Can a Global Model Explain the Local
Cross-Section of Equity Returns?∗
Greg Buchak
University of Chicago
June 9, 2015
Abstract
I examine global integration of Size, Book-to-Market, and Momentum
anomalies by testing whether a model capable of pricing the returns of
portfolios sorted without consideration of country is capable of pricing
within-country variation in analogous sorts. Previous models that reject
global integration suffer from a joint-hypothesis problem because they
jointly reject global integration and the candidate global model being
the correct global model. This paper sidesteps the latter problem by
using a global model, which, by construction, prices the global cross
section. Therefore, rejections of the global model in local markets is
due to the local markets being non-integrated. I find evidence that the
anomalies in North America, Europe, and Japan are integrated, while
the anomalies in Asia-Pacific are not. My results imply that economic
models explaining the anomalies should be global in nature but allow
for some regional frictions.
∗
Thanks to Eugene Fama, Tobias Moskowitz, Bryan Kelly, and students of the Spring
2015 Finance Research Seminar for their very helpful comments.
1
Introduction
The CAPM cannot explain the cross-section of equity returns when sorted
on size, value, and momentum. As Fama and French (2012) show, this is
a worldwide phenomenon, occurring when sorts are performed both within
country and globally, without respect to the equity’s home country. In this
paper, I study whether these empirical regularities that exist separately in
many countries are all parts of singular, integrated size, book-to-market, or
momentum effects, or whether these effects around the world represent many
disconnected, local phenomena. That is, can a model capable of explaining
returns of sorts formed without respect to nationality—global sorts—explain
returns of sorts formed within-country—local sorts?
Fama and French (2012) ask and answer a similar question by testing
whether a global three- or four-factor model can explain the cross-section
of local sorted portfolios. By running Gibbons/Ross/Shanken (GRS) Gibbons et al. (1989) tests with local portfolios formed on size/book-to-market
or size/momentum as left-hand-side variables, and either a global or local factor model as right-hand-side variables, Fama and French find near-universal
rejection of both the global and local factor models in local markets. Global
integration stands little chance to pass this test, however, because the global
three- and four-factor models fail to pass GRS tests on the corresponding
global sorts themselves. This points to the difficulty in testing global integration using an ex-ante model arising from a joint-hypothesis problem: if a global
model fails to price local equities, is it because markets are not integrated, or
because the candidate global model is not the right global model?
The literature contains multiple examinations of global financial integration and asset pricing integration in particular, using varied approaches.
Karolyi and Stulz (2003) review the literature, which has focused on evaluating international versions of the CAPM, international versions of general
multi-factor models, and international versions of consumption-based asset
pricing models. The report that tests of global CAPM or multi-factor models tend to have limited success in explaining local cross-sectional variation.
2
Griffin (2002), for instance, finds that local factors explain more time-series
variation than global factors. Hou et al. (2011) search to find global factors
or firm characteristics that best explain stock returns. Comparing the relative
success of global and local versions of the models, they find that local models
produce smaller errors. In testing directly whether whether global models can
explain local cross-sections, they suffer from the same joint-hypothesis problem
in that all candidate global models tend to do badly in pricing global equities
unconditional on the country.
Some papers also depart from the traditional GRS asset-pricing paradigm
and test integration in different ways. Guesmi and Nguyen (2011) examine
the time-variation in covariances of local market returns to global returns and
document that at the country index level, integration has been increasing in
Latin America but not elsewhere. This study does not, however, examine the
cross-sectional variation of returns within countries, which is my object here.
Finally, in a strategy that departs from more traditional asset pricing tests,
Bekaert et al. (2013) examine integration in a model-free way by considering
convergence in valuation differentials among countries before and after integration events, such as joining the European Union. My test is more traditional,
but attacks the joint-hypothesis problem in a different way.
Here, I address the question of global integration by sidestepping identification the correct ex-ante global model. Unlike Fama and French (2012), who
test integration using a specified global factor model, I do not attempt to identify the correct ex-ante global model. Instead, I ask, given some model that
prices the global cross-section ex-post, could that model have priced the crosssection in local markets? This approach allows for a direct test of whether a
successful global model would have succeeded locally. Given global and local
five-by-five sorts on size and book-to-market or size and momentum, I calculate the ex-post tangency portfolio from among the global sorts only. By
construction, this “model” will produce a cross section of ex-post αi ’s that
are identically zero among the global sorts. I then use this global tangency
portfolio as the candidate model in the right-hand-side of GRS tests of local
five-by-five sorts. Failing this second GRS test means that there is cross3
sectional variation in the local sorts that the global model cannot explain,
which I interpret as a rejection of integrated pricing.
I find that global ex-post tangency portfolios formed on size and bookto-market or size and momentum price local cross-sectional variation in North
America, Europe, and Japan, but consistently fail to do so in Asia-Pacific
(ex Japan). My results are robust to alternative construction methodologies
and specifications of the test. This suggests that with the exception of the
Asia-Pacific region, the empirical regularities of size, book-to-market, and momentum returns are globally integrated, and that the correct global model will
be sufficient for these regions.
The question is of academic and practical interest for several reasons.
From an academic standpoint, the answer helps focus research on the appropriate candidate explanations of size, book-to-market, and momentum effects.
If global effects explain local effects, it means that local anomalies are driven
by common exposure to a global state variable—or a behavioral bias—that impacts all localities alike. On the other hand, if the global model cannot explain
local cross-sections, it means that local state variables—or behavioral biases—
that do not impact global markets must drive the local regularities. From a
practical standpoint, this test quantifies the portfolio-management value of
including many local factors rather than a few global factors. If global fiveby-five cross-sectional variation absorbs the local five-by-five cross-sectional
variation, having both local and global versions of models that seek to explain
this variation may have limited marginal value for the purposes of portfolio
construction. The GRS test directly measures the Sharpe Ratio benefit of
additional local information.
My paper proceeds as follows: In Section 2, I describe in detail the test
specification and data, in Section 3, I present my main results, in Section 4, I
present alternative constructions and robustness checks, in Section 5, I discuss
the results’ economic significance, and in Section 6, I conclude.
4
2
2.1
Data and Tests
Data
Data are from Kenneth French’s website and run monthly from July 1990
through February 2015. For most test specifications, the right-hand-side variables are weighted combinations of five-by-five global sorts—sorts made without regard to an equity’s home country—on (1) size and book-to-market and
(2) size and momentum. For many tests I also include local value-weighted
market returns as a right-hand-side asset to absorb level differences in regional returns. The four left-hand-side test regions are (1) North America, (2)
Europe, (3) Japan, and (4) Asia-Pacific, i.e., Asia-ex-Japan. The composite
regional portfolios include the following countries. North America includes (1)
the United States and (2) Canada. The European portfolio includes (1)Austria, (2) Belgium, (3) Denmark, (4) Finland, (5) France, (6) Germany, (7)
Greece, (8) Italy, (9) Ireland, (10) the Netherlands, (11) Norway, (12) Portugal,
(13) Spain, (14) Sweden, (15) Switzerland, and (16) the United Kingdom. The
Asia-Pacific portfolio includes (1) Australia, (2) New Zealand, (3) Hong Kong,
and (4) Singapore. For each of the four test regions, I use similarly constructed
five-by-five sorts on size/book-to-market and size/momentum. In secondary
tests, I also use two-by-three sorts of size/book-to-market and size/momentum,
and Fama-French three-factor and four-factor models.
The object of the paper is to understand local, cross-sectional variation
in equity returns. Table 1 shows monthly CAPM α’s of five-by-five sorts.
As documented at length in the literature, small stocks, high book-to-market
stocks, and stocks with higher 2-12 returns tend to have high average returns
relative to the CAPM’s predictions. This effect is strong globally with the
possible exception of Japan. A particularly vexing portfolio in previous asset
pricing tests has been the smallest, lowest book-to-market portfolio, with returns consistently too low relative to the CAPM and Fama-French Three- and
Four-Factor models.
To illustrate the relationship of the local sorts’ returns to the global sorts’
returns, Table 2 shows the time-series correlation of CAPM residuals for each
5
Table 1: Monthly CAPM α for global and local sorts. Each element in the table represents
the αi from the following time-series regression: rti = αi + βi rtm + it , where rti is portfolio’s
excess return and rtm is the excess return of the local value-weighted market index.
Low
Size/Book-to-Market
2
3
4
High
Global
Small
2
3
4
Big
-0.34
-0.34
-0.23
-0.09
-0.05
-0.09
-0.05
-0.07
-0.01
-0.02
0.21
0.11
0.05
0.07
0.03
0.28
0.18
0.13
0.12
0.07
0.57
0.27
0.24
0.16
-0.05
-0.50
-0.51
-0.39
-0.40
-0.45
0.23
0.07
0.04
0.01
-0.08
0.43
0.18
0.14
0.15
0.02
0.71
0.39
0.20
0.16
0.16
0.97
0.57
0.37
0.39
0.16
North America
Small
2
3
4
Big
-0.44
-0.61
-0.06
-0.07
-0.04
-0.21
-0.19
-0.13
-0.04
0.00
0.15
0.09
0.12
0.17
0.01
0.18
0.12
0.12
0.13
0.07
0.48
0.19
0.27
0.20
-0.20
-0.56
-0.58
-0.44
-0.46
-0.44
0.25
0.20
0.05
0.12
-0.04
0.47
0.22
0.27
0.27
-0.01
0.75
0.33
0.33
0.19
0.16
0.90
0.50
0.37
0.42
0.23
Europe
Small
2
3
4
Big
-0.54
-0.26
-0.18
-0.04
-0.06
-0.18
-0.06
0.03
0.05
0.03
-0.02
0.07
0.01
0.06
0.02
0.10
0.21
0.06
0.04
0.09
0.28
0.28
0.18
0.14
-0.02
-0.94
-0.84
-0.56
-0.51
-0.51
-0.08
-0.12
-0.09
-0.02
-0.11
0.19
0.24
0.17
0.15
0.13
0.58
0.43
0.30
0.26
0.18
1.18
0.93
0.58
0.60
0.21
Japan
Small
2
3
4
Big
0.04
-0.25
-0.18
-0.29
-0.17
0.02
-0.16
-0.15
-0.01
-0.00
0.14
0.00
-0.08
-0.05
0.00
0.19
0.13
0.00
0.13
0.21
0.33
0.15
0.22
0.19
0.44
0.25
0.00
0.03
0.00
-0.05
0.32
0.07
-0.10
0.05
-0.13
0.29
0.10
0.04
-0.03
-0.17
0.38
0.17
0.09
-0.02
0.00
0.10
0.06
0.12
0.12
0.06
Asia Pacific
Small
2
3
4
Big
-0.16
-0.74
-0.56
-0.04
-0.12
-0.26
-0.61
-0.29
0.14
0.11
0.03
-0.23
-0.03
-0.12
0.11
0.31
-0.06
-0.00
0.12
0.03
0.79
0.16
0.02
0.24
0.14
-0.52
-1.29
-1.09
-0.68
-0.05
0.24
-0.09
-0.24
0.03
-0.18
0.62
0.12
0.11
0.14
0.20
1.12
0.33
0.43
0.24
0.24
0.89
0.42
0.32
0.34
0.28
6
Low
Size/Momentum
2
3
4
High
Table 2: Correlation of monthly excess returns between local sort (i, j) and global sort
(i, j). Excess returns are rti − αi − β i rtm = ˆit from Table 1.
Low
Size/Book-to-Market
2
3
4
High
Low
Size/Momentum
2
3
4
High
North America
Small
2
3
4
Big
0.80
0.81
0.81
0.85
0.89
0.77
0.76
0.81
0.72
0.70
0.76
0.80
0.73
0.72
0.81
0.73
0.76
0.75
0.78
0.82
0.74
0.76
0.77
0.83
0.84
0.83
0.85
0.88
0.91
0.88
0.65
0.76
0.80
0.81
0.88
0.61
0.67
0.70
0.71
0.87
0.68
0.73
0.71
0.76
0.86
0.78
0.86
0.89
0.92
0.91
Europe
Small
2
3
4
Big
0.77
0.70
0.69
0.70
0.72
0.71
0.59
0.55
0.65
0.61
0.71
0.59
0.61
0.66
0.50
0.69
0.68
0.76
0.72
0.65
0.64
0.68
0.73
0.69
0.71
0.77
0.78
0.80
0.80
0.78
0.71
0.72
0.73
0.71
0.73
0.73
0.72
0.71
0.71
0.63
0.69
0.66
0.64
0.69
0.75
0.69
0.67
0.64
0.68
0.83
Japan
Small
2
3
4
Big
0.27
0.36
0.36
0.37
0.50
0.24
0.25
0.38
0.32
0.48
0.26
0.25
0.39
0.42
0.38
0.34
0.38
0.51
0.52
0.32
0.39
0.51
0.51
0.40
0.30
0.53
0.58
0.57
0.52
0.54
0.58
0.56
0.61
0.58
0.42
0.51
0.52
0.51
0.50
0.48
0.49
0.46
0.45
0.34
0.50
0.35
0.38
0.40
0.46
0.53
Asia Pacific
Small
2
3
4
Big
0.41
0.22
0.23
0.02
0.24
0.37
0.38
0.24
0.07
0.09
0.49
0.22
0.23
0.16
-0.03
0.49
0.29
0.30
0.32
0.16
0.46
0.26
0.19
0.15
0.23
0.48
0.42
0.37
0.34
0.26
0.41
0.37
0.38
0.39
0.36
0.43
0.32
0.37
0.33
0.22
0.45
0.35
0.22
0.27
0.35
0.54
0.36
0.41
0.42
0.46
local market’s (i, j)th sort to the corresponding global (i, j)th sort. These correlations are relatively high in North America and Europe and relatively lower
for Japan and Asia Pacific, though still in general significantly positive. I take
this as preliminary, informal evidence that these anomalies are related globally.
Correlations, however, are insufficient to conclude that regions are integrated
under my criterion. My test requires that the local portfolios’ premia be explainable by the same model that explains the global premia.
7
2.2
Testing Approach
Write the returns for sort i in country c at time t as
g
rict =
J
X
c
g g
βicj
r̃jt
+
j=1
J
X
c
c
βicj
r̃jt
+ ict
(1)
j=1
g J g
c J c
Where r̃jt
is a set of global factors and r̃jt
is a set of country-specific
j=1
j=1
local factors. Global factors, by definition, will be those factors necessary to
explain cross-sectional variation in portfolios sorted without conditioning on
the equity’s home country; local factors will be the residuals necessary so that
within-country sort i has E[ict ] = 0 for all i.
g J g
A country is integrated if the set of global factors r̃jt
alone is suffij=1
cient to explain cross-sectional variation of its within-country sorts. Formally,
country c is integrated if for all test assets i in country c,
g
"
E rict −
J
X
#
"
g g
r̃jt = E
βicj
j=1
c
J
X
#
c
c
βicj
r̃jt
=0
(2)
j=1
If the true global factors are observed, a GRS test with those factors as righthand-side variables will test jointly whether (2) is true for each i in c. The
true global factors are, however, not observed. Instead, a test must use a
g
candidate global model consisting of J0g proxy factors with returns r̂jt
and
g
factor exposures bijc . The test above becomes

E rict −
g
J0
X



J0g
Jg
Jc
X
X
X
g
g g
g
c
c
bgijc r̂jt
=E
βicj
r̃jt +
βicj
r̃jt
−
bgijc r̂jt
j=1
j=1
j=1
(3)
j=1


" Jc
#
g
J0g
J
X
X g g X g g
c
c
βicj
r̃jt
=E
βicj r̃jt −
bijc r̂jt  +E
j=1
|
j=1
j=1
{z
global model error
}
Tests, such as Fama and French (2012), which use a candidate three- or fourfactor global model to test for global integration may reject because the global
8
model error is non-zero. In fact, a test of (3) on global sorts fails when using
a global model, suggesting that the global model error is significant.
To test directly (2), one needs a model providing zero global model error.
Rather than looking for those factors ex-ante, we can choose a particular factor that will guarantee this condition ex-post: the ex-post tangency portfolio
formed by the global sorts with return rtτ . This will, by the definition of the
global factors, price them. That is, we will have,
g
rjt
= βj rtτ + jt
E[jt ] = 0
Then, for each local sort, we have
c
rict = βiτ rtτ +
J
X
c
c
βicj
r̃jt
+ ict
(4)
j=1
Then a GRS test with the global tangency portfolio as the global model will
test (2).
2.3
Test Specification
Letting r̂git be the twenty-five-by-one vector of in-sample average excess returns
of the global sorts and Σ̂ be the twenty-five-by-twenty-five in-sample covariance
matrix of the global sorts, the tangency weights are:
ŵ =
Σ̂−1 r̂git
10 Σ̂−1 r̂git
(5)
The returns of the global tangency portfolio are the rtτ = rt0 ŵ, which become
the right-hand side factor in a test of (4). Equivalently, we run the following
regression for each i ∈ {1, · · · , 25}:
rict = αic + βic rtτ + ci,t
9
(6)
and test whether the α’s from (6) are jointly zero, both for all regions at once,
and for each region separately. The GRS test here can be understood as testing
whether the ex-post global tangency portfolio formed only from global sorts is
the tangency portfolio formed from both the local sorts and global tangency
portfolio. That is, I test whether there is idiosyncratic information in the local
sorts that aids in portfolio construction.
Over the test period, equities at the regional level exhibited drastic differences in average returns. Explaining country-level differences is not this
paper’s objective. Consequently, in most tests, I absorb level differences by
including in the right-hand-side of the regression the local value-weighted marc
. With this modification, (6) becomes:
ket, rm,t
c
c
ri,t
= αic + βic rtτ + βim rm,t
+ ci,t
(7)
(7) has the interpretation of testing whether a global model explains local
variation not explained by a local CAPM. To interpret the testing methodology, the test of (4) asks whether a model that successfully prices global
cross-sections correctly prices local cross sections. Failure to reject (4) suggests that the candidate market is integrated in the sense that a model able
to price the global sorts—one that spans the global tangency portfolio—would
be sufficient to price local sorts. This test does not identify the ex-ante global
model, but tells us that if we had it, it would work locally. My focus is to
ask the conditional question: If I had the right global model, would it work
locally?
3
Main Results
Table 3 shows the GRS tests using the CAPM, Fama-French Three- and FourFactor Models, and the global ex-post tangency portfolio as right-hand side
variables. The portfolios’ α’s are shown in the appendix. The left-hand side
variables are the five-by-five local sorts from each of the regions (twenty-five
portfolios from each of four regions for one-hundred LHS portfolios in total).
10
Table 3: GRS test of all four regions (North America, Europe, Japan, Asia-Pacific) as LHS
portfolios on different global models as the right-hand side variables. Each line represents a
different candidate model: CAPM uses the global market return. FF3 uses the global FamaFrench Three-Factor Model. FF4 uses the Fama-French Four-Factor Model. 5x5 Global
Tangency uses the ex-post global tangency portfolio formed on sorts of the corresponding
column: For Size/BM, the global tangency is formed from the 5x5 global Size/BM sorts.
For Size/Mom, the global tangency is formed from the 5x5 global Size/Momentum sorts.
For Size/Bm/MOM, the global tangency is formed from the 50 portfolios composed of the
5x5 Size/BM and 5x5 Size/Mom sorts. Each test is whether the αic 100
i=1 from the regression
c
ri,t
= αic + βic · M ODELt + i,t are jointly zero.
CAPM
FF3
FF4
Global Tangency
Size/BM
GRS p(GRS)
2.36
0.00
2.37
0.00
2.26
0.00
1.36
0.04
Size/Mom
GRS p(GRS)
3.44
0.00
3.35
0.00
3.14
0.00
1.79
0.00
Size/BM/MOM
GRS p(GRS)
2.55
0.00
2.48
0.00
2.29
0.00
1.22
0.14
The GRS test strongly rejects the CAPM, Fama-French Three- and FourFactor Models at the 5% significance level.
For the ex-post tangency portfolio, the test narrowly rejects that the
Size/Book-to-Market α’s from the ex-post tangency portfolio are zero at the
5% level, rejects that the Size/Momentum α’s from the ex-post global tangency
portfolio are zero at the 5% level, and fails to reject all 200 regional α’s from
the ex-post global tangency portfolio composed of both Size/Book-to-Market
sorts and Size/Momentum sorts are all zero at the 5% level. This suggests that
local size/book-to-market effects are not entirely integrated with the global
phenomenon, but are more integrated than the CAPM or other factor models
suggest. Size/Momentum is more strongly local in nature, though again, less
so than the factor models suggest.
While Table 3 shows the results using the ex-post model to jointly explain
all regions at once, it is informative to break up by region to test which regions’
anomalies are integrated and which are not. To that end, the remainder of the
paper considers disaggregated regions. Table 4 shows the results for each region
separately. In addition to running the GRS test with the global tangency being
the only right-hand side variable as in Equation (6), columns labeled “with
market” also show the results when including local markets as right-hand side
variables as per Equation (7). This inclusion removes region-specific level
11
effects and focuses the analysis on within-region cross-sectional variation, but
does not dramatically impact the bottom-line results. For the remainder of
this paper, GRS tests will include the local market returns as RHS variables.
Table 4: Regionally disaggregated GRS tests using the ex-post global tangency portfolio
as the RHS variable and local 5x5 sorts as the left-hand side variable. The Size/BM column
uses only the global Size/BM tangency to price local Size/BM sorts; the Size/BM with
market column also includes the local market portfolio as a RHS variable in the GRS test.
Similarly, the Size/Mom column uses only the global Size/Mom tangency to price local
Size/Mom sorts; the Size/Mom with market column also includes the local market portfolio
as a RHS variable in the GRS test. These test whether the {αt } from Equations (6) and
(7), respectively, are jointly zero. σα is the standard deviation of the portfolio α’s, and R2
is the average R2 from Equations (6) and (7) in each test.
Size/BM
Global
North America
Europe
Japan
Asia Pacific
Size/BM, with market
2
GRS
p(GRS)
σα
R
0.75
0.74
0.68
2.11
1.00
0.80
0.81
0.88
0.00
0.00
0.10
0.09
0.13
0.25
0.03
0.02
0.02
0.02
0.05
Size/Mom
Global
North America
Europe
Japan
Asia Pacific
GRS
p(GRS)
σα
R2
0.76
0.75
0.58
2.24
1.00
0.80
0.81
0.95
0.00
0.00
0.09
0.09
0.12
0.25
0.83
0.75
0.83
0.79
0.78
Size/Mom, with market
2
GRS
p(GRS)
σα
R
1.10
1.74
1.10
2.41
1.00
0.34
0.02
0.34
0.00
0.00
0.11
0.21
0.21
0.46
0.04
0.01
0.03
0.05
0.06
GRS
p(GRS)
σα
R2
1.03
1.74
0.90
2.60
1.00
0.43
0.02
0.60
0.00
0.00
0.12
0.22
0.27
0.45
0.80
0.72
0.80
0.75
0.77
Table 4 reveals a common pattern present for nearly every subsequent
test: we fail to reject integratedness for North America, Europe, and Japan
for Size/Book-to-Market and Size/Momentum, but strongly reject integratedness for Asia-Pacific. These tests show that global cross-sectional variation
in returns to Size/Book-to-Market returns explain well local cross-sectional
variation in North America, Europe, and Japan and suggests that the effects
in these countries are integrated with the global effects. The cross-sectional
variation in the corresponding portfolios in Asia-Pacific, on the other hand,
appears to be local and idiosyncratic. This is surprising because per Table 1,
12
the five-by-five Size/BM sorts in Asia-Pacific exhibit the same patterns that
α’s are decreasing in size and increasing in Book-to-Market that is present
globally and in other regions.
A similar pattern holds for Size/Momentum sorts: North America and
Japan pass the global test and Europe barely fails, suggesting integration in
these effects. As before, Asia-Pacific strongly fails the test. Overall, the global
portfolio is less successful with local size/momentum than with local size/bookto-market as seen by comparing the p-values from the two tests. Despite the
weaker results, it is interesting that the results mostly favor integration for
momentum. Momentum may present a behavioral “under-reaction” story, but
interestingly, the returns to exploiting this anomaly seem to be linked globally.
That is, if a behavioral bias is present, the returns to exploiting the local biases
are globally linked, or alternatively, the marginal investor causing the bias is
the same everywhere.
Table 4 shows the standard deviation of the local portfolios’ α’s and the
mean R2 . First, consider the R2 s for the “without market” and “with market.”
Including the local market portfolio as a right-hand-side variable dramatically
increases the R2 . This is to be expected as market returns are almost always
the first principal component in explaining asset returns, and hence, including
the main principal component increases the R2 of a model. Including the local
market does not, however, greatly impact the GRS test results or the variance
of the α’s. With respect to σα , the standard deviation is relatively low in
North America, Europe, and Japan as compared to Asia-Pacific, suggesting
that the global model explains much of the variance in these α’s in those three
regions but not in Asia-Pacific.
Table 3 shows that the ex-ante models fail globally when run on all regions
simultaneously. It is instructive, however, compare the ex-ante and ex-post
models locally. Table 5 shows performance of the the five-by-five global tangency portfolio, and alternative construction of the tangency portfolio formed
on three-by-two sorts, and to the candidate ex-ante models: the global CAPM,
and the global Fama-French Three- and Four-Factor models. I include the
global “region” as a test LHS set of portfolios in order to illustrate the role
13
of the joint-hypothesis problem in rejecting global integration using a global
model that cannot price the global cross-section. The ex-post global tangency
portfolio, by construction, passes the global GRS test perfectly. This is not
an economically meaningful statement, but rather falls mechanically out of
mean-variance mathematics. More importantly, the GRS test strongly rejects
the other candidate global models in the global region. This illustrates the
important point: when these models also fail in North America, Europe, and
Asia-Pacific, one cannot on this basis alone conclude that these regions are
not globally integrated. Rather, one must conclude that the candidate global
model is not the right global model and should not be surprised when these
models fail locally.
Table 5 also shows the results using an ex-post tangency portfolio formed
from a three-by-two sort, rather than a richer five-by-five sort. While this
coarser ex-post tangency portfolio performs better in terms of explaining local
cross-sectional variation than the ex-ante models, it still performs much worse
than the five-by-five sort. This suggests two important things: First, simply,
it shows that there is indeed information in the finer five-by-five sorts not
captured in the coarser three-by-two sort. As discussed earlier, the smaller,
lowest book-to-market portfolio has very low α that may be missed by the
three-by-two sorts, and constructing the tangency from the five-by-five sort
accounts for this variation.
Second, more subtly, it illustrates the joint-hypothesis problem, showing
that the success of the five-by-five ex-post tangency locally is not merely about
using ex-post information to form an efficient portfolio. Rather, being able to
properly capture global cross-sectional variation is important for being able
to capture local cross-sectional variation: Even with another ex-post optimal
portfolio, if that model cannot price the global cross-section, it has trouble
pricing local sorts as well. That is, the test for global integratedness relies
crucially on, at a minimum, having a passable global model.
To summarize: (1) We fail to reject the global ex-post model in North
America, Europe, and Japan for both local size/book-to-market, and in North
America and Japan for size/momentum sorts. (2) We strongly reject the ex14
Table 5: GRS test of five candidate models in each region: global tangency formed on
corresponding 5x5 sort, global tangency formed on corresponding 3x2 sort, global CAPM,
and global Fama-French Three- and Four-factor models. Each GRS test also includes the
local market as an RHS variable. return.
GRS
Size/BM
p(GRS)
σα
R2
GRS
Size/Mom
p(GRS)
σα
R2
Global
5x5
3x2
CAPM
FF-3
FF-4
2.17
3.89
3.74
3.46
1.00
0.00
0.00
0.00
0.00
0.00
0.10
0.20
0.13
0.13
0.83
0.84
0.82
0.96
0.96
3.49
5.36
5.62
4.69
1.00
0.00
0.00
0.00
0.00
0.00
0.13
0.38
0.41
0.19
0.80
0.82
0.78
0.88
0.95
North America
5x5
3x2
CAPM
FF-3
FF-4
0.76
1.64
3.04
2.78
2.19
0.80
0.03
0.00
0.00
0.00
0.09
0.13
0.24
0.17
0.15
0.75
0.76
0.75
0.89
0.90
1.03
2.68
4.21
3.82
3.29
0.43
0.00
0.00
0.00
0.00
0.12
0.16
0.37
0.41
0.17
0.72
0.74
0.62
0.69
0.77
Europe
5x5
3x2
CAPM
FF-3
FF-4
0.75
1.29
1.66
1.50
1.57
0.81
0.17
0.03
0.06
0.05
0.09
0.11
0.18
0.12
0.12
0.83
0.84
0.83
0.90
0.91
1.74
3.58
5.65
5.26
4.35
0.02
0.00
0.00
0.00
0.00
0.22
0.26
0.51
0.56
0.32
0.80
0.81
0.65
0.73
0.77
Japan
5x5
3x2
CAPM
FF-3
FF-4
0.58
0.81
1.23
1.01
1.02
0.95
0.73
0.21
0.45
0.44
0.12
0.12
0.21
0.13
0.14
0.79
0.80
0.80
0.84
0.83
0.90
1.23
1.07
1.05
1.26
0.60
0.21
0.38
0.41
0.19
0.27
0.27
0.17
0.17
0.18
0.75
0.76
0.26
0.34
0.36
Asia Pacific
5x5
3x2
CAPM
FF-3
FF-4
2.24
2.54
3.50
3.26
2.74
0.00
0.00
0.00
0.00
0.00
0.25
0.27
0.32
0.30
0.27
0.78
0.78
0.78
0.81
0.82
2.60
3.82
4.78
4.76
3.93
0.00
0.00
0.00
0.00
0.00
0.45
0.41
0.53
0.57
0.42
0.77
0.77
0.48
0.55
0.57
15
post model in Asia-Pacific for both sorts and weakly reject it in Europe for
size/momentum sorts. (3) The ex-post model uniformly explains the cross
section of average returns better than the ex-ante models, which highlights
the joint-hypothesis problem’s role in these tests.
I now move on to robustness tests and other ways of constructing the test
statistics.
4
Robustness Tests
In this section, I consider several alternative specifications and show that my
results are robust. First, a mechanical explanation for why the global expost tangency portfolio is successful in local markets, independently of the
anomalies’ integratedness, is a composition effect in that the global portfolios
mechanically partially include the local test assets. In Section 4.1, I rule this
out. Second, I test the effectiveness of using ex-post tangency weights to form
an ex-ante model out of sample, and in Section 4.2, I show that while ex-post
weights do not produce a reliable out-of-sample model, they still do better than
the ex-ante models at explaining the cross section of average returns. Third,
I consider two alternative test specifications using a generalized method of
moments approach and a constrained optimization approach, and in Sections
4.3 and 4.4 I show that these approaches deliver similar results.
4.1
Ruling out a mechanical relationship
First, even though the returns of these anomalies may in fact be segmented, a
tangency portfolio formed on global sorts still be able to explain cross-sectional
variance in region c simply because the global sort includes equities from region
c. That is, there may simply be a mechanical explanation for why the global
ex-post tangency portfolio is successful in explaining local cross-sections is a
composition effect. To be concrete, if, for instance, the global portfolios are essentially United States portfolios, the global ex-post tangency portfolio should
mechanically price the North American cross-section because the global portfo-
16
Table 6: GRS tests using true global sorts, under “Global” rows, and using a global tangency
portfolio constructed from local sorts where the current local test market is excluded, under
“Excluding Home” row.
GRS
Size/BM
p(GRS)
σα
R2
GRS
Size/Mom
p(GRS)
σα
R2
Global
Global
Excluding Home
0.02
0.44
1.00
0.99
0.00
0.05
0.83
0.83
0.01
0.45
1.00
0.99
0.00
0.07
0.80
0.79
North America
Global
Excluding Home
0.76
1.71
0.80
0.02
0.09
0.20
0.75
0.75
1.03
1.23
0.43
0.21
0.12
0.21
0.72
0.71
Europe
Global
Excluding Home
0.75
1.18
0.81
0.25
0.09
0.11
0.83
0.83
1.74
1.97
0.02
0.00
0.22
0.32
0.80
0.79
Japan
Global
Excluding Home
0.58
0.87
0.95
0.65
0.12
0.13
0.79
0.79
0.90
1.03
0.60
0.43
0.27
0.35
0.75
0.75
Asia Pacific
Global
Excluding Home
2.24
2.29
0.00
0.00
0.25
0.34
0.78
0.78
2.60
2.54
0.00
0.00
0.45
0.53
0.77
0.76
lios are essentially North American portfolios. On the other hand, if common
global factors drive local returns, if we construct the tangency portfolio to test
region c from all regions excluding c and the non-excluded regions span the
global factors, this portfolio that excludes the test region should still be able to
price the test region if pricing is integrated. Table 6 compares the results when
the global tangency portfolio is constructed from the global sorts, and when
the global tangency portfolio is constructed from combined regional sorts that
exclude the current test market.
The results are as follows: First, as expected, the tests point more towards
rejection, suggesting that the composition effect exists. Second, only in the
case of Size/Book-to-Market in North America does a previous “no-reject” become a “reject,” and other rejections or failures to reject are not affected. This
may not be surprising—a global model without North America—in particular,
without the United States—loses a huge percentage of its market capitaliza-
17
tion and is therefore not much of a global model at all. The other robustness
tests exhibit little impact. From this I conclude that while the composition
effect impacts the results, tests support the conclusion that (1) local regions
excluding one can span the global factors, and (2) the global factors can price
the excluded local market. The tests support global integration.
4.2
Out-of-sample tests
Having ruled out a mechanical force that drives the results, I now consider the
results of out-of-sample tests. I first form the global ex-post tangency portfolio
using means and covariances estimated from odd (even) months only, and then
perform the GRS tests on the unused even (odd) months, both globally and
locally. Next, I form the global tangency portfolio using an expanding window,
excluding the current month, to estimate the r̂ and Σ̂ to get the weights. The
results are in Table 7. The tests show two things. First, the ex-post global
tangency portfolio fares much out of sample than it does in sample, with the
even/odd models and the expanding window model failing to price the global
sorts out of sample.
Second, and more importantly for the goals of this paper, the results in
Table 7 line up with earlier results favoring “integration” in the sense that when
the candidate model can price global assets the candidate model can also price
local assets. On the other hand, when the candidate global model cannot price
the global assets, the candidate model has a much harder time pricing local
assets. Finally, it is worth noting that even though the expanding-window
tangency portfolio fails to price the global sorts, it performs much better than
the CAPM, Three- and Four-Factor models as shown in 5. Moreover, among
the integrated region-factors, it performs significantly better in local sorts than
the traditional ex-ante models.
Next, I consider two alternative approaches to running the test. The first
approach is to use generalized method of moments to estimate a stochastic
discount factor to price the fifty local and global assets, but constructed from
the twenty-five global assets. Because this system is over-identified, the pric-
18
Table 7: GRS tests using out of sample tangency portfolios. “All” uses the in-sample
tangency portfolio. “Odd” constructs the tangency portfolio using odd months and then
runs the test on even months. “Even” constructs the tangency portfolio using even months
and runs the test on odd months. “Expanding” constructs the tangency weights by looking
at all data in previous months starting from 1990, with a 5-year burn-in period.
GRS
Size/BM
p(GRS)
σα
R2
GRS
Size/Mom
p(GRS)
σα
R2
Global
All
5x5 Odd
5x5 Even
Expanding
3.15
3.70
2.56
1.00
0.00
0.00
0.00
0.00
0.24
0.19
0.17
0.83
0.82
0.86
0.83
1.40
2.90
2.86
1.00
0.12
0.00
0.00
0.00
0.26
0.31
0.15
0.80
0.78
0.81
0.81
North America
All
5x5 Odd
5x5 Even
Expanding
0.76
1.86
1.28
1.73
0.80
0.01
0.19
0.02
0.09
0.35
0.20
0.20
0.75
0.75
0.78
0.76
1.03
2.75
2.13
1.68
0.43
0.00
0.00
0.03
0.12
0.36
0.32
0.20
0.72
0.70
0.74
0.73
Europe
All
5x5 Odd
5x5 Even
Expanding
0.75
1.13
1.38
1.36
0.81
0.32
0.13
0.12
0.09
0.21
0.12
0.18
0.83
0.83
0.84
0.83
1.74
2.34
2.91
3.15
0.02
0.00
0.00
0.00
0.22
0.30
0.51
0.30
0.80
0.79
0.81
0.81
Japan
All
5x5 Odd
5x5 Even
Expanding
0.58
1.21
0.90
0.70
0.95
0.24
0.61
0.85
0.12
0.26
0.15
0.15
0.79
0.79
0.80
0.74
0.90
1.02
1.14
1.35
0.60
0.45
0.31
0.13
0.27
0.25
0.29
0.28
0.75
0.75
0.76
0.72
Asia Pacific
All
5x5 Odd
5x5 Even
Expanding
2.24
1.64
2.50
2.74
0.00
0.04
0.00
0.00
0.25
0.37
0.44
0.29
0.78
0.78
0.79
0.80
2.60
2.31
2.67
4.20
0.00
0.00
0.00
0.00
0.45
0.57
0.58
0.43
0.77
0.76
0.78
0.79
19
ing restrictions will not be exactly satisfied, but we ask whether the deviations
from zero are jointly statistically significant. The second approach is to allow
the weights to have more latitude by asking the portfolios weights to simultaneously price the twenty-five local assets and the twenty-five global assets, but
not requiring the global pricing to be perfect. These are relevant tests because
the right ex-ante model need not exactly price the global sorts ex-post, but
rather must come to within a statistical threshold. In the next two sections, I
give results from two implementations of these tests.
4.3
GMM test
Given a stochastic discount factor mt and excess returns rti , standard asset
pricing implies the following moment condition:
0 = E[mt rti ] ∀i
(8)
To test global integration, I assume that mt is a linear function of the twentyfive global sorts. Let rgt , be the return vector of these global sorts, and let w
be the vector of weights on each sort, to be determined in a GMM problem.
Then, the candidate stochastic discount factor, as a function of the weights
w, is:
mt (w) = 1 − (rgt − E[rgt ])0 w
(9)
For each region, I search for a vector of weights such that the twenty-five
regional pricing restrictions and the twenty-five global pricing restrictions are
jointly satisfied. This gives fifty moment conditions for each region in which I
run the test, whose sample analogs, for each region c, are given by the following
20
Table 8: GMM test statistics with moment conditions and objective functions described
in Equations (9)–(13). p(J) < 0.05 means we reject at the five-percent level that weights
can be found to jointly satisfy the moment conditions.
North America
Europe
Japan
Asia Pacific
Size/BM
J
p(J)
27.69 0.32
32.75 0.14
19.23 0.79
56.00 0.00
Size/Mom
J
p(J)
31.25 0.18
62.17 0.00
30.71 0.20
77.86 0.00
equation:


rtc,1 mt (w)


..


.




T
T
c,25
X


r
m
(w)
1X c
1
t
t
c


≡
ĝt (w) ∈ R50
ĝ (w) =
g,1


T t=1  rt mt (w)  T t=1


..


.


g,25
rt mt (w)
(10)
The GMM objective is
Jc = T × min ĝ c (w)0 V̂ ĝ c (w)
w
(11)
Asymptotically, under the null hypothesis that there exist weights that satisfy
the moment conditions, Jc is distributed Jc ∼ χ225 . I implement the test using
two-stage GMM, where the first-stage weighting matrix is the identity, and the
second-stage weighting matrix is the efficient weighting matrix using weights
w1 estimated from the first stage:
= Ω̂−1
T
1X c
ĝ (w1 )ĝtc (w1 )0
Ω̂ =
T i=1 t
V̂
The J-statistics and asymptotic probabilities are shown in Table 8.
21
(12)
(13)
The results in Table 8 are broadly consistent with previous findings: We
fail to reject global integration for Size/BM sorts in North America, Europe,
and Japan, and strongly reject for Asia-Pacific. Moreover, we fail to reject
for momentum in North America and Japan, but reject in Europe and Asia
Pacific. Compared to the earlier GRS tests I report, the GRS tests do not
take into account noise inherent in estimating the global tangency portfolio
ex-post. In other words, I fit the global tangency portfolio to noise and then
assume that this is the true global tangency portfolio. In this case, the GMM
test takes into account the fact that the global pricing is also estimated with
error, and (1) does not require a perfect fit on the global moment conditions,
and (2) takes into account the noise in fitting those moment conditions.
4.4
Constrained optimization
The final test I run, similar in motivation to the GMM test above, is a constrained optimization. I do not require the ex-post model to perfectly price
the global sorts, as I do in the earlier GRS tests, but rather constrain the
probability of rejecting the global model in the global region to be below a
certain probability threshold. The intuition here is that if we had the correct
global model ex-ante, it should pass an ex-post GRS test at a certain significance level, though it need not be perfect. To that end, I look for weights on
the twenty-five portfolios to minimize the pricing errors in the local regions,
but subject to the constraint that it pass a GRS test on the global assets at
a 5% significance level. This effectively expands the set of candidate models
that have the opportunity to price the local assets, and makes rejection of the
model more difficult. Specifically, I run the following optimization:
w = arg min GRS(local, global · w)
w
s.t
0.05 ≤ GRS(global, global · w)
22
(14)
Table 9: GRS test statistics for the constrained optimization problem described in (14).
The first column of each group is the GRS probability on the local market given the optimized weights. The second column is the GRS probability on the global market given the
optimized weights. The third and fourth columns are the standard deviations of the α’s and
average R2 s in the local market.
North America
Europe
Japan
Asia Pacific
p(GRSlocal )
0.98
1.00
1.00
0.02
Size/BM
p(GRSglobal )
1.00
0.90
0.97
0.92
σα
0.06
0.05
0.08
0.22
R2
0.75
0.83
0.79
0.78
North America
Europe
Japan
Asia Pacific
p(GRSlocal )
0.89
0.12
1.00
0.00
Size/Mom
p(GRSglobal )
0.84
1.00
0.27
0.98
σα
0.08
0.18
0.10
0.35
R2
0.72
0.80
0.75
0.77
The results of this optimization are given in Table 9.
Table 9 confirms the earlier pattern that North America, Europe, and
Japan are globally integrated, while Asia Pacific is not. Since the test presented here is easier to pass than previous tests, the most surprising result is
not that North America, Europe, and Japan pass, but that Asia Pacific still
fails. The interpretation here is that there exists a model, constructible from
global factors, that could have simultaneously passed GRS tests in the global
sorts and in the three regions that do pass, but there do not exist weights
such that this portfolio could have passed in Asia-Pacific. In there words, here
does not exist a global model—at least one constructible from the candidate
global portfolios—that would have worked in Asia-Pacific, strongly rejecting
integration.
5
Discussion
The results paint a consistent story that Size/Book-to-Market and Size/Momentum
anomalies in North America, Europe, and Japan are global in nature, and that
23
the anomalies in Asia-Pacific are not. Formally, I fail to reject that in these
three regions, the average returns, of portfolios sorted on Size/book-to-market
and Size/Momentum, unexplained by the ex-post global model, are jointly
different from zero. Although I do not purport to identify the correct global
model ex-ante, I do conclude that if one had the correct ex-ante global model,
that global model should also price local cross sections in North America,
Europe, and Japan, and would fail to do so in Asia-Pacific. I interpret this
as evidence that the anomalies examined here in North America, Europe, and
Japan are globally integrated in the sense that once we can explain the anomalies at a global level, the same model can explain them at a local level as well.
By contrast, it appears that that even the correct global model will fail locally
in Asia-Pacific.
Out of sample tests of the ex-post tangency portfolio demonstrate that it
is not a great proxy for the true ex-ante tangency portfolio. It is interesting,
however, that this portfolio has better out-of-sample performance in explaining
cross-sectional α’s than either the Fama-French Three- of Four-Factor models.
One explanation for this increased effectiveness of the ex-post model globally
is that the ex-post tangency portfolio is allowed to take positions in portfolios
that traditionally give the Fama-French models trouble: in particular, the very
small, low value portfolios that tend to have extreme α’s not well-captured by
the factor models. Since these portfolios tend to have very low α’s in many
regions, this additional degree of freedom may allow the asset pricing tests
to have more success. Indeed, the α’s in the appendix generally confirm that
when the ex-post models pass and the ex-ante models fail, it is because the
ex-post models give lower α’s for the “corners” of the sorts.
Once we believe that these anomalies are globally integrated in most regions, this can help us understand possible economic explanations. For instance, in trying to understand whether these are risk or behavioral effects,
this paper shows that we should consider only risk or behavioral effects that are
global in nature. For instance, the candidate risk-based explanation must be a
factor that impacts equities globally rather than locally. The candidate behavioral explanation must be able to explain why the returns to taking advantage
24
of these behavioral misses are correlated and connected across countries, rather
than idiosyncratic effects. The results also demonstrate that it is important to
consider think carefully about decompositions of returns into cash flow components and changes to the stochastic discount factor. If we measure the cash
flows of the sorted portfolios in the integrated markets versus Asia-Pacific and
see that they are largely correlated, it implies that different stochastic discount factors must be pricing these cash flows, which suggests market frictions
or segmentation that make the marginal investor in these regions different.
6
Conclusion
In this paper I gave global integration of asset pricing phenomenon a fighting chance. Previous literature that rejects global integration suffers from a
joint hypothesis problem in that global models that fail to price local asset
cross-sections also fail globally. We cannot on the basis of these tests reject
global integration alone, but rather must jointly reject global integration and
the candidate model. To address this problem, I asked a different question
that bypasses this problem: given a model that prices the global cross section,
can that model price local cross sections? Importantly, rather than specify an
ex-ante model or attempt to explain in economic terms what the right ex-ante
global model should look like, I allow the data to provide the ex-post global
model that mechanically prices the global cross-section, and test whether this
ex-post global model can price local cross-sections. Across a number of model
constructions and test specifications, I consistently find evidence supporting global integration of Size/Book-to-Market anomalies and Size/Momentum
anomalies in North America, Europe, and Japan. Asia-Pacific, on the other
hand, consistently evades explanation by global asset pricing models.
From an academic standpoint, this result is interesting for several reasons.
First, it suggests that searches for explanations of these phenomena should be
global in nature but leave room for financial frictions or segmentation that
cuts Asia-Pacific out of the mainline global risk factors. Second, it narrows
the focus to factors where there are material differences or barriers between
25
Asia-Pacific and the rest of the world. For instance, explanations must be
related to factors to which equities in North America, Europe, and Japan are
commonly exposed, but to which equities in Asia-Pacific (Asia-ex-Japan) are
either not exposed to, or to another set of factors to which only Asia-Pacific
equities are exposed. From a practitioner standpoint, it calls into question the
marginal value of strategies that exploit Size, Book-to-Market, and Momentum
premia in separate “strategies” for North American, European, and Japanese
equities, rather than simply a single global portfolio for each one. Further
research could consider individual countries rather an aggregated regions and
run similar tests.
26
References
Bekaert, G., Harvey, C. R., Lundblad, C. T., and Siegel, S. (2013). The european
union, the euro, and equity market integration. Journal of Financial Economics,
109(3):583–603.
Fama, E. F. and French, K. R. (2012). Size, value, and momentum in international
stock returns. Journal of financial economics, 105(3):457–472.
Gibbons, M. R., Ross, S. A., and Shanken, J. (1989). A test of the efficiency of a given
portfolio. Econometrica: Journal of the Econometric Society, pages 1121–1152.
Griffin, J. M. (2002). Are the fama and french factors global or country specific?
Review of Financial Studies, 15(3):783–803.
Guesmi, K. and Nguyen, D. K. (2011). How strong is the global integration of emerging market regions? an empirical assessment. Economic Modelling, 28(6):2517–
2527.
Hou, K., Karolyi, G. A., and Kho, B.-C. (2011). What factors drive global stock
returns? Review of Financial Studies, 24(8):2527–2574.
Karolyi, G. A. and Stulz, R. M. (2003). Are financial assets priced locally or globally?
Handbook of the Economics of Finance, 1:975–1020.
27
28
7
7.1
Appendix
Model αs
Table 10: Global CAPM
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.20
0.17
0.15
0.14
0.14
0.16
0.12
0.12
0.11
0.13
0.14
0.12
0.09
0.10
0.11
0.13
0.08
0.07
0.09
0.11
0.10
0.06
0.06
0.08
0.11
0.20
0.13
0.12
0.12
0.19
0.19
0.12
0.11
0.11
0.17
0.17
0.11
0.09
0.09
0.17
0.18
0.10
0.08
0.08
0.16
0.17
0.09
0.07
0.08
0.17
North America
Small
2
3
4
Big
0.31
0.24
0.22
0.19
0.19
0.25
0.22
0.17
0.14
0.17
0.23
0.16
0.12
0.13
0.14
0.20
0.11
0.10
0.12
0.13
0.11
0.07
0.08
0.11
0.14
0.26
0.16
0.16
0.19
0.26
0.24
0.15
0.15
0.15
0.28
0.22
0.14
0.12
0.14
0.25
0.23
0.12
0.10
0.10
0.24
0.20
0.12
0.09
0.10
0.22
Europe
Small
2
3
4
Big
0.19
0.16
0.14
0.14
0.14
0.17
0.13
0.13
0.13
0.15
0.16
0.12
0.11
0.12
0.14
0.12
0.09
0.09
0.11
0.14
0.11
0.08
0.07
0.09
0.14
0.20
0.14
0.13
0.13
0.20
0.20
0.14
0.12
0.13
0.18
0.19
0.12
0.11
0.12
0.17
0.20
0.11
0.10
0.11
0.16
0.20
0.11
0.08
0.10
0.17
Japan
Small
2
3
4
Big
0.31
0.25
0.23
0.21
0.21
0.25
0.21
0.20
0.18
0.19
0.21
0.17
0.16
0.15
0.17
0.16
0.13
0.12
0.13
0.16
0.14
0.11
0.11
0.13
0.20
0.29
0.22
0.20
0.21
0.26
0.26
0.19
0.18
0.18
0.21
0.23
0.17
0.16
0.15
0.20
0.21
0.15
0.13
0.13
0.18
0.24
0.14
0.11
0.12
0.20
Asia Pacific
Small
2
3
4
Big
0.33
0.25
0.23
0.23
0.26
0.21
0.20
0.18
0.20
0.24
0.20
0.18
0.17
0.16
0.21
0.16 0.13
0.15 0.10
0.15 0.11
0.1529 0.14
0.20 0.22
0.27
0.21
0.20
0.24
0.28
0.25
0.19
0.18
0.19
0.27
0.24
0.16
0.14
0.16
0.24
0.23
0.16
0.13
0.13
0.22
0.28
0.17
0.12
0.14
0.23
Table 11: FF3
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.09
0.08
0.07
0.06
0.06
0.06
0.05
0.05
0.04
0.05
0.07
0.06
0.06
0.06
0.06
0.07
0.06
0.06
0.06
0.06
0.05
0.05
0.05
0.05
0.07
0.15
0.07
0.07
0.07
0.13
0.16
0.07
0.06
0.06
0.11
0.16
0.08
0.06
0.07
0.12
0.17
0.08
0.06
0.07
0.13
0.17
0.08
0.06
0.08
0.16
North America
Small
2
3
4
Big
0.19
0.15
0.14
0.12
0.11
0.15
0.14
0.12
0.10
0.11
0.15
0.12
0.10
0.10
0.10
0.13
0.09
0.09
0.10
0.09
0.08
0.07
0.07
0.08
0.10
0.21
0.11
0.11
0.13
0.19
0.22
0.12
0.12
0.12
0.21
0.22
0.12
0.10
0.12
0.20
0.22
0.10
0.09
0.10
0.21
0.21
0.11
0.09
0.10
0.20
Europe
Small
2
3
4
Big
0.14
0.12
0.11
0.11
0.11
0.13
0.11
0.10
0.10
0.11
0.12
0.10
0.10
0.09
0.10
0.10
0.09
0.07
0.10
0.11
0.08
0.08
0.07
0.09
0.12
0.18
0.12
0.11
0.11
0.18
0.18
0.12
0.10
0.11
0.15
0.17
0.11
0.09
0.10
0.15
0.20
0.10
0.09
0.10
0.14
0.20
0.11
0.07
0.10
0.16
Japan
Small
2
3
4
Big
0.29
0.23
0.22
0.19
0.20
0.24
0.20
0.19
0.17
0.18
0.20
0.16
0.15
0.15
0.16
0.16
0.12
0.12
0.13
0.15
0.12
0.10
0.10
0.12
0.18
0.27
0.20
0.19
0.19
0.24
0.24
0.18
0.16
0.17
0.20
0.22
0.16
0.15
0.15
0.19
0.21
0.14
0.13
0.13
0.18
0.24
0.14
0.10
0.12
0.19
Asia Pacific
Small
2
3
4
Big
0.31
0.24
0.21
0.21
0.24
0.20
0.19
0.17
0.19
0.23
0.19
0.17
0.17
0.15
0.21
0.16
0.15
0.15
0.14
0.20
0.13
0.10
0.11
0.14
0.21
0.26
0.20
0.19
0.23
0.27
0.24
0.18
0.17
0.18
0.26
0.24
0.15
0.13
0.15
0.22
0.23
0.16
0.13
0.13
0.22
0.28
0.17
0.12
0.14
0.23
30
Table 12: FF4
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.09
0.08
0.07
0.06
0.06
0.06
0.05
0.05
0.05
0.05
0.07
0.06
0.06
0.06
0.06
0.07
0.06
0.06
0.06
0.06
0.05
0.05
0.05
0.05
0.07
0.09
0.07
0.07
0.07
0.10
0.07
0.06
0.06
0.06
0.06
0.07
0.07
0.07
0.06
0.07
0.08
0.06
0.06
0.06
0.07
0.08
0.06
0.06
0.05
0.08
North America
Small
2
3
4
Big
0.19
0.16
0.15
0.13
0.11
0.15
0.15
0.13
0.10
0.11
0.16
0.13
0.10
0.11
0.10
0.14
0.10
0.10
0.11
0.09
0.08
0.07
0.07
0.09
0.10
0.16
0.11
0.11
0.12
0.16
0.16
0.11
0.12
0.12
0.15
0.14
0.10
0.11
0.11
0.15
0.14
0.09
0.09
0.09
0.14
0.13
0.09
0.09
0.08
0.13
Europe
Small
2
3
4
Big
0.15
0.13
0.11
0.11
0.12
0.13
0.11
0.10
0.10
0.11
0.13
0.10
0.10
0.09
0.11
0.10
0.09
0.07
0.10
0.11
0.08
0.08
0.07
0.09
0.12
0.15
0.11
0.11
0.11
0.17
0.13
0.11
0.10
0.11
0.14
0.13
0.10
0.09
0.10
0.14
0.15
0.09
0.09
0.09
0.12
0.15
0.09
0.07
0.08
0.11
Japan
Small
2
3
4
Big
0.30
0.24
0.22
0.20
0.20
0.24
0.21
0.19
0.17
0.18
0.21
0.16
0.15
0.15
0.16
0.16
0.12
0.12
0.13
0.15
0.12
0.10
0.10
0.13
0.19
0.26
0.20
0.19
0.20
0.25
0.22
0.18
0.17
0.18
0.21
0.19
0.15
0.16
0.15
0.19
0.18
0.14
0.13
0.13
0.17
0.21
0.13
0.11
0.11
0.17
Asia Pacific
Small
2
3
4
Big
0.32
0.25
0.22
0.22
0.25
0.21
0.19
0.18
0.20
0.23
0.20
0.18
0.17
0.16
0.21
0.16
0.16
0.15
0.15
0.21
0.14
0.11
0.12
0.15
0.21
0.26
0.21
0.20
0.23
0.27
0.24
0.18
0.17
0.18
0.26
0.23
0.16
0.14
0.16
0.22
0.22
0.16
0.13
0.13
0.20
0.28
0.17
0.12
0.14
0.22
31
Table 13: 5x5 Tangency
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.38
0.35
0.32
0.29
0.27
0.38
0.34
0.30
0.29
0.29
0.37
0.34
0.30
0.29
0.30
0.36
0.31
0.30
0.30
0.32
0.30
0.29
0.29
0.29
0.36
0.44
0.29
0.26
0.26
0.34
0.46
0.31
0.28
0.28
0.36
0.46
0.33
0.29
0.28
0.37
0.45
0.32
0.28
0.29
0.36
0.43
0.32
0.28
0.28
0.36
North America
Small
2
3
4
Big
0.35
0.28
0.25
0.22
0.21
0.28
0.25
0.20
0.16
0.19
0.27
0.18
0.14
0.15
0.16
0.23
0.12
0.12
0.14
0.15
0.13
0.08
0.09
0.13
0.16
0.31
0.20
0.19
0.22
0.31
0.29
0.18
0.18
0.19
0.33
0.27
0.17
0.15
0.17
0.30
0.27
0.15
0.12
0.12
0.29
0.24
0.14
0.11
0.12
0.26
Europe
Small
2
3
4
Big
0.22
0.18
0.17
0.16
0.16
0.19
0.15
0.15
0.16
0.17
0.18
0.14
0.13
0.14
0.16
0.14
0.11
0.10
0.13
0.16
0.13
0.09
0.08
0.11
0.16
0.24
0.17
0.16
0.15
0.23
0.24
0.17
0.15
0.15
0.21
0.23
0.15
0.13
0.14
0.20
0.24
0.13
0.12
0.13
0.19
0.24
0.13
0.09
0.12
0.21
Japan
Small
2
3
4
Big
0.36
0.29
0.27
0.25
0.26
0.30
0.25
0.24
0.21
0.23
0.25
0.20
0.19
0.19
0.21
0.19
0.15
0.14
0.16
0.20
0.16
0.13
0.12
0.15
0.23
0.36
0.28
0.26
0.26
0.33
0.32
0.24
0.23
0.23
0.27
0.28
0.21
0.20
0.20
0.25
0.26
0.19
0.16
0.16
0.23
0.29
0.17
0.13
0.14
0.24
Asia Pacific
Small
2
3
4
Big
0.38
0.29
0.27
0.26
0.30
0.24
0.23
0.21
0.23
0.27
0.23
0.21
0.20
0.19
0.25
0.19
0.18
0.17
0.17
0.24
0.16
0.12
0.13
0.17
0.26
0.33
0.25
0.23
0.28
0.32
0.31
0.23
0.21
0.22
0.31
0.29
0.20
0.17
0.19
0.28
0.28
0.19
0.16
0.16
0.27
0.34
0.21
0.14
0.17
0.28
32
Table 14: 3x2 Tangency
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.35
0.32
0.30
0.27
0.24
0.35
0.31
0.27
0.25
0.25
0.34
0.31
0.27
0.26
0.26
0.34
0.27
0.26
0.26
0.28
0.27
0.26
0.26
0.27
0.33
0.38
0.25
0.21
0.21
0.27
0.40
0.27
0.23
0.22
0.27
0.40
0.28
0.24
0.22
0.28
0.40
0.28
0.24
0.23
0.28
0.38
0.27
0.24
0.24
0.30
North America
Small
2
3
4
Big
0.31
0.25
0.23
0.20
0.20
0.25
0.23
0.18
0.14
0.17
0.24
0.17
0.12
0.13
0.15
0.20
0.11
0.11
0.12
0.14
0.12
0.08
0.09
0.12
0.15
0.27
0.17
0.16
0.18
0.25
0.25
0.16
0.15
0.15
0.26
0.23
0.15
0.13
0.14
0.24
0.23
0.13
0.10
0.10
0.23
0.20
0.12
0.10
0.11
0.23
Europe
Small
2
3
4
Big
0.19
0.17
0.15
0.14
0.15
0.17
0.14
0.14
0.14
0.15
0.16
0.13
0.12
0.12
0.14
0.13
0.10
0.09
0.12
0.15
0.12
0.08
0.07
0.10
0.15
0.21
0.15
0.14
0.13
0.20
0.21
0.15
0.13
0.12
0.17
0.19
0.13
0.11
0.12
0.17
0.20
0.11
0.11
0.11
0.15
0.20
0.12
0.08
0.10
0.18
Japan
Small
2
3
4
Big
0.33
0.27
0.25
0.23
0.23
0.28
0.23
0.22
0.19
0.21
0.23
0.18
0.17
0.17
0.19
0.17
0.14
0.13
0.14
0.17
0.15
0.12
0.12
0.13
0.21
0.31
0.24
0.22
0.22
0.28
0.27
0.21
0.20
0.20
0.23
0.23
0.19
0.18
0.17
0.21
0.22
0.16
0.14
0.14
0.19
0.24
0.14
0.12
0.12
0.21
Asia Pacific
Small
2
3
4
Big
0.35
0.27
0.25
0.24
0.28
0.22
0.21
0.19
0.22
0.26
0.21
0.19
0.18
0.17
0.23
0.17
0.16
0.16
0.15
0.22
0.14
0.11
0.12
0.15
0.24
0.29
0.22
0.21
0.25
0.29
0.27
0.21
0.18
0.20
0.27
0.25
0.17
0.15
0.16
0.24
0.24
0.17
0.14
0.14
0.23
0.30
0.18
0.13
0.15
0.25
33
Table 15: Expanding Tangency
Size/Book-to-Market
Size/Momentum
Low
2
3
4
High
Low
2
3
4
High
Global
Small
2
3
4
Big
0.40
0.37
0.34
0.31
0.28
0.39
0.35
0.31
0.30
0.30
0.38
0.35
0.31
0.30
0.30
0.37
0.31
0.30
0.30
0.32
0.31
0.28
0.30
0.30
0.37
0.46
0.31
0.27
0.28
0.36
0.48
0.32
0.29
0.29
0.38
0.47
0.34
0.30
0.29
0.38
0.47
0.33
0.29
0.29
0.37
0.45
0.33
0.29
0.29
0.36
North America
Small
2
3
4
Big
0.39
0.30
0.27
0.23
0.22
0.31
0.27
0.21
0.17
0.20
0.29
0.19
0.14
0.16
0.17
0.25
0.13
0.13
0.15
0.16
0.13
0.09
0.10
0.14
0.17
0.31
0.20
0.20
0.23
0.32
0.28
0.18
0.18
0.19
0.34
0.27
0.17
0.15
0.17
0.31
0.27
0.15
0.12
0.13
0.29
0.24
0.15
0.12
0.12
0.27
Europe
Small
2
3
4
Big
0.23
0.19
0.17
0.16
0.16
0.21
0.16
0.15
0.16
0.18
0.20
0.14
0.14
0.14
0.16
0.15
0.11
0.10
0.13
0.16
0.14
0.10
0.08
0.11
0.17
0.24
0.16
0.15
0.15
0.23
0.24
0.16
0.16
0.15
0.21
0.22
0.15
0.13
0.14
0.21
0.23
0.13
0.13
0.13
0.19
0.24
0.14
0.10
0.12
0.21
Japan
Small
2
3
4
Big
0.37
0.29
0.27
0.23
0.24
0.31
0.24
0.24
0.20
0.23
0.26
0.20
0.19
0.18
0.21
0.19
0.14
0.14
0.16
0.20
0.17
0.13
0.13
0.15
0.24
0.34
0.25
0.23
0.24
0.31
0.30
0.22
0.21
0.22
0.26
0.27
0.20
0.19
0.19
0.25
0.25
0.18
0.16
0.16
0.22
0.29
0.17
0.14
0.15
0.24
Asia Pacific
Small
2
3
4
Big
0.37
0.30
0.26
0.26
0.31
0.25
0.23
0.22
0.23
0.28
0.23
0.21
0.20
0.19
0.26
0.18
0.18
0.18
0.18
0.25
0.17
0.12
0.14
0.17
0.27
0.33
0.25
0.23
0.27
0.31
0.31
0.24
0.21
0.22
0.31
0.30
0.20
0.17
0.18
0.27
0.29
0.19
0.17
0.16
0.27
0.35
0.20
0.15
0.16
0.29
34