4
Test Structure
a in log size,
If we assume linearity of the excess return
generalized to include
^^^it
-
^01 *
=
«Ft^
a
12
(constant)
\h ^%t
'
Vt^
*
5^
and
Yq.,
S
T).
5
a
may be
size measure:
measure of size for asset
^'^
^
^i
1
where
(1)
1
\t
*
s
.... N; t B
1,
1,
...t T
applicable to the period (1. T)
parameters.
Y^i are scalar
will be assumed constant for any asset
over any time period (1,
1
returns and 5^ is cross sectional,
Since the relation between excess
it is not possible to estimate a separate
cross sectional
and
y^^
information can be Incorporated
can be considered
aopflrntoly for
obtnlned by (soy)
0L3.
oaoh asset
in
a
The
for each asset.
two ways.
and an
1.
Then In n second step
y^^
estimate
Yq and
common
First.
-^
Y^
(1)
oan be
onn be
estimated from the cross sectional regression:
\-\^h\'\
'
=
'
"
a common Y^ and
Alternately, since S^ varies across equations,
"'.
Y^
may
Unrelated Regression Model (SURM)
be estimated directly using the Seemingly
as follows:
-10-
>,
-11-
In the error term
c.
In this case t,
.
'it =
where
Is
v.
and V.
l.i.d.
for t
7
c..
would be autocorrelated
SURM model (1), cross-sectionally random
with E (c..
Only firm size is allowed aa
oommon
fixed
e.
)
=0,
identification,
effects
firm-specific
estimated by firm-specific y
estimated
there
if
will
are
.
fixed
be
impounded
estimation
(cross-sectionally)
of
(5)
fl,
where
X
e
the
SURM
by
identical
is
1)
is
second
specified
step (3)
as estimation
than
other
and
(5).
and
any
1^4
Yq
Subject to
CO.
could
size
effects
random
.
equation
of a
be
other
than
Rj,)'
in
(
(5)
i_
in
or
(5).
by
(1)
equation
can be written
1?;)]"^
and
n
Since the souroe of the
.
error of a.,
with a covariance matrix
each equation
to
element of [(\_
covariance matrix of the disturbances ?
(3)
in
The covariance matrix
(1,
Likewise,
(1980)].
regressors are Identical
estimation of (1).
X
effects
the
in
e.^^
(1)
in
in
e
(v^ V^)
Finally, a more general SURM model (4) can
.
size in the model [e.g., Baltagl
Since the
= E
t / a in
cross-sectional fixed effect.
a
aeourity-specific time-varying effects will be in
be
e^^)
have been ruled out.
y.
they are ruled out by the assumption E(e..
Any
,
Then even wi th
by assuming non-autocorrelated disturbances
Thus,
3.
i^
'^
•••
time-dependent error component.
(orthogonal)
the
,
*^=^'
'if
assumed cross sectionally random but time-Invariant element
an
is an
*
^'i
could be written as:
we
use GLS regression
estimated
from
£
n^^
in
OLS
as
the
in
the
the residuals of
the first step regressions in (1).
The
results
this
from
those of the SURM (H)
,
and
two-step procedure are virtually identical
we
present only the latter.
to
However, the two-
step procedure is comparable with the SURM only if GLS (not GLS) regression
Is applied
in the
second step.
We provide OLS second step results for com-
-12-
m
at lea.t
ond
pa.i3on.
Tor
tM.
result.
application^^ the .i,.peoirication
errors in the second step.
severely understated standard
2.5
Preliminary Estimates
Th. two-3tep prooedur. (1)
t..
tl„e
»arle.
(3)
return,
monthly
of
and
the
=nd
for
SURH (*)
»=re applied
portfolio, of
ten
all
to
po»ibl.
June 1979 (150
oo»p.nlea fro» January 1967 to
„.„ber, of the aa„pl. of 566
the ,toc.a aocordlng to
were forced by ranging
„onth3>. The ten portfolios
containing the
1967. portfolio one
„.r.et value of equity at June 30.
the S.»M for («)
OLS for equation (3) and
,„euest atoc.3. The r.euU, for
.onth
•
period
January
to
,967
,979.
June
flr=t la the
The
different perloda."
are given In Table « for
The
full
,50
Subperlod, I motion taUe.
prior to
bias in the Relnganum sa»ple
aocount of the potential survivorship
Finally, sine,
at the end of ,975.
•„76 by breaking the overall period
over long
e^„ of («> .ay be untenable
.tatlonarity In (at least) the
»onth subperlods of
period into the three 50
periods, we brea. the total
detects a statistically algnlfloant
The 3UR« procedure
Subperiod, 11.
June ,979. (including
the period May ,975 to
negative size effect only for
January
Over the earlier subperlod
,979).
the period January ,976 to Juno
„67
to fecenber
positive,
,975.
the
point
estimates
though only a two-step procedure
of
using
the
size
OLS In
effect
the
are
T,
second
step
shows It to bo significant.
The results of Table
comparison
of
the
results
!
r.ls.
with
two
those
Interesting
reported
by
questions.
Banz
(,98,.
First,
a
Table
,)
corresponds to
,966-,975 which most closely
reveals that over the subperlod
reports a negative size
to December ,975. he
,967
January
subperlod
„„.
procedure yield.
By contrast, our SU«H
-,.55.
effect with e t-st,ti,tlo of
-13-
positive, but insignificant, point estimates for the subperiod January 1967
to December
Is
1975.
significant.
actually
t-statistlc
(overstated)
The
Second,
point
though
OLS in this subperiod
for
estimates
the
in
period
January 1967 to December 1975 are positive, those for the approximate two
subperiod breakdown January
February 1971 and March 1971 to April
This loose analysis serves only to underscore
are both negative.
1975
to
1967
general impression that the subperlods in Table
>i
are "different".
a
We now
consider the possible instability of the size effect in more detail.
3.
STATIONARITY OF THE EXCESS RETURNS
3.1
The Techniques and Estimates
Two techniques were used here to study the statlonarity properties of
the
expected
size
effect
premium
discount.
or
One
recursive
the
Is
residuals technique associated with Brown, Durbin and Evans (1975) and the
other
is
a
Kalman
filtering
Since the results were
technique.
similar,
only those from the Kalman filtering technique are discussed in detail.
The Kalman
filter
technique requires speolflcation of the stochastic
process generating the ex
here
are
generated
18
under
ante
the
excess return
series
that
assumption
a
The
{*iJf«
follows
a
results
random walk.
I.e.,
"it =
-i.t-i*"it
"it
'-'-'
^^'^
V
^^^
We return to this stochastic process assumption later, noting only at this
point
that diagnostic
checks
suggest
it
is
sufficiently well
Conditional on the specified process, the technique estimates
a
specified.
time series
of stochastic coefficients which has been filtered from the noise or error
term as in signal extraction.
It also
tests whether the resulting series
-lU-
ia
stochastic
or
reflects
Just
a
constant
The results indicate that the assumption of a non-stochastic
parameter.
150 month period
over the
around
variation
sampling
<»,
most seriously violated for the smallest and
is
A
largest
portfolios;
that
the
excess
returns
firm) six portfolios arc positively correlated
and
tenth
Inrgast
(the
portfolios;
firm)
of the
{"^j^J
19
(smaller
as are those of the ninth
,
that
first
returns
excess
the
of the
first six portfolios ore negatively correlated with those of the ninth and
tenth; and that the excess returns of the seventh and eighth portfolios are
fact best described as a
not well specified by the random walk, but are in
constant
Jensen
Black,
ranked on
and
over
a
the
Soholes
and
those with
P.,
residual variances
time
month
150
(1972)]
higher
a
(1).
in
period.
have
B.
tend
to
l^.^l
in
have higher
Since higher
to be
are
errors
standard
firms tend
)
[e.g.
portfolios
when
smaller
"extracts" a component of
(1), we likewise find higher standard errors of (a.^}
The above correlation estimates for (5.^)
for small firms.
writers
Previous
that
found
firms, ^ and since the Kalman Filter estimate [a.
th e residual
20
and
(o..) would
also be consistent with high oovorianoes between the residuals of adjacent
Black, Jensen and Soholes (1972) portfolios.
Figure
a.,
superimposes the time series of the estimated excess returns
2
for portfolios
month
1
January
period
smallest
(comprising the smallest firms),
portfolio
firm
5
displays
others.
9)
a
The
there existed
to
1967
although with different
portfolio
21
is
June
1979.
repeated
variance.
for
The
basic
each
Portfolio
10
5 and
whose returns are negatively correlat
and
•
150
shown by
the
through
6,
pattern
portfolio
^
10 over the
to
2
a
lesser degree
vlth Portfolios
1
and
time series pattern which is somewhat of a mirror image of the
plots
a
suggest
that
from
say
January
1969
to
December
1973
relatively stable positive relation between excess return
.
-15-
felatively stable
1979 there was a
fro. January 197^ to June
and Size, and
return and size.
negative relation between excess
3.2
Nonstationarity
Statistical Significance of the
establish
TO
the
that
expected
excess
returns
.
size
to
attributed
be integrated with
requires significance tests to
effects are nonstationary
of which use the
T.o tests are conducted, both
the preceding analysis.
Ansley.
program VPAR written by Craig
first test is based on
•me
the
of
variance
(portfolio)
u,,
in
For
N.
1=1
(6)
of
statistic formed by taking the ratio
a
'
variance
the
to
the large
of
In
e^.
each
for
(1)
portfolio, this statistic is
firm
asymptotically normal
terms of the statistics
2.23 which is significant In
For the small firm portfolio
level. ^^
distribution at the conventional 5%
ends of the size range
Thus for portfolios at both
the statistic is 2.01.
hypothesis that the
would reject at the 51. level the
a sampling theorist
excess* return
around
series lS,l^ is merely variation
a
The
constant «,.
relatively
small firm portfolio where the
rejection is interesting for the
the
series (e^)^ (i.e.. noise) increases
large variance of the disturbance
Further, as one would expect.
a^.
difficulty of extracting the variation
la},
for
smallest
and
the
.
the difference
largest
firms
portfolios of the
between the returns of the
is
not
constant
time,
through
with
test
a
statistic of 2.20.^^
If.
An
equivalent
in
a
above
given
zero
large firms,
for
a
test
is
subpericd.
(say)
small
given
the
lower
firms
confidence
by the
and
conclusion (ba.ed on
a
Intervals for
{a,,}
confidence
Interval
for
upper
interval
below
the
Joint
Bonferonni
{a^,}
zero
were
for
Interval) would be
and
significantly positive for small firms
that. ex ante excess returns wore
•
firms.
Significantly negative for large
Alternatively, the bands switching
.
-16-
,
,
.„te
„ro
V'
wo.l.
totcU. ..low zero
to
for
«ce»= returns,
'V-'
i...
..
'^"^^':J:,\T!:.t
bna
-
'
,„„„,,„„ t.n.3
Hron from
pnor..^^.^
zero
above
region
a
viciioiJR.iv.'-on
8^69119
s-ia
datri.
zero after that
*a
„oe«
„„
J-e
to
„H-an.
1979.
January
^o
,9.
tbe
out prev.oualy.
,.te.
^^",;
<3,
the
or
m
^^o
•
to
=,
.
^^
eat.ate
^^^^
t.
which
interval, luring
an
„,„,.ber'
190V
January
^,^,^,^
only vaUa
over
per.o. Oan.ar,
-;- ^^ a
reasonably
e.ea3 r.u.a ^e.
^
".^^.r-
er..t
la
„
,
subperloda
.9,9.
June
It
SURH C)
SUBH
the
t.a
<„
However,
.Ut.onary-<aU.u.
.„,
over t.e
P.. size
portrouo, .anKe.
by portfoU
a .,
returns earnod
nf'
'i';
the
pa
^^^
^^^
^^,^
are
^'-.a' «^rlable
i
variable coefficient
size
the
including
,n,orporating
=»^ows
which Figur^.^^a^
1975.
u
to December
.067 ^o
., 19of
a
January
^^^ results showed
ec ^^
e
aUe^
.
poaltlve an. negative
perioda of both
.
IT
^
.
,
—
.lgninoa-..lUve
eneot while
the
S..
^^ ^^
""
=
^^^
^
^=>"-'""-^^'"::o;T:25;S-o^-^-
—
waa negative
,. that the errect
,„„,
^„^
which a atatlonar,
,9V9 .ur.ng
<- -°
' '«-^-- .„bpprl„.a
whUh
Subperloda")
are
have
given
^
-^ "'"""rrZtt
-^'
-
^^ ^^^^^^
negatlve,,Ue,_err^t
'
^"""
a
"''"°t:
atatlonary al.e
In
Table
,
along
er
e
rt
e
<
he
.r.tha
two
..re.eter.lne.
with ,th^^^OU
^^^^^
^^^
-17-
comparison.
period
SURM shows
The
t-statlstlc
(the
for
positive
strong
a
coefficient
slope
the
opposite effect in the latter period (t(Y
effect
size
Y
Is
3.11)
and
the
Note that while the
is -3.17).
)
first
the
in
positive effect clearly shows up more strongly than in the Subperiods I-II
of
Table
results.
does
so
1,
the
The economic
25%
1973 small
annum,
per
25
also
holds
this
for
the
OLS
significance of the phenomenon can be seen from the
firms over large
excess returns earned by small
to December
effect;
negative
firms had
while
from
From January 1969
firms.
ante negative excess returns of about
ex
January
June
igY'J'to
they had
1979
ex
ante
M
were
positive excess returns of more than 25X per annum.
estimates
The
generated
after
"pro-teat"
estimates
ore
Table
in
stationarlty tests to "pick"
using
those
Consequently,
Subperiods
Predetermined
the
for
the
subperiods.
since
biased
the
sample
period was selected after soarohlng through the data, and the Predetermined
Subperiods
section
Table
of
instability of these effects.
^
not
is
fully
Information
about
constant.
However,
that
problems of (econometric)
5.
evidence of
further
no
the
Also, the procedure in this section of Table
efficient.
parameters
contains
^
procedure
efficient
An
(other
procedure
than
would
Y
which
)
would
assumed
are
to
the
be
and
unresolved
which
implies a
formidable
pose
"pool"
identification and diagnostics.
POTFNTIAL EXPLANATIONS OF THE STOCHASTIC "SIZE EFFECTS"
We
stable
appear
note
first
premium for
to
explanation
be
of
that
small
any
firms is at best
inconsistent
size
explanation of size
effects
estlmntcd bota coefficients.
with
in
It
the
terms
Incomplete.
simple
of
effects
27
version
non-trading
is certainly
true
The results also
of
Roll's
induced
(1981)
biases
that the Kalman
in
filter
-18-
would pick up any autocorrelation In the market model residuals Induced by
non-trading,
firms
but
the
predicted
is
explanation
Roll's
sufficient
oould
instability
constant
of a
be
to
in
sign
generalized
be
the
attributable
return
ante "excess"
ex
pattern
non-trading
of
Presumably
results
our
fit
to
magnitude.
and
small
(say)
to
allowing
by
across
firms,
but
there are still at least three factors which weigh against the explanation.
First, the size effoot.'i show up both in daily excess returns computed using
Second, the results
Scholes and Williams (1977) betas and in monthly data.
hold when the large firm portfolio is related to the value-weighted market
where
Index
recursive
residuals
reported)
(not
autocorrelation
substantial
should
problem
non-trading
the
which
not
do
would
minimal.
be
occur
general
in
Finally,
the
exhibit
the
estimated
If the
unstable because of this hypothesized instability in non-trading.
We now discuss some potential
stochastic
explain
a
First, can
apparent
the
the
beta
were
28
explanations that are consistent with
a
zero-beta factor of the Black (1972) model
excess return relative
Sharpe-Lintner model?
the
to
Suppose (1) is replaced by the unrestricted market model
=
^t
If the
holds,
"l *
^iM
^^Mt
^=^
* '^it
Sharpe-Lintner CAPM holds,
a'
it
=
(1
-
0'
In
)
E(R,,^)
«'
lu
where
=
t =
":
(1
-
H:(R,J
6'
IM
)
R_^
r
•••
If the
will- be an
instead
the
Sharpe-Lintner model
artificially induced "excess return" of
If it were the case that small
one and vice versa
only change
sign
for
the
Black model
return on an
lX,
1.x,
asset whose return is orthogonal to that of the market Index.
model holds but
^7)
"f
t
expected
the
is
;
1'
imposed on (8)
is
(
If the
1
- ^.'^^(^(R.J
In
Zt
,
Black
there
- R-..).
ft
firms had an average ^.^ substantially above
large
firms,
across slzo groups
If
the
artificial
(E(R^)
-
Rp)
excess return can
changes sign through
-19-
But
tlme.
must
E(R_)
mean-varianoe efficient
Macbeth
study,
(1973)
negative
greater
be
portfolio
equal
or
six
R_
to
- Rf)
In
the
ante
ex
Fame
and
(insignificantly)
was
subperiods (January
the
If
boundary l3 concave.
estimate of (E(L)
the
only one of the
for
than
to
1961
June
1968)
which they examined.
sampling error
The
enormous.
the
In
implied by such
January
period
positive ftxoons roturnn of
small
1979
per month or 2Q% per onnum
firms
earned
(and henoe, by
Rp exceeded R_ where R_ is the point estimate of Z(B.y^))
this explanation,
The estimated
,'',06t
Juno
to
197^^
factor explanation is
zero-beta
a
for
B.^,
in
period
this
are
and
1.15
1.10
small
for
and
.
large
For the zero-beta explanation to account for
firm portfolios respectively.
the 2.06J per month actual difference for five years, R_ must have exceeded
the
true E(R
January
1969
)
December
to
per annum (actual
As
a
Section
3
of
al
39.09J per month or 51'<2t per annum.
by
S,|.
further
where
chock,
],i\Q
very
similar
the subperiod
over
R_ la
Ii3%
and 0.9'<)«
repeated
we
excess of E(R-)
implied
the
wna replacod by the
a
appears
are
1973 the
For
to
a'
that
recursive residuals analysis in
in
of
The time series behavior
(8).
a
,
which
implies
that
the
nonstationarity of the excess expected return is not very sensitive to the
particular version of the CAPM used.
that the observed
and
Soholes
sl'/e
(1972)
Thus,
even though it seems possible
effect might somehow be related to the Black, Jensen
finding
of
positive
(negative)
alphas
for
low (high)
beta stocks, the connection is far from clear.
Another possibility is thnt the CAPM holds with respect to stocks, but
that
the
stock
betas
are
not
constant
through time because of an option
effect of debt [see, for example, Galai and Masulis (1976)].. Since size is
strongly negatively related to debt-equity ratios,
29
several specifications
.
-20-
of
returns.
However,
allowance
an
for
a
stochastic
"excess"
expected
nonatatlonary
generate
will
non-constancy
beta
the
each
for
»
size
portfolio does not seem to remove the apparent size effect In the expected
excess return a..
with several different models of °^^ ^.
We experimented
but none
had much of an
Gibbons
(1980,
nonstatlonarlty "does not seem
The
.
robustness is consistent with
concludes
who
79),
p.
a
Impact on
the
'causing'
be
to
parameter
apparent
that
rejection of the CAPM
."
restriction
Alternatively, the selection of the market proxy might account for our
results [Roll (1977)].
Suppose the Sharpe-Lintner model holds when applied
portfolio of all assets M»
to the value-weighted market
use, denoted
Then,
.
of
convexity
strict
given
efficient
the
can
It
R_.
be
[Roll
shown
(1977.
the
set,
Implied by the CAPM based on M, that is, a^/(1 - ^^^)
observed
the
proxy we
efficient frontier but it differs from M»
the
also on
is
M,
and
,
of
"Rp"
will differ from the
,
TJS)]
p.
value
that
the
post
ex
r
regression fitted using proxy M will be:
where
B,„_
ZM"
the
is
beta generated by the regression of the returns on the
zero-beta
asset which is
with respect
first term on the right hand
(1).
If
firms,
6,,,
in
any ex
exceeds one
for
to
M,
on
smnll
If
ob.'iervable,
to
evaluate.
explain
the
true market
H*
side constitutes a "spurious intercept"
small
firms and
less than one
is
for
The
.
<^^
in
large
post variability of this spurious Intercept will again be in
The spurious intercept will
opposite dlrootlona for .imeU and Ifirwo firms.
bo
the
the
common oUmont
&
l;t
.Miniill
.
Unfortunately
so the Importance of the mnrkct proxy error
But
excesss
in
(9)
P^,^,
is
not
is difficult
the order of magnitude of this proxy error required to
returns
is
the
same
as
that
of
the
zero-beta
error
.
•-21-
implausibly high.
oonsldered above, I.e.,
Addlfionally, our results imply
more than ex post variability of the spurious intercept in (9).
alternative is that the atoohaatlo expected excess return is
A flrml
symptomatic
of
the
explanation
X(t)
follows (say)
The
V.
where
as
a
omitted
is
is
a
catch-all,
in
we
consider
a
the
Since
CAPM.
the
concrete
few
suppose the traditional CAPM omits a state variable from
First,
intertemporal
denoted
misspeciflcation
underlying
an
misspeciflcation
examples.
30
asset
capital
a
function
of
pricing
model
time.
Through
mean reverting (elastic)
term
related
term contributes to
to
a
in
tho
size
time
If
the
Merton
state
(1973)f
variable
random walk with central tendancy
CAPM might be of the
ranking.
of
(ICAPM)
[E^(X(t))
form
- w]
6
j<
[E^(X(t))
0,
-
V]
this omitted
nonstationary expected excess return which Is related
to si ze
Our conc.luftfonM with rfflpnct to tho nniiBtntlonnrlcy of the size effects
do not appear sensitive to the exact form of the stochastic process.
31
For example, we have evidence of a lag twelve factor in the diagnostics
from the random walk fit for the small firm portfolio.
The seasonal
factor appears to be robust with respect to choice of the market index, and
consistent with the evidence of
a
"January effect" reported by Keim (1981).
If a dummy variable for January is included in our Kalman filter,
it is
significant for both small and large firms and both value-weighted and
-22-
In
Indicea.^^
ecually-weightcd .arket
all
significant Instability
case.,
allowing for the January dummy variable.
in the .ize effect remains after
repeated
We
the
HYSE from
takes
possible
to
are
and
1978,
to
shown
speculate on
tho
with
here
but
unexpected inflation effects.
only
Importance
this
On
from
AMEX firms
all
confirmed,
are
here
associated
procedures using
stochastic estimation
magnitude
the
on
Which
1926
reported
resulta
the
later
the
period,
the other hond
It
is
effect of variables
size
example,
for
.
effect
L.U.a-4^5€^.
the
since
on
T^e
1973.
nonstaj^lonary^ size
the
for
to
1963
firms
all
expected
and
the data set of NYSE-iLMEX
post-1950's
sequence of realizations over the
returns constitutes only one
of a number
"relatively unique" realizations
period m Which there may be
provide
the data sot may simply not
If .o
of highly colimear variable..
.
the source of the size effect.
enough degrees of freedom to pinpoint
«
6.
SUMMARY
Wc
believe
that
there
are
three
related anomalies in .tuck returns.
between excess returns and
of
size.
The
transform
distribution of firm size.
new
here
results
relation
First, we have shown that the
firm size oan be regarded
is
important
Second, we
because
have shovm
as
linear in the log
of
skewness
that the
constant through time.
returns attributable to size are not
shown
can
that different estimation methodolosles
conclusions about the size effects.
^ZE^
size-
concerning
lead
ex
in
the
ante excess
Third, we have
to
different
.
FOOTNOTES
By "excess return" we mean ths extent to which the rate of return
deviates from that predicted by the trac'itional capital asset pricing model
(CAPM).
2
In
(1972)
Section 5. we examine whether our results are affected if the Black
CAPM is considered ra'^her than the Shnrpo-Lintnor version.
'Banz (1979, ^93'^) does net work with
Black (1972) version of the CAPM.
(1)
directly since he uses the
L
See Danz (1979, p.
12)
and Figure
1
below.
5
For construction details, see Center for Research in Security Prices
Briefly, all available securities are ranked into ten portfolios
(1980).
according to their Scholes-Williams (1977) betas calculated on daily
returns.
A security's day by day excess return is then calculated as the
raw return less the equally weighted average return for that day for the
portfolio into which the security is ranked.
Note that this method is
excess return and a
between
relation
potentially biased against finding any
variable such as size that is correlated with beta since the mean daily
return for each portfolio is taken out of these excess returns.
•
The companies had 35 quarters of complete data for earnings,
dividends, and prices from June 1967 through December 1975 on a quarterly
For
Compustat tape.
All companies had fiscal years ending on December 31.
The numbers
the analysis in this paper 566 of the 577 companies were used.
differ because ten companies were not contained on the CRSP dally master
and return tapes and because Reinganum was unable to find the earnings
announcements for one multinational company, Unilever Ltd.
563 of the 566 are in the CRSP daily excess returns file,
9 of the 563 have no excess return data since January 6, 1976.
Itowever
although
o
In fact,
AMEX/NYSE fipns are "large" relative to the size (market
value of equity) that many appear to have in mind when discussing small
firms.
If one really wished to focus on transaction/information costs, OTC
stocks would seem more appropriate all else equal.
[See for example,
'»Drexel Burnham Aide Seeks Bargains Among Snail Firms That Others Shun,"
The Wall Street Journal
Tuesday, June 12, 1979, p.6].
,
q
Note
for
example the overall ratio of negative to non-negative
averages in Table 3.
As a check, the excess returns for all securities on
the excess returns tape for this period were also -alculated; there were
the a>'2rage for all
952635 negative and 826958 non-negative averages, ar.
securities for this period was close to zero (O.OOOOH
i
;
Since many earnings announcements
after December 1975, we also examined
with similar results.
would be made in the three months
excess returns from April 1976 on
2
f
.
-2'\-
The size
and
return variables for each portfolio are equally
The results for the
weighted averages of the component securities.
expanded sample are virtually identical to those in Figure 1.
1
incorporates only one
It Is important to understand that (2)
particular size effect variable, albeit the one which has been the focus of
It is possible that size
attention in the studies mentioned earlier.rankings given in (?) by the average size over time, could be "priced", as
could temporal deviations of firm size around this mean.
,
A common y
o£ y, must be imposed to allow estimation, although in
If a common
ge
both pnrnmatora need not be common across all assets.
o«norol
nocpsaary
is
to
la
imposed,
of
avoid
singularity
In the
equality
two y^.
Y
must be restricted to allow any
regroasor matrix, and at least onO more y
degrees of freedom in a orooy seotlonal regression such as (3)«
11
Although this specification is formally necessary to achieve
consistency between the two-step GLS and SURM estimation, It does not seem
overly restrictive since the CAPM as a null hypothesis also rules out any
cross sectionally random but time-invariant component of a...
15
As discussed
In
footnote 13-
In a related application, Fama (1976, p. 336) notes that the
parameters obtained in Fama and MacBeth (1973), by cross sectional OLS
portfolio betas, have
regression of portfolio returns on (estimated)
Gauss-Markov minimum variance properties only if the cross-sectional error
terms are i .1 .d
.
Estimation of CD is lty''t'He full information maximum likelihood (FIML)
procedure of TSP using the Gauas method [see Hall and Hall (1979)].
^fi^tf'
18
All recursive residual results are available on request. The Kalman
filter results arc obtained using programs written by Craig F. Ansley which
are described In Ansley
(1979) (1980),
•
IQ
Under the assumption thot a. follows a random walk, the "correlations"
of the [a.
are not defined.
However, the stationary first differences of
the {a.,.r for the first six size portfolios are also highly mutually
correlated (.96 >^ p 2. '85, where p = the correlation coefficient), and
strongly negatively correlated with those of portfolio 10 (-.57 > P >.
For portfolio 9, the IS..) are e.itimated as close to constant, and
-.7'!).
the first dlfforenooa i\re not siRnl ioantly correlated with those of the
other portfolios.
]
20
Note that given the positive correlation amongst the returns of the
first six size-ranked portfolios, there must be scxne negative correlation
between one or more of them and the remaining four portfolios, or between
the returns on the remaining four (since excess returns must sum crosssectlonally to zero).
The cross-sectional constraint does not imply,
however, the particular cross-sectional correlation pattern observed here.
.
-25-
2lKote that the estimate o^^^^-^^J
%. ' n BlL\^. Jensen and
10 is
portfolios i = 1
They ^^lll\°l%l^^,'llJ,-^,^^^^^
average of
Scholes (1972. p. 109).
7/°/^; j^^;/;,;;^^'. since the portfolios.
for
size
^'-.t^-^^^^.^j;^: ,,Tiud
ud^ th^int;rrnediate
n
our
walk assumption.
i;%nTi';risTuU°;c°onsrstent With our rando.
their
Ve
^h.
least
for
the case of a
results by Garbado
(1977)
-"?- -,^\,n°^a^;:;tTtr
Uhf:s:ut^°)
P °^
that the a
„,ai
-
«"Sgest
^^,^^ ,,,
likelihood ratio te=t '^'^
coefficient,, aaZ^
[J'^^^^^'ZlZ
hypothesis of constancy of ^he
^^^^^ hypothesis
approximation as the
Pagan (1980) establishes
thirty-one
^"^^^VrV/f
size
°"%^;
increases, even with a sample
Variance ratio (in a
^^Jf .^"Jy
and
-^P,\°
consistency
stochastic
the
h^d' onJ; when the
standardized form) but '^^^ f^l^t.^~.
identified.
coefficient version of (1) is
.
^.^^tf
^
omitted.
when the seasonal dummy is
two
^'T.o..h constancy over tnese
less
be
induced bias (squared) mlBht
from
result
estimators ul.lch would
period.
partltionlne of the overall
f
^Slnce
the
period
January
.969
t^n"e7feft"ls'^r?p?icated'\rerr
the size effect
to
an'd'
^'^';„'°J,^"',f,',rean"
"^="
^''^^/"'^tl^^tlon
(pre-test)
^^-^^ ^^f ^ .^^s
co'nf/r.ed
the
Uncf o'/ tM
of
d'au"
apparent
a
finer
i Tl'f
swltoHlns
in
data.
found using monthly
unra^iUa;
example might assist ^^0^
econometrics
If the null
Some introductory
J°'^l\, ''riods.
^^^'''°°^'
and l^^fwar^^
prebetween
parameters
changes in
^^^ parameters are
v^th^p^^^^^^^^^^
^Sn
Intermediation.
could be reduced by financial
.
.Jt-nrvatr^f t%r:aT:::e-:i^\^.irdrtinr
arrib^i:
beta
the ran. °-»;
^"^^.-"^^Lr^au";" M^Pie'Trarsform
and the <''="-'-°'"\ "''„,,
(size)
equity
of
,alue
'jV" 'of the relation is
(T/S^lfhaye .1=0 e^a^med the
"Table
. .l.es
::airiror;r;ifst"''c;;rlst\e-::dr^r
debt-equity ratios.
relation between size and
-26-
1
However, one logical possibility is that ex ante shifts in the
relation between the true index and (say) the NYSE index Induce (ex ante)
shifts in
^2.H»'
•3 1
Monte Carlo experiments (with a sample size as small as 35), Cooley
and Prescott (1973) also found that the random walk intercept model is
robust, in terms of mean square forecast error, to specifications of the
The alternatives considered were a model
intercept's stochastic process.
with a small probability of a large change in the intercept in each period,
and a model with a constant change in the intercept each period.
In
^ The
January dummy variable coefficient, its t-statlstic, and the test
Btatlatlc for inHtablllcy in tlu; size effect with the January dummy, and the
test statistic for infltnblllty In Che size effect without the January dutar.y,
rcBpcctlvely, arc
-0.11, -3,12, 2.31, 2.23 for the large-firm portfolio and
valuc-welnlited m/irkut liidox;
-0.037, -A. 85, 1.93, 1.70 for the large firm
portfolio and i.M|nnUy-woliihtuii market Index;
0.100, 7.60, 2.47, 2.01 for
the omall-flrm portfolio and value-weightod index; 0.053, 5.96, 1.84 and
0.37 (10~^) for tlie Bmall-firm portfolio and equally-weighted Index; 0.111,
7.46, 2.48, and 2.20 for the difference between small and large firm
portfolios and value-weighted Index; and 0.090, 6.76, 2.27 and 1.80 for the
difference between small and large portfolios and the equally-weighted index.
:
33
Indeed,
case of the small-firm portfolio and the equallyindex, the instability in the size effect becomes
statistically significant only after the January effect is included.
weighted
in
market
the
.
-27-
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(I)
The stoclc
In
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for thelt average daily returns. ii.'
REINCANUM
January 1976 - Dacombar
TOTAL
ALL STOCKS
SAMPLE
197.8
July 1962 - Deceober 1978
V
/able
RoHull" of
|ior( fxl lo
llio
l(.'liir..iiiiiin
.liiiiii
\')l''i
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rci'i .'I'lIiMi
tlio
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of moiitlily rxi'nm rcliiniH on
furnud liy illvlilliu-,
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(It) tin- imliiui l.iilii January \')U\ to
Mul'iiurli'ilii, ,in<l
J.iiuury 1^74 to Juno I^IH ovur wliUh proUmlnjtry tests
to be conatmu.
).
In
ii'v'ii
iinJ
tho nlza uffoct
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O^ - Yq + Yj^S^ + n^
1
-
1,...,10
(3)
whore
R^
S.
market Index
"
VaIuc
••
log flzo of portfolio
wclp.liccd
1
.
Hodel for SL'RMi
«^tt-
•
nvcrnno
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IS'i? to
V "Yq* VVVc'^Vi^St
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• utLiiitnl K% rantxm wilka (or
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t lie
)
for pvirtfolloa 1, ], and 10
1967 lo June 1979
I'crloJ .Kinuury
''lt*''lM'^a-^c>*'u
(«)
"it " °i.t-i*"u
E(u|^) - al
Ktjj.u^^) •
U*k
Adjunied
for
»n
t,T - 1,...,T,
(1)
1
" 1,5,10
7849 U05:Sf
*S
'..
Date Due
W^ 7,W
Lib-26-67
HD28.M414 no.l379- 82
Marsh. Terry
74533,7.
,.
3
A/New
D^BKS
TDfiO
evidence on the nat
00.142875
0D2 153
TflT