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A Latent Factor Model with Global, Country, and
Industry Shocks for International Stock Returns
Robin Brooks and Marco Del Negro
Working Paper 2002-23b
May 2004
Working Paper Series
A Latent Factor Model with Global, Country and
Industry Shocks for International Stock Returns∗†
Robin Brooks
Marco Del Negro
Research Department
Research Department
International Monetary Fund
Federal Reserve Bank of Atlanta
First Draft: October 2002
This Draft: May 2004
JEL CLASSIFICATION: G11, G15
KEY WORDS: Diversification; Risk; International financial markets; Industrial structure
∗
Robin Brooks: International Monetary Fund, 700 19th Street N.W., Washington D.C. 20431, Tel: (202)
623-6236, Fax: (202) 623-4740. rbrooks2@imf.org. Marco Del Negro (corresponding author): Federal Reserve
Bank of Atlanta, 1000 Peachtree Street NE, Atlanta GA 30309-4470. Tel: (404) 498-8561, Fax: (404) 498-8956.
Marco.DelNegro@atl.frb.org.
†
The paper previously circulated under the title “International Diversification Strategies.” We wish to thank,
without implication, Bernard Dumas, Wayne Ferson, Linda Goldberg, Andrew Karolyi, Ashoka Mody, Cesare
Robotti, Geert Rouwenhorst, Jay Shanken, Steve Smith, Dan Waggoner, and participants at the Atlanta Fed,
2003 European Finance Association meetings (Glasgow), Georgia Tech, INSEAD, 2002 LAMES (Saõ Paulo),
Spring 2003 System Committee in International Economics, for comments. We are also grateful to Young Kim
for excellent research assistance and Iskander Karibzhanov for translating some of our code into C. The views
expressed in this paper do not necessarily reflect those of the Federal Reserve Bank of Atlanta, the Federal
Reserve System or the IMF.
A Latent Factor Model with Global, Country and Industry
Shocks for International Stock Returns 0
Abstract
We estimate a latent factor model that decomposes international stock returns into
global, country- and industry-specific shocks and allows for stock-specific exposures to
these shocks. We find that across stocks there is statistically and economically significant
dispersion in the exposures, which is partly explained by the extent to which firms operate internationally and firm size. We show that portfolios consisting of stocks with low
exposures to country-specific shocks achieve substantial variance reduction relative to the
global market, both in- and out-of-sample. The shock exposures are thus a stock-selection
device that can be successfully used for international diversification strategies.
0
The paper previously circulated under the title “International Diversification Strategies.”
1
I
Introduction
We investigate the exposures of individual stocks to global, country- and industry-specific
shocks. The methodological contribution of this paper is to estimate a latent factor model
that allows for stock-specific exposures to these shocks for a large panel of international stock
returns. Allowing for stock-specific exposures has intuitive appeal. After all, it seems plausible that multinationals have higher exposures to global shocks, for example, than firms that
operate only domestically. Allowing for stock-specific exposures has practical appeal as well.
The exposures represent a stock-selection device that is potentially useful for international
diversification. If – as has been documented by Heston and Rouwenhorst (1994) and Griffin
and Karolyi (1998) among others – country-specific shocks are on average the most important
source of international return variation, why not construct portfolios consisting of stocks with
low country shock exposures in the first place? To the extent that country exposures differ significantly from stock to stock, these portfolios may deliver substantial reductions in volatility
relative to the global benchmark.
Building on Heston and Rouwenhorst (1994), our model decomposes stock returns into (i) a
global factor common to all stocks; (ii) country-specific factors that capture common variation
within countries; (iii) industry-specific factors that capture common variation within industries;
and (iv) idiosyncratic variation in each stock. As in the latent factor model literature (see Cho
et al. 1986 and Heston et al. 1995), our factors are unobserved. However, borrowing from
the identified vector autoregression (VAR) literature (see Sims 1986 and Bernanke 1986) in
macroeconomics, we identify our factors as global, country- and industry-specific by imposing
restrictions on the factor exposures of individual stocks, called exposures below. For example,
we identify the U.S. country factor by assuming that only U.S. stocks have non-zero exposures
to that factor. In other words, we sort stocks into groups according to observable characteristics
(country and industry affiliation) and identify a given factor by assuming that it affects only
stocks in that group. Conceptually, our identifying assumption, which groups stocks according
2
to observed characteristics, is similar to that in Heston and Rouwenhorst (1994) who create
portfolios of stocks based on their country and industry affiliations. We depart from their
approach in that we estimate the degree to which stocks belong to their assigned portfolios.
Our shock exposures capture both observed and unobserved firm-level characteristics that
determine the true exposure to a factor.
We estimate our model for 1,965 stocks from 21 countries in the period January 1985 to
February 2002, and for a larger sample of 3939 stocks in 33 countries from January 1990 to
February 2002. Our results can be summarized as follows. First, we document that dispersion
in exposures to global, country and industry shocks is economically and statistically significant.
Second, we show that shock exposures relate to observed firm-level characteristics, such as
the degree to which companies operate internationally and their size. We find however that
observables explain only a small fraction of the cross-section of the exposures, suggesting
that the shock exposures capture more information than that captured by our list of firmlevel variables. Third and most importantly, we show that a portfolio consisting of stocks
with low exposure to country shocks delivers substantial variance reduction relative to the
global market portfolio, both in- and out-of-sample. Countries in this portfolio are present
in the same proportion as in the global market portfolio, so that variance reduction results
not from a particular country tilt, but purely from lower exposure to country-specific shocks.
We construct similar low country exposure portfolios for individual countries and show that
in virtually all cases these low exposure portfolios are less volatile than the respective country
benchmark, again both in- and out-of-sample.
The remainder of the paper is organized as follows. Sections II and III describe the model
and the data, respectively. Section IV contains the results. Section V concludes.
3
II
The model
This section describes our model. Let us denote by Rnt the excess return over a riskless
benchmark on stock n in period t, where n ranges from 1 to N and t from 1 to T . Let us
index countries with the letter c (c = 1, .., C) and industries with the letter i (i = 1, .., I). The
model can then be written as:
Rnt = µn + βnG ftg +
C
X
C c
βnc
ft +
c=1
I
X
I i
βni
ft + ²nt ,
(1)
i=1
where µn represents the expected excess return on stock n, ftg , ftc and fti represent the global
market factor, the country factor c, the industry factor i, and ²nt represents the idiosyncratic
shock to the return of stock n, all in period t. The factors are unobservable random variables,
i.e. latent factors. In order to identify the factors as global, country, and industry shocks we
impose the following set of zero restrictions on the exposures (βs):
βnG
C
βnc
I
βni
= unconstrained,
unconstrained
=
0
unconstrained
=
0
if stock n belongs to country c,
otherwise,
(2)
if stock n belongs to industry i,
otherwise.
As Chan et al. (1998) argue, latent factor model have the disadvantage that it is difficult to
give economic interpretation to purely statistical factors. Our model attempts to address this
deficiency via the identifying restrictions.
Model (1) can be written in stacked form as:
Rt = µ + Bft + ²t ,
(3)
where µ denotes the N × 1 vector of mean returns, and ft the K × 1 vector of factors (K is the
total number of factors), B is a N × K matrix of βs, and ²t the N × 1 vector of idiosyncratic
shocks. Without restrictions (2), model (3) is not identified: if we replace B with BΩ and ft
4
with Ω0 ft , where Ω is an orthonormal matrix, the likelihood is unchanged. However, in our
model the zero restrictions (2) on B are enough to pin down the rotation matrix Ω: only if Ω
is equal to the identity matrix, will BΩ have the same zero restrictions as B. With Ω pinned
down, the factors are “identified.”
We estimate model (1) via maximum likelihood and make the following distributional
assumptions:
ft à N (0, I),
(4)
where I is a K × K identity matrix, and:
²t à N (0, Σ) for all t and n,
(5)
where Σ is diagonal (idiosyncratic shocks are cross-sectionally uncorrelated) with elements σn2 .
In expression (4), the assumption that the factors have unitary variance is purely a normalization assumption. However, the assumption that the factors are uncorrelated is not without
loss of generality here, given that the rotation matrix Ω has already been pinned down by the
zero restrictions (2). We discuss this issue later in greater detail.
The use of zero restrictions implies that we cannot use standard packages to obtain the
maximum likelihood estimates. The restrictions also imply that other methods for estimating
APT models (Connor and Korajczyk (1986)) are not immediately applicable. A value-added
of this paper is that we provide a method for estimating model (1) via maximum likelihood
for large cross-sections, using the EM algorithm.1 The first step of the EM algorithm follows
the intuition that if the factors were observable the exposures could be estimated by means
of OLS – equation by equation (see also Marsh and Pfleiderer, 1997). In the next step, an
estimate of the factors is obtained by taking their conditional expectation given the data
and the parameters from the previous step. Specifically, at each step q of the algorithm, the
5
estimate of the non-zero loadings for stock n are given by the formula:
P
P
βqn = [ Tt=1 M n IE q−1 [ ft ft0 ]M n0 ]−1 Tt=1 M n IE q−1 [ ft ](Rnt − µn ),
(6)
where the matrix M n simply selects the factors that are relevant for stock n (i.e., the appropriate country and industry factor). Details on the derivation of expression 6 and of the
conditional expectations IE q−1 [ft ] and IE q−1 [ ft ft0 ] are given in the appendix. This iterative
procedure is guaranteed to increase the likelihood at each step and hence delivers maximum
likelihood estimates. Since formula (6) is computed stock by stock, the approach is computationally feasible even for very large cross-sections.
One may question the orthogonality assumption (4): Are the country factors for Germany
and France really uncorrelated? Note that from a Bayesian point of view assumption (4) can
be seen as a prior on the vector of factors ft . Upon convergence, the EM algorithm delivers
a (conditional) posterior distribution for the factors, which is normal with first and second
moments equal to IE[ft |B, Σ, Rt ] and IE[ft ft 0 |B, Σ, Rt ], respectively. From the average of the
posterior second moments matrix of the factors, computed as
T
X
IE[ft ft 0 |B, Σ, Rt ]/T,
(7)
t=1
one can assess whether the orthogonality assumption is borne out by the data or not.
III
The data
Our data are based on Brooks and Del Negro (forthcoming) who collect monthly U.S. dollardenominated returns for 9,679 stocks in 42 countries from January 1985 to February 2002 from
Datastream International. They show that the sample provides a good depiction of the global
stock market, both in terms of coverage within and across countries and in terms of overall
coverage, which comes close to matching the market capitalization of the known universe of
stocks.2
6
We modify this data in two ways. First, we balance the data because our maximum
likelihood algorithm does not allow for missing observations. Since this step may result in
survivorship bias and reduce coverage, we do this at two points in time, to ensure that our
results are robust to this step. One balanced sample consists of stocks with continuous returns
for the entire sample. It contains 1,965 stocks in 21 countries. Another begins in January 1990
and contains stocks with continuous returns for the rest of the sample. It covers 3,939 stocks
in 33 countries. Second, we follow common practice in the literature, see Ferson and Harvey
(1994) and Heston et al. (1995), and estimate our model for excess returns. We compute
these as the difference between individual stock returns and the return on a three month U.S.
Treasury Bill, which we obtain from Datastream International. Although both samples cover
fewer stocks than the original data, Table B-1 in the appendix shows that they are comparable
in terms of the means and standard deviations of the equal-weighted excess return across all
stocks. In particular, there is little indication of a systematic bias in the standard deviation of
excess returns, which is important because we focus on return variability and comovement.
To identify country and industry shocks, we use the country and industry affiliations for
each stock from Datastream International. With regard to industry affiliation, Datastream has
six levels of disaggregation. At each level, there are more disaggregated industry definitions,
up to the most disaggregated classification, level 6. We use the most disaggregated level in our
benchmark estimation.3 To estimate the country and industry shocks in a meaningful way,
our balanced data must provide a reasonable representation of international stock markets,
both in terms of country and industry coverage. The data appendix reviews this coverage and
shows that it is representative.4
IV
Results
To our knowledge, this is the first paper that studies firm-level exposures to global, country,
and industry shocks. Since firms differ in the degree to which they operate across countries
7
and industries, it seems a priori plausible that such differences be reflected in their exposures
to common shocks. The next section investigates whether firm-level heterogeneity in the exposures is statistically and economically significant. Section IV-B will show how the exposures
relate to observable firm-level information, such as size, book-to-market, and the degree to
which firms operate internationally. In section IV-C we show that these exposures represent
valuable information for bottom-up international diversification strategies: knowledge of the
exposures makes it possible to construct global and country portfolios that are less volatile
than their respective benchmarks, both in- and out-of-sample. Finally, section IV-D documents that our model reproduces well-known findings in terms of variance decomposition for
international stock returns, and considers some robustness checks.
IV-A
Firm-Level Heterogeneity in the Exposures to Global, Country, and
Industry Shocks
We first test whether allowing for stock-specific exposures to global, country- and industryspecific shocks helps significantly in explaining comovement in international stock returns. We
implement this test by estimating a restricted version of our model, in which all stocks within
a country (industry) are restricted to have the same exposure to that country (industry) shock.
In addition, we impose the restriction that all stocks (across countries and industries) have
the same exposure to the global shock. By testing the restricted versus the unrestricted model
we learn whether stock-level heterogeneity in shock exposures is statistically important, i.e.,
whether the benefit in terms of fit outweighs the cost of estimating additional parameters.
5
We compare the goodness of fit across the restricted and unconstrained models using both
the likelihood ratio statistic and the Bayesian Information Criterion (BIC), which we compute
following Kass and Raftery (1995):
BIC = L −
ln(T )
× (# of free parameters)
2
where L is the log-likelihood at the peak.6
(8)
8
The value of the likelihood ratio statistic, equal to twice the difference between the loglikelihood values at the peak for the unrestricted and the restricted models, is equal to 38475.8
and 59394.8 for the 1985-2002 and 1990-2002 samples, respectively. The degrees of freedom are
5768 and 11677, respectively. In both samples, the difference in likelihoods is large enough that
the restricted model is soundly rejected, with the p-value essentially zero. We reach the same
conclusions using the more conservative BIC. In summary, we find strong statistical evidence
in favor of the unrestricted model.
We now investigate whether the differences across stocks in their exposures to global,
country- and industry-specific shocks are economically large. Table 1 describes the distribution
of global, country, and industry exposures within each portfolio (global, country and industry).
The first column shows the number of stocks in each portfolio. The second column shows the
mean exposures, while the third shows their standard deviations. The mean exposure to
global and industry shocks amounts to 2 percent, while it measures 6 percent on average
across countries. This result points to the fact that country shocks are the dominant source
of variation for international stock returns – a result that we will revisit in section IV-D.
The standard deviation of the exposures measures the degree of firm-level heterogeneity. The
standard deviations of exposures to global and industry shocks are about 2 percent, compared
with 1.5 percent on average for country exposures. However, these standard deviations may
be biased upward, because the factor exposures are estimated with error. To address this
question, we decompose the dispersion of the exposures into that associated with estimation
error and that associated with variation in the underlying “true” betas. The expected value
of the standard deviation of estimated exposures for a portfolio of N stocks is equal to:
N
1 X 2
β̂i −
IE[
N
i=1
Ã
N
1 X
β̂i
N
i=1
!2
N
1 X 2
] = IE[
βi −
N
i=1
Ã
N
1 X
βi
N
i=1
!2
] + IE[
N
N N
1 X 2
1 XX
ηi − 2
ηi ηj ]
N
N
i=1
i=1 j=1
(9)
where β̂ and β are the estimated and true exposure, respectively, and η is the sampling error.
Expression 9 shows that the estimated standard deviation is equal to the true standard devia-
9
tion plus a bias term. Using the asymptotic (normal) distribution of our maximum likelihood
estimator, we can compute this bias and correct for it, which we do in the fourth column of
Table 1. This column shows that stock-level heterogeneity in the exposures persists after the
bias-correction: heterogeneity in our shock exposures is not simply a result of sampling error.
To investigate whether the dispersion of shock exposures is economically large, we divide the
global, country and industry portfolios into high and low exposure portfolios. A high country
(global, industry) exposure portfolio, for example, is a portfolio with stocks whose exposure to
that country (global, industry) shock is higher than the mean exposure. Low country (global,
industry) exposure portfolios are portfolios whose stocks have exposures to country (global,
industry) shocks below their respective means. We then contrast the differences in exposure
of high and low exposure portfolios to the respective factor. Assuming that the distributions
of the betas is normal – the skewness and kurtosis numbers in columns (7) and (8) show that
this is approximately the case – this difference equals:
Z
Z
∞
βϕ(µ, σ)dβ −
µ
µ
σ
βϕ(µ, σ)dβ = 2 √ ,
2π
−∞
(10)
where ϕ indicate the Normal distribution with mean µ and standard deviation σ. The results of
this exercise are reported in columns (5) and (6), using the actual and bias-corrected standard
deviations, respectively. After the bias-correction, this difference is 1.4, 1, and 1.3 percent
on average for global, country, and industry portfolios. This implies, for instance, that a
high country exposure portfolio will have an exposure to country shocks that is one percent
higher – after bias-correction – than the exposure of a low-beta-portfolio (on average across
countries).
In summary, we have found that differences across individual stocks in their exposures to
global, country- and industry-specific shocks are both statistically significant and economically
relevant.
10
IV-B
What Are Stock-Specific Exposures Capturing?
We have shown that stock-specific exposures to global, country- and industry-specific shocks
are empirically and economically important. We now investigate what these exposures are
capturing, by relating them to observable characteristics that measure (i) the extent to which
firms operate across countries; (ii) their size; (iii) their book-to-market ratio; and (iv) the
presence of financial constraints.
To measure the extent to which firms operate across countries, we use two variables. First,
we construct a dummy variable for traded versus non-traded goods firms using the classification in Griffin and Karolyi (1998). This classification uses industrial affiliation as opposed to
firm-specific information. To address this deficiency, we construct an additional variable that
measures the average importance of foreign sales for each company over our sample. To this
end, we download annual firm-level data from Worldscope on the percentage of foreign sales
and make sample averages.
Separately, we download annual data from Worldscope for year-end market capitalizations
and the book-to-market ratio for each firm and construct sample averages for these variables.
To measure financial constraints, we follow Lamont et al. (2001) and Kaplan and Zingales
(1997) and use the following five variables: cash flow to total capital ratio, Tobin’s Q, debt
to total capital ratio, dividends to total capital ratio, and cash holdings to capital ratio. We
download annual data for each variable from Worldscope and construct sample averages of the
KZ index for each stock.
Table 2 reports multivariate regressions of the global (Panel A), country-specific (Panel
B) and industry-specific (Panel C) stock market exposures (in percent) from the 1985 to 2002
sample on our list of observables. The regression results using the shock exposures from the
1990 to 2002 sample are qualitatively similar and therefore omitted for brevity. In place of
the KZ index, which is based on regression results for the U.S., we adopt the more general
specification of including each of the constituent variables separately. We omit Q from the list
11
of KZ constituent variables because the book-to-market ratio enters our regression separately.
Because Table 1 and work by Ferson and Harvey (1998) suggest that there are substantial
differences across countries in mean shock exposures, we also include country dummies in our
regressions. We report t-ratios in parentheses, which we compute using robust standard errors
(White 1980), the associated adjusted R2 , and the number of observations in each regression.
Panel A shows that global shock exposures are positively and significantly related to measures of international activity. For traded goods companies, exposure to the global shock is on
average 0.62 percent per month higher than for non-traded goods firms. Separately, a company
raising its foreign sales ratio by 10 percentage points raises its exposure to the global shock by
2 percentage points per month.7 Among the other explanatory variables, exposure to global
shocks is negatively and significantly related to size. Since we take the natural log of market
capitalization, the size coefficient is readily interpretable: if market capitalization doubles in
size, the exposure to the global shock falls by 0.21 percent per month. This negative relation
could be an indication that large firms are less risky, ceteris paribus. Meanwhile, there is little
evidence of a systematic link between the book-to-market ratio and the KZ index constituents,
with the exception of the cash flow to total capital ratio and the dividends to total capital
ratio. Global shock exposures are significantly positively related to the former, while they are
significantly negative related to the latter. Since these variables have opposite signs, there is
little indication that global shock exposures are systematically related to financial constraints.
Panel B shows that there is little evidence that country shock exposures are systematically related to measures of international activity. The traded goods dummy has a positive
coefficient, while the foreign sales ratio has a negative coefficient that is borderline significant. Instead, country exposures are significantly negatively related to size, while they are
positively and significantly related to book-to-market ratio. In this dimension, our results parallel Avramov and Chordia (2001) who find that comovement among U.S. stocks is negatively
associated with size but positively associated with the book-to-market ratio. Among the KZ
12
constituents, there is evidence that country exposures are increasing with leverage (the debt
to capital ratio has a significant positive coefficient) and declining with liquidity (the cash to
capital ratio has a significant negative coefficient), suggesting that more financially constrained
companies are more exposed to these shocks. However, the coefficient on cash flow to total
capital ratio is significant and goes in the opposite direction, which suggests that the link
between country exposures and measures of liquidity should be interpreted with caution.
Panel C shows that there is little systematic association between industry shock exposures
and observables. The one robust pattern that stands out is that industry shock exposures
are positively and significantly associated with firm size, consistent with Chan et al. (1999)
who for U.S. data find that correlations across stocks within industries are higher among large
firms.
Overall, we find that (i) global shock exposures are positively associated with measures of
international activity and negatively related to size; (ii) country shock exposures are negatively
related to size, positively related to the book-to-market ratio, and increasing in leverage; and
(iii) industry shock exposures are positively associated with size. However, it is important to
note that the observable characteristics fail by a substantial margin to explain these exposures:
the adjusted R2 of the regressions in panels A, B, and C measure only 27, 57 and 15 percent
respectively, and about half of the explanatory power is due to country fixed effects. This
suggests that the shock exposures contain information that is not readily available: there is
substantial heterogeneity in the exposures which cannot be explained with observable characteristics. To put it differently, it is not the case that the low (high) country exposure portfolios
described in the previous section are simply portfolios containing large (small) firms, or firms
that are not (are) financially constrained. Indeed, if we divide firms within each country into
large and small firms (firms with market cap higher–or lower–than the median), we find that
on average low and high country exposure portfolios contain the same proportion of large and
small firms. The same finding roughly applies to the other observables as well.
13
IV-C
Diversification Strategies Using Stock-Specific Exposures
We have shown that there is heterogeneity across stocks in their exposures to common shocks,
which in part reflects differences in observed characteristics of the underlying companies. We
now explore whether we can use these exposures for risk reduction strategies. We do this by
constructing bottom-up low exposure portfolios with respect to global, country and industryspecific shocks. We then examine how effective these portfolios are relative to their respective
benchmark in terms of volatility and returns, both in- and out-of-sample.
We construct low exposure portfolios in the following way. With respect to the global
shock, we construct an equal-weighted portfolio of stocks with global shock exposures below
the median, which we call the low global exposure (LGE) portfolio. By construction this
portfolio contains half the stocks in our sample. With respect to country shocks, within each
country we determine which stocks have country exposures above and below their respective
country median. We form a global low country exposure (LCE) portfolio that equal-weights
across countries all stocks that have shock exposures below the country median. This is a global
portfolio consisting of half the stocks from each country. We compare the performance of the
LGE and global LCE portfolios to that of the equal-weighted global portfolio. Furthermore,
we construct country portfolios that include all stocks within a given country with country
exposures below the respective country median, which we call local low country exposure (LCE)
portfolios. We compare the performance of these portfolios to the respective equal-weighted
country portfolio. With respect to industry-specific shocks, we follow the same procedure
and form global and local low industry exposure (LIE) portfolios. For reasons of symmetry,
we also construct the respective high exposure portfolios at the global, country and industry
levels – the HGE, the global and local HCE portfolios, and the global and local HIE portfolios.
Our global LCE and LIE portfolios share an important characteristic: in each we impose
that countries and industries be present in the same proportion, respectively, as in the overall
sample. With respect to the global LCE portfolio, this means that risk reduction relative to
14
the global market is not the result of a particular country tilt, but reflects the incremental gain
from picking stocks with low country exposures. The same holds with respect to the global
LIE portfolio. This is of course not true for our LGE portfolio. Here, country and industry
composition depend entirely on the distribution of the global shock exposures.
Based on shock exposure estimates for the 1985 to 2002 sample, Table 3 shows in- and
out-of-sample volatility of the equal-weighted global market portfolio, and of the respective
low and high exposures portfolios. The results for the shorter sample from 1990 to 2002 are
qualitatively similar and therefore omitted for brevity. Column (1) shows the variance of the
benchmark (B), column (2) shows the variance of the corresponding low exposure portfolio
(L), while the number in parentheses in column (3) shows the variance reduction (increase, if
positive) relative to benchmark ( L−B
B ). All figures (variances and variance reduction) are in
percent. Column (4) shows the variance of the corresponding high exposure portfolio (H), with
column (4) showing the variance increase over the respective benchmark ( H−B
B ). Columns (6)
through (10) repeat this format for out-of-sample results, based on re-estimating our model
from January 1985 through February 2000 and examining the risk reduction properties of the
low exposure portfolios in the period March 2000 to February 2002.
The first three rows of Table 3 compare the volatility of the global market to that of the LGE
and HGE portfolios (row 1), the global LCE and HCE portfolios (row 2), and the global LIE and
HIE portfolios (row 3). The Table shows that in-sample the global LCE portfolio performs best
in terms of risk reduction: its variance is 27 percent below that of the global market portfolio.
Next in line is the LGE portfolio, whose variance is 17 percent below that of the global market,
while no risk reduction is associated with the global LIE portfolio. Why is it that the global
LCE portfolio does best in terms of risk reduction? Section IV-A sheds some light on this:
the means of country shock exposures are higher than those for the global or industry shocks.
This means that country-specific shocks are more important drivers of international return
variation than global or industry shocks, a result we will confirm in the next section using
15
variance decompositions. Taken together with the fact that there is substantial heterogeneity
in country shock exposures, as demonstrated in Section IV-A, the global LCE portfolio is
best positioned to reduce volatility relative to the global market portfolio. Importantly, these
results carry over to out-of-sample. Relative to the market during the period from March 2000
to February 2002, the global LCE portfolio yields a reduction in volatility of 31 percent, while
the LGE portfolio delivers a reduction in volatility of 26 percent. Again, no risk reduction is
associated with the global LIE portfolio.
The remaining rows of Table 3 compare the volatility of our low and high exposure portfolios
with that of the respective country or industry benchmarks. For countries, we show the
volatilities of the local LCE and HCE portfolios for each country, as well as the equal-weighted
average across countries. For industries, we only show the equal-weighted average across
industries to save space. We have seen that at the global level our LCE portfolio deliver risk
reduction relative to the global market. Do this result carry over to the country level? The
answer is yes, both in- and out-of-sample: our local LCE portfolios have lower variance than
the respective country benchmarks. Only for a handful of countries for which we have very few
stocks in our sample (like South Africa) this is not the case. On average across countries, local
LCE portfolios deliver an in-sample reduction in volatility of 26 percent relative to the country
portfolio, while high exposure portfolios boost volatility by 43 percent. Out-of-sample, there
is a 25 percent reduction in volatility associated with local LCE portfolios on average. Again,
note that this finding is due to the heterogeneity in exposures: under the null that all stocks
have the same “true” country exposure, we would simply be randomly picking half the stocks
in each country. It would be quite surprising to find that both in- and out-of-sample (almost)
all of these portfolios have lower variance than their respective benchmark. Finally, the last
row in Table 3 confirms that there is little if any systematic risk reduction associated with the
local LIE portfolios.
We next examine the robustness of our risk reduction result over different sub-periods.
16
Table 4 shows the variance reduction relative to the global market portfolio of the LGE portfolio
(column 1), the global LCE portfolio (column 2) and the global LIE portfolio (column 3) for
eight two-year sub-periods, working backward from the end of the sample. This exercise shows
that the global LCE portfolio is the most robust in terms of risk reduction. For all sub-samples
the variance of this portfolio is less than that of the global market, though the gains range
from 37 to 15 percent. Not so for the LGE portfolio where much of the risk reduction appears
to be driven by the later part of the sample.
Figure 1 examines the risk reduction properties of our low exposure portfolios using the
graphical representation in Heston and Rouwenhorst (1994) and Solnik (1975). It gives the
average portfolio variance as the number of stocks in a given portfolio increases from one
to forty, expressed as a percentage of the average variance of all individual stocks in our
sample. The solid lines in Figure 1 show the diversification benefit associated with the global
market portfolio (line 3), the average country portfolio (line 5), and the average industry
portfolio (line 6). All averages are equal-weighted. The global benchmark has a variance of
16 percent relative to the average stock. The country benchmark has a variance of 34 percent
(on average across countries) relative to the average stock. The industry benchmark delivers a
risk reduction of 27 percent (on average across industries) relative to the average stock. These
figures restate the well-known finding that diversifying across countries (within an industry)
is more promising for risk reduction than diversifying across industries (within a country).
The value added of this paper is represented by the dashed lines – lines (1), (2), and (4).
Line (4) shows that risk reduction for the local LCE portfolios (on average across countries) is
25 percent. This shows that one does not need to diversify internationally to achieve higher risk
reduction than that delivered by the average country portfolio. Just by selecting stocks with
low country exposure one obtains roughly the same level of volatility. Lines (1) and (2) show
that the LGE and global LCE portfolios have a lower variance than the global benchmark,
consistently with Table 3. A bottom-up strategy that picks stocks according to their country
17
or global exposures beats the global benchmark in terms of risk reduction. Of course, for this
to be the case the exposures need to be estimated quite precisely. Using the shorter sample
(1990-2002) we find that all the above results hold, except for the out-of-sample performance
of the LGE portfolio. This is perhaps because global exposures, which are harder to estimate
than country exposures (due to the lower variability of global relative to country shocks), are
imprecisely estimated in the shorter sample. The global and local LCE portfolios however still
outperform their respective benchmarks for the 1990-2002 sample, both in- and out-of-sample.
We have shown that global and local LCE portfolios and the LGE portfolio successfully
achieve risk reduction relative to the global market portfolio. But does this risk reduction
come at the cost of lower expected returns? In other words, is there no difference in riskadjusted terms between holding low exposure portfolios and holding the market? Table 5
examines this question. The rows of Table 5 correspond to those in Table 3. Column (1)
and (2) show the in-sample mean monthly return in percent for the benchmark (B) and for
the corresponding low exposure portfolio (L), while the number in parentheses in column (3)
shows the difference between the two (L-B). Columns (4) and (5) show the corresponding
figures for the high exposures portfolios. Columns (6) through (10) give the in-sample Sharpe
ratios, while columns (11) through (15) show out-of-sample means for the period March 2000
to February 2002, based on shock exposure estimates for the shortened sample from January
1985 to February 2000. The in-sample mean returns on the LGE, global LCE, and global LIE
portfolios are 7, 5 and 1 basis points per month in excess of the global benchmark, while
the HGE, global HCE, and global HIE all do worse than the market. The in-sample Sharpe
ratios of these portfolios are 2, 3, and 0 basis points per month over the global market. On
a risk-adjusted basis, the LGE and global LCE portfolios thus outperform the market. Outof-sample the low global and country exposure portfolios do better than the market, to the
tune of 64 and 10 basis points per month. This is noteworthy given that the out-of-sample
period spans the bursting of the tech-bubble and is thus a challenging period during which
18
to examine out-of-sample behavior of our low exposure portfolios. This is evidence that our
bottom-up approach to construct international portfolios provides diversification gains, even
on a risk-adjusted basis.
IV-D
Variance Decomposition Results and Robustness Tests
Are country- or industry-specific shocks more important in explaining the behavior of national
stock markets? Heston and Rouwenhorst (1994) and Griffin and Karolyi (1998) among others
conclude that country shocks are much more important, using cross-sectional regressions of international returns on country and industry dummies. We use our methodology to re-examine
this result. Although we allow for firm-specific exposures to the different shocks, the impact of
country shocks on a country portfolio, for example, depends on the average country exposure,
which should roughly correspond to the standard deviation of the country shock in the Heston
and Rouwenhorst model. Hence we expect to find a similar answer for the variance decomposition as in the existing literature. However, in contrast to the existing literature, we control
for global shocks in addition to country- and industry-specific shocks. Does this throw off the
balance between country- and industry-specific shocks? Finally, the literature focuses on the
relative importance of country versus industry shocks in country and industry portfolios. We
ask how important these shocks are for the average stock and for the global market portfolio.
Based on the 1985 to 2002 sample, the first row in Table 6 reports the average variance
decomposition across country portfolios, while the second row shows the average variance decomposition across industry portfolios. The first column gives the actual variance in percent
per month squared of the average country and industry portfolios. The second column gives
the percentage of that variance explained by the global shock, the third column gives the percentage due to country-specific shocks, while the fourth column gives the percentage associated
with industry-specific shocks. On average, we find that country shocks explain 81 percent of
variation in our country portfolios, while industry shocks explain barely one percent. Mean-
19
while, we find that country shocks explain 32 percent of the volatility for the average industry
portfolio, while industry shocks explain only 11 percent. In short, we replicate the result in
the literature: country shocks are much more important than industry shocks for international
returns. That said, Heston and Rouwenhorst (1994) and Griffin and Karolyi (1998) find that
country shocks explain virtually all of the variation in national stock markets. Our number is
lower because model (1) includes a global shock, which explains close to 12 percent of the variation in the average country portfolio. In summary, we replicate the existing result, with the
caveat that accounting for global shocks reduces the importance of country shocks in national
stock markets somewhat.
The third row in Table 6 gives the variance decomposition for the average across individual
stocks. Country shocks account for 32 percent at this level, followed by industry shocks at
7 percent and global shocks at 6 percent. The fourth row in Table 6 gives the variance
decomposition for the global market portfolio. Here country shocks are still the most important
source of variation, at 28 percent, followed by 21 percent for the global shock.
In Table 6 we show the results for the ex-ante variance decomposition, that is, assuming
that the factors are orthogonal. As discussed in section II, ex-post the factors may not turn
out to be orthogonal. Table 7 shows the median ex-post correlations among factors for the
1985-2002 sample (the 1990-2002 results are very similar). The diagonal elements of the matrix
show the average correlation within each class of factors and the off-diagonal elements show the
average correlation across different classes of factors. The average correlation among country
factors is .3, and all other average correlations are less than .1. These results do not point
out substantial mis-specifications in the baseline model. Most importantly, the off-diagonal
elements are virtually zero, suggesting that the orthogonality assumption used in the variance
decomposition is not far-fetched.
Our decision to focus on global, country- and industry-specific shocks is based on Heston
20
and Rouwenhorst (1994) and Griffin and Karolyi (1998). We now test the robustness of our
results with respect to other sources of common variation. We augment our model with: (i) an
additional global factor–orthogonal to the first one–which may capture different kinds of global
shocks, e.g. oil versus interest rate shocks; (ii) an emerging markets factor, to allow for comovement associated with emerging markets; (iii) an ADR factor that captures common variation
associated with stocks that have ADRs; (iv) regional factors for Developed Europe, Emerging
Europe, Developed Americas, Emerging Americas, Developed Asia, Emerging Asia, to capture
regional comovement; and (v) factors associated with small cap stocks (a size factor), high
book-to-market ratio stocks (value stocks) and high KZ index stocks (financing constrained
stocks). In some instances (depending on the sample) regional factors, the size factor, book-tomarket factor and the KZ factor add significantly to the explanatory power of the model: they
have higher BICs than our baseline model. However, they do not add significantly to explaining
international return variation according to additional variance decomposition results (available
upon request). In other words, our model consisting of global, country- and industry-specific
shocks provides a parsimonious specification with which to investigate international return
comovement.
We consider one final robustness test, which is motivated by recent papers that argue that
the relative importance of country versus industry shocks in international returns has been
changing over time.8 We modify the model to allow the variances of global, country and
industry shocks to vary over four pre-specified sub-periods of equal length. Estimating this
version of the model for the 1985 to 2002 sample, variance decompositions show that the global
shock increases in importance during the last sub-period (1997:11 to 2002:02), while country
and industry shocks are roughly stable over time. Overall, however, these changes leave the
results qualitatively unchanged.
21
V
Conclusion
The paper presents a methodological contribution, as it estimates a latent factor model that
allows for stock-specific exposures to global, country and industry shocks on a large panel of
international stock returns. These firm-level exposures capture observed and unobserved heterogeneity across firms. We make the following empirical contributions. First, we quantify the
extent to which stock-specific exposures to global, country and industry shocks are important,
both statistically and economically. Second, we explore whether exposures to global, country
and industry shocks are systematically related to observed characteristics of firms, such as
their extent of their international operations, size, book-to-market ratio, and whether they are
financially constrained. Finally, we explore whether stock-specific shock exposures can serve
as a stock-selection device to construct portfolios that successfully diversify risk. We find that
this is the case at the global as well as the country level, both in- and out-of-sample.
Our model can be extended is several ways, including: (i) having regime-switching in the
variance of the factors (see Ang and Bekaert 2002); (ii) having time variation in the exposures.
We leave these extensions for future research.
22
References
Avramov, D. and T. Chordia, 2001. “Characteristic Scaled Betas.” Unpublished working
paper, University of Maryland.
Ang, A., Bekaert, G., 2002. “International Asset Allocation with Regime Shifts.” The Review
of Financial Studies, Fall 15, 1137-1187.
Bernanke, B., 1986. “Alternative Explanations for the Money-Income Correlation.” CarnegieRochester Series on Public Policy 25, Autumn, 49-99.
Bodnar, G., Dumas, B. and R. Marston. 2002. “Pass-Through and Exposure.” Journal of
Finance 57 (1), 199-231.
Brooks, R., and M. Del Negro, Forthcoming. “The Rise in Comovement across National Stock
Markets: Market Integration or IT Bubble?” Journal of Empirical Finance. (WP version
available as FRB Atlanta WP 2002-17a: http://www.frbatlanta.org/filelegacydocs/wp0217a.pdf).
Chan, L.K.C., Karceski, J., and J. Lakonishok, 1998. “The Risk and Return from Factors.”
Journal of Financial and Quantitative Analysis 33 (2), 159-187.
Chan, L.K.C., Karceski, J., and J. Lakonishok, 1999. “On Portfolio Optimization: Forecasting
Covariances and Choosing the Risk Model.” Review of Financial Studies 12 (5), 937-974.
Cho, D., Eun, C., and L. Senbet, 1986. “International Arbitrage Pricing Theory: An Empirical Investigation.” Journal of Finance XLI, 313- 329.
Connor, G., and R. Korajczyk, 1986. “Performance Measurement with the Arbitrage Pricing
Theory: A New Framework for Analysis” Journal of Financial Economics 15, 373-94.
Ferson, W., and C.Harvey, 1994. “Sources of Risk and Expected Returns in Global Equity
Markets.” Journal of Banking and Finance 18, 775 - 803.
Ferson, W., and C. Harvey, 1998. “Fundamental Determinants of national Equity market
Returns: A Perspective on Conditional Asset Pricing.” Journal of Banking and Finance
21, 1625-1665.
Gelman, A., Carlin, J., Stern, H., and D. Rubin, 1994. “Bayesian Data Analysis.” Chapman
& Hall, London.
23
Griffin, J., and R. Stulz, 2001. “International Competition and Exchange Rate Shocks: A
Cross-Country Industry Analysis of Stock Returns” The Review of Financial Studies 14,
215 - 241.
Griffin, J., and G. Karolyi, 1998. “Another Look at the Role of Industrial Structure of
Markets for International Diversification Strategies.” Journal of Financial Economics
50, 351-373.
Heston, S., and K. Rouwenhorst, 1994. “Does Industrial Structure Explain the Benefits of
International Diversification?” Journal of Financial Economics: 36, 3-27.
Heston, S., Rouwenhorst, K., and R. Wessels, 1995. “The Structure of International Stock
Returns and the Integration of Capital Markets” Journal of Empirical Finance 2, 173197.
Ince, O. and R. Porter. 2004. ”Individual Equity Return Data from Thomson Datastream:
Handle with Care!” Unpublished working paper, Warrington College of Business, University of Florida.
Kass, R., and A. Raftery, 1995, “Bayes Factors.” Journal of the American Statistical Association, June 90, 773-795.
Lehmann, B., and D. Modest, 1985. “The Empirical Foundations of Arbitrage Pricing Theory
I: The Empirical Tests.” NBER Working Paper No.1725.
Marsh, T., and P. Pfleiderer, 1997. “The Role of Country and Industry Effects in Explaining Global Stock Returns.” Unpublished working paper, U.C. Berkeley and Stanford
University.
Sims, C., 1986. “Are Forecasting Models Usable for Policy Analysis? ” Federal Reserve Bank
of Minneapolis Quarterly Review 10, Winter, 2-16.
White, H. 1980. “A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct
Test for Heteroskedasticity.” Econometrica 48(4), 817-838.
24
Notes
1
Lehmann and Modest (1985) also use the EM algorithm to estimate factor models.
Our model extends
their work to the case where the restrictions (2) are present.
2
Using U.S. dollar-denominated returns may lump nominal currency movements into the country-specific
shocks, potentially biasing up the importance of these shocks in our model. We investigate the magnitude
of this bias by using local currency returns instead and find it to be negligible, consistent with Heston and
Rouwenhorst (1994).
3
Griffin and Karolyi (1998) argue that broad industrial classifications may lump together heterogeneous
industries, which may bias downward the importance of industry-specific shocks. Against this background,
Griffin and Stulz (2001) use the level 6 industry definitions from Datastream in their study on the importance of
industry-specific and exchange rate shocks in stock returns. See http://www.datastream.com/product/investor/index.htm
for a description of the Datastream Global Market Indices.
4
Ince and Porter (2004) report that individual equity return data for the U.S. from Datastream is sometimes
inaccurate. However, they find that these problems are concentrated among smaller size deciles. They are thus
not likely to materially affect our results. Moreover, as we document below, our variance decomposition results
are consistent with earlier papers that use different data sources.
5
Formally, in the restricted version of our model the restrictions (2) are replaced with:
βnG
=
C
βnc
=
I
βni
6
=
βG
8
>
< βcC
if stock n belongs to country c
>
: 0
8
>
< βiI
if stock n belongs to industry i
>
: 0
otherwise.
otherwise,
(11)
Recall that in the factor model the variance-covariance matrix of excess returns is a sufficient statistics for
the likelihood. Also, at the peak of the likelihood the variance of the idiosyncratic term is by construction such
that the model exactly fits the variance of each individual stock. Hence the model that maximizes the BIC is
the one that best captures the co-movements among stocks, and does so parsimoniously.
7
The positive association between the global shock exposures and the measures of international activity
is consistent with Bodnar et al. (2002) who argue that foreign sales tend to increase a firm’s exposure to
international shocks.
8
See Brooks and Del Negro (forthcoming) for a review of this literature.
25
Table 1: Distribution of the Exposures to Global, Country-, and Industry-Specific Shocks
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
bias
bias
# of Stocks Mean St. Dev. ( corrected
) H-L ( corrected
) Skewness Kurtosis
Global Factor 1965
2.04
1.93
(1.78)
1.54
(1.42)
0.92
4.4
451
280
54
100
31
529
89
41
9
26
20
51
14
56
16
16
42
45
18
54
23
4.07
4.95
5.33
4.49
6.88
7.23
3.79
5.87
5.85
4.74
4.67
8.80
6.06
4.03
5.82
5.92
4.33
10.31
9.45
8.73
5.60
1.35
1.21
1.38
1.04
1.29
1.89
1.14
1.09
1.35
0.85
0.85
2.39
1.10
1.27
1.60
1.29
0.89
2.44
2.18
2.61
1.52
(1.21)
(1.05)
(1.23)
(0.92)
(1.16)
(1.79)
(0.94)
(0.95)
(1.21)
(0.72)
(0.67)
(2.30)
(0.66)
(1.18)
(1.28)
(1.16)
(0.71)
(2.19)
(2.09)
(2.54)
(1.32)
1.08
0.96
1.10
0.83
1.03
1.51
0.91
0.87
1.07
0.68
0.68
1.91
0.88
1.01
1.28
1.03
0.71
1.95
1.74
2.08
1.21
(0.97)
(0.84)
(0.98)
(0.73)
(0.93)
(1.43)
(0.75)
(0.76)
(0.97)
(0.57)
(0.54)
(1.84)
(0.52)
(0.94)
(1.02)
(0.92)
(0.57)
(1.75)
(1.67)
(2.02)
(1.05)
-0.16
0.18
-0.53
-0.13
-0.59
-0.10
-0.20
-0.80
-0.23
-0.45
-0.90
-0.36
0.56
-0.33
0.10
-0.92
-0.37
0.38
1.72
-0.35
-0.24
2.7
3.1
2.5
2.6
2.9
2.9
3.8
3.3
1.6
1.8
4.4
2.8
2.5
2.3
2.5
3.4
2.2
2.8
7.1
3.1
1.8
Average
across
Countries
94
6.04
1.46
(1.30)
1.17
(1.04)
-0.18
3.0
Average
across
Industries
22
2.14
2.02
(1.68)
1.61
(1.34)
0.42
2.8
Country Factors
U.S.
U.K.
France
Germany
Italy
Japan
Canada
Australia
Austria
Belgium
Denmark
HongKong
Ireland
Netherlands
Norway
Sweden
Switzerland
Korea
Malaysia
Singapore
S.Africa
Notes: The Table describes the distribution of global, country, and industry exposures within the global, country and
industry portfolios. The columns show: (1) number of stocks in each portfolio; (2) mean ; (3) standard deviations;
(4) biased-corrected standard deviations (see formula (9) for bias-correction); (5) difference in exposure of high and low
exposure portfolios to the respective shock (computed using the formula 2 St.√Dev. ); (6) same as previous column using
2π
bias-corrected standard deviations; (7) skewness; (8) kurtosis. The table is based on estimates for the 1985 to 2002 sample.
26
Table 2: Regressions of Exposures on Observables
A
B
C
Global Exposures Country Exposures Industry Exposures
Cash Flow/Total Capital
Cash/Total Capital
Debt/Total Capital
Dividends/Total Capital
Book-to-Mkt
Market-Cap
International Sales
Traded/Non Traded Dummy
(1)
(2)
(3)
(4)
(5)
(6)
Coef
t-ratios
Coef
t-ratios
Coef
t-ratios
0.018
0.001
-0.006
-0.045
-0.014
-0.209
0.024
0.618
(4.39)
(1.24)
(-1.50)
(-3.25)
(-0.06)
(-3.47)
(8.20)
(5.58)
0.008
-0.002
0.029
0.014
1.479
-0.157
-0.005
0.076
(2.30)
(-2.10)
(8.26)
(1.27)
(6.20)
(-3.43)
(-1.66)
(0.80)
-0.006
0.001
-0.008
-0.008
0.439
0.326
0.004
0.473
(-1.265)
(1.296)
(-1.500)
(-0.373)
(1.346)
(4.463)
(1.195)
(3.050)
adj R2
0.251
n
946
adj R2
0.555
n
946
adj R2
0.158
n
945
Notes: Columns (1), (3), and (5) report the OLS coefficients for the cross-sectional regressions where the independent
variable is global, country, and industry exposures, respectively. All regressions include country fixed effects. The
exposures are measured in percent. Columns (2), (4), and (5) report the associated t-ratios in parentheses, which we
compute using robust standard errors (White 1980). The last row in the table reports for each regression the adjusted
R2 and the number of observations. The table is based on estimates for the 1985 to 2002 sample. The regressors are
constructed as follows. We download annual data from Worldscope for year-end market capitalizations, book-to-market
ratio, capital, debt, cash holdings, cash flow, and dividends for each firm. The variables Cash Flow/Total Capital,
Cash/Total Capital, Debt/Total Capital, Dividends/Total Capital, Book-to-Mkt, and Market-Cap are constructed as the
average for each company over the sample period of ratio of cash flow over total capital, cash holdings over total capital,
debt over total capital, dividends over total capital, book-to-market, and market capitalization respectively. The variable
Market-Cap enters the regression in logs. The International Sales variable is constructed as the average over the sample
period of ratio of foreign sales over total sales, which is downloaded from Worldscope. The Traded/Non Traded Dummy
dummy is constructed by aggregating up the Datastream level 6 industry dummies, following the classification in Griffin
and Karolyi (1998), and is equal to one for traded goods industries and zero otherwise.
27
Table 3: Portfolio Variances of Low versus High Exposure Portfolios relative to Benchmark
(1)
(2)
B
L
(3)
(4)
In-sample
( L−B
H
B )
(5)
(6)
( H−B
B )
B
(7)
(8)
(9)
Out-of-sample
L ( L−B
H
B )
(10)
( H−B
B )
Global Portfolios
global
country
industry
19.4 16.1 (-17)
19.4 14.2 (-27)
19.4 19.3 (-0)
27.9
26.3
20.0
(44)
(35)
( 3)
19.9 14.8 (-26)
19.9 13.8 (-31)
19.9 20.0 ( 0)
30.7
28.3
20.5
(54)
(42)
( 3)
(-37)
(-26)
(-27)
(-30)
(-24)
(-26)
(-18)
(-19)
(-6)
(-12)
(-10)
(-31)
(-26)
(-30)
(-17)
(-0)
(-19)
(-23)
(-20)
(-37)
(30)
34.1
42.9
50.1
33.8
78.9
79.8
32.2
61.9
60.9
34.5
35.4
138.9
95.8
31.0
84.8
64.1
34.6
172.3
143.8
138.4
66.1
(56)
(37)
(42)
(50)
(36)
(40)
(36)
(27)
(42)
(33)
(35)
(43)
(75)
(52)
(58)
(23)
(38)
(41)
(38)
(53)
(17)
21.7
33.6
30.9
16.3
44.0
54.4
22.0
32.6
26.0
18.1
21.1
46.9
47.8
16.6
24.2
64.1
28.5
141.3
35.2
46.8
54.8
(-24)
(-21)
(-31)
(-32)
(-15)
(-34)
(-8)
(-4)
(43)
(-11)
( 6)
(-24)
(-1)
(-35)
(-16)
(35)
(-34)
(-30)
(-19)
(-29)
(35)
37.3
44.2
46.8
28.9
52.8
80.7
32.9
38.0
38.0
25.8
36.3
63.7
61.1
26.5
34.8
50.8
47.9
202.5
48.3
71.2
58.7
(72)
(31)
(52)
(78)
(20)
(48)
(49)
(17)
(46)
(43)
(72)
(36)
(28)
(59)
(44)
(-21)
(68)
(43)
(37)
(52)
( 7)
42.5 31.4 (-26)
60.6
(43)
37.9 28.3 (-25)
55.6
(47)
Country Portfolios
U.S.
U.K.
France
Germany
Italy
Japan
Canada
Australia
Austria
Belgium
Denmark
HongKong
Ireland
Netherlands
Norway
Sweden
Switzerland
Korea
Malaysia
Singapore
S.Africa
Average
across
Countries
21.8
31.3
35.2
22.6
57.9
56.9
23.7
48.7
43.0
26.0
26.1
96.9
54.6
20.4
53.7
51.9
25.2
122.3
104.0
90.6
56.3
13.7
23.2
25.9
15.8
43.9
42.0
19.3
39.3
40.5
23.0
23.6
66.5
40.2
14.4
44.8
51.9
20.4
94.3
82.8
56.8
73.3
16.5
26.7
21.3
11.1
37.3
36.0
20.1
31.1
37.0
16.0
22.3
35.8
47.4
10.8
20.3
86.6
18.8
98.3
28.5
33.0
73.9
Industry Portfolios
Average
40.4 44.1 ( 9)
55.6 (38)
33.2 36.0 ( 8) 48.3 (45)
across
Industries
Notes: Column (1) shows the variance of the corresponding benchmark portfolio (B) – that is, the global portfolio for
rows 1 through 3, county portfolios for the following twenty-two rows, industry portfolios for the last row. Column (2)
shows the variance of the corresponding low exposure portfolio (L). The global low exposure portfolios are considered in
rows 1-3 and are constructed as follows (see also text): The low global exposure (LGE) portfolio contains half the stocks
in the sample, those with global shock exposures below the median. The global low country exposure (LCE) portfolio
also contains half the stocks in the sample, those with country shock exposures below the median for each country. The
global low industry exposure (LIE) portfolio contains stocks with industry shock exposures below the median for each
industry. The local low exposure portfolios are considered in the remaining rows and are constructed as follows (see
also text): For countries, the local low country exposure (LCE) portfolio contains half the stocks in each country, those
with country shock exposures below the median. For industries, the local low industry exposure (LIE) portfolio contains
half the stocks in each industry, those with industry shock exposures below the median. Column (3) shows the variance
). Column (4) shows the variance of the
reduction (increase, if positive) relative to the corresponding benchmark ( L−B
B
corresponding high exposure portfolio (H). High exposure portfolios are constructed as the mirror image of low exposure
portfolios. Column (5) shows the variance increase over the respective benchmark ( H−B
). Columns (6) through (10)
B
repeat this format for out-of-sample results. Portfolios are constructed based on estimates from January 1985 through
February 2000 and the variances are computed for the period March 2000 to February 2002. All figures (variances and
variance reduction) are in percent. The table is based on estimates for the 1985 to 2002 sample.
28
Table 4: Variance Reduction (Increase) Relative to Global Benchmark Portfolio: Sub-sample
Analysis
Portfolios:
LGE
LCE
LIE
Sub-Periods:
3/00-2/02
-38.00 -27.59 2.83
3/98-2/00
-23.94 -37.36 2.57
3/96-2/98
-25.73 -32.95 -5.92
3/94-2/96
-17.44 -14.71 -4.22
3/92-2/94
-8.80 -20.27 -5.88
3/90-2/92
-7.35 -27.91 -8.34
3/88-2/90
3.35 -22.15 -1.85
3/86-2/88
-33.69 -25.80 6.65
Notes: The table show the variance reduction in percent relative to the global benchmark for three low exposure portfolios
during eight different subsamples. The low global exposure (LGE) portfolio contains half the stocks in the sample, those
with global shock exposures below the median. The global low country exposure (LCE) portfolio also contains half the
stocks in the sample, those with country shock exposures below the median for each country. The global low industry
exposure (LIE) portfolio contains stocks with industry shock exposures below the median for each industry. The table is
based on estimates for the 1985 to 2002 sample.
29
Table 5: Portfolio Mean Returns and Sharpe Ratios of Low versus High Exposure Portfolios
relative to Benchmark
(1) (2)
(3)
(4)
(5)
In-sample Mean
B
L
(L-B) H
(6) (7)
(8)
(9)
(10)
(11) (12)
In-sample Sharpe Ratio
(H-B)
B
L
(L-B) H
(H-B)
(13)
(14)
(15)
Out-of-sample Mean
B
L
(L-B)
H
(H-B)
Global Portfolios
global
country
industry
0.35 0.42 (0.07) 0.28 (-0.07) 0.08 0.10 (0.02) 0.05 (-0.03) -0.71-0.08 (0.64) -1.35(-0.64)
0.35 0.40 (0.05) 0.30 (-0.05) 0.08 0.11 (0.03) 0.06 (-0.02) -0.71-0.61 (0.10) -0.81(-0.10)
0.35 0.36 (0.01) 0.33 (-0.02) 0.08 0.08 (0.00) 0.07 (-0.00) -0.71-0.70 (0.01) -0.71 (0.00)
Country Portfolios
0.68 0.66 (-0.02) 0.70 (0.02)
0.52 0.58 (0.06) 0.46 (-0.06)
France
0.62 0.56 (-0.06) 0.68 (0.06)
Germany
0.42 0.51 (0.09) 0.33 (-0.09)
Italy
0.22 0.49 (0.27) -0.03(-0.25)
Japan
-0.14-0.00 (0.13) -0.27(-0.13)
Canada
0.38 0.49 (0.10) 0.28 (-0.10)
Australia
0.45 0.42 (-0.03) 0.47 (0.02)
Austria
0.56 0.39 (-0.17) 0.70 (0.14)
Belgium
0.63 0.49 (-0.13) 0.76 (0.13)
Denmark
0.49 0.52 (0.03) 0.46 (-0.03)
HongKong
0.51 0.52 (0.01) 0.51 (-0.01)
Ireland
0.91 1.32 (0.41) 0.50 (-0.41)
Netherlands 0.77 0.84 (0.07) 0.70 (-0.07)
Norway
0.18 0.49 (0.31) -0.13(-0.31)
Sweden
0.73 0.68 (-0.04) 0.77 (0.04)
Switzerland 0.37 0.15 (-0.22) 0.59 (0.22)
Korea
0.50 0.44 (-0.06) 0.56 (0.06)
Malaysia
0.11 0.15 (0.05) 0.06 (-0.05)
Singapore
0.11 0.24 (0.13) -0.03(-0.13)
S.Africa
0.07 0.07 (0.00) 0.07 (-0.00)
U.S.
U.K.
Average
across
Countries
0.15 0.18 (0.03) 0.12 (-0.03)
0.09 0.12 (0.03) 0.07 (-0.02)
0.10 0.11 (0.01) 0.10 (-0.01)
0.09 0.13 (0.04) 0.06 (-0.03)
0.03 0.07 (0.04) -0.00(-0.03)
-0.02-0.00 (0.02) -0.03(-0.01)
0.08 0.11 (0.03) 0.05 (-0.03)
0.06 0.07 (0.00) 0.06 (-0.00)
0.09 0.06 (-0.02) 0.09 (0.00)
0.12 0.10 (-0.02) 0.13 (0.01)
0.10 0.11 (0.01) 0.08 (-0.02)
0.05 0.06 (0.01) 0.04 (-0.01)
0.12 0.21 (0.09) 0.05 (-0.07)
0.17 0.22 (0.05) 0.13 (-0.04)
0.02 0.07 (0.05) -0.01(-0.04)
0.10 0.10 (-0.01) 0.10 (-0.00)
0.07 0.03 (-0.04) 0.10 (0.03)
0.05 0.05 (-0.00) 0.04 (-0.00)
0.01 0.02 (0.01) 0.01 (-0.01)
0.01 0.03 (0.02) -0.00(-0.01)
0.01 0.01 (-0.00) 0.01 (-0.00)
0.46 0.76 (0.30)
-0.97-1.12(-0.15)
-0.93-0.44 (0.49)
-0.47-0.09 (0.38)
-1.78-1.89(-0.11)
-1.65-1.77(-0.12)
-0.15-0.14 (0.01)
0.12 0.68 (0.57)
-0.16 0.14 (0.30)
-0.36-0.33 (0.03)
-0.00-0.22(-0.22)
-0.57-0.64(-0.07)
-0.76-0.86(-0.10)
-0.78-0.62 (0.16)
-0.53-0.85(-0.32)
-1.64-2.54(-0.90)
-2.14-1.20 (0.94)
0.32 0.40 (0.08)
-1.54-1.10 (0.44)
-0.99-0.63 (0.36)
-0.88 0.63 (1.51)
0.17 (-0.29)
-0.82 (0.15)
-1.41(-0.49)
-0.85(-0.38)
-1.67 (0.11)
-1.53 (0.12)
-0.15(-0.01)
-0.42(-0.54)
-0.40(-0.24)
-0.39(-0.03)
0.21 (0.22)
-0.51 (0.07)
-0.66 (0.10)
-0.94(-0.16)
-0.21 (0.32)
-0.75 (0.90)
-3.08(-0.94)
0.24 (-0.08)
-1.98(-0.44)
-1.35(-0.36)
-2.27(-1.39)
0.35 0.40 (0.05) 0.30 (-0.05) 0.07 0.09 (0.02) 0.05 (-0.02) -0.71-0.61 (0.10) -0.81(-0.10)
Industry Portfolios
Average
0.35 0.36 (0.02) 0.33 (-0.01) 0.07 0.07 (-0.00) 0.06 (-0.01) -0.71-0.71 (0.00) -0.71 (0.00)
across
Industries
Notes: Column (1) shows the mean return for the corresponding benchmark portfolio (B) – that is, the global portfolio
for rows 1 through 3, county portfolios for the following twenty-two rows, industry portfolios for the last row. Column (2)
shows the mean return of the corresponding low exposure portfolio (L). The global low exposure portfolios are considered
in rows 1-3 and are constructed as follows (see also text): The low global exposure (LGE) portfolio contains half the stocks
in the sample, those with global shock exposures below the median. The global low country exposure (LCE) portfolio
also contains half the stocks in the sample, those with country shock exposures below the median for each country. The
global low industry exposure (LIE) portfolio contains stocks with industry shock exposures below the median for each
industry. The local low exposure portfolios are considered in the remaining rows and are constructed as follows (see also
text): For countries, the local low country exposure (LCE) portfolio contains half the stocks in each country, those with
country shock exposures below the median. For industries, the local low industry exposure (LIE) portfolio contains half
the stocks in each industry, those with industry shock exposures below the median. Column (3) shows the difference
between columns (2) and (1) (L-B). Column (4) shows the mean return of the corresponding high exposure portfolio
(H). High exposure portfolios are constructed as the mirror image of low exposure portfolios. Column (5) shows the
difference between columns (4) and (1) (H-B). Columns (6) through (10) are constructed analogously for the in-sample
Sharpe Ratios. Columns (11) through (15) are constructed analogously for the out-of-sample mean returns. Portfolios
are constructed based on estimates from January 1985 through February 2000 and the mean returns are computed for
the period March 2000 to February 2002. Mean returns figures are in percent. The table is based on estimates for the
1985 to 2002 sample.
30
Table 6: How Much Do Global, Country, and Industry Factors Matter? Variance Decomposition
Actual Variance Global Country Industry-6
average across country portfolios
42.5
11.86
81.16
0.81
average across industry portfolios
33.2
15.03
31.88
10.73
average across stocks
124.7
5.82
32.09
7.31
global market portfolio
19.4
21.44
28.34
0.29
Notes: Sample: 1985-2002. All portfolios, as well as the averages, are constructed using equal weights. All figures are
in percent. Actual variances are: averages of the variances of individual stocks, variance of the global market portfolio,
and averages of the variances of country and/or industry portfolios. The variance decomposition figures are obtained as
follows. Eq. (3) implies that the return for any portfolio with a N × 1 vector of weights ω is given by
P
PI
0 C c
0 I i
0
ω 0 Rt = ω 0 µ + ω 0 BG ftg + C
c=1 ω Bc ft +
i=1 ω Bi ft + ω ²t ,
Using the factor orthogonality assumption, the variance of the portfolio is:
P
PI
0 C 2
0 I 2
0
var(ω 0 Rt ) = (ω 0 BG )2 + C
i=1 (ω Bi ) + var(ω ²t ),
c=1 (ω Bc ) +
where BG is the column of the B matrix corresponding to the global factor, BC
c is the column corresponding to the
country factor c, and so on. The variance decomposition obtains by dividing the summands in the right hand side of the
last equation by the actual (sample) variance of the portfolio. We discuss in the text the fact that the factors may not be
ex post orthogonal.
31
Table 7: Factor Correlation (median)
Global Country Industry-6
Global
Country
Industry-6
1.000
0.000
-0.000
0.343
0.027
0.006
Notes: Sample: 1985-2002. The table shows the ex-post factor correlation averaged across periods (see formula (7))
for each class of factors (global, country, and industry). The diagonal elements show the median covariance across
factors within the same class, and the off-diagonal terms show the median correlation across classes. The correlations are
computed fixing the mean of the factor at zero.
32
Figure 1: The Diversification Gains Associated With Low Global and Country Exposure Portfolios
100
90
80
70
60
50
40
(6)
34 % limit − country benchmark
30
(3)
27 % limit − industry benchmark
(4)
25 % limit − within countries LCE
(3)
16 % limit − global benchmark
(2)
13 % limit − global LGE
(1)
11 % limit − global LCE
20
10
0
10
20
30
40
Notes: The figure examines the risk reduction properties of the low exposure portfolios using the graphical representation
in Heston and Rouwenhorst (1994) and Solnik (1975). It gives the average portfolio variance as the number of stocks
in a given portfolio increases from 1 to 40, expressed as a percentage of the average variance of all individual stocks in
our sample. The solid lines in Figure 1 show the diversification benefit associated with the global market portfolio (line
3), the average country portfolio (line 5), and the average industry portfolio (line 6). The dashed lines – lines (1), (2),
and (4) – show the diversification benefit associated with the low exposure portfolios. The low global exposure (LGE)
portfolio contains half the stocks in the sample, those with global shock exposures below the median (line 2). The global
low country exposure (LCE) portfolio also contains half the stocks in the sample, those with country shock exposures
below the median for each country (line 1). The country low country exposure (LCE) portfolio contains half the stocks
in each country, those with country shock exposures below the median. Line (4) is computed using the equal-weighted
average of the variance of country LCE portfolios across countries.
33
Appendix not intended for publication
A
Computational Appendix
Let us rewrite model (1) as:
R̃t = Bft + ²t ,
(A1)
where R̃t = Rt − µ are the de-meaned returns. The variance-covariance matrix of ²t , called Σ, is diagonal, following
assumption (4). The log-likelihood function of B and Σ is given by:
L(B, Σ|R̃) = −
where Ω = BB 0 + Σ and S =
PT
t=1
R̃t R̃t0
.
T
T
T
ln |Ω| − tr{Ω−1 S}
2
2
(A2)
It is hard to maximize the likelihood for (A1) by brute force when N is
very large, given that the number of parameters is 4 × N . The application of the EM algorithm simplifies the problem
considerably from a numerical point of view (see Lehmann and Modest 1985). The EM follows the intuition that if the
factors were observable B could be estimated by means of OLS - equation by equation (given that Σ is diagonal). The
EM algorithm is an iterative procedure where at each step the factors ft are replaced by their conditional expectations
given the observations and the value of the parameters obtained at the end of the previous step, and then applies standard
regression tools to obtain a new estimate of the parameters until convergence. Each iteration of the algorithm is bound
to increase the likelihood, so that convergence to a -possibly local- maximum is guaranteed.
Given the distributional assumptions, and assuming a flat prior on all the parameters of interest (except for the
factors), the logarithm of the joint posterior distribution of B, Σ and f , given the observations R̃, is equal to the likelihood
of the model if the factors were observable
ln pdf (B, Σ, f |R̃) ∝ −
T
T
T
1X
1X 0
ln |Σ| −
(R̃t − Bft )0 Σ−1 (R̃t − Bft ) −
f ft
2
2 t=1
2 t=1 t
(A3)
Let us call (B q , Σq ) the values of the parameters obtained at the end of the previous (q th ) iteration of the algorithm. The
first step of the EM algorithm involves taking the expectation (E) of (A3) with respect to the conditional distribution of
f given (B q , Σq ) and R̃. If we denote by IE q [.] the expectation taken with respect to the conditional distribution of f
given (B q , Σq ) and y, we can write the E-step as:
IE q [ln pdf (B, Σ, f |R̃)] = −
T
T
X
X
IE q [ft ]R̃t0
IE q [ft ft0 ] 0
T
{ln |Σ| + tr{Σ−1 [S − 2B(
) + B(
)B ]}.
2
T
T
t=1
t=1
(A4)
One obtains that (see Lehman and Modest 1985):
T
X
IE q [ft ]R̃t0
T
t=1
=
(IK + B 0 Σ−1 B)−1 B 0 Σ−1 S
(A5)
T
X
IE q [ft ft0 ]
T
t=1
=
(IK + B 0 Σ−1 B)−1
(A6)
+(IK + B 0 Σ−1 B)−1 B 0 Σ−1 SΣ−1 B(IK + B 0 Σ−1 B)−1 ,
where we drop the superscript q for convenience. The second step consists in maximizing (M) the resulting expression
with respect to (B, Σ). Since the joint maximization is complicated, we split the (M) step into a number of conditional
maximization (CM) (see Gelman et al. 1994). Each substep consists in maximizing (A4) with respect to a first set of
parameters, keeping all other parameters constant at the level attained at the end of the previous substep. When N
34
Appendix not intended for publication
is large and restrictions (2) are present, it is convenient to divide the maximization or (A4) with respect to B into N
maximizations - equation by equation. In order to see how this can be accomplished, let us note that A4 can be rewritten
as (considering the case N = 2 to simplify the notation):
P
IE q [ft ft0 ] 0
IE q [ln pdf (B, Σ, f |R̃)] = tr{Σ−1 [S − 2BN1 + B T
B ]}
t=1
T
22
3
2 3
2 3
3
1
0
0
B 1 7 PT IE q [ft ]R̃1,t
S 11 S 12 7
B
0
P
P
I
E
[f
]
R̃
I
E
[f
f
]
q
t
6
6
6
6
7
7
q t t
T
2,t
= tr{Σ−1 44
+4 5 T
[B 10 B 20 ]5}
5− 24 5
t=1
t=1
t=1
T
T
T
120
22
2
2
S S
B
B
2
3
0
0
P
P
I
E
[f
]
R̃
I
E
[f
f
]
q
t
q t t
T
1,t
11
1
+ B1 T
B 10 /σ12
...
t=1
t=1
6 S − 2B
7
T
T
7
= tr 6
0
4
5
0
P
P
I
E
[f
]
R̃
I
E
[f
f
]
q
t
q t t
T
T
2,t
2
20
2
22
2
+B
B
/σ2
...
S − 2B
t=1
t=1
T
T
where B n and R̃n,t are the 1×K vector of loadings and the demeaned return, respectively, for stock n . Since the likelihood
depends only on the trace, we can ignore the off-diagonal terms. Call β n the K n × 1 vector of non-zero parameters in B n ,
and M n the K n × K matrix that maps B n into β n (i.e,, the matrix that “picks” the non-zero loadings): M n B n0 = β n .
Then:
n
βq+1
=[
T
T
0
X
M n IE q [ft ft0 ]M n0 −1 X M n IE q [ft ]R̃n,t
]
T
T
t=1
t=1
(A7)
Note that M n simply selects the factors that are relevant to stock n. Call Bq+1 the outcome of the N maximizations.
The last CM step consists in maximizing (A4) with respect to Σ, for given Bq+1 , obtaining:
diag(Σ) = diag(S − 2Bq+1 N1 + Bq+1
T
X
IE q [ft ft0 ] 0
Bq+1 ).
T
t=1
(A8)
Convergence is reached whenever the mean squared gradient is less than 10−4 . Note that the (M) step is essentially
OLS equation by equation, as in Marsh and Pfleiderer (1997). But the (E) step - where one obtains the estimates of ft
given the betas - is different from the cross-sectional GLS estimator used in Marsh and Pfleiderer (1997). The former
is IE q [ft ] = (IK + B 0 Σ−1 B)−1 B 0 Σ−1 R̃t while the latter is fˆt GLS = (B 0 Σ−1 B)−1 B 0 Σ−1 R̃t . Hence the two algorithms
will not in general converge to the same estimator. The difference between the (E)-step estimator and the GLS estimator
arises because the (E)-step estimator treats ft as a random variable with prior variance Ik while the GLS estimator treats
the factor(s) as unknown- but fixed- coefficients.
Next we discuss the computation of the variance covariance matrix of the estimates. The information matrix, written
for ease of notation as a function of Ω = BB 0 + Σ, is equal to:
IE[d2 L] = −
T
vec(dΩ)0 (Ω−1 ⊗ Ω−1 )vec(dΩ).
2
(A9)
Given that dΩ = dBB 0 + BdB 0 + dΣ, and that we know the mapping between β (the non-zero parameters in B) and
B, we can use formula (A9) to compute the (ith , j th ) element of the information matrix with respect to the unrestricted
elements of B and Σ. The inverse of the information matrix is the asymptotic variance-covariance matrix of our estimates.
B
Data Appendix
This appendix explains construction of the data for which we estimate the model and review its country and industry
composition. The purpose of this exercise is to show that our data provide a reasonable representation of international
Appendix not intended for publication
35
equity markets, so that our model delivers meaningful estimates of country- and industry-specific shocks.
We derive our data from Brooks and Del Negro (forthcoming, BD hereafter) who cover monthly total US dollardenominated stock returns and market capitalizations for 9,769 stocks in 42 developed and emerging markets from January
1985 to February 2002, downloaded from Datastream International. Their sample includes all constituent stocks in the
Datastream Global Market Indices for these countries as of March 2002 and is augmented with a list of active and inactive
stocks for each country derived from Worldscope. This step ensures that the data include stocks that become inactive over
time, due to bankruptcy or merger for example. The data include dummy variables that identify the country affiliation of
each stock. It also includes 110 industry dummies, which measure industry affiliation according to the most disaggregated
industry definition provided by Datastream International, level 6. We balance the data at two different points in time,
in order to check the robustness of our results to changes in data coverage. The first sample consists of all stocks with
continuous returns for the entire sample period (1,965 stocks in 21 countries). The second begins in January 1990 and
contains stocks with continuous returns through the remainder of the sample (3,939 stocks in 33 countries).
Table B-2 gives a snapshot of the data as of December 2000. For each of the balanced panels it provides by country
the number of stocks, the market capitalization in billions of U.S. dollars, and the capitalization share in percent of
the sample total. It gives the same breakdown by industry, using the level 3 industry classification, which groups level
6 industries into 10 broad sectors. Table A1 provides the same breakdown for the original data by BD, to put into
perspective how balancing affects coverage. Finally, it provides the same decomposition for the known universe of stocks
in each country, according to the Global Stock Markets Factbook (2001), to give an assessment of the degree to which
the various datasets provide a realistic representation of international stock markets.
In December 2000, the smaller of our balanced panels has a total market capitalization of 15,334 billion U.S. dollars,
while this number amounts to 21,834 billion U.S. dollars for the balanced panel that begins in January 1990. This compares
to a total market capitalization of 31,486 billion U.S. dollars in the original BD data, almost 99 percent of total market
capitalization in our 42 countries, according to the Factbook. While the original data thus provide fairly exhaustive
coverage in capitalization terms, the balanced data are clearly more limited in this regard. However, although balancing
shrinks coverage within countries and industries, the balanced data remain fairly representative in their coverage across
countries and industries. For example, in the smaller of our balanced samples, the U.S. makes up 52 percent of total
market capitalization, marginally above the 50 percent in the original BD data and the same order of magnitude as the 47
percent in the Factbook. The same goes for other advanced countries, whose relative importance in the balanced data is in
line with the Factbook. The major omission from the balanced data are emerging markets. With the exception of Korea,
Malaysia and South Africa, for example, the smaller of our balanced samples contains no emerging markets. Coverage of
emerging markets is somewhat better in the balanced sample that begins in January 1990. In terms of coverage across
industries, the balanced data are comparable to the original BD dataset. The largest level 3 sector in their sample is
made of of financial sector stocks, which account for almost 24 percent of their sample in market capitalization terms.
This is also true for the balanced data, where financials make up 23 and 24 percent of market capitalization, respectively.
While balancing the original BD data thus clearly reduces coverage, the balanced data still provide a reasonable
representation of international stock markets, in terms of the relative importance of countries and industries.
36
Appendix not intended for publication
Table B-1: Descriptive Statistics for Equal-Weighted Excess Return Across Stocks
Panel A. Balanced Sample 1985 - 2002
Full Sample
Balanced Sample
0.30
0.34
Standard Deviation 4.32
4.38
Time-Series Mean
Panel B. Balanced Sample 1990 - 2002
Time-Series Mean
Full Sample
Balanced Sample
-0.18
-0.13
Standard Deviation 4.18
4.02
Notes: The table shows that the balanced samples used for estimation are comparable to the original data in Brooks and
Del Negro (forthcoming), in terms of the time-series means and standard deviations of the equal-weighted excess return
across all stocks. In particular, there is little indication of a systematic bias in the standard deviation of excess returns,
important because we focus on the covariance structure of excess returns. The original data from Brooks and Del Negro
(forthcoming) cover 9,679 stocks in 42 countries from January 1985 to February 2002. The Balanced Sample 1985 - 2002
consists of stocks with continuous returns for the entire sample and contains 1,965 stocks in 21 countries. The Balanced
Sample 1990 - 2002 begins in January 1990 and contains stocks with continuous returns for the rest of the sample. It
covers 3,939 stocks in 33 countries.
37
Table B-2:
Comparing Our Balanced Data to Brooks and Del Negro (forthcoming) and Reality
Balanced Sample
1985 - 2002
Balanced Sample
1990 - 2002
Full Sample
S&P Stock Market
Factbook 2001
No. of Mkt Cap Share No. of Mkt Cap Share No. of Mkt Cap Share No. of Mkt Cap Share
Firms (Bill. $) (%)
Firms (Bill. $) (%)
Firms (Bill. $) (%)
Firms (Bill. $) (%)
World
United States
United Kingdom
France
Germany
Italy
Japan
Canada
Australia
Austria
Belgium
Denmark
Hong Kong
Ireland
Netherlands
Norway
Sweden
Switzerland
Korea
Malaysia
Singapore
South Africa
Philippines
Luxembourg
New Zealand
Spain
Finland
Thailand
Taiwan
Argentina
Greece
Mexico
Portugal
Turkey
Chile
Brazil
India
Indonesia
China
Peru
Poland
Colombia
Czech Republic
1965
451
280
54
100
31
529
89
41
9
26
20
51
14
56
16
16
42
45
18
54
23
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Basic Industries
287
General Industries
320
Cyclical Consumer Goods119
Non-Cyclical Consumer 226
Goods
Cyclical Services
306
Non-Cyclical Services
42
Utilities
92
Information Technology 73
Financials
406
Resource Industries
94
15334
7977
1440
505
696
258
2002
418
159
4
59
51
378
35
412
21
172
579
52
19
61
36
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
100
52.02
9.39
3.30
4.54
1.68
13.06
2.73
1.04
0.03
0.39
0.33
2.46
0.23
2.68
0.14
1.12
3.78
0.34
0.12
0.40
0.24
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
3939
703
530
167
181
104
783
198
81
28
59
52
96
45
87
27
50
105
61
98
77
46
16
0
27
73
27
35
44
9
31
19
21
9
50
0
0
0
0
0
0
0
0
21834
11220
2121
963
773
406
2516
564
196
12
73
57
442
59
437
27
198
710
82
53
84
70
13
0
5
323
196
13
87
5
41
24
21
10
33
0
0
0
0
0
0
0
0
100
51.39
9.71
4.41
3.54
1.86
11.52
2.58
0.90
0.06
0.33
0.26
2.02
0.27
2.00
0.12
0.91
3.25
0.38
0.24
0.38
0.32
0.06
0.00
0.02
1.48
0.90
0.06
0.40
0.02
0.19
0.11
0.10
0.04
0.15
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
8969
1309
986
363
404
219
1189
424
207
82
127
109
208
79
183
103
145
185
168
168
164
189
71
31
69
135
106
78
165
73
109
117
59
72
92
189
153
77
152
60
60
31
59
31486
15577
3114
1438
1043
750
3052
792
323
33
131
116
669
81
688
69
305
830
136
97
164
153
24
41
23
387
250
29
270
51
86
106
76
223
54
102
110
6
48
4
18
3
13
100
49.47
9.89
4.57
3.31
2.38
9.69
2.52
1.02
0.11
0.41
0.37
2.13
0.26
2.18
0.22
0.97
2.64
0.43
0.31
0.52
0.49
0.08
0.13
0.07
1.23
0.79
0.09
0.86
0.16
0.27
0.34
0.24
0.71
0.17
0.32
0.35
0.02
0.15
0.01
0.06
0.01
0.04
714
1876
557
3124
4.65
12.23
3.63
20.37
558
581
215
422
998
2208
689
3730
4.57
10.11
3.16
17.08
1059
1140
457
990
1254
2838
853
4558
3.98
9.01
2.71
14.48
1610
814
525
1522
3443
1150
10.50
5.31
3.43
9.92
22.45
7.50
594
102
160
178
935
194
2148
1611
730
2956
5236
1527
9.84
7.38
3.34
13.54
23.98
6.99
1455
376
345
815
1925
407
3724
2891
1107
4933
7414
1915
11.83
9.18
3.52
15.67
23.55
6.08
37188
7524
1904
808
1022
291
2561
3977
1330
97
174
225
779
76
234
191
292
252
1308
795
418
616
230
54
144
1019
154
381
531
127
329
179
109
315
258
459
5937
290
1086
230
225
126
131
31852
15104
2577
1447
1270
768
3157
841
373
30
182
108
623
82
640
65
328
792
172
117
153
205
52
34
19
504
294
29
248
166
111
125
61
70
60
226
148
27
581
11
31
10
11
100
47.42
8.09
4.54
3.99
2.41
9.91
2.64
1.17
0.09
0.57
0.34
1.96
0.26
2.01
0.20
1.03
2.49
0.54
0.37
0.48
0.64
0.16
0.11
0.06
1.58
0.92
0.09
0.78
0.52
0.35
0.39
0.19
0.22
0.19
0.71
0.46
0.08
1.82
0.03
0.10
0.03
0.03
Notes: The table provides by country and by industry (Level 6) the number of stocks, the market capitalization in billions
of U.S. dollars, and the capitalization share in percent of the sample total as of December 2000 for each balanced sample
(1985 - 2002 and 1990 - 2002), for the full sample of Brooks and Del Negro (forthcoming), and the known universe of
stocks from the S&P Stock Market Factbook (2001).
@FedFRASER