pure geographic stock returns indexes - insead€¦ · pure geographic stock returns indexes...
TRANSCRIPT
Pure Geographic Stock Returns Indexes∗
Bernard Dumas, INSEAD, NBER and CEPRTymur Gabuniya, INSEAD
Richard Marston, Wharton School of the University of Pennsylvania and NBER
November, 2014
Abstract
∗Dumas’s work and Gabuniya’s work received the support of a grant from the INSEADResearch Fund. We are grateful to Humberto Gomez, a Master of Science student at theUniversity of Lausanne, who was instrumental in setting up the MatLab programs for anearlier draft of this paper and helped us greatly in this research. We are also very grateful toPhilippe Piette of WVB, who generously provided the data.
1
“What is changing is that corporations are becoming more andmore global in their business activities through increased exportsand cross-border M&A.”Diermeier and Solnik (2001)
What does it mean for a firm to belong to a country? In most statisticalstudies of international stock returns, a firm is classified as belonging to a coun-try (i.e., included in the country index) if it is listed in the stock market ofthat country. This classification scheme ignores the operations of the firm. Thedistinction between place of stock trading and place of business is becomingmore and more relevant as a growing number of firms have activities abroad.For instance, the index of the Amsterdam stock exchange, where many “Dutch”multinationals are traded, is not representative of the risks and returns attachedto investing in operations physically taking place in The Netherlands.We propose, instead, to allocate firms to “geographic zones”according to the
place where they conduct business and we re-measure zone returns in keepingwith that classification scheme. Such a re-construction should be relevant forat least four purposes. First, financial analysts making an investment decisionoften do so because they want to take a view concerning the economy and growthprospects of a zone. An analyst, or an investor he or she advises, may see highergrowth prospects in that zone than other investors do and may accordingly wantto pursue a strategy of investing in companies that do business in the zone.Investing in the corresponding country’s stock market index is not a clean wayto implement that strategy if and when the companies traded on the country’sexchange conduct a good deal of business abroad. The stocks making up thatindex are not a clean representation of the risks and return of doing businessin the zone. Second, an investor may want to invest in a target zone, but fearthe form of trading taking place in the corresponding country’s stock market(insider trading, preferential trades and other corrupt practices). In that case,“investing by proxy” may be a good alternative. The investor can invest incompanies of another country that do a lot of business in the target country,hedging away the business that these companies conduct at home. Foreigncompanies as opposed to the country’s companies can serve to invest in a targetzone. Third, corporations contemplating a capital investment in a production ordistribution facility in a zone not their own need to be told what risk premium isapplicable to the risk of operating facilities in that zone, not what risk premiumapplies to the risk of being listed in the corresponding country.1
Fourthly and most importantly for research purposes, our undertaking shouldenhance the meaningfulness of cross-country correlation studies. It is true inmany models of international financial-market equilibrium that, everything elseequal, the cross-country correlation is higher if the financial markets are inte-grated (i.e., movements of capital take place between countries the same waythey do within countries) than if they are segmented (see Dumas, Harvey andRuiz (1993)). For that reason, correlations have been used to measure diver-
1We are not denying that there may also exist risk premia for being listed in one country.See Froot and Dabora (). At present, we do not have a model to explain such a phenomenon,if it exists.
2
sification potential (Solnik (1974), Heston and Rouwenshorst (1994), Cavagliaet al. (2004)) and to measure the degree of integration between markets. Thecatch, however, is the phrase “everything else equal”. As time goes by, the com-position of country indexes evolves not just as to the list of firms included in theindex but also as to the locus of the business conducted by the firms that areincluded. For that reason, we may be deluding ourselves when we observe thatcross-country correlations increase and conclude that the international financialmarket is growingly integrated. Diermeier and Solnik (2001) put it plainly inthe quote we used as epigraph. The phenomenon they point at may be a form ofintegration of the market for goods and services but it should not be confusedwith integration of the financial markets. What we take to be a growing commonmovement between two given country assets may in fact be simply a change inthe nature of these assets. In terms of financial theory, the distinction we aremaking is a key one: it is the distinction between cash flows to be discountedand the state prices used to discount them. Integration of financial markets is aphenomenon by which the state prices become more similar to each other whencompared across investors of different countries. If cash flows become more andmore similar, this does not qualify as integration. In fact, cash-flow similaritiesmake it harder to determine whether or not state prices are becoming moresimilar.While empirical studies are not all in agreement (see Heston and Rouwen-
horst (1994) vs. Cavaglia et al. (2004)), one is tempted to think that the correla-tion of stock returns has been growing over time. The graphs in Table 4, PanelsC and D below, bear witness to that fact. Cavaglia et al. (2004) shows thatthe increased correlation between countries is, indeed, due to country factorsbecoming more correlated as opposed to industry factors becoming more so.And Goetzmann et al. (2005) shows that the correlation has been rising mostlybetween “core” (basically, developed) countries as opposed to other countries.Both observations militate in favor of an interpretation in terms of increasedintegration of financial markets. But Heston and Rouwenhorst (1994) indicatethat, even between European financial markets, which are presumably by nowfairly well integrated, country factors as opposed to industry factors offer thehiger diversification potential. Any rise or fall in correlations can be variouslyinterpreted either as evidence concerning market integration or as an industryeffect or an industry bubble (such as the IT bubble of the end of the 90’s) orsimply as the result of the gradual redefinition of stock market indexes. It is thegoal of this paper to throw light on and deconstruct these effects.The balance of the paper is organized as follows. Section 1 contains the
“literature review.” Section 2 describes the statistical model to identify purecountry indexes while Section 3 displays some features of the dataset, especiallythe dataset on geographic segment allocations of firms. The statistical techniquethat will serve to estimate the parameters of the model, namely the EM algo-rithm, is reviewed in Section 4. Results are given in Section 5 at yearly frequencyand in Section 6 at daily frequency. Section 7 states the conclusions.
3
1 The study of Diermeier and Solnik (2001)
The literature review on our subject contains primarily one item: the article byJeff Diermeier and Bruno Solnik, published in the Financial Analysts Journal.In it, the authors study monthly stock returns of 1,213 individual companieslisted in eight large country stock markets (France, Germany, Italy, Japan,Netherlands, Switzerland, United Kingdom and United States) from July 1989to January 1999. The statistical model they estimate is the following:
Ri = αi + βiIdom +∑reg
γi,regIreg +∑reg
δi,regCreg + εi (1)
where Ri is the return on firm i, Idom is the return on the firm’s domesticindex, Ireg are the returns on three regional indexes computed in their localcurrencies and Creg are the returns on the three regional currencies measuredin the domestic currency. The term “domestic” refers to the country to whichfirm i belongs, as explained in the next paragraph. The coeffi cients βi, γi,regand δi,reg are called the “exposures”to he various factors.Diermeier and Solnik have available to them data on stock returns, of course,
but also data on the shares of activity (i.e., revenues) of firms in the domesticcountry and in the three regions of the globe that they have chosen to consider.The statistical analysis can be described as being in three stages. First, theyconstruct a “pure”market index for each country as a value-weighted averagereturn of firms with mostly domestic activities.2 From these they also calculateregional returns as value-weighted average of country returns for countries thatbelong to a “region”. There are three regions: Asia, Europe and the UnitedStates.Second, they apply the factor model (1) (i.e., run times-series OLS regres-
sions) to estimate exposures. The typical result of that stage is represented inTable 1, which reproduces their Table 5. While the regression (1) is run for everySwiss firm of their dataset, Figure represents the average result for all Swissfirms. The average Swiss firm has an exposure of .45 to the pure Swiss factor,exposures equal to .07, .38 and .15 to the three regional factors and exposuresof 0.10, .10 and −.07 to the three regional currencies. The sum of the exposuresis greater than 1, which indicates a slightly leveraged plan on the set of factors.As a more specific example of the second-stage result, Diermeier and Solnik
cite the example of SmithKline Beecham, a “British”multinational, which hasstock market exposures equal to .17, .08, .31 and .55, respectively, to the UKpure factor, to the Asia factor, to the Europe ex UK factor and to the NorthAmerican factor.In the third stage of their study, they ask the key question that motivates
the whole undertaking: do these exposures resemble the share of revenues thatSmithKline Beecham receives from the various geographic segments? The firm
2The thresholds of domestic operations for inclusion in the domestic index was differentfor different ocuntries: 75% for France, 85% for Germany, 85% for Italy, 99% for Japan, 65%for the Netherlands, 60% for Switzerland, 99% for the UK and 99% for the United States.
4
Factor Average ExposureDomestic (β) 0.45entry entryPanel A: Foreign market ( γ)Asia 0.07Europe 0.38United States 0.15Total 0.60Panel B: Foreign currency ( δ)Asia 0.10Europe 0.10United States -0.07Total 0.13
Table 1: Exposures for a value-weighted portfolio of Swiss stocks, July1989-January 1999 data (Table 5 in Diermeier and Solnik (2001)).
makes 8% of its sales to the UK, 12% to Asia, 23.5% to Europe ex UK and46.1% to North America. It seems that the stock market is broadly aware of thegeographic distribution of the activities of the firm. Admittedly, it should notbe the share of sales that would serve to measure the geographic distribution.That is at best a proxy. One should use the share of value that arises from thecash flows generated in the geographic segments. That data, unfortunately, isnot available.To address the same question that has been asked à propos SmithKline
Beecham in a more systematic way, Diermeier and Solnik, in the third stage oftheir work, relate in the cross-section the stock market exposures of firms to thegeographic distribution of their sales. An example of the result of that stage isFigure 1, which reproduces their Figure 1. The same conclusion emerges fromthis figure as it does from tables and statistical tests of equality conducted intheir paper: the stock market broadly reflects the activities of firms.
2 The statistical model to be used
The study by Diermeier and Solnik hits the nail on the head very well butit has two drawbacks. First, it uses a limited number of firms in a limitednumber of countries. Developing countries in particular are not covered. Secondand more importantly, it is implemented in stages. The stage that serves todefine pure country indexes only uses the firms that have a large share of theiractivity at home and a later stage relates the stock market exposures (mostlyof the other firms) to their share of foreign activity. From the point of viewof statistical theory, it would be vastly more effi cient to use all firms to doeverything in a single stage. That is, knowing the geographic distribution ofactivities of each firm, one should use all the information on all firms to identifythe pure country factors. A Malaysian firm, to the degree that it conducts
5
Figure 1: Regression between International Market Exposure and theDegree of International Activity: Netherlands, July 1989-January 1999data (Figure 1 in Diermeier and Solnik (2001)).
6
operations in Switzerland, should also help in identifying the pure Swiss countryfactor.If information were available giving for each firm its share of activity to
each country, the goal we just stated could simply be reached by inverting ahuge matrix (hoping it is not singular). That matrix inversion would directlyconstruct the country returns from the company returns, although even then onewould have to cope with the fact that there are more companies than countries.In practice, as we shall see, the information about geographic segments is not asrich. We have at our disposal only fragmentary information about the activitiesof each firm. Because the country returns cannot be measured directly, theyhave to be considered as latent (or unobserved) factors and the estimation hasto be viewed as an exercise in factor analysis.A note on vocabulary is warranted before we go on. We call “segments”such
as “countries” or “regions” the geographic entities that are variously referredto by firms in their annual reports, as transcribed in the database. There are“Rest of” segments, which are regions in which firms sell without explicitlygiving the name of a country: “Rest of Europe”, “Rest of the world”are typicalexample. These are defined as containing countries that the firm has not referredto explicitly among its segments. Typically, there is also a country referred tovariously as “Home”or “Domestic sales”.By way of contrast we call “zones”a set of standard geographic entities that
are uniform across all the firms of our sample to reflect the operating risks theytake, in the form of a factor model . We have zones for all the major countries(see Table 3 below) and we have five “Rest of”zones that contain the countriesfor which we have not defined a specific zone: Rest of Europe, Rest of America,Rest of Middle East/Asia, Rest of Africa, Rest of Asia/Oceania. For instance,a firm may have an ISIN starting with US (because the USA is its country oflisting) and indicate that its only geographic segment is Zimbabwe. We classifyits sales as being made in the “Rest of Africa”zone. As another example, thedefinition of “Home”is simply the zone that the firm’s annual report refers towhen it refers to “Domestic sales.”We compile a “dictionary” that serves todecipher the meaning of the segments posted in the various annual reports interms of standardized zones.The following model, adapted from Brooks and Del Negro (2004), describes
the statistical structure from which pure geographic-zone index returns C willbe calculated:
Rt = B × Ct + et (2)
where Rt is the realization at time t of the vector of time-series demeanedreturns (all measured in a common currency) for N stock securities, B is theN ×K matrix that contains the exposures of all firms to all K geographic-zonefactors, Ct is the realization at time t of the K × 1 vector of zero-mean returnsof zone factors, K being the number of zones and et is the realization at time tof the vector of unsystematic residuals of the stock returns.Without further specification, the model would not be viable because C is
not observed so that B is not identified. We impose enough constraints on B to
7
be able to calculate C as a latent factor. We use the fragmentary informationwe have on firms’activities to set some of the elements of the B equal to thecorresponding shares of activities. Specifically, the constraints are as follows:
•∑Kk=1 bi,k = 1.
Assumption: We impose that the sum of the exposures of a firm equal 1.That assumption is questionable as firms could be leveraged, both finan-cially and operationally so that their total risk would be more than whatis captured simply by sales. Summation to 1 is an auxiliary assumptionwe make.
•∑kεj bi,k = Ai,j .
Assumption: The sum of the exposures of a specific firm in a specificgeographic zone is equal to the percentage A of activities in that zone.Here again, we exploit whatever information about country and regionalsales is given in annual reports. In particular, bi,k=i′s country = Ai,k=i′s countryi:the firm’s stock market exposure to one particular stock market called“home” is assumed equal to the share of sales made at home. The homecountry does not play a unique role, as the more general form of the re-striction shows.3
It is debatable whether exposures to zone stock markets and share of ac-tivities should be set strictly equal to each other or, more broadly, related(perhaps linearly related) to each other. Equality —as opposed to just anincreasing relationship —is an auxiliary assumption we make.
• bi,k > 0. No negativity for the exposures of each firm to the pure indexcountry stock returns.
As a result of these assumptions and restrictions, some of the elements ofthe matrix B are “observed.”The others have to be estimated. At this point,we further assume that the exposures bi,k, whether observed or estimated areconstant over time within a year.We now need data from annual reports in order to implement these restric-
tions.
3 The dataset on firms’geographic segments
We elect to study all firms that are part of the Datastream world stock marketindex and have sales data available. As of 19/05/2006, there were 6,385 suchfirms. Our study covers the years 1999-2005 included, during which the actualnumbers of firms available after some filtering are as indicated in Table 3 below.Such a choice presents some drawbacks as China, for instance, has very fewfirms in that index. The choices made by Datastream are based on accessiblityof the stock for foreign investors.
3For “Rest of” zones, the equality constraint is replaced by an inequality because theinformation in the annual reports often refers to one of the several countries of the zone.
8
1999 2000 2001 2002 2003 2004 2005
1300
1400
1500
1600
1700
1800
1900
2000
2100
Year
Num
ber o
f mul
tinat
iona
l com
pani
es
.
Figure 2: Number of multinational companies: We define multinationalcompanies as those that sell to more than one geographic zone, as implied bythe known exposures (fractions of sales) in our sample.
The World Vest Base (WVB) database transcribes annual report informationfor a very large number of firms worldwide. The owner of the database providedus with data on sales’ geographic distribution of firms that are part of theDatastream world index for the period 1999-2005. Figure 2 shows the growingnumber of firms in our sample with international activities.The selection and filtering of the geographic-segment data is explained in
detail in Appendix A. It left a number of stocks growing from 2019 in 1999 to3455 in 2005. Their distributions across country are displayed for year 2005 inTable 2. The United States and the United Kingdom are the countries withthe largest numbers of firms. The firms from the developed countries have salesthat are more diversified than those from developing countries. When a countryhas few firms, that does not imply that the corresponding zone’s pure index iscomputed on the basis of these firms only. The algorithm takes into considerationthe returns of the firms from other countries that sell in the designated zone.For instance, Venezuela has few firms in our reduced sample, but all the firmsof the sample that sell in that country also contribute to the calculation of thepure country index of the Rest of America zone. Moreover, all the firms that sellin that zone also contribute somewhat to the calculation of the Rest-of-Americaindex. Such is the virtue of the algorithm.For each firm, the fraction of home sales is calculated as the ratio of home
sales to total sales of the same year. The fraction of region sales for eachfirm is calculated as the ratio of the specific region sales over the total sales.The regional restrictions were the most complex step to be implemented in theempirical application, due to the non standardized description of regions in thesales database.
9
Region Country 1999 2000 2001 2002 2003 2004 2005
Europe Belgium 19 NA 42 46 NA NA 54
Switzerland 47 56 64 65 NA 68 73
Germany 113 140 161 168 167 165 175
Spain 52 68 68 67 75 78 80
Finland NA NA NA 39 NA NA NA
France 120 135 163 173 183 192 181
UK 258 267 284 277 311 322 334
Italy 39 55 61 77 73 73 69
Netherlands 60 64 72 79 86 82 78
Norway NA NA NA NA NA NA 28
Sweden 33 40 45 43 53 59 57
Rest of Europe 147 189 207 195 352 306 234
America Brazil 44 50 51 52 53 54 59
Canada 77 85 97 103 106 110 116
Mexico 34 37 28 33 NA 39 39
USA 400 621 655 664 676 672 728
Rest of America 80 89 79 83 127 110 128
Middle East/Asia India NA NA NA 42 NA 61 53
Rest Middle East/Asia 51 58 73 48 94 47 49
Africa South Africa 28 NA 38 NA NA NA 43
Rest of Africa 1 31 2 41 41 42 1
Asia/Oceania Australia 65 70 79 91 93 122 130
Hong Kong 35 32 28 35 38 49 49
Japan 145 207 302 341 380 334 333
Malaysia 61 63 69 74 75 79 79
New Zealand 22 26 28 34 NA 40 41
Singapore 42 50 57 65 72 75 81
Thailand NA NA NA NA NA NA 40
Taiwan NA NA NA NA NA NA 46
Rest of Asia/Oceania 45 63 83 87 144 137 62
Table 2: Number of stocks per country in 2005. Here the stocks areassigned to each country according to the ISIN code (i.e., place of listing). Thecountries correspond to zones NA means that the country zone was includedinto the Rest of zone.
10
4 Implementation: the EM algorithm
The statistical implementation of the model (2) is very close in spirit to that ofBrooks and Del Negro (2004). These authors considered a similar factor modelwith country factors (and a world factor, which we do not have and do not needhere) but with restrictions on the exposures contained in the matrix B thatdiffer from ours. They fix the share of activity abroad at 0 and they force thecountry factors to be independent of each other so that all common movementsin countries take place through the world factor.We have described in Section 2 our own set of restrictions. These do not allow
us to assume that zones are independent of each other (which is the reason forwhich we do not need a world factor). We call ωωᵀ the covariance matrix ofthe zone factors C and we assume that C and ε are independent of each other.The diagonal variance-covariance matrix of ε is denoted D. Were it not forthe constraints on B, zone factors could be redefined to be orthogonal to eachother, yielding a coeffi cient matrix equal to Bω, so that we would be back witha standard factor analysis model:
Rt = b× ct + et
b = Bω
Ct = ωct
The separate identifications of B and ω, therefore, is based entirely on theconstraints on B, while ω is chosen freely.4
Assumption: All random variables are multivariate IID normal.The full log-likelihood is as in Lehmann and Modest (2005):
L (B,D, ω|R) =−NT
2ln (2π)− T
2ln |Σ| − T
2trace
(SΣ−1
)(3)
=−NT
2ln (2π)− T
2ln |Σ| − 1
2
∑t
Rᵀt Σ−1Rt
where:
R = [Rt; t = 1, ..T ]
Σ = BωωᵀBᵀ +D
S =1
TRRᵀ
Direct maximization yields a system which is nonlinear and hard to solve. In-stead, we use an iterative method called the EM algorithm, which was firstproposed to solve missing-data problems by Dempster, Laird and Rubin (1977)
4Unfortunately, we are not able to provide suffi cient conditions for identification. Belowwe describe the efforts made to bring about identification. They consist in adjusting thenumber of zone factors in such a way that the rank of BωωᵀBᵀ is full. When a zone must beabandoned, it is merged into a “Rest of” zone.
11
and then applied to latent-factor models by Rubin and Thayer (1982).5 Ourchallenge is to extend the technique to the case in which the factors are not in-dependent of each other and, instead, there are restrictions on the exposures (orloadings, in the language of factor analysis), giving rise to a covariance matrixof zone factors.If the latent factors c were observed, the log likelihood would be:
LL = −T2
N∑j=1
lnDj −1
2
T∑t=1
N∑j=1
(Rj,t −Brow jωct)2
Dj(4)
In the E step of the algorithm, we calculate the expectation of LL given theobservations R, at the currently estimated values of the parameters. I.e., weintegrate (4) over C. Neglecting constants, this gives:
E [LL|R] = −T2
N∑j=1
lnDj
−1
2
N∑j=1
1
Dj
{T∑t=1
(Rj,t)2 − 2
T∑t=1
Rj,tE [cᵀt |Rt]ωᵀ (Brow j)ᵀ
+Brow jω
T∑t=1
E [ctcᵀt |Rt]ωᵀ (Brow j)
ᵀ}
The suffi cient statistics that are contained in the likelihood function are:∑Tt=1 E [cᵀt |Rt] and
∑Tt=1 E [ctc
ᵀt |Rt]. We compute them in Appendix B, on the
basis of the model. After substitution, the result is:
E [LL|R] = −T2
N∑j=1
lnDj (5)
−T2× trace
{D−1 [S − 2XωᵀBᵀ +BωY ωᵀBᵀ]
}where we have defined:
X , SΣ−1Bω = SD−1Bω(I + ωᵀBᵀD−1Bω
)−1(6)
Y , ωᵀBᵀΣ−1 (S − Σ) Σ−1Bω + I
=(I + ωᵀBᵀD−1Bω
)−1 [ωᵀBᵀD−1SD−1Bω
(I + ωᵀBᵀD−1Bω
)−1+ I](7)
In the M part of the algorithm, the function (5) is maximized at each step withrespect to D, B and then ω (from updated B and D), keeping X and Y fixed.
5E stands for “expectation”and M for “maximization”. The meaning of that riddle becomesclear below.
12
At the next E step, X and Y are recomputed from (6) and (7), ω is updatedagain and the M part is run anew.6
The first-order condition with respect to B need not be written as, in anycase, that optimization is to be done under the restrictions of Section 2, bymeans of a numerical optimization algorithm. The first-order condition withrespect to D is simply:
D = diagonal [S − 2XωᵀBᵀ +BωY ωᵀBᵀ]
Fortunately, the optimization with respect to the elements of B and D canbe performed individually firm by firm.7 That is the major benefit of the EMalgorithm.Our last task is to write the first-order condition with respect to ω. Differ-
entiating Equation (5) with respect to ωi,j , we get:
trace{D−1
[−2X
(ωi,j
)ᵀBᵀ +Bωi,jY ωᵀBᵀ +BωY
(ωi,j
)ᵀBᵀ]}
= 0
where ωi,j stands for the single-entry matrix of same size as ω with zeros every-where except for a number 1 in location i, j. Working out the effect of thesingle-entry matrix, we get:8
−(2XᵀD−1B
)j,i
+(BᵀD−1BωY
)i,j
+(Y ωᵀBᵀD−1B
)j,i
= 0
−(XᵀD−1B
)ᵀ+(BᵀD−1BωY
)= 0
−(XᵀD−1B
)ᵀY −1 +BᵀD−1Bω = 0(
BᵀD−1B)−1 (
XᵀD−1B)ᵀY −1 = ω
During the execution of the EM steps, the full likelihood (3) is calculated pe-riodically to verify that it keeps increasing. That calculation is computationallyintensive.We define the “Expected true”zone indexes to be
E [Ct|Rt] = ω(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1Rt (8)
except for the fact that Rt, in this definition, contains the individual firm re-turns, not demeaned. This definition allows us to obtain a measure of the zoneindexes at any frequency we wish.Using daily stock returns, we perform the estimation year by year (from
1999 to 2005) assuming that all exposure coeffi cients, which are the elements ofthe matrix B, are constant within a year. The elements that, by virtue of therestrictions, are set equal to geographic shares of sales, are set equal to eachyear’s shares of sales. The convergence of the EM algorithm is considered to beachieved whenever the mean squared gradient is lower than 2× 10−4.
6ω is updated again to remedy the fact that we do not optimize jointly B, D and ω.7These optimization, except for the restrictions, are analogous to a time-series regression
run for each firm.8 trace
(AJi,jB
)= (AᵀBᵀ)i,j = (BA)j,i. See Petersen and Pedersen (2007), page 41.
13
The choice of the number of the geographic zones that are responsible forthe common variation in individual stock returns involves considerable trial anderror. We can start with the set of zones with at least one single known ex-posure and then reduce the number of zones gradually each time eliminatingzones with the smallest number of single known exposures until the rank of thecommunality of the variates (the rank of BωωᵀBᵀ) is full. Practical implemen-tation of the zones selection procedure, however, showed that it is much moretime-economizing to start with the zones with a much larger number of knownsingle exposures.We have a distribution of the number of individual firm expo-sures. One way is to calculate the median number of single exposures per zoneand eliminate any zone that has a number of single known exposures less thanthe median. The preliminary choice of the number of the geographical zones ismade based on 500 iterations. Once the convergence is reached, we again checkwhether the rank of the communality is full and reduce the number of zonesif needed. Table 3 summarizes the structure of the statistical model that wasfinally selected.In the next section, we examine the output (especially the estimated correla-
tions) of the EM algorithm year by year. In the subsequent section we reprocessthe zone factors on a daily basis to discover their finer properties.
5 Geographic factors: year-by-year results
Below, the figures in Table 5 and following compare across countries the secondmoments of the EM indexes with those of the Datastream-type indexes, whichwe call “observed.” The “observed”Datastream-type indexes are identical tothe indexes distributed by Datastream except that we re-compute them using,not the complete set of stocks that are included in them, but the slightly re-duced set of stocks that have passed our filtering (see above). They come in twoversions: the “equally weighted” and the “value-weighted” versions. For eachyear, we obtain a cross-section of standard deviations and correlations of indi-vidual country returns and country pairs and display the mean and the medianin Table 4.We display the two versions in those figures in order to show thatthey have similar second-moment properties, so that it is all right to continueour comparison between EM and Datastream indexes on an equally-weightedbasis only.EM indexes also come in two versions, although the two versions are shown
below to be almost identical. The second moments of the indexes are thosegiven directly by the variance covariance matrix ωωᵀ. We call these the “Real-ized true”moments of the indexes. For those the variance-covariance matrix isavailable but not the time-series of the indexes. We also obtain a time series offiltered or expected values of the indexes from formula (8) above and call these“Expected true”indexes.
14
Year 1999 2000 2001 2002 2003 2004 2005
Panel A: Dynamics of the number of stocks
Number of stocks 2019 2497 2837 3023 3199 3331 3455Min number of stocks 1 26 2 33 38 39 1Max number of stocks 400 621 655 664 666 672 728
Panel B: Dynamics of the number of the geographical zones across regions
Overall 25 23 25 26 20 24 29Europe 10 9 10 11 8 9 11America 5 5 5 5 4 5 5
Middle East/Asia 1 1 1 2 1 2 2Africa 2 1 2 1 1 1 2
Asia/Oceania 7 7 7 7 6 7 9
Panel C: Dynamics of the composition of the rest of zones
Rest of Europe 17 18 17 16 19 18 16Rest of America 9 9 9 9 10 9 9
Rest of Middle East/Asia 5 5 5 4 5 4 4Rest of Africa 2 3 2 3 3 3 2
Rest of Asia/Oceania 7 7 7 7 8 7 5
Panel D: Common set of zones
Europe France, Netherlands, Germany, Spain, Italy,United Kingdom, Sweden
Asia/Oceania Australia, Malaysia, Singapore, Hong Kong, JapanAmerica Brazil, USA, Canada
Table 3: The structure of the estimation output
15
Panel A: Standard deviations of individual stocks and “observed” indexes
Mean Median
1999 2000 2001 2002 2003 2004 20050.005
0.01
0.015
0.02
0.025
0.03
0.035
Year
Valu
e
Equa lly weightedValue weightedIndiv idual
1999 2000 2001 2002 2003 2004 20050.005
0.01
0.015
0.02
0.025
0.03
Year
Valu
e
Equa lly weightedValue weightedIndiv idual
Panel B: Std. dev. of indiv. stocks and “observed” indexes, for common zones
Mean Median
1999 2000 2001 2002 2003 2004 20050.005
0.01
0.015
0.02
0.025
0.03
0.035
Year
Valu
e
Equal ly weightedValue weightedIndiv idual
1999 2000 2001 2002 2003 2004 20050.005
0.01
0.015
0.02
0.025
0.03
Year
Valu
e
Equa lly weightedValue weightedIndiv idual
Panel C: Correlations of “observed” indexes
Mean Median
1999 2000 2001 2002 2003 2004 20050.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
Year
Valu
e
Equa lly weightedValue weighted
1999 2000 2001 2002 2003 2004 2005
0.16
0.18
0.2
0.22
0.24
0.26
0.28
Year
Valu
e
Equa lly weightedValue weighted
Panel D: Correlations of “observed” indexes, for common zones
Mean Median
1999 2000 2001 2002 2003 2004 20050.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
Year
Valu
e
Equa lly weightedValue weighted
1999 2000 2001 2002 2003 2004 2005
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
Year
Valu
e
Equa lly weightedValue weighted
Table 4: Comparative second-moment properties of Datastream-type“observed” indexes. For each year, we obtain a cross-section of standarddeviations and correlations of individual country returns and then calculate themean and the median. The common country indexes are listed as “common setof zones”in Table 3.
16
Panel A: Standard deviations of “observed”and EM indexes
Mean Median
1999 2000 2001 2002 2003 2004 20050.008
0.01
0.012
0.014
0.016
0.018
0.02
Year
Valu
e
Real ized trueEx pec ted trueEqually weighted
1999 2000 2001 2002 2003 2004 20057
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12x 10 3
Year
Valu
e
Real ized trueEx pec ted trueEqually weighted
Panel B: Std. dev. of “observed”and EM indexes, for common zones
Mean Median
1999 2000 2001 2002 2003 2004 20050.007
0.008
0.009
0.01
0.011
0.012
0.013
0.014
Year
Valu
e
Real ized trueEx pec ted trueEqually weighted
1999 2000 2001 2002 2003 2004 20056
7
8
9
10
11
12x 10 3
Year
Valu
e
Real iz ed trueEx pec ted trueEqual ly weighted
Panel C: Correlations of “observed”and of EM indexes
Mean Median
1999 2000 2001 2002 2003 2004 2005
0.15
0.2
0.25
0.3
0.35
Year
Valu
e
Real ized trueEx pec ted trueEqually weighted
1999 2000 2001 2002 2003 2004 2005
0.15
0.2
0.25
0.3
0.35
Year
Valu
e
Real ized trueEx pec ted trueEqually weighted
Panel D: Correlations of “observed”and of EM indexes, for common zones
Mean Median
1999 2000 2001 2002 2003 2004 2005
0.15
0.2
0.25
0.3
0.35
Year
Valu
e
Real iz ed trueEx pec ted trueEqual ly weighted
1999 2000 2001 2002 2003 2004 2005
0.15
0.2
0.25
0.3
0.35
Year
Valu
e
Real iz ed trueEx pec ted trueEqual ly weighted
Table 5: Comparison of second-moment properties of “observed”Datastream vs. EM indexes. For each year, we obtain a cross-section ofstandard deviations and correlations of individual country (for observed indexes)or zone (for EM indexes) returns and then calculate the mean and the median.The common country indexes are listed as “common set of zones”in Table 3.
17
5.1 Comparison of standard deviations
The figures in Table 5, Panels A and B reveal that, for most countries, it is thecase that the EM indexes are less volatile than “observed”indexes. This comesas a surprise since the observed indexes are diversified across zones while theEM indexes, by construction, are not. One interpretation is that some of thevolatility of developed-country stock exchange indexes (which are more numer-ous in our sample) arises from their firms’ involvement in developing countryzones, which are more volatile. Indeed, most of the increase in internaliza-tion illustrated in Figure 2 reflects the penetration of developing markets bydeveloped-country firms.
5.2 Comparison of correlations
The figures in Table 5, Panels C and D displays mean and median statisticsof the pairwise correlations of EM zone indexes on the one hand and observedDatastream country indexes and the other. The difference tends to be negative:EM indexes are less correlated across countries than are Datastream indexes.This is consistent with the hypothesis that some of the correlation betweentraditional stock market indexes arises from the interpenetration of corporateactivity across countries. We have achieved the principal goal of our exercise,which was to deconstruct that interpenetration.
6 Geographic factors: daily index returns9
We now investigate the degree to which the stochastic processes of the twotypes of indexes differ from each other. For that, we collect daily realizations ofobserved and EM indexes and try to fit a statistical stochastic process of theGARCH. Before we do that, however, visual inspection of the figures in Table 6containing the series and their volatilities and correlations reveal the presence ofasymmetry in individual constitutent stocks. Volatilities and correlations seemto increase after an episode of negative returns. To accomodate that feature,we now fit a TARCH process to volatilities and an asymmetric DCC (ADCC)process to correlations.
6.1 Conditional variance: TARCH analysis
Threshold ARCH is a GARCH model of Zakoian (1994) with a different adjust-ment of conditional standard deviation depending on whether the last innovationin the mean equation is above or below a threshold (in our case 0). We fit aTARCH(1,1,1) meaning that the model includes one lag in the mean equationand one lag in the variance equation whether the last innovation is positive or
9Common holidays and non-synchronous returns affect the correlation structure of thedaily international stock returns. Appendix A explains the way this matter was (imperfectly)handled.
18
Standard deviation Correlation
05/1999 05/2000 04/2001 03/2002 03/2003 02/2004 01/2005 12/20050
0.005
0.01
0.015
0.02
0.025
Date
Valu
e
M eanM edian
05/1999 05/2000 04/2001 03/2002 03/2003 02/2004 01/2005 12/20050
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Date
Valu
e
M eanM edian
Table 6: Evidence of the presence of asymmetric effect in volatilitiesand correlations. For each country we form an equally weighted portfolio ofindividual stocks used in the estimation. The figures represent the mean and themedian of rolling standard deviations and correlations of zone portfolio returns(with a window of 100 observations). The vertical lines indicate the dates onwhich at least 80% of individual stocks used in the estimation have negativereturns.
negative:
h1/2t = a+ b× |εt−1|+ c× 1 [εt−1 < 0]× |εt−1|+ d× h1/2t−1
On that basis, we obtain daily univariate estimates of conditional variancefor each zone factor. The figures in Table 7 are analogous to those of Table 5except that the latter covered the entire world and that they show conditionaldaily variances —including asymmetric effects —instead of unconditional yearlyones. Panel A displays mean variances in various zones belonging to three areas(Europe, America and Asia/Oceania). We see again that the “true”geographicindex variance tends to be lower than that of the “observed”Datastream in-dexes.The zones of Europe come out as an exception. To investigate that exception
further, we ask which pattern the single-country zones included in the Europeanarea follow. In most single-country zones, the worldwide average pattern obtainsrather than the European pattern, except in the two zones we display in PanelB: Italy and above all Sweden.The zones of America deserve a specific investigation as America is the area
for which we have the largest number of firms. Within that area, we find (inPanel B) a strong difference between Brazil on the one hand and the USA andCanada on the other. In the two latter countries the worldwide pattern holdsvery strongly presumably because firms of these countries have foreign activitiesin highly risky zones. In Brazil, however, the variances of the two kinds of indexesare practically equal. Within the Asia/oceania area, Malaysia and Hong Kongdeserve a special notice as the worldwide pattern of difference holds in thesecountries particularly.
19
Panel A: “observed”and EM indexes in selected areas
Europe America
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
1
2
3
4
5
6
7
x 10 4
Date
Valu
eTrueObserv ed
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x 10 3
Date
Valu
e
TrueObserv ed
Asia/Oceania
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10 4
Date
Valu
e
TrueObserv ed
Panel B: “observed”and EM indexes in selected single-country zonesItaly Sweden
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.2
0.4
0.6
0.8
1
1.2
x 10 3
Date
Valu
e
TrueObserv ed
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.5
1
1.5
2
2.5
3
3.5
4x 10 3
Date
Valu
e
TrueObserv ed
USA Canada
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
1
2
3
4
5
x 10 4
Date
Valu
e
TrueObserv ed
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.5
1
1.5
2
2.5
3
3.5x 10 4
Date
Valu
e
TrueObserv ed
Malaysia Hong Kong
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x 10 3
Date
Valu
e
TrueObserv ed
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
1
2
3
4
5
6
x 10 4
Date
Valu
e
TrueObserv ed
Table 7: Univariate variance processes Panel A: main areas formed bythe common set of zones. Panel B: selected zones. TARCH(1,1,1) processesare fitted to each individual series. The figures represent the means of theconditional variances. The vertical lines indicate the dates when at least 80 %of individual stocks used in the estimation have negative returns.
20
6.2 Conditional correlation: ADCC analysis
We are, of course, mostly interested in the dynamics of pairwise correlationsof zone indexes since these may be interpreted as an indication of financial-market integration. Once the variance behavior of a zone index has been fittedby TARCH, its time series can be standardized and used to study the dy-namics of pairwise correlation. Asymmetric Dynamic Conditional Correlation(ADCC(1,1,1)) is obtained for any pair of zone indexes by fitting the followingmodel (Cappiello et al. (2006)) for the 2× 2 matrix process Q:
Qi,j,t = Q̄i,j × (1− α− β)− γN̄i,j + α× εi,t−1εj,t−1+βQi,j,t−1 + γ × (1 [εi,t−1 < 0]× εi,t−1 × 1 [εj,t−1 < 0]× εj,t−1) ; i, j = 1, 2
and then estimating the conditional correlation as:
Qi,j,t√Qi,i,t
√Qj,j,t
We show as Figure 3 the mean outcome for the entire set of zones thatare common to all years. Given the previous result of Table 5, Panel C, weare not surprised to find that the average of the pairwise correlations for “true”EM indexes is below that for “observed”Datastream-type indexes. Surprisingly,however, while the “observed”average correlation is approximately constant, the“true”average correlation is rising. That might, indeed, indicate that financialmarkets are getting more and more integrated.To resolve this paradox, we examine separately several areas of the world
and then some individual zones. In the figures of Table 8, two features are im-mediately apparent. First, for correlations within Europe, both the traditional,“observed”correlations and the EM correlations are rising in tandem, implyinga rise in the integration of European financial markets, over and above any risein intra-European trade activity. Second,. the rise in the “true”correlations ac-companied by flat “observed”correlations occurs between America and Europeand within America itself.Continuing our investigation, we split the “America” area into individual
zones. The figures in Table 9, Panel A display the correlations of the “Brazil”zone, showing that the “true”correlations for Brazil are rising, which we inter-pret as a rising integration of the financial market for firms that have businessactivity in Brazil with the financial market for firms that do business elsewhere.Panel B of the same table shows that simultaneously a growing integration intrade exposures to Brazil is taking place. At the same time, however, PanelA shows that there is no trend in the “observed”correlations of firms listed inBrazil.
7 Conclusion
We have identified “pure”or “true”stock index returns based on the geographiczones in which firms have their activity, as opposed to traditional or “observed”
21
01/1999 01/2000 01/2001 01/2002 01/2003 12/2003 12/2004 12/20050
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Date
Valu
e
TrueObserved
Figure 3: ADCC, all countries which are members of the commonset of zones. We use the ADCC(1,1,1) model with the TARCH(1,1,1) toestimate conditional correlations for each possible pair of countries, separatelyfor “True”and “Observed”indexes. The figures represent the means of all theconditional correlations. The vertical lines indicate the dates when at least 80% of individual stocks used in the estimation have negative returns.
22
Europe-Europe
Europe-America
Europe-Asia/Oceania
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Date
Value
True
Obse
rved
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
0.250.
3
0.35
Date
Value
True
Obse
rved
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
0.25
Date
Value
True
Obse
rved
America-America
America-Asia/Oceania
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
0.250.
3
0.35
Date
Value
True
Obse
rved
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.02
0.04
0.06
0.080.
1
0.12
0.14
Date
Value
True
Obse
rved
Asia/Oceania-Asia/Oceania
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
0.250.
3
0.350.
4
0.45
Date
Value
True
Obse
rved
Table8:ADCCcorrelations,mainregionsformed
bythecommon
setofzonesWeusetheADCC(1,1,1)modelwith
theTARCH(1,1,1)toestimateconditionalcorrelationsforeachpossiblepairofcountries.Thefiguresrepresentthemeansof
theconditionalcorrelationsineachgroup.Theverticallinesindicatethedateswhenatleast80%ofindividualstocksusedin
theestimationhavenegativereturns.
23
PanelA:“observed”andEM
indexes
Brazil-Europe
Brazil-America
Brazil-Asia/Oceania
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
0.250.
3
0.350.
4
0.45
Date
Value
True
Obse
rved
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
Date
Value
True
Obse
rved
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0
0.050.
1
0.150.
2
Date
Value
True
Obse
rved
PanelA:EM
indexesandnumberoftradeexposures
Brazil-Europe,trade
Brazil-America,trade
Brazil-Asia/Oceania,trade
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0.1
0.150.
2
0.250.
3
0.350.
4
0.45
Date
Value
True
Trad
e
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0.050.
1
0.150.
2
Date
Value
True
Trad
e
01/1
999
01/2
000
01/2
001
01/2
002
01/2
003
12/2
003
12/2
004
12/2
005
0.16
0.17
0.18
0.190.
2
0.21
0.22
0.23
0.24
Date
Value
True
Obse
rved
Table9:Dynam
icsoftheconditionalcorrelationforBrazil,common
setofzones
WeusetheADCC(1,1,1)model
withtheTARCH(1,1,1)toestimateconditionalcorrelationsforeachpossiblepairofcountries.Thefiguresrepresentthemeans
oftheconditionalcorrelationsforBrazilwiththezonesineachgroupandthenumberofestimatedexposuresmorethan0.0001
totheBrazilzone(rescaled).Theverticallinesindicatethedateswhenatleast80%ofindividualstocksusedintheestimation
havenegativereturns.
24
indexes constructed according to the country stockmarket in which they arelisted.While “observed” indexes, on a yearly or on a daily basis, exhibit almost
no upward trend in their correlations, we have been able to show that thecorrelations of the “true”indexes do trend upward, especially for the Americanarea, with Brazil being the clear case of growing financial-market integrationthat we have been able to identify.
25
BibliographyBrooks, R. and M. Del Negro, 2004, “A Latent Factor Model with Global,
Country and Industry Shocks for International Stock Returns,” working pa-per, Federal Reserve Bank of Atlanta, previously circulated as: “InternationalDiversification Strategies,”2002-23b.Brooks, R. and M. Del Negro, 2006, “Firm-Level Evidence on International
Stock Market Comovement,”Review of Finance, 10, 69-98.Cappiello, L., Engle, R.F. & Sheppard, K., 2006,.“Asymmetric Dynamics
in the Correlations of Global Equity and Bond Returns,”Journal of FinancialEconometrics, 4, 537—572.Cavaglia, S., J. Diermeier, V. Moroz, and S. De Zordo 2004, “Investing in
Global Equities,”The Journal of Portfolio Management, 30, 3, 88-94.Dempster, A. P., N. M. Laird and D. B. Rubin, 1977, “Maximum Likelihood
from Incomplete Data via the EM Algorithm,”Journal of the Royal StatisticalSociety, B39, 1-38.Diermeier, J. and B. Solnik, 2001, “Global Pricing of Equity,” Financial
Analysts Journal, Vol. 57, No. 3, July/August 2001.Dumas, B., C. R. Harvey and P. Ruiz, 2003, “Are Correlations in Interna-
tional Stock Returns Justified by Subsequent Changes in National Outputs?”The Journal of International Money and Finance, 22, 777-811.Goetzmann, W. N., L. Li and K. G. Rouwenhorst, 2005, “Long-Term Global
Market Correlations,”Journal of Business,.78, 1-38.Heston, S. L. and K. Geert Rouwenhorst, 1994, “Does Industrial Structure
Explain the Benefits of International Diversification,”Journal of Financial Eco-nomics, 36, 3-27Lehmann, B. N. and D. M. Modest,.2005, “Diversification and the Optimal
Construction of Basis Portfolios,”Management Science, 51, 581-598.Petersen, and Pedersen, 2007, The Matrix Cookbook, on the web.Rubin, D. B. and D. T. Thayer, 1982, “EM Algorithms for ML Factor Analy-
sis,”Psychometrika, 59, 69-76.Solnik, B., 1974, “Why Not Diversify Internationally Rather than Domesti-
cally?”Financial Analysts Journal, 30, 48-52+54Zakoian, Jean-Michel, 1994, “Threshold heteroskedastic models,”Journal of
Economic Dynamics and Control, 18, 931—955.
26
Appendixes
A Description of revenue data and stock returns
A.1 Revenue data
To facilitate comparison with the conventional country stock market indexeswe restrict the analysis on the stocks that are part of the Datastream worldindex. The proxy for foreign sales activity of firms is the geographical breakdownof revenues. World’Vest Base has extensive information about sales activitiesof a large number of companies (98% of global capitalization in 2003) and isour source of the revenue breakdown information. The data on geographicaldistribution of sales was obtained in 2007 and since then ISIN has changedfor many stocks. This has created a diffi culty in matching the returns datawith the sales data. Using the names and ISINs downloaded in previous years,we managed to match the old ISINs with the new ISINs for 402 out of 463problematic stocks.In some cases, it is impossible to infer a geographical assignment of revenues
reported in WVB. Additionally there is a number of negative, zero and missingrevenue values. To circumvent the first problem, we restrict the analysis to alist of 937 assignments which unambiguously refer to countries or geographi-cal regions. We address the second problem by eliminating the records whichcorrespond to negative, zero or missing revenue values.A firm may restate or reclassify its revenues within a year. Once the infor-
mation is available, WVB updates the records on the revenue breakdown and,at the same time, retains the old records. As the result, a given firm may havemultiple records. Therefore, using the data without additional filtering maybias the geographical distribution of the sales activities. We try to solve thisproblem by a filtering procedure elaborated below.According to the WVB Data Manual, multiple records may exist due to a
change in the accounting standards, in the income statement format, due toreclassification of items or to changes in the fiscal year end. Consequently, forfiltering we use information on the report type, currency and on the fiscal yearend. Using the report type code, we select the report that belongs to I, IIpriority groups in Table 10: consolidated and covering a period of 12 monthsand consolidated and covering a period of less than 12 months. Table 11 showsthat we cover vast majority of the revenue observations available across all theyears.In order to keep the maximum number of records, for a given firm we choose
the most detailed financial record type. Moreover, if the number of recordsgroped by the financial data header is the same, we chose the one which ispreferred according to the rule set up in Table 10 (the most preferred reporttype being at the top of the list and the least preferred at the bottom). Wethen look at the firms which have multiple records corresponding to the samegeographical regions and choose the records with the latest fiscal year end date.
27
Some firms have multiple records from the same report type and fiscal yearend date but stated in different currencies. For these firms we keep revenuesstated in one arbitrarily chosen currency. Finally, to control for a possibilitythat a firm without multiple records referring to the same geographical segmentreports revenues in different currencies, we convert revenue data into US $.
A.2 Returns
For each stock we need to obtain daily total return index in both, the USAdollar and the home currency. While the estimation is done using the dollarreturns, the home currency returns are used for filtering. The reason behindthis choice is that our filtering requires calculation of the number of non-zeroreturns and the dollar non-zero returns may be a result of the exchange ratevariation rather than the price variation.Other problems which affect the measured correlation structure of the daily
international stock returns are common holidays and non-synchronous returns.We remove holidays common to most of the countries: based on the dollarreturns, we count the number of zero returns for each day and remove thewhole string of returns for this day if the number of zero returns exceeded thethird of the total number of stocks. Furthermore, in order to mitigate the effectof non-synchronous returns we assign the geographical regions to time zonesand adjust the timing of returns of the regions which are suffi ciently far fromeach other. In this way, tomorrow returns on the stocks from North and SouthAmerica are matched with yesterday returns on the stocks from Asia Pacificand Australasia and today returns from the other regions.Finally, we give 20 days for the stock market to digest the information about
the geographical distribution of sales of a firm. To elaborate, each calendar yearconsists of slightly less than 260 trading days. We divide a year into sub-periodsof 20 trading days (with the last two sub-periods overlapping) and require allstocks in the sample to have at least one non-zero return within each of thesub-periods with zero-returns calculated on the basis of home currency prices.
B Suffi cient statistics of the geographic analysis
Based on the model:Rt = Bω × ct + et
where the covariance matrix of c is the identity matrix, we compute∑Tt=1 E [cᵀt |Rt] ,∑T
t=1 E [ctcᵀt |Rt]:
E [ct|Rt] = cov [ct, Rᵀt ] [var [Rt]]
−1Rt
= ωᵀBᵀ [BωωᵀBᵀ +D]−1Rt
=(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1Rt
The last equality follows from the Woodbury formula:
[BωωᵀBᵀ +D]−1
= D−1 −D−1Bω(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1
28
Priority
WVBheader
Description
IC
Consolidatedreportcovering12-monthsperiod
CR
Consolidatedreportcontainingrestateddata
CS
Consolidadolegislacaosecretaria(Brazil)
CC
Consolidadoem
moedadepodreaquisitivoconstante(Brazil)
IConsolidatedreportmeetinginternationalstandardscoveringa12-monthsperiod
IRConsolidatedreportmeetinginternationalstandardsandcontainingrestateddata
IIIP
Consolidatedreportmeetinginternationalstandardscoveringaperiodlessthan12months
CP
Consolidatedreportcoveringaperiodlessthan12months
CU
Consolidatedreportwithpreliminary/summarydata
CQ
Quarter/Interimreport
Table10:Recordtypepriority:Multiplerecordsareeliminatedaccordingtothepriorityruledescribedinthistable(from
themostpreferredinthefirstrowtotheleastpreferredinthelastrow).
29
Year 1999 2000 2001 2002 2003 2004 2005Initial 6807 8400 9331 11006 12826 15905 16128Type I 6541 8012 8863 9302 10369 12499 10755Type II 64 94 134 1336 2080 2421 5072
Type I+Type II 6605 8106 8997 10638 12449 14920 15827
Table 11: Number of revenue observations in each priority group, forall stocks: The first row gives the number of revenue observations after elimi-nation of negative, zero or missing revenue values. All the other rows—numberof revenue observations in each priority group.
which implies:
ωᵀBᵀ [BωωᵀBᵀ +D]−1
= ωᵀBᵀD−1 − ωᵀBᵀD−1Bω(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1
=[I − ωBᵀD−1Bω
(I + ωᵀBᵀD−1Bω
)−1]ωᵀBᵀD−1
but: (I + ωᵀBᵀD−1Bω
)−1+ ωᵀBᵀD−1Bω
(I + ωᵀBᵀD−1Bω
)−1= I
so that:
ωᵀBᵀ [BωωᵀBᵀ +D]−1
= ωᵀBᵀD−1 − ωBᵀD−1Bω(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1
=(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1
the great benefit of this transformation being that the matrix to be inverted isonly as large as the number of countries, as opposed to being as large as thenumber of firms.Next, we handle the second suffi cient statistic
∑Tt=1 E [ctc
ᵀt |Rt]:
E [ctcᵀt |Rt] = var [ct|Rt] + E [ct|Rt] · E [cᵀt |Rt]
var [ct|Rt] = I − cov [ct, Rᵀt ] [var [Rt]]
−1 cov [Rt, cᵀt ]
= I − ωBᵀ [BωωᵀBᵀ +D]−1Bω
= I −(I + ωᵀBᵀD−1Bω
)−1ωᵀBᵀD−1Bω
=(I + ωᵀBᵀD−1Bω
)−1so that:
E [ctcᵀt |Rt] =
(I + ωᵀBᵀD−1Bω
)−1 [I + ωᵀBᵀD−1RtR
ᵀtD
−1Bω(I + ωᵀBᵀD−1Bω
)−1]
30