Economic Determinants of the Correlation Structure Across

Economic Determinants of the CorrelationStructure Across International Equity Markets

Kevin Bracker and Paul D. Koch

This study investigates whether, how, and why the matrix of correlations across interna-tional equity markets changes over time. A theoretical model is proposed to specifypotential economic determinants of this correlation structure. The empirical validity of thiseconomic model is investigated by employing daily returns for different national stockindexes, from 1972 through 1993, to construct a quarterly time series of the correlationmatrix. This quarterly time series is used to investigate the stability of the correlationmatrix over time, and to estimate the economic model. The model is then applied togenerate out-of-sample forecasts of the correlation structure. © 1999 Elsevier Science Inc.

Keywords:International market integration

JEL classification:F36, G15.

I. IntroductionThe degree of capital market integration varies substantially across different pairs ofnational markets and over time. During some periods, there is less association between theeconomic health of countries in different sectors of the world economy. At such times,national equity markets tend to follow their own paths, and experience less comovementwith other markets. At other times, there is greater association between the economicperformance of different sectors of the world, especially across countries that are moreinterdependent through trade, and we observe greater comovement across respectivenational equity markets. Recently, we have witnessed occasional contagion effectswhereby certain groups of national equity markets rally or crash together in response toeconomic and/or political events, exhibiting correlations that are temporarily very high.

This discussion describes evolution in the nature and extent of global capital marketintegration, characterized by dramatic changes over time in the correlation matrix across

Economics, Finance, and Banking (K.B.), Pittsburg State University, Pittsburg, Kansas, School of Business(P.D.K.), University of Kansas, Lawrence, Kansas, USA

Address correspondence to: Dr. P. D. Koch, School of Business, University of Kansas, Lawrence, KS 66045.

Journal of Economics and Business 1999; 51:443–471 0148-6195 /99 /$–see front matter© 1999 Elsevier Science Inc., New York, New York PII S0148-6195(99)00021-1

national equity returns. This study focuses on understanding, modeling, and forecastingdynamic movements in the correlation structure.1 We hypothesize that greater economicintegration across countries should lead to greater capital market integration. Interdepen-dence through trade and capital flows implies interdependence in investors’ valuationdecisions across national equity markets. In this light, we suggest that pairs of countriesexperiencing greater economic integration should also experience greater comovement intheir respective capital markets. Furthermore, as the nature and extent of economicintegration across countries changes over time, we may expect concomitant changes in thedegree of comovement across respective capital markets.

This study addresses whether, how, and why the correlation structure changes overtime by testing the stability of the correlation matrix over different periods, and bymodeling potential economic determinants of the correlation structure. The economicmodel is then applied to generate out-of-sample forecasts of the correlation structure. Abetter understanding of dynamic movements in the correlation structure is critical, giventhat this structure reflects the nature and extent of global market integration, and given itsimpact on the risk-return performance of international equity portfolios. In light of therecent precarious situation in world financial markets, the ability to predict changes incorrelation patterns should be of interest to market participants, national policy makers,and regulatory bodies involved in monitoring and managing global market behavior.2

This study employs daily returns on ten national stock indexes, from 1972 through1993, to construct a quarterly time series of the correlation matrix. This time seriesembodies any variation in the correlation structure experienced over the sample, and isused to test formally the hypothesis that the correlation matrix does not change over time.Results reveal substantive changes over both short and long time horizons throughout this22-year sample period. Augmented Dickey–Fuller tests indicate that almost all of thesetime series of pairwise correlations contain no unit root. This outcome alleviates concernabout possible spurious relationships with economic variables. This battery of stabilitytests provides evidence of the dramatic evolution in the correlation matrix, and motivatesthe need to further analyze potential economic determinants of the correlation structure.

We seek to gain a better understanding of these dynamic changes in the correlationstructure by modeling economic factors that may influence the degree of comovementacross equity markets. We hypothesize that the degree of integration across internationalcapital markets at any point in time depends upon the degree of economic integration

1 Throughout this study, the termcorrelation structurerefers to the matrix of correlations across returns indifferent national equity indexes. This correlation structure characterizes the nature and extent of integrationacross different pairs of national equity markets. Dynamic changes in the correlation structure imply changes inthe risk-return performance of international equity portfolios over time. For a summary of the role of thecorrelation structure in international diversification, see Haugen (1990, pp. 134–136), Isimbabi (1992), andHatch and Resnick (1993). For more related work on the nature and extent of international market integration,see Arshanapalli and Doukas (1993), Bachman et al. (1996), Eun and Resnick (1984), Kasa (1992), and Longinand Solnik (1995).

2 Changes in the correlation matrix, along with changes in expected returns and variances across nationalequity markets, imply shifts in the international opportunities locus over time. The “optimal” international equityportfolio is traditionally constructed from a given estimated opportunities locus as the portfolio with the highestSharpe (1966) measure. Computation of both the opportunities locus and the Sharpe measure requires forecastsof expected returns, variances, and correlations across national equity markets. As global capital markets evolvewith changing economic conditions, these three dimensions of the international opportunity set should vary overtime, making this forecasting task a difficult endeavor. In this light, market participants and regulators shouldbe interested in the ability to predict changes in the correlation structure.

444 K. Bracker and P. D. Koch

across the countries involved.3 We develop a theoretical model in which each time seriesof pairwise correlations may depend upon factors that characterize and influence theextent of economic integration across the two markets in question. The set of all suchpairwise equations is then estimated both as a pooled regression and as a system ofseemingly unrelated regressions that describes potential economic determinants of thecorrelation structure. This analysis is conducted on the subset of seven countries for whichcomplete economic data are available. Results indicate that the degree of internationalintegration (the magnitude of the correlations) is positively associated with (1) worldmarket volatility and (2) a trend; while it is negatively related to (3) exchange ratevolatility, (4) term structure differentials across markets, (5) real interest rate differentials,and (6) the return on a world market index.

This analysis sheds light on the economic forces that influence the correlation structureover time, and thus the evolution in global capital market integration. For example, theseresults corroborate a priori expectations that divergent macroeconomic behavior acrosscountries tends to be associated with divergent behavior across national equity markets,resulting in lower market correlations. In addition, the influence of economic factors (1)and (6) listed above indicate an increase in comovement across international equitymarkets when world markets are more volatile and/or falling. This outcome suggests adecline in the risk-reduction benefits of international diversification at the very time whenthese benefits are needed most.

Finally, the economic model is employed to generate out-of-sample forecasts of thecorrelation structure. The forecast performance of this model dominates that of otheratheoretical models which ignore economic determinants of the correlation structure. Thisoutcome adds credence to the validity of the theoretical economic model, and suggests thatmodeling the correlation structure holds promise in assisting managers to exploit inter-national investment opportunities.

The paper proceeds as follows. Section II reviews the literature on stability in thecorrelation structure, discusses the data, and presents the stability tests. Section IIIdevelops the theoretical model describing economic determinants of the correlationstructure, and Section IV estimates the model. Section V compares the forecast perfor-mance of this model with respect to alternative forecasting techniques. A final sectionsummarizes and concludes.

II. Literature Review, Stock Market Data, and Stability Tests

Literature Review

The existing literature offers mixed evidence on the stability of the correlation structure.Several earlier studies by Panton et al. (1976), Watson (1980), and Philippatos et al.(1983) all find support for stable relationships across national equity markets. In contrast,Makridakis and Wheelwright (1974), Haney and Lloyd (1978), Maldonado and Saunders(1981), Fischer and Palasvirta (1990), Madura and Soenen (1992), Wahab and Lashgari(1993), and Longin and Solnik (1995) argue that the relationships are unstable. Fallingsomewhere between these two camps, Kaplanis (1988) suggests that correlations arestable while covariances are unstable; Marcus et al. (1991) suggest that the holding period

3 For other studies that motivate this hypothesis, see Arshanapalli and Doukas (1993), Bachman et al. (1996),Bodurtha et al. (1989), Campbell and Hamao (1992), and Roll (1992).

Determinants of International Correlation 445

analyzed has a bearing on the correlation structure observed; and Meric and Meric (1989)find instability for shorter periods, offset by stability over longer periods.

The disparity in results across this literature is presumably attributable to the widerange of sample periods and sampling frequencies examined, as well as different meth-odologies employed.4 Interestingly, most recent studies tend to find greater instability,suggesting that interrelationships among national stock markets may have undergone asubstantive change during the 1980s.

Stock Market Data

The data include Morgan Stanley Capital International’s daily closing stock index valuesfor ten national markets (Australia, Canada, Germany, Hong Kong, Japan, Mexico,Singapore, Switzerland, the United Kingdom, and the United States), as well as dailybilateral exchange rates between the US dollar and the nine other currencies, from 1972through 1993.5 The analysis is applied to daily returns for every national market,denominated both in US dollars and in the home currency.

Tests for Stability in the Correlation Matrix

From these time series of daily stock returns, we compute the correlation between everypair of national markets over each quarter during this 22-year sample period. The resultingtime series of 88 quarterly observations on the correlation matrix reveals the nature andextent of changes in the correlation structure over time. This quarterly time series is thenemployed to test for stability in the correlation structure, using a Jennrich (1970) test forthe equality of two correlation matrices.6

Table 1 presents the results of three different approaches for investigating the hypoth-esis that the correlation matrix does not change over time. First, we examine the nullhypothesis that the correlation matrix does not change from one quarter to the next, acrosseach of the 88 quarters in the sample period. Panel A of Table 1 presents the frequencyof rejections out of these 87 tests comparing consecutive quarterly correlation matrices. Ifthe correlation matrix is truly constant, we would expect approximatelynine rejections outof 87 tests at the 0.10 level of significance (along withfive rejections at the 0.05 level andone rejection at the 0.01 level). Panel A reveals that stability is rejected far more

4 Different studies in this literature employ data: (1) taken over different sample periods which sometimesextend prior to the early 1970s, when world equity and currency markets were more stable, and (2) taken atdifferent frequencies (daily, weekly, monthly, quarterly, or annually).

5 Morgan Stanley Capital International (MSCI) data on individual firm stocks around the world representapproximately 60% of the world’s total market capitalization. From these data MSCI constructs national stockindexes that are fully comparable. The ten national equity markets chosen for this analysis account forapproximately 90% of the world’s total market capitalization.

6 Appendix A provides the formal Jennrich (1970) test. Data for Mexico are unavailable prior to 1988.Therefore, stability tests where both periods tested begin after 1988 use all ten countries. Otherwise only ninecountries are employed.

Following Meric and Meric (1989), a Box M test is also applied to test for equality across correlationmatrices. The Jennrich (1970) test is the more conservative approach, designed specifically for testing equalityof correlation matrices, while the Box M test is primarily used for investigating covariance matrices. In addition,all stability tests have been performed using both Pearson and Spearman correlation estimates. Results are robustacross both stability tests and both correlation measures, as well as across data using both US dollar returns andhome currency returns. For brevity, we report only results for the Jennrich test on Pearson correlations using USdollar-denominated data. All results are available upon request.


Table 1. Jennrich Tests for Equality of Two Correlation Matrices.a

Pearson correlations are estimated from daily returns in U.S. dollars across ten national markets,for each quarter throughout the 22-year period, 1972–1993. The result is a time series thatembodies quarterly movements in the correlation matrix across national equity markets.

The Jennrich (1970) test is applied to investigate the equality of:A: consecutive quarterly correlation matrices;B: nonconsecutive quarterly correlation matrices (one, two, and three quarters apart);C: consecutive correlation matrices estimated over time intervals longer than one quarter.

This Table presents the frequency of rejections in applying these Jennrich (1970) tests.b

Panel A: Frequency of Rejections Across Consecutive Quarterly Correlation Matrices

Panel A10% Level

(1 2 p) 5 .105% Level

(1 2 p) 5 .051% Level

(1 2 p) 5 .01

Number of times that equality is rejected outof 87 Tests

39*** 29*** 13***

Panel B: Frequency of Rejections Across Non-Consecutive Quarterly Correlation Matrices

Panel B: 10% Level 5% Level 1% Level

1 Quarter apart (86 tests) 42*** 28*** 18***2 Quarters apart (85 tests) 47*** 33*** 16***3 Quarters apart (84 tests) 50*** 34*** 23***

Panel C: Frequency of Rejections Across Consecutive Time Intervals Greater than One Quarterin Length

Panel C: 10% Level 5% Level 1% Level

Semiannual (43 tests) 33*** 22*** 16***Annual (21 tests) 20*** 19*** 16***Biannual (10 tests) 10*** 10*** 9***Five-and-one-half years (3 tests) 3 3 3Eleven years (1 test) 1 1 1

a Jennrich (1970) provides a Chi-square statistic to test the equality of two correlation matrices; details appear in Appendix A.b Results are robust with respect to the use of Spearman correlations, home currency returns, and the alternative Box M test

for equality across correlation matrices.*** Indicates that the number of rejections is greater than expected by chance 1% of the time, within a binomial framework

that bases the probability of acceptance for each individual test atp (5 .90, .95, and .99), and the probability of rejection at 1-p(5 .10, .05, and .01). Therefore, (12 p) represents the significance level for each individual test. This binomial analysis is notapplied to the last two rows in Panel C due to the small number of tests conducted in each case.

To elaborate, this binomial framework considers the outcome of each test to be drawn independently from a binomialdistribution. If we consider an acceptance of the null in one trial to be a success, and a rejection to be a failure, then at the 10%level of significance the probability of success for every test is .90 and the probability of failure is .10. By viewing the sequenceof tests this way, we can determine the ‘‘critical values’’ for the number of rejections expected at the 10%, 5%, and 1%significance levels. Given a probability of success atp 5 .90 (the 10% level of significance), with 87 independent tests thereis less than a 1% chance of rejecting the null more than16 times. Similarly, given a probability ofp 5 .95, there is less thana 1% chance of rejecting the null more than10 times. Finally, given a probability of success atp 5 .99, there is less than a 1%chance of rejecting the null more than4 times. Since the numbers in each cell of Panel A are greater than these respective‘‘critical values,’’ the number of rejections indicated in each cell of Panel A is greater than would be expected 1% of the time,under this binomial exercise.


frequently than is expected by chance (39 rejections at the 0.10 level,29 rejections at the0.05 level, and13 rejections at the 0.01 level).

We can more formally address the issue of whether the number of rejections docu-mented in Table 1 is “significantly greater than expected” (at each level of significance),by considering the outcome of every test as being drawn independently from a binomialdistribution. If we consider an acceptance of the null in one trial to be a success, and arejection to be a failure, then at the 10% level of significance the probability of successfor that trial is .90 and the probability of failure is .10. By viewing the sequence of teststhis way, we can determine the “critical values” for the number of rejections expected atthe 10%, 5%, and 1% significance levels. Given a probability of success atp 5 .90 (the10% level of significance), with 87 independent tests there is less than a 1% chance ofrejecting the null more thansixteentimes. Similarly, given a probability of success atp 5.95, there is less than a 1% chance of rejecting the null more thanten times. Finally, givena probability of success atp 5 .99, there is less than a 1% chance of rejecting the null morethanfour times. Once again, because the numbers in Panel A of Table 1 are greater thanthese respective “critical values,” the number of rejections in each cell of Panel A isgreater than would be expected 1% of the time, under this binomial exercise.

Next, we consider the possibility that these quarterly correlation matrices may changemore slowly over time, by investigating the equality of correlation matrices that are one,two, and three quarters apart. Panel B of Table 1 presents the frequency of rejectionsacross nonconsecutive quarterly correlation matrices, and once again reveals far morerejections than could be expected by chance. Finally, Panel C considers the stability of thecorrelation matrix computed over longer time intervals of 6 months, 1 year, 2 years, 51⁄2years, and 11 years. Results in Panel C uniformly indicate instability over these longertime periods as well.

While our results are consistent with most recent studies on this issue, these latterfindings contrast with the results of Meric and Meric (1989), which suggest someunderlying long-run stability in the correlation matrix. This contrast is important. If therewere truly long-run stability, then the short-run instability documented in Panels A and Bof Table 1 would not be critical for long-term investors. Instead, our results overwhelminglyindicate significant changes in the correlation matrix over both short and long time horizons.

These results provoke the question as tohow andwhy the correlation structure variesover time. Before we proceed to address this question by developing the economic model,we must investigate whether each time series of pairwise correlations contains a unit root.If changes in the correlation structure contain a unit root (follow some form of randomwalk), then our model of the correlation structure may result in identifying spuriousrelationships with economic variables (Enders 1995).

In this light, we perform Augmented Dickey–Fuller (ADF) tests on each time series ofquarterly pairwise correlations. Results are provided in Table 2 and indicate rejection ofthe unit root hypothesis for 18 of the 21 time series of pairwise correlations, at the .10level or better. Because the vast majority of these time series have no unit root, it isreasonable to proceed with our attempt to model how and why the correlation structurechanges over time.

III. Modeling Economic Determinants of Correlation StructureDevelopment of the economic model begins with reference to prior work that analyzespotential macroeconomic determinants ofexpected returnsin a national equity market


(See, for example, Gultekin, 1993; Solnik, 1983; Chen et al., 1986; Bodurtha et al., 1989;Campbell and Hamao, 1992). These studies suggest the following country-specific factorsas possible determinants of national equity returns:

IND it 5 industrial production growth in countryi during quartert;INFL it 5 inflation in countryi during quartert;

Table 2. Augmented Dickey-Fuller (ADF) Tests

This table presents the results of ADF tests that investigate the existence of a unit root in thetime series of quarterly correlation coefficients for each pair of national equity markets.

The regression model used to conduct the ADF test is specified as follows:

Dcorrt 5 a0 1 gcorrt21 1 b1 (Dcorrt21) 1 b2 (Dcorrt22) 1 «t.

Under H0: g 5 0, there is a unit root. Therefore a rejection of Ho:g 5 0 indicates no unit root.Results indicate no unit root (rejection of Ho) for 18 of the 21 time series of correlations.

Lagged changes in the correlation coefficient beyond two quarterly lags are insignificant.This analysis includes 88 quarters, but only 85 observations are used due to the two quarterly lags.Statistical significance is based on critical values generated from a conservative sample size of 50observations (significance levels based on a sample size of 100 do not alter the results.)

We have also conducted unit root tests using: (i) an expanded model that includes adeterministic trend, and (ii) an abbreviated model that omits both the trend and the intercept.Comparison of results across the three models indicates that the model specified above(including an intercept but omitting a trend) dominates the other specifications. The trend isalmost never significant, while the intercept is almost always significant. Results for bothalternative models are largely consistent with those presented here, indicating that most timeseries of pairwise correlations exhibit no unit root.

Country Pair g (t ratio)

Australia/Canada 20.729 (24.007)***Australia/Germany 20.400 (22.904)*Australia/Japan 20.341 (22.648)*Australia/Switzerland 20.255 (22.197)Australia/United King. 20.415 (23.155)**Australia/United States 21.099 (25.624)***Canada/Germany 20.455 (22.966)**Canada/Japan 20.520 (23.463)**Canada/Switzerland 20.514 (23.299)**Canada/United King. 20.384 (22.990)**Canada/United States 20.381 (22.634)*Germany/Japan 20.311 (22.700)*Germany/Switzerland 20.324 (22.971)**Germany/United King. 20.299 (23.162)**Germany/United States 20.619 (23.868)***Japan/Switzerland 20.255 (22.473)Japan/United King. 20.332 (22.917)*Japan/United States 20.671 (23.868)***Switzerland/Unit. King. 20.233 (22.473)Switzerland/United St. 20.748 (24.379)***United King./United St. 20.551 (23.963)***

* Indicates significance at the .10 level; ** at the .05 level; and *** at the .01 level.


INT it 5 real interest rate in countryi during quartert;LOSHit 5 term structure premium in countryi during quartert;7

SIZEit 5 share of total world market capitalization in countryi during quartert;

The economic rationale for hypothesizing that these variables are among the determinantsof national stock returns is grounded in the discounted cash flow model. The first fourvariables listed above represent different aspects of a country’s macroeconomic perfor-mance that affect expected cash flows and/or discount rates in that national market, andthus have a bearing on the market’s expected returns. The last variable listed above ismotivated by the well-documented size effect within a national market, whereby higherdiscount rates are demanded for smaller firms (Keim 1983; Reinganum 1983). Potentialexplanations for this firm-size effect include greater information costs, transaction costs,and less liquidity associated with trading equity in smaller firms. By extension, we suggestthat the relative size of a national equity market may also have a bearing on that country’sequity returns, due to greater information costs, transaction costs, and less liquidityassociated with trading equity in smaller national markets.

While this discussion serves to motivate potential macroeconomic determinants of thefirst moment (expected returns) across different elements in the vector of national equityreturns, we are interested in modeling determinants of the second moment (the correlationstructure).8 As a first step, we postulate that the extent of comovement between a pair ofnational markets may depend upon the extent to which these five macroeconomicvariables diverge across the two markets, as follows:

r ijt 5 b0 1 b1uIND i 2 IND jut 1 b2uINFL i 2 INFL jut 1 b3uINT i 2 INT jut

1 b4uLOSHi 2 LOSHjut 1 b5uSIZEi 2 SIZEjut 1 e ijt , (1)

where

rijt 5 estimated correlation between daily returns in countriesi and j during quartert;eijt 5 disturbance term, assumed to beiid N(0, s2).

Generally if there is greater divergence in macroeconomic behavior across countries, weexpect less comovement across equity markets, implying negative coefficients forb1–b5.

9

7 The term structure premium is defined as the difference between long term and short term government bondrates in countryi during quartert (Chen et al., 1986). This difference is a measure of the premium demandedfor long term investments in a country.

8 For other work that motivates and investigates this issue, see Arshanapalli and Doukas (1993), Bachmanet al. (1996), Bodurtha et al. (1989), Campbell and Hamao (1992), Roll (1992), and Bracker et al. (1999).

9 The recent convergence in various aspects of macroeconomic behavior across European Union membercountries (such as interest rates and inflation rates), as specified in the Maastricht treaty, serves to illustrate andmotivate the spirit of our theoretical specification in Equation 1.

The empirical validity of this specification requires some attention. Note that Equation 1 constrains eachmacroeconomic variable to enter the model as the absolute differential in behavior across marketsi andj, duringquartert. It is conceivable, however, that each variable in countryi has a substantive influence on the correlation,independent of the analogous factor in countryj. That is, it may be more appropriate to allow each factor in bothcountries to enter the regression model independently, rather than in the form of an absolute differential, asfollows:

r ijt 5 a0 1 a1IND it 1 a2IND jt 1 a3INFL it 1 a4INFL jt 1 a5INT it 1 a6INT jt 1 a7LOSHit 1 a8LOSHjt

1 a9SIZEit 1 a10SIZEjt 1 e ijt .

The model sp


In addition to motivating the potential influence of the above five macroeconomicdifferentials on rijt , the discounted cash flow model further motivates expansion ofEquation 1 to incorporate five additional macroeconomic variables that may also directlyinfluence international correlations.

First, bilateral trade conditions may impact national equity index returns for a givenpair of trading partners (Bodurtha et al., 1989). As exports from economyi to economyj (Xij) increase, higher cash flows should be expected into countryi. In contrast, theimports of economyi from economyj (Mij) may have the opposite effect on cash flowsfor countryi’s firms. In this light, changes in the absolute magnitude of the trade balancebetween two economies (uXij 2 Mij u) should positively influence one economy and stockmarket, while negatively influencing the other, to exert a negative impact on the corre-lation, rijt . The extent of this impact, however, should reflect the degree of importance ofthis bilateral trade activity with respect to aggregate economic activity in each nationalmarket involved. If this bilateral trade balance reflects a small proportion of GrossDomestic Product (GDP) forone of the two countries, it is not likely to exert muchinfluence on that country’s stock market. On the other hand, if the trade balance representsa large proportion of GDP foronecountry, we may expect a substantive impact on stockreturns in that country, and thus a substantive response inrij . Furthermore, ifuXij 2 Mij urepresents a large proportion of GDP forboth countries, we would expect a greaterresponse inrij . In this light we construct the following variable to incorporate the potentialinfluence of the trade gap onrij , from the point of view of both countries:

GAPij 5 ~uXij 2 Mij u!/GDPi 1 ~uXij 2 Mij u!/GDPj.

Second, this trade gap variable may not adequately reflect the full impact of bilateraltrade conditions on the correlation structure, because a given pair of countries may have a tradegap near zero while engaging in a substantial amount of trade. In this light, we also incorporatea second variable to account for the total amount of trade across the two countries:

TRADEij 5 ~Xij 1 Mij!/GDPi 1 ~Xij 1 Mij!/GDPj.

This variable emphasizes the notion that both exports and imports have a role in wealthcreation, and may thus influence the interaction of national equity markets. Appendix Bprovides further insight into this specification of the GAPij and TRADEij variables.

Third, absolute changes in the bilateral exchange rate (uXRCHij u) may influence thetrade conditions discussed above, and may thus also influence national equity returns inboth countries. If the exchange rate changes by a larger percentage, then we expect alarger adjustment in bilateral trade conditions (with a lag) in favor of the depreciating

The model specified in (1) imposes the following constraints on the above model:

a1 5 2a2(5b1); a3 5 2a4(5b2); a5 5 2a6(5b3); a7 5 2a8(5b4); a9 5 2a10(5b5).

These restrictions have been tested jointly in the regression model for each bilateral correlation in our sample,and are rarely rejected (5 times out of 21 equations at the .05 level of significance, and never at the .01 level).The fact that these constraints are not empirically binding supports our postulate that it is the differential behavioracross markets that influences the correlation structure. The theoretical and empirical validity of these constraintstherefore compels us to focus on Equation 1 as the appropriate model for analysis and interpretation regardingthese five economic influences.


country. This observation suggests a potential indirect negative influence of absoluteexchange rate changes on the correlation, through their impact on the trade gap.10

Fourth, volatility in the bilateral exchange rate (XRSDij) represents another source ofuncertainty which may dampen economic and equity market integration. If all countries’returns are calculated on a US Dollar basis, each national return contains both an equityreturn component and a currency return component. As currency rates become morevolatile, the currency component becomes more important relative to the equity returncomponent. In this case, higher exchange rate volatility may be expected to dampen thecorrelation between different pairs of national equity market returns, denominated in USDollars. This dampening effect should be less important across home currency returns thatabstract from exchange rate movements.

Fifth, overall volatility across the world’s stock markets may influence the level ofdiscount rates commanded around the world. As the variance of a world equity index(WLDVOL) increases, investors around the world may demand higher rates of return tocompensate this risk, resulting in higher correlations across different pairs of national equitymarkets. Erb et al. (1994), Farrell (1997), Longin and Solnik (1995), and Solnik et al. (1996)all argue that world market volatility is an important determinant of correlations acrossnational markets.

Finally, in addition to the above ten macroeconomic variables, we incorporate severaladditional factors that appear in anecdotal discussions of potential influences on thecorrelation structure, although they do not enter explicitly into the discounted cash flowmodel. First, Erb et al. (1994) observe that correlations tend to be low during times ofgeneral upward trends in world equity valuation, but that correlations become higher whenstock returns are declining around the world. This observation suggests asymmetricbehavior of the correlation structure in times of rising versus falling markets worldwide.In this light, the return on a world market portfolio (WLDMKT) may exhibit a negativeassociation with the correlation structure over time. Second, evolution toward globalcapital market integration would suggest that correlations are trending upward over timedue to factors such as greater interdependence across national economies, improvedtelecommunications technology, global deregulation of markets, more cross-listing ofsecurities, growth in multinational activities, increasing international diversification, andso forth (Bachman et al., 1996; Kaplanis, 1988; Kasa, 1992; Longin and Solnik, 1995).This possibility is accounted for by incorporating a trend in the regression model. Third,these correlations were dramatically higher during October 1987 and October 1989, twoperiods of increased world market volatility (Roll, 1989). Thus, two dummy variables areincorporated to account for potentially aberrant behavior during the fourth quarters of1987 and 1989, respectively. Fourth, it is well-documented that different national equitymarkets have experienced seasonal patterns in market activity and valuation. Meric andMeric (1989) further document that the correlation matrix is less stable during the summer

10 It is important to note that the three variables,uINFL i 2 INFL ju, uINT i 2 INT ju, anduXRCHij u, are relatedthrough international parity conditions (Eiteman et al., 1998). In fact, if purchasing power parity and interest rateparity were to hold perfectly in all periods, these three variables would be collinear. This situation may givecause for concern about potential multicollinearity in the regression model. However, we observe that it isdeviations frompurchasing power parity and interest rate parity that should induce capital flows and trade flowsacross countries. Hence these deviations may have a bearing on the extent of comovement across internationalequity markets. This observation provides another economic rationale for including all three variables in ourmodel (see Bodurtha et al., 1989).


months. In this light, we account for potential seasonality in the correlation matrix byadding quarterly dummy variables.

The final regression model incorporates all of these influences, as follows:

r ijt 5 b0 1 b1uIND i 2 IND jut 1 b2uINFL i 2 INFL jut 1 b3uINT i 2 INT jut

1 b4uLOSHi 2 LOSHjut 1 b5uSIZEi 2 SIZEjut 1 b6GAPijt 1 b7TRADEijt

1 b8uXRCHij ut 1 b9XRSDijt 1 b10WLDVOL t 1 b11WLDMKT t 1 b12TREND

1 b13OCT871 b14OCT891 b15Q11 b16Q21 b17Q31 e ijt ; (2)

wherei 5 country 1 to 6;j 5 country (i 1 1) to 7; t 5 quarter 1 to 88;

IND it 5 Growth in industrial production in countryi during quartert;INFL it 5 Inflation Rate in countryi during quartert;

INT it 5 Real interest rate (long term government rate-inflation rate) duringquartert;

LOSHit 5 Spread between long and short term bond rates in countryi duringquartert;

SIZEit 5 Percent of world equity market share in marketi during quartert;GAPijt 5 (uXij 2 Mij ut)/GDPit 1 (uXij 2 Mij ut)/GDPjt;

TRADEijt 5 (Xij 1 Mij)t/GDPit 1 (Xij 1 Mij)t/GDPjt;XRCHijt 5 Percent change in bilateral exchange rate during quartert;XRSDijt 5 Standard Deviation in daily bilateral exchange rate during quartert;

WLDVOL t 5 Standard deviation of daily world stock market index return duringquartert;

WLDMKT t 5 Percent change in world stock market index during quartert;TREND 5 Nonlinear trend, ln(t);

OCT87(89) 5 Dummy variable equal to 1 in fourth quarter of 1987(1989);Q1, Q2, Q3 5 Seasonal Dummy variables for the first three quarters.

Appendix C discusses data sources for the variables specified above.

IV. Estimation of the Economic ModelFor three countries (Hong Kong, Mexico, and Singapore), daily stock market data and/orquarterly macroeconomic data are unavailable over a considerable portion of the sampleperiod. Hence, these three countries are not included in the regression analysis. Theremaining seven countries offer 21 distinct pairwise correlations. A different time seriesregression Equation 2 is specified for each such pairwise correlation.

These 21 regression equations are estimated in two different ways. First, all equationsare estimated as a pooled sample, constraining all regression coefficients (except theintercept) to be identical across all 21 equations. This pooled approach offers a potentialgain in power in analyzing the statistical significance of each economic determinant of thecorrelation structure. Second, we also estimate the 21 equations as a system of seeminglyunrelated regressions (SURs). The latter approach incorporates possible contemporaneouscorrelation across regression error terms, and allows the individual parameter estimates tovary across different pairs of countries.


Table 3. Results of Pooled Time Series and Cross-Sectional Regression Analysisa

Equation 2 specifies the economic model describing quarterly time series movements in thecorrelation structure. The dependent variable is the bilateral correlation between national marketsi and j, computed from daily returns over each quarter throughout the period, 1972–1993.Complete quarterly economic data are available for seven countries. Hence, we specify adifferent equation to determine the correlation between each of the 21 possible pairs of theseseven national markets. This table presents results of estimating this model as a pooled system,constraining the influence of each economic variable to be identical across all 21 equations,while allowing the intercept to vary.The second column presents results using U.S. dollar-denominated returns. The third columnpresents results using home currency returns.

Variable US Dollar Based Returns Home-Currency Returns

Interceptb 20.1780*** 20.0992**(24.337) (22.396)

0.0117 0.0832uIND i-IND ju (0.198) (1.433)

20.0136 20.1363uINFL i-INFL ju (20.033) (20.341)

20.0124*** 20.0049uLOSHi-LOSHju (23.239) (21.429)

0.0014** 0.0008uSIZEi-SIZEju (2.217) (1.422)

20.0019 20.0090***uINT i-INT ju (20.696) (23.890)

20.0040 20.0024uGAPij u (21.064) (20.707)TRADEij 20.0007 20.0010

(20.481) (20.775)0.0433 0.1603

uXRCHij z (0.409) (1.520)XRSDij 28.7695*** 28.4812***

(25.941) (25.873)WLDMKT 0.0711 20.1399***

(1.535) (23.066)WLDVOL 37.2466*** 33.9431***

(14.859) (14.853)TREND 0.0589*** 0.0482***

(10.672) (10.053)OCT87 20.2532*** 20.2410***

(24.659) (24.675)OCT89 20.0094 0.0775**

(20.272) (2.274)Q1 0.0111 0.0095

(1.168) (0.986)Q2 0.0093 20.0143

(0.884) (21.376)Q3 0.0369*** 0.0285***

(3.825) (2.894)AUSCAN 20.0229 20.1029**

(20.445) (22.354)AUSGER 0.0398 20.0297

(0.798) (20.700)


Results of the Pooled Regression Model

Pooled regression results are presented in Table 3 for both US Dollar-based data andhome-currency returns. First, the goodness of fit statistics indicate that these economicdeterminants offer substantive explanatory power regarding time series movements in thecorrelation structure. Second, the intercept varies substantively across many of the 21

Table 3. (Continued)

Variable US dollar Based Returns Home-Currency Returns

AUSJAP 0.0652 0.0029(1.682) (0.090)

AUSSWT 0.0696 0.0243(1.294) (0.532)

AUSUK 0.0365 20.0631(0.808) (21.652)

AUSUS 20.1517*** 20.1873***(23.900) (25.682)

CANGER 0.0208 20.0594(0.408) (21.372)

CANJAP 0.0421 20.0948**(20.955) (22.538)

CANSWT 0.0648 0.0059(1.230) (0.131)

CANUK 0.0868 0.0313(1.914) (0.817)

CANUS 0.4141*** 0.4416***(5.366) (6.597)

GERJAP 0.1487*** 20.0467(3.122) (21.149)

GERSWT 0.4487*** 0.2120***(10.563) (5.935)

GERUK 0.2062*** 0.0338(5.145) (1.009)

GERUS 20.1199*** 20.1458***(22.897) (24.150)

JAPSWIT 0.1840*** 20.0187(4.011) (20.480)

JAPUK 0.1030** 20.0414(2.263) (21.071)

JAPUS 20.0959** 20.1346***(22.249) (23.710)

SWITUK 0.2346*** 0.0749**(5.361) (2.032)

SWITUS 20.0783 20.0889**(21.921) (22.568)

Adjusted R2 0.4251 0.4410F value 37.890*** 40.353***

a A Cochrane–Orcutt transformation is employed to correct for first-order autocorrelation in the error term in (11). Resultsare generally robust without this transformation.

b This coefficient presents the intercept for the corrrelation between the US and UK markets. The remaining 20 pairwisecorrelations are allowed to have intercepts that deviate from the US-UK intercept. The 20 coefficients that are presentedfollowing the model’s main parameters represent the difference between the intercept for each pair of markets and the interceptfor the US and UK markets.

* Indicates significance at the .10 level;** Indicates signficance at the .05 level;*** Indicates significance at the .01 level.


pairwise equations in the model. Third, economic variables that are significant at the 0.01level for both US Dollar and home-currency returns include exchange rate volatility,world market volatility, the time trend, the October 1987 dummy variable, and a seasonaldummy for the third quarter. For US Dollar-denominated returns, two additional signif-icant explanatory variables appear in the differential in the term structure premium(uLOSHi 2 LOSHju) and the size differential (uSIZEi 2 SIZEju). Alternatively, for home-currency returns, additional significant factors include the real interest differential(uINT i 2 INT ju), the world market return (WLDMKT), and the October 1989 dummy. Weelaborate on the individual influence of each regression variable below in the discussionof the SUR model.

Results of the Seemingly Unrelated Regressions (SUR) Model

Estimation of the SUR model yields 17 parameter estimates for each of the 21 regressionequations. To summarize the nature and strength of these results, we present the frequencythat each parameter estimate takes on a (significant) positive or negative value across all21 equations in Tables 4 and 5, for US Dollar-denominated returns and for home-currencyreturns, respectively.11

These SUR results largely corroborate the pooled results. First, for US Dollar returnsin Table 4, economic variables that enter significantly across a substantial number ofcountry pairs include (in order of their importance) world market volatility, the trend,exchange rate volatility, the real interest differential, the size differential, total trade, andthe October 1987 dummy. For home currency returns in Table 5, analogous results pointto world volatility, the trend, the real interest differential, the October 1987 dummy, thedirection of world market returns, total trade, exchange rate volatility, and the termstructure differential. We elaborate below.

The dominant factor in this analysis is world equity market volatility, which issignificantly positive in all 21 equations. This is a reasonable result, given that greatervolatility in the world index is likely to be associated with worldwide shocks that affectmany markets (see Solnik et al. 1996).12

Exchange rate volatility also displays a stable dampening effect on the correlationsacross national equity markets. For US Dollar returns in Table 4, the coefficient for thisvariable is negative in 19 of 21 equations, and is statistically significant in 12 equations.In home currency returns (Table 5), this coefficient is negative in 17 equations but

11 We have also re-estimated the model over two subsamples of equal length, to investigate the stability ofparameter estimates across this 22-year sample period. With the exception of the time trend, results are generallyrobust and are available upon request. The unstable trend in the correlation structure is discussed below.

12 The return variance for any portfolio is, by definition, positively related to the return variances of itscomponents, as well as the covariances across all possible pairs of components. Due to the influence ofcovariances, we may expect a “definitional” relationship between world market volatility and the correlationstructure acrossall national markets.

In order to investigate whether the effect we document is more than just such a definitional relationship, wehave re-applied the analysis using an alternative measure of world market volatility which incorporates only thevariances of our subset of the seven individual national components in the world portfolio (and ignores theircovariances, as well as all other countries in the world market portfolio). Our alternative measure is the averagestandard deviation across the seven national markets in our analysis. Overall results are robust with respect tothis alternative specification. While the coefficient of this specification for world market volatility declinesslightly in magnitude, it is still highly significant (at the .0001 level in the pooled regression, and at the .05 levelin twelve of the 21 SUR equations). This result indicates more than just a definitional relationship between worldmarket volatility and the correlation structure.


significant in only three equations. Thus, as expected, this dampening effect appears lessimportant for a portfolio that abstracts from exchange rate movements. It is noteworthythat the pooled results in Table 3 reveal a significant negative influence of exchange ratevolatility, regardless of the adjustment for exchange rate movements.

The trend is positive in 20 equations, and significant in 12, on a US Dollar basis inTable 4. Similar results obtain for home-currency returns. This outcome supports a trendtoward greater integration over time. On the other hand, re-estimation of the model overtwo equal subperiods reveals that this trend is predominantly in the first 11-year subpe-riod.13

13 The specification of the trend variable as the natural log of the quarter (1 through 88) implies a morepronounced trend over the first half of the 22-year sample period. When estimated over the second 11-yearsubperiod (1983–1993), this trend appears slightly negative for US dollar returns, or neutral for home-currencyreturns. A linear trend and piecewise linear trends (allowing the slope to change in the fourth quarter of 1985or the fourth quarter of 1987) have also been investigated. The overall results (available upon request) are robustwith respect to each specification. The observation that the trend is statistically significant when estimated overthe entire collection of correlations in the pooled model and over the entire sample period, while it is insignificantfor many bilateral pairs in the SUR model and over two 11-year subsamples, is consistent with the results ofBachman et al. (1996), Kasa (1992), and Solnik et al. (1996).

Table 4. Results of Seemingly Unrelated Regression (SUR) Analysis: US Dollar-Based Data

Equation 2 specifies the economic model describing quarterly time series movements in thecorrelation structure. The dependent variable is the bilateral correlation between national marketsi and j, computed from daily returns over each quarter throughout the period, 1972–1993.Complete quarterly economic data are available for seven countries. Hence, we specify adifferent equation to determine the correlation between each of the 21 possible pairs of theseseven national markets. This table presents results of estimating this system as a SUR model.For brevity, we provide the frequency that the parameter estimate for each economic variabletakes a (significant) positive or negative value, across all 21 equations.

Variable

Number ofPositive

Coefficients

Number ofCoefficientsSignificantly

Positive at the5% Level

Number ofNegative

Coefficients


Negative at the5% Level

INTERCEPT 8 1 13 1uIND i-IND ju 11 0 10 1uINFL i-INFL ju 10 1 11 1uLOSHi-LOSHju 8 1 13 2uSIZEi-SIZEju 9 2 12 5uINT i-iNT ju 9 2 12 6uGAPij u 9 1 12 2TRADEij 3 0 18 4uXRCHij u 12 1 9 1XRSDij 2 0 19 12WLDMKT 10 0 11 1WLDVOL 21 21 0 0TREND 20 12 1 0OCT87 0 0 21 4OCT89 6 2 15 0Q1 14 0 7 1Q2 12 0 9 1Q3 18 0 3 0


The real interest differential reveals a negative influence on the correlation structure,especially for home currency returns. These results suggest that, as real interest ratesdiverge across two markets, stock returns also tend to diverge resulting in a lowercorrelation.

Greater trade flows between countries reflect greater economic integration, which maybe expected to result in greater equity market integration. However, Tables 4 and 5 reveala tendency for the total trade variable to display a negative influence on the correlation,which is counterintuitive. In contrast, the pooled regression in Table 3 reveals nosignificant relationship.

There is mixed evidence for the argument that two markets should display lesscorrelation if they are more divergent in size. For US Dollar returns in Table 4, thecoefficient on the size differential is negative 12 times (with five coefficients significant),while it is positive nine times (with two significant). Results are similar for home currencyreturns in Table 5. These results lean toward the argument that markets more disparate insize have lower correlations. In contrast, the pooled regression in Table 3 suggests apositive relationship between the size differential and the correlation structure.

The world market return exhibits a tendency for a negative relationship with thecorrelation structure, but only for returns stated on a home currency basis. In Table 5, this

Table 5. Results of Seemingly Unrelated Regression (SUR) Analysis: Home Currency Data

Equation 2 specifies the economic model describing quarterly time series movements in thecorrelation structure. The dependent variable is the bilateral correlation between national marketsi and j, computed from daily returns over each quarter throughout the period, 1972–1993.Complete quarterly economic data are available for seven countries. Hence, we specify adifferent equation to determine the correlation between each of the 21 possible pairs of theseseven national markets. This table presents results of estimating this system as a SUR model.For brevity, we provide the frequency that the parameter estimate for each economic variabletakes a (significant) positive or negative value, across all 21 equations.

Variable

Number ofPositive

Coefficients


Positive at the5% Level

Number ofNegative

Coefficients


Negative at the5% Level

INTERCEPT 8 1 10 1uIND i-IND ju 13 0 8 1uINFL i-INFL ju 11 2 10 1uLOSHi-LOSHju 11 0 10 3uSIZEi-SIZEju 8 1 13 2uINT i-INT ju 5 1 16 6uGAPij u 12 1 9 2TRADEij 6 0 15 4uXRCHij u 11 1 10 1XRSDij 4 0 17 3WLDMKT 4 0 17 4WLDVOL 21 21 0 0TREND 19 10 2 0OCT87 1 0 20 4OCT89 10 6 11 1Q1 13 1 8 1Q2 10 0 11 0Q3 13 1 8 1


coefficient is negative 17 times, and is significant four times. This negative coefficient isreinforced in the pooled analysis of Table 3. This result is consistent with the observationsof Erb et al. (1994), suggesting greater comovement across international equity marketswhen world markets are declining.

Finally, during the global crash of October 1987 international market correlationsincreased dramatically, leading us to expect a positive coefficient for the October 1987dummy variable. Instead, this dummy coefficient is consistently negative across allequations, indicating that the fitted model over-estimates the correlation structure duringthis period. These high-fitted values are attributable to the strong positive associationbetween world market volatility and the correlation for all 21 pairs of countries, combinedwith the extraordinarily large value for world volatility experienced in the fourth quarterof 1987.14 When the model is re-estimated without the October 1987 dummy, thecoefficient on world volatility declines by 25% in the pooled regression, and is reducedin most equations of the SUR model. In this light, the October 1987 dummy should beviewed as a tool that allows the model to reveal the influence of world volatility (and theremaining economic factors) under normal market conditions, abstracting from the aber-rant behavior during the fourth quarter of 1987. This is a desirable feature of our modelin light of the forecasting goals we address next.

In summary, the estimated pooled and SUR models indicate the following. Forcorrelations estimated with both US dollar and home currency returns:

(1) world market volatility (WLDVOLij) is positively associated with therij ;(2) a positive trend (TREND) in therij appears in the first half of this 22-year period,

from 1972 to 1982, while no trend appears from 1983 to 1993; and(3) exchange rate volatility (XRSDij) has a dampening effect on therij .

In addition, to a lesser extent, for US dollar returns:

(4) term structure differentials (LOSHi 2 LOSHj) are negatively related torij ;

while for home currency returns:

(5) real interest differentials (INTi 2 INT j) are negatively related torij ; and(6) the world market return is negatively associated with therij .

V. Forecasting the Correlation StructureThe above analysis improves our understanding of how and why the correlation structurechanges over time, and thus contributes to the dialogue on global integration. A stringenttest of the validity of any such economic model is its out-of-sample forecasting ability. Inthis light, we compare the forecasting ability of our economic model with that of fouratheoretical forecasting models.15

14 The mean correlation across the twenty-one pairs of national equity markets during the fourth quarter of1987 is 2.55 standard deviations above the mean for the entire 88-quarter sample period, for US dollar returns(and 3.29 standard deviations above the mean for home-currency returns). However, the volatility of worldmarket returns (WLDVOL) during this quarter is greater still, at 6.73 standard deviations above the mean.

15 See, for example, Erb et al. (1994), Eun and Resnick (1984 and 1992), and Kaplanis (1988).


Forecasting Models

Consider first the SUR economic model specified in Equation 2. Assuming that therelation in Equation 2 also holds in periodt 1 1, and taking conditional expectations, weobtain the one-step-ahead forecast:

E@r ijt 11uI t# 5 b0 1 b1Et@uIND i 2 IND jut11uI t# 1 b2Et@uINFL i 2 INFL jut11uI t#

1 b3Et[ uINT i 2 INT jut11uI t] 1 b4Et@uLOSHi 2 LOSHjut11uI t#

1 b5Et[ uSIZEi 2 SIZEjut11uI t] 1 b6Et@GAPijt 11uI t#

1 b7Et[TRADEijt 11uI t] 1 b8Et@uXRCHij ut11uI t# 1 b9Et@XRSDijt 11uI t#

1 b10Et[WLDVOL t11uI t] 1 b11Et@WLDMKT t11uI t# 1 b12TREND

1 b13OCT871 b14OCT891 b15Q11 b16Q21 b17Q31 Et@eijt11uIt#;(3)

whereEt[X,t11uIt] represents the expectation in periodt of variableX, in period t 1 1,conditional on information available in periodt, and Et[eijt11uIt] 5 0. Observe that allvariables in Equation 2 with time subscripts appear in the forecast, Equation 3, asconditional expectations. If we replace the conditional expectation of each such right-hand-side variable in Equation 3 with its actual value in periodt, we obtain a forecast ofthe correlation generated from this regression model. This approach is taken here togenerate forecasts with the economic model.16

Initially the SUR economic model in Equation 2 is re-estimated over the first 16 yearsof the 22-year sample period (1972–1987). The resulting 21 fitted equations in this modelare then employed in Equation 3 to generate out-of-sample one-step-ahead forecasts of thecorrelation structure. By updating the observations and re-estimating the model in Equa-tion 2 each quarter, a set of 24 one-step-ahead forecasts of the correlation matrix isgenerated over the 6-year holdout sample period (1988–1993).17

16 This is a conservative approach to implement our economic model to generate forecasts. This approachwould be appropriate, for example, if each right-hand-side macroeconomic variable followed a random walk. Analternative approach would be to use other forecasting methodology to generate one-step-ahead quarterlyforecasts of each conditional expectation on the right-hand side of Equation 3 to employ in forecastingrijt11.Such an effort remains the subject of future work.

17 Tables 4 and 5 indicate that the estimation results for the economic model vary substantially acrossdifferent pairs of countries. This result implies that different economic variables are important to differingdegrees for different pairs of countries. By employing the SUR model to forecast the correlations (rather thanthe pooled model), we allow different sets of economic variables to be more important in forecasting thecorrelations for different pairs of countries.

In addition, the economic forecasting model is fitted using Fisher transformations of the correlation as thedependent variable [z 5 .5*ln(1 1 rijt /1 2 rijt)] to ensure that all forecasted correlations range between21 and11. Estimation results are generally robust with respect to those presented in Tables 4 and 5, and are availableupon request.

Finally, we wish to address the use of a long (16-year) estimation period for the economic and ARIMAforecasting models. First, the model parameters are generally robust when we split the sample into twoeleven-year sample periods (1972–1982 and 1983–1993). Hence, the choice of sample period has little bearingon the estimation results. Second, due to the large number of variables specified in the economic model, arelatively long estimation period is desirable to have sufficient degrees of freedom to appeal to the asymptoticproperties of the estimation technique. For example, it would be unfeasible to employ annual periods forestimation and forecasting, due to the reduction in degrees of freedom associated with the use of such longerperiods.


The forecasting performance of theeconomicmodel is then compared with that of fouralternative models. We discuss these alternative models in order, from the simplest to themore complex. The first model represents a one-step-ahead forecast ofno changefrom theprevious quarter’s observation on the correlation matrix. While this naive model mayseem like a weak challenge, McNees (1992) has shown it to perform well in predictingmany economic variables. A second alternative model, similar to the first, sets theone-step-ahead forecast of each bilateral correlation equal to thehistorical averageoverthe past eight quarterly observations. The third challenger model employs an“empiricalBayes” approach similar to the top performing model for Kaplanis (1988). This modeleffectively generates a correlation forecast for every pair of markets each quarter thatregresses toward the global mean across all 21 correlation forecasts (see Kaplanis, 1988,for details). These first three models offer interesting alternatives, since most of themean-variance optimization studies in this literature implicitly assume that future corre-lations are the same as correlations in the recent past (Eun and Resnick 1992; Levy andSarnat 1970; Odier and Solnik 1993). The final alternative model is an ARIMA model foreach time series of pairwise correlations. While the structure of each ARIMA forecastingmodel proves to be stable across the estimation and holdout periods, the ARIMA modelparameters are re-estimated for each quarter throughout the holdout sample with updateddata.18

Forecasting Results

The performance of these five models is evaluated on the basis of minimizing the MeanSquared Error (MSE) across the set of 21 forecasts every quarter. Each model is appliedto forecast correlations estimated on both a US Dollar basis and a home currency basis.Tables 6 and 7 present the MSE across all 21 forecasts in the correlation structure for eachquarter of the 6-year holdout period, as well as for this entire 6-year period and three2-year subperiods. In addition, due to the influence of the October 1987 international crashon the forecasts for the first quarter of 1988, we provide results for the subperiod coveringthe second quarter of 1988 through 1993.

Results for the first quarter of 1988 reveal that, while the forecast performance of thehistorical average model and ARIMA model is not greatly affected by the October 1987crash, the same cannot be said for the remaining three models. This result is not surprising,as both the no-change and the empirical Bayes models rely on the previous quarter’scorrelation matrix to forecast the current quarter’s matrix. Similarly, the economic modelforecast is also heavily influenced by the value of world market volatility in the previousquarter. This behavior leads to forecasts from these three models that dramaticallyover-estimate the correlation matrix for the first quarter of 1988. In contrast, the historicalaverage model and ARIMA model are not as greatly affected by the aberrant behaviorduring the fourth quarter of 1987, as this quarter’s correlation matrix makes up only a

18 The exponentially weighted model employed in J. P. Morgan’sRiskMetrics(1996) package to forecastvariances and covariances is a special case of the general class of ARIMA models. Our application of thisapproach fits the unique ARIMA model that is appropriate for each pairwise time series of correlations andemploys it to forecast. Details on the fitted ARIMA models employed in this application, along with theirdiagnostics, are available upon request.

Of the twenty-one ARIMA models estimated in this study, 14 models have a statistically significant AR(1)term. When a laggedrijt21 term is added to the economic model, regression results are not significantly altered.


portion of the information set employed by these two models to forecast the first quarterof 1988.

The aberrant behavior of the forecasts for the first quarter of 1988 leads us to focus onthe forecasting results for the subperiod from the second quarter of 1988 through 1993,presented in the last row of Tables 6 and 7. Over this subperiod, the weakest forecast

Table 6. Mean Squared Errors for Forecasts: US Dollar-Based DataFive models are used to generate forecasts of the correlation matrix for next quarter, as follows:

(1) The No Change model employs the correlation from the previous quarter as the forecast;(2) The Historical Average model uses the average correlation over the previous eight quarters;(3) The Empirical Bayes approach regresses each bilateral correlation toward the global mean

across all correlations of the previous quarter;(4) The ARIMA model is used to forecast 1-step-ahead;(5) The fitted values from the economic model are used to project 1-step-ahead;

The estimation period for models (4) & (5) is 1972–1987; the holdout forecast period is 1988–1993.By updating data and re-estimating models (4) & (5) each quarter over the 6-year holdout sample, wegenerate a set of 24 one-step-ahead forecasts of the correlation matrix for each model.

This table presents the MSE across the 21 forecasts in the correlation matrix:(i) for each quarter throughout the 6-year holdout sample period, 1988–1993;(ii) for the entire 6-year holdout sample period 1988–1993;(iii) for three 2-year subperiods, 1988–1989, 1990–1991, and 1992–1993; and(iv) for the subperiod from the second quarter of 1988 through 1993.

NoChange

HistoricalAverage

Empirical BayesApproach

ARIMAModel

EconomicModel

1988: Q1 0.23582 0.05622 0.23255 0.05966 0.34584Q2 0.04593 0.01039 0.03563 0.01523 0.01705Q3 0.01656 0.02020 0.01684 0.01275 0.01520Q4 0.02621 0.02219 0.02334 0.01590 0.02274

1989: Q1 0.02742 0.01169 0.03019 0.01845 0.01234Q2 0.02828 0.02915 0.02753 0.02540 0.02960Q3 0.01825 0.02046 0.01341 0.02479 0.02258Q4 0.10241 0.05733 0.10197 0.06236 0.05325

1990: Q1 0.03070 0.01098 0.02533 0.01093 0.01358Q2 0.03904 0.02867 0.02539 0.02031 0.02637Q3 0.02732 0.03096 0.02013 0.03326 0.02371Q4 0.02643 0.02412 0.02206 0.01970 0.01950

1991: Q1 0.02423 0.03718 0.02301 0.04201 0.01917Q2 0.02754 0.02911 0.03018 0.02598 0.03092Q3 0.08582 0.07538 0.07409 0.08491 0.05707Q4 0.12171 0.01972 0.12367 0.03298 0.03729

1992: Q1 0.02047 0.02133 0.02505 0.02366 0.01518Q2 0.04106 0.01474 0.03215 0.01751 0.02139Q3 0.09475 0.07905 0.09166 0.04455 0.09985Q4 0.05275 0.04678 0.03858 0.01891 0.02324

1993: Q1 0.02023 0.04538 0.01818 0.03098 0.02411Q2 0.05255 0.05408 0.04891 0.04602 0.03560Q3 0.03758 0.04817 0.03152 0.03565 0.05326Q4 0.03979 0.01312 0.03195 0.01456 0.01693

Avg MSE 1988–1989 0.06261 0.02845 0.06018 0.02932 0.06482Avg MSE 1990–1991 0.04785 0.03201 0.04298 0.03376 0.02845Avg MSE 1992–1993 0.04490 0.04033 0.03975 0.02898 0.03620Avg MSE 1988–1993 0.05179 0.03360 0.04764 0.03069 0.04316Avg MSE 1988Q2–1993 0.04378 0.03262 0.03960 0.02943 0.03000


performance in terms of MSE is given by the naive model of no change from the previousquarter, for both US dollar and home currency returns. The empirical Bayes approach doesnot fare much better, ranking as the fourth best model for US dollar returns and the thirdbest model for home currency returns. The ARIMA model performs well for US dollar

Table 7. Mean Squared Errors for Forecasts: Home Currency Data

Five models are used to generate forecasts of the correlation matrix for next quarter, as follows:(1) The No Change model employs the correlation from the previous quarter as the forecast;(2) The Historical Average model uses the average correlation over the previous eight quarters;(3) The Empirical Bayes approach regresses each bilateral correlation toward the global mean

across all correlations of the previous quarter;(4) The ARIMA model is used to forecast 1-step-ahead;(5) The fitted values from the economic model are used to project 1-step-ahead;

The estimation period for models (4) & (5) is 1972–1987; the holdout forecast period is 1988–1993.By updating data and re-estimating models (4) & (5) each quarter over the 6-year holdout sample, wegenerate a set of 24 one-step-ahead forecasts of the correlation matrix for each model.

This table presents the MSE across the 21 forecasts in the correlation matrix:(i) for each quarter throughout the 6-year holdout sample period, 1988–1993;(ii) for the entire 6-year holdout sample period 1988–1993;(iii) for three 2-year subperiods, 1988–1989, 1990–1991, and 1992–1993; and(iv) for the subperiod from the second quarter of 1988 through 1993.

NoChange

HistoricalAverage

Empirical BayesApproach

ARIMAModel

EconomicModel

1988: Q1 0.18207 0.04718 0.17842 0.05782 0.14023Q2 0.03717 0.03166 0.02877 0.07376 0.05535Q3 0.01447 0.02179 0.01400 0.04268 0.02686Q4 0.03869 0.02325 0.03457 0.02115 0.02596

1989: Q1 0.02746 0.01462 0.02643 0.02236 0.01900Q2 0.04085 0.04554 0.03917 0.04323 0.02241Q3 0.02063 0.02607 0.01358 0.01493 0.01664Q4 0.12879 0.07718 0.12431 0.08159 0.09777

1990: Q1 0.04058 0.01172 0.03416 0.02967 0.02957Q2 0.03356 0.02441 0.02341 0.04069 0.01615Q3 0.05459 0.06773 0.04112 0.08063 0.05529Q4 0.03111 0.02706 0.03252 0.04661 0.01734

1991: Q1 0.01729 0.04554 0.01297 0.07746 0.03167Q2 0.04325 0.02921 0.04360 0.04297 0.03477Q3 0.07832 0.06299 0.06410 0.12861 0.07363Q4 0.07186 0.01206 0.07740 0.01590 0.01796

1992: Q1 0.02660 0.03601 0.02509 0.01858 0.01979Q2 0.06685 0.02081 0.05317 0.03585 0.04029Q3 0.04976 0.02867 0.03596 0.02324 0.02587Q4 0.04929 0.04835 0.04821 0.02709 0.02726

1993: Q1 0.02118 0.06143 0.01728 0.03878 0.02317Q2 0.03718 0.09656 0.02644 0.05821 0.03868Q3 0.04247 0.04261 0.03431 0.03072 0.03205Q4 0.05033 0.01918 0.04074 0.02017 0.02869

Avg MSE 1988–1989 0.06127 0.03591 0.05741 0.04469 0.05053Avg MSE 1990–1991 0.04632 0.03509 0.04116 0.05782 0.03455Avg MSE 1992–1993 0.04296 0.04420 0.03515 0.03158 0.02948Avg MSE 1988–1993 0.05018 0.03840 0.04457 0.04470 0.03818Avg MSE 1988Q2–1993 0.04445 0.03802 0.03875 0.04413 0.03375


returns (ranking first), but does considerably worse (fourth) for returns on a homecurrency basis. This outcome suggests that time series regularities may be more stable andpredictable for correlations across exchange rates than across equity index returns. Thehistorical average over the previous eight quarters consistently offers reasonably accurateforecasts that are worthy of note, ranking third for US dollar returns and second for homecurrency returns. Finally, the economic model performs well under both US dollar returns(ranking a close second) and home currency returns (ranking first). While these results arenot necessarily overwhelming, the economic model does dominate in its ability to forecastduring this holdout period. Similar results over the shorter 2-year holdout samples solidifythe relative dominance of the economic model, especially for home currency forecasts.

After documenting that the economic model dominates the other forecasting models inconsistently minimizing MSE, we can gain additional insight into the forecast perfor-mance of each model by conducting a Theil decomposition of the MSE (Madalla, 1977).The Theil decomposition partitions the MSE into three components which sum to one:bias (Um), regression (Ur), and disturbance (Ud). Besides displaying a forecast profile withthe minimum MSE, the optimal forecasting model should ideally yield values ofUm andUr that approach zero whileUd approaches one. Such a forecast profile would typifyforecast errors that fluctuate randomly about zero, and thus display little tendency for themodel to systematically over- or under-estimate the actual values.

These Theil decompositions are presented in Tables 8 and 9 for US dollar returns andhome currency returns, respectively. Again we focus on the forecast period covering theholdout sample that excludes the first quarter of 1988, presented in the last column of

Table 8. Theil Decomposition of Forecast MSE: US Dollar-Based Data

The Theil Decomposition partitions the MSE of a set of forecasts into 3 additive components:(i) Um 5 bias component;(ii) Ur 5 regression component; and(iii) Ud 5 disturbance component.

For the optimal forecasting model, the bias and regression components should approach zerowhile the disturbance component should approach one (Madalla, 1977, pp. 343–345).

This table presents the Theil Decomposition of the MSE for each forecasting model, over theentire holdout sample period and over various subperiods from the holdout sample.

1988–1993 1988–1989 1990–1991 1992–1993 1988Q2–1993

No Change Um 0.0033640 0.0309081 0.0013259 0.0002840 0.0000814Ur 0.2459002 0.2576696 0.3095426 0.2124636 0.2119127Ud 0.7507358 0.7114224 0.6891315 0.7872524 0.7880059

Historical Average Um 0.0023420 0.0201028 0.1173250 0.3099923 0.0024740Ur 0.0158330 0.0036388 0.0463054 0.0057332 0.0187329Ud 0.9818250 0.9762583 0.8363696 0.6842745 0.9787931

Empirical Bayes Um 0.0086228 0.0436621 0.0000000 0.0022806 0.0006563Ur 0.1434385 0.1704913 0.2125032 0.0802900 0.0996870Ud 0.8479386 0.7858466 0.7874968 0.9174294 0.8996567

ARIMA Model Um 0.0127502 0.0060997 0.1749821 0.0329274 0.0181984Ur 0.0111766 0.0000253 0.0489162 0.0241418 0.0119014Ud 0.9760732 0.9938750 0.7761018 0.9429309 0.9699002

Economic Model Um 0.0265206 0.0124100 0.0005429 0.1322994 0.0048421Ur 0.1225076 0.2622355 0.0764341 0.0555171 0.0391367Ud 0.8509718 0.7253545 0.9230230 0.8121835 0.9560212


Tables 8 and 9. Table 8 indicates that for US dollar returns the economic, historicalaverage, and ARIMA models offer similarly desirable forecast profiles in terms of theirTheil decomposition. However, the performance of the ARIMA model drops somewhatfor home currency returns in Table 9, leaving the economic and historical average modelsas the dominant performers in terms of Theil decomposition.

V. Summary and ConclusionsPrevious studies yield mixed evidence regarding the stability of the correlation structureacross international equity markets. We employ daily data on national stock returns since1972 to generate a quarterly time series in the correlation matrix. Stability tests on thistime series overwhelmingly indicate that the correlation matrix changes substantivelyacross both short and long time intervals, throughout the sample period 1972–1993. Thisoutcome focuses attention on the issue of how and why the correlation structure changesover time.

A theoretical model specifying potential economic determinants of the correlationstructure is proposed and estimated, using both US dollar denominated returns andhome-currency returns. Results indicate the following behavior. For both US dollarreturns and home-currency returns:

(1) world market volatility (WLDVOLij) is positively associated withrij ;

Table 9. Theil Decomposition of Forecast MSE: Home Currency Data

The Theil Decomposition partitions the MSE of a set of forecasts into 3 additive components:(i) Um 5 bias component;(ii) Ur 5 regression component; and(iii) Ud 5 disturbance component.

For the optimal forecasting model, the bias and regression components should approach zerowhile the disturbance component should approach one (Madalla, 1977, pp. 343–345).

This table presents the Theil Decomposition of the MSE for each forecasting model, over theentire holdout sample period and over various subperiods from the holdout sample.

1988–1993 1988–1989 1990–1991 1992–1993 1988Q2–1993

No Change Um 0.0030100 0.0272836 0.0014379 0.0004003 0.0000055Ur 0.2752795 0.2651327 0.3703286 0.2967222 0.2381793Ud 0.7217105 0.7075837 0.6282335 0.7028775 0.7618152

Historical Average Um 0.0032052 0.0040444 0.2828797 0.3304780 0.0054201Ur 0.0525208 0.0063974 0.0573262 0.0404575 0.0540870Ud 0.9442740 0.9895583 0.6597941 0.6290645 0.9404929

Empirical Bayes Um 0.0080154 0.0402257 0.0000005 0.0022005 0.0009322Ur 0.1614739 0.1756970 0.2633517 0.1233563 0.1140647Ud 0.8305107 0.7840773 0.7366478 0.8744432 0.8850031

ARIMA Model Um 0.0948682 0.0749593 0.5119004 0.0378351 0.0892753Ur 0.0543140 0.0184368 0.0422331 0.0741522 0.0583557Ud 0.8508178 0.9066039 0.4458665 0.8880127 0.8523689

Economic Model Um 0.0044154 0.0012217 0.1685946 0.0693845 0.0219598Ur 0.0955003 0.1768599 0.0883174 0.0615846 0.0452333Ud 0.9000844 0.8219183 0.7430881 0.8690309 0.9328069


(2) a positive trend (TREND) in therij appears in the first half of this 22-year period,from 1972 to 1982, while no trend appears from 1983 to 1993; and

(3) exchange rate volatility (XRSDij) has a dampening effect onrij .

In addition, to a lesser extent, for US dollar returns:

(4) term structure differentials (LOSHi 2 LOSHj) are negatively related torij ;

and for home-currency returns:

(5) real interest differentials (INTi 2 INT j) are negatively related torij ; and(6) the world market return is negatively associated withrij .

This analysis and evidence contributes to our understanding of the dynamics of globalintegration. For example, these results corroborate our theoretical expectations thatdivergent behavior across nations in several macroeconomic variables tends to be asso-ciated with divergent behavior across national equity markets, resulting in lower marketcorrelations. In addition, it is noteworthy that the first and last factors listed above indicatean increase in comovement across international equity markets when world markets aremore volatile and/or falling. This result implies a decline in the risk-reduction benefits ofinternational diversification at the very time when these benefits are needed most (Erb etal., 1994, Solnik et al., 1996).

Finally, we employ the economic model to generate out-of-sample short-term forecastsof the correlation matrix. The forecast performance of this model is seen to dominate thatof four alternative forecasting techniques that ignore economic determinants of thecorrelation structure. This evidence adds credence to the validity of the theoreticaleconomic model, and suggests a potential opportunity for portfolio managers to improveupon their portfolio selection techniques.

On the other hand, while these results document an improvement in the ability togenerate short term forecasts, the issue of longer term forecasting capabilities is unre-solved. As a result, the question of whether this economic-based forecasting model can beemployed to generate “optimal” ex ante international portfolios with enhanced ex postperformance remains for future inquiry.

We gratefully acknowledge support from University of Kansas GRF Grant #3710. We also wish to acknowledgethe helpful comments of William Beedles, Ken Cogger, Ali Fatemi, Peter Gillett, O. Maurice Joy, GeorgePinches, J. Jay Choi, and two anonymous referees. Morgan Stanley provided the data. These do not necessarilyreflect the views of Morgan Stanley. Please do not quote without permission.

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Appendix A: Jennrich Chi-Square Test for Equality of CorrelationMatricesThe following statistic will have a chi-square distribution with (k 2 1)p(p 2 1)/2 degreesof freedom

x2 5 Oi51

k S1

2tr ~Zi

2! 2 dg9~Z9i!S21dg~Zi!D ; (A1)

where fori 5 1 to k, Ri are the sample correlation matrices forp countries of sample sizeni and


ni 5 number ofobservations used to estimate sample correlation matrix in countryi;

n 5 n1 1 · · ·1 nk; (A2)

p 5 number of countries;

k 5 number ofp 3 p correlation matrices tested;

R# 5 ~n1R1 1 · · ·1 nkRk!/n 5 ~r# ij!; (A3)

S5 ~d ij 5 r ij rij!; (A4)

d ij 5 Kronecker’s delta; (A5)

Zi 5 ÎniR#21~Rj 2 R# !. (A6)

See Jennrich (1970) for details.

Appendix B: Construction of Trade Gap and Total Trade VariablesThe two variables measuring bilateral trade activity are constructed as follows:

GAPij 5 ~uXij 2 Mij u!/GDPi 1 ~uXij 2 Mij u!/GDPj;

TRADEij 5 ~Xij 1 Mij!/GDPi 1 ~Xij 1 Mij!/GDPj.

In order to motivate the rationale for constructing the GAPij variable as the sum of thebilateral trade balance, from the standpoint of both countries involved, we discuss threepossible outcomes:

(1) uXij 2 Mij u is large proportionate to GDP in both countries. If the bilateral tradedeficit (surplus) has divergent economic impacts on the two national economiesinvolved, then the two national equity markets are driven in different directions,leading to a negative correlation.

(2) uXij 2 Mij u is large proportionate to GDP in one country, but not the other. Theformer country’s economy and stock market are affected by a relatively largedeficit (surplus) while the latter country feels little pressure in its economy orequity market. This situation should lead to a negative correlation, but to a lesserextent than case one.

(3) uXij 2 Mij u is small proportionate to GDP in both countries. Neither countryexperiences much economic pressure from this bilateral trade balance, leading tolittle correlation across equity markets.

The value of GAPij will be small in case three, larger in case two, and largest in case one.As the GAPij variable increases in value, moving towards case one, correlations areexpected to become more negative.

Next consider three situations for the total trade variable (TRADEij):

(1) (Xij 1 Mij) is large proportionate to GDP in both countries. In this case, totalbilateral trade activity is substantive relative to total economic activity in bothcountries, leading to a high correlation across equity markets.


(2) (Xij 1 Mij) is large proportionate to GDP in one country, but not the other. In thiscase, total trade flows should have a substantive impact in one country, but not theother. That is, one country’s economy and stock market are highly dependent uponthe other, but not vice versa, resulting in a medium correlation across equitymarkets.

(3) (Xij 1 Mij) is small proportionate to GDP in both countries. In this case total tradeflows should have little impact on either country’s economy or stock market,resulting in a low correlation.

Again, the value of TRADEij will be small in case three, larger in case two, and largestin case one. Thus, as TRADEij increases in magnitude, we move toward case one and thecorrelation should become larger.

We have also estimated a variation of the economic model which includes the fourvariables, (Xij 1 Mij)/GDPi, (Xij 1 Mij)/GDPj, uXij 2 Mij u/GDPi, and uXij 2 Mij u/GDPj,separately rather than aggregated into the variables, GAPij and TRADEij . Results (avail-able upon request) are qualitatively similar to those presented here.

Appendix C: Data Sources for Regression Model VariablesInternational Monetary Fund,International Financial Statistics, 1972–1994:

IND it 5 Growth in industrial production in countryi during quartert; derived fromindex of industrial production (Line 66).

INFL it 5 Inflation rate in countryi during quartert; derived from Consumer PriceIndex (Line 64).

INT it 5 Real interest rate in countryi during quartert; derived by subtractinginflation rate from long term bond rate (Line 61).

LOSHit 5 Spread between long- (Line 61) and short-term (Line 60) bond rates incountry i during quartert.

GDPit 5 Gross Domestic Product (Line 99) of countryi during quartert.

International Monetary Fund,Direction of Trade, 1972–1994:a

Xijt 5 Exports from countryi to countryj during quartert.Xjit 5 Exports from countryj to countryi during quartert.Mijt 5 Imports to countryi from countryj during quartert.Mjit 5 Imports to countryj from countryi during quartert.

Morgan Stanley Capital International:b

rijt 5 Pearson correlation estimated from daily equity index returns forcountriesi and j over quartert.

XRCHijt 5 Percent change in bilateral exchange rate between countriesi and jduring quartert.

XRSDijt 5 Standard deviation of daily exchange rate between countriesi and jduring quartert.

a While theoreticallyXijt should equalMjit , these values are not necessarily equal with our data, due torecording methods for exports versus imports.

b Morgan Stanley provided the daily equity index values and daily exchange rates (with the US dollar) fromwhich these variables are derived.


WLDVOLt 5 Standard deviation of daily returns on World Market Index duringquartert.

WLDMKTt 5 Percent change in value of the World Market Index over quartert.


Documents

Economic Determinants of the Correlation Structure Across