Modeling Economic Determinants of Correlation Structure

frequently than is expected by chance 39 rejections at the 0.10 level, 29 rejections at the 0.05 level, and 13 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 binomial distribution. 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 that trial is .90 and the probability of failure is .10. By viewing the sequence of 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 at p 5 .90 the 10 level of significance, with 87 independent tests there is less than a 1 chance of rejecting the null more than sixteen times. Similarly, given a probability of success at p 5 .95, there is less than a 1 chance of rejecting the null more than ten times. Finally, given a probability of success at p 5 .99, there is less than a 1 chance of rejecting the null more than four times. Once again, because the numbers in Panel A of Table 1 are greater than these respective “critical values,” the number of rejections in each cell of Panel A is greater than would be expected 1 of the time, under this binomial exercise. Next, we consider the possibility that these quarterly correlation matrices may change more 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 rejections across nonconsecutive quarterly correlation matrices, and once again reveals far more rejections than could be expected by chance. Finally, Panel C considers the stability of the correlation matrix computed over longer time intervals of 6 months, 1 year, 2 years, 5 1 ⁄ 2 years, and 11 years. Results in Panel C uniformly indicate instability over these longer time periods as well. While our results are consistent with most recent studies on this issue, these latter findings contrast with the results of Meric and Meric 1989, which suggest some underlying long-run stability in the correlation matrix. This contrast is important. If there were truly long-run stability, then the short-run instability documented in Panels A and B of Table 1 would not be critical for long-term investors. Instead, our results overwhelmingly indicate significant changes in the correlation matrix over both short and long time horizons. These results provoke the question as to how and why the correlation structure varies over 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 random walk, then our model of the correlation structure may result in identifying spurious relationships with economic variables Enders 1995. In this light, we perform Augmented Dickey–Fuller ADF tests on each time series of quarterly pairwise correlations. Results are provided in Table 2 and indicate rejection of the unit root hypothesis for 18 of the 21 time series of pairwise correlations, at the .10 level or better. Because the vast majority of these time series have no unit root, it is reasonable to proceed with our attempt to model how and why the correlation structure changes over time.

III. Modeling Economic Determinants of Correlation Structure

Development of the economic model begins with reference to prior work that analyzes potential macroeconomic determinants of expected returns in a national equity market 448 K. Bracker and P. D. Koch 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 factors as possible determinants of national equity returns: IND it 5 industrial production growth in country i during quarter t; INFL it 5 inflation in country i during quarter t; Table 2. Augmented Dickey-Fuller ADF Tests This table presents the results of ADF tests that investigate the existence of a unit root in the time 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: Dcorr t 5 a 1 gcorr t2 1 1 b 1 Dcorr t2 1 1 b 2 Dcorr t2 2 1 « t . Under H : 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 50 observations 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 a deterministic 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 is almost never significant, while the intercept is almost always significant. Results for both alternative models are largely consistent with those presented here, indicating that most time series of pairwise correlations exhibit no unit root. Country Pair g t ratio AustraliaCanada 20.729 24.007 AustraliaGermany 20.400 22.904 AustraliaJapan 20.341 22.648 AustraliaSwitzerland 20.255 22.197 AustraliaUnited King. 20.415 23.155 AustraliaUnited States 21.099 25.624 CanadaGermany 20.455 22.966 CanadaJapan 20.520 23.463 CanadaSwitzerland 20.514 23.299 CanadaUnited King. 20.384 22.990 CanadaUnited States 20.381 22.634 GermanyJapan 20.311 22.700 GermanySwitzerland 20.324 22.971 GermanyUnited King. 20.299 23.162 GermanyUnited States 20.619 23.868 JapanSwitzerland 20.255 22.473 JapanUnited King. 20.332 22.917 JapanUnited States 20.671 23.868 SwitzerlandUnit. King. 20.233 22.473 SwitzerlandUnited 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. Determinants of International Correlation 449 INT it 5 real interest rate in country i during quarter t; LOSH it 5 term structure premium in country i during quarter t; 7 SIZE it 5 share of total world market capitalization in country i during quarter t; The economic rationale for hypothesizing that these variables are among the determinants of national stock returns is grounded in the discounted cash flow model. The first four variables listed above represent different aspects of a country’s macroeconomic perfor- mance that affect expected cash flows andor discount rates in that national market, and thus have a bearing on the market’s expected returns. The last variable listed above is motivated by the well-documented size effect within a national market, whereby higher discount rates are demanded for smaller firms Keim 1983; Reinganum 1983. Potential explanations for this firm-size effect include greater information costs, transaction costs, and less liquidity associated with trading equity in smaller firms. By extension, we suggest that the relative size of a national equity market may also have a bearing on that country’s equity returns, due to greater information costs, transaction costs, and less liquidity associated with trading equity in smaller national markets. While this discussion serves to motivate potential macroeconomic determinants of the first moment expected returns across different elements in the vector of national equity returns, we are interested in modeling determinants of the second moment the correlation structure. 8 As a first step, we postulate that the extent of comovement between a pair of national markets may depend upon the extent to which these five macroeconomic variables diverge across the two markets, as follows: r ijt 5 b 1 b 1 uIND i 2 IND j u t 1 b 2 uINFL i 2 INFL j u t 1 b 3 uINT i 2 INT j u t 1 b 4 uLOSH i 2 LOSH j u t 1 b 5 uSIZE i 2 SIZE j u t 1 e ijt , 1 where r ijt 5 estimated correlation between daily returns in countries i and j during quarter t; e ijt 5 disturbance term, assumed to be iid N0, s 2 . Generally if there is greater divergence in macroeconomic behavior across countries, we expect less comovement across equity markets, implying negative coefficients for b 1 –b 5 . 9 7 The term structure premium is defined as the difference between long term and short term government bond rates in country i during quarter t Chen et al., 1986. This difference is a measure of the premium demanded for long term investments in a country. 8 For other work that motivates and investigates this issue, see Arshanapalli and Doukas 1993, Bachman et 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 member countries such as interest rates and inflation rates, as specified in the Maastricht treaty, serves to illustrate and motivate the spirit of our theoretical specification in Equation 1. The empirical validity of this specification requires some attention. Note that Equation 1 constrains each macroeconomic variable to enter the model as the absolute differential in behavior across markets i and j, during quarter t. It is conceivable, however, that each variable in country i has a substantive influence on the correlation, independent of the analogous factor in country j. That is, it may be more appropriate to allow each factor in both countries to enter the regression model independently, rather than in the form of an absolute differential, as follows: r ijt 5 a 1 a 1 IND it 1 a 2 IND jt 1 a 3 INFL it 1 a 4 INFL jt 1 a 5 INT it 1 a 6 INT jt 1 a 7 LOSH it 1 a 8 LOSH jt 1 a 9 SIZE it 1 a 10 SIZE jt 1 e ijt . The model sp 450 K. Bracker and P. D. Koch In addition to motivating the potential influence of the above five macroeconomic differentials on r ijt , the discounted cash flow model further motivates expansion of Equation 1 to incorporate five additional macroeconomic variables that may also directly influence international correlations. First, bilateral trade conditions may impact national equity index returns for a given pair of trading partners Bodurtha et al., 1989. As exports from economy i to economy j X ij increase, higher cash flows should be expected into country i. In contrast, the imports of economy i from economy j M ij may have the opposite effect on cash flows for country i’s firms. In this light, changes in the absolute magnitude of the trade balance between two economies uX ij 2 M ij u should positively influence one economy and stock market, while negatively influencing the other, to exert a negative impact on the corre- lation, r ijt . The extent of this impact, however, should reflect the degree of importance of this bilateral trade activity with respect to aggregate economic activity in each national market involved. If this bilateral trade balance reflects a small proportion of Gross Domestic Product GDP for one of the two countries, it is not likely to exert much influence on that country’s stock market. On the other hand, if the trade balance represents a large proportion of GDP for one country, we may expect a substantive impact on stock returns in that country, and thus a substantive response in r ij . Furthermore, if uX ij 2 M ij u represents a large proportion of GDP for both countries, we would expect a greater response in r ij . In this light we construct the following variable to incorporate the potential influence of the trade gap on r ij , from the point of view of both countries: GAP ij 5 ~ uX ij 2 M ij uGDP i 1 ~ uX ij 2 M ij uGDP j . Second, this trade gap variable may not adequately reflect the full impact of bilateral trade conditions on the correlation structure, because a given pair of countries may have a trade gap near zero while engaging in a substantial amount of trade. In this light, we also incorporate a second variable to account for the total amount of trade across the two countries: TRADE ij 5 ~X ij 1 M ij GDP i 1 ~X ij 1 M ij GDP j . This variable emphasizes the notion that both exports and imports have a role in wealth creation, and may thus influence the interaction of national equity markets. Appendix B provides further insight into this specification of the GAP ij and TRADE ij variables. Third, absolute changes in the bilateral exchange rate uXRCH ij u may influence the trade conditions discussed above, and may thus also influence national equity returns in both countries. If the exchange rate changes by a larger percentage, then we expect a larger 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: a 1 5 2a 2 5b 1 ; a 3 5 2a 4 5b 2 ; a 5 5 2a 6 5b 3 ; a 7 5 2a 8 5b 4 ; a 9 5 2a 10 5b 5 . 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 behavior across markets that influences the correlation structure. The theoretical and empirical validity of these constraints therefore compels us to focus on Equation 1 as the appropriate model for analysis and interpretation regarding these five economic influences. Determinants of International Correlation 451 country. This observation suggests a potential indirect negative influence of absolute exchange rate changes on the correlation, through their impact on the trade gap. 10 Fourth, volatility in the bilateral exchange rate XRSD ij represents another source of uncertainty 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 equity return component and a currency return component. As currency rates become more volatile, the currency component becomes more important relative to the equity return component. In this case, higher exchange rate volatility may be expected to dampen the correlation between different pairs of national equity market returns, denominated in US Dollars. This dampening effect should be less important across home currency returns that abstract from exchange rate movements. Fifth, overall volatility across the world’s stock markets may influence the level of discount 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 to compensate this risk, resulting in higher correlations across different pairs of national equity markets. 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 across national markets. Finally, in addition to the above ten macroeconomic variables, we incorporate several additional factors that appear in anecdotal discussions of potential influences on the correlation structure, although they do not enter explicitly into the discounted cash flow model. First, Erb et al. 1994 observe that correlations tend to be low during times of general upward trends in world equity valuation, but that correlations become higher when stock returns are declining around the world. This observation suggests asymmetric behavior 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 negative association with the correlation structure over time. Second, evolution toward global capital market integration would suggest that correlations are trending upward over time due to factors such as greater interdependence across national economies, improved telecommunications technology, global deregulation of markets, more cross-listing of securities, growth in multinational activities, increasing international diversification, and so 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, two periods of increased world market volatility Roll, 1989. Thus, two dummy variables are incorporated to account for potentially aberrant behavior during the fourth quarters of 1987 and 1989, respectively. Fourth, it is well-documented that different national equity markets have experienced seasonal patterns in market activity and valuation. Meric and Meric 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 j u, uINT i 2 INT j u, and uXRCH ij u, are related through international parity conditions Eiteman et al., 1998. In fact, if purchasing power parity and interest rate parity were to hold perfectly in all periods, these three variables would be collinear. This situation may give cause for concern about potential multicollinearity in the regression model. However, we observe that it is deviations from purchasing power parity and interest rate parity that should induce capital flows and trade flows across countries. Hence these deviations may have a bearing on the extent of comovement across international equity markets. This observation provides another economic rationale for including all three variables in our model see Bodurtha et al., 1989. 452 K. Bracker and P. D. Koch months. In this light, we account for potential seasonality in the correlation matrix by adding quarterly dummy variables. The final regression model incorporates all of these influences, as follows: r ijt 5 b 1 b 1 uIND i 2 IND j u t 1 b 2 uINFL i 2 INFL j u t 1 b 3 uINT i 2 INT j u t 1 b 4 uLOSH i 2 LOSH j u t 1 b 5 uSIZE i 2 SIZE j u t 1 b 6 GAP ijt 1 b 7 TRADE ijt 1 b 8 uXRCH ij u t 1 b 9 XRSD ijt 1 b 10 WLDVOL t 1 b 11 WLDMKT t 1 b 12 TREND 1 b 13 OCT87 1 b 14 OCT89 1 b 15 Q1 1 b 16 Q2 1 b 17 Q3 1 e ijt ; 2 where i 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 country i during quarter t; INFL it 5 Inflation Rate in country i during quarter t; INT it 5 Real interest rate long term government rate-inflation rate during quarter t; LOSH it 5 Spread between long and short term bond rates in country i during quarter t; SIZE it 5 Percent of world equity market share in market i during quarter t; GAP ijt 5 uX ij 2 M ij u t GDP it 1 uX ij 2 M ij u t GDP jt ; TRADE ijt 5 X ij 1 M ij t GDP it 1 X ij 1 M ij t GDP jt ; XRCH ijt 5 Percent change in bilateral exchange rate during quarter t; XRSD ijt 5 Standard Deviation in daily bilateral exchange rate during quarter t; WLDVOL t 5 Standard deviation of daily world stock market index return during quarter t; WLDMKT t 5 Percent change in world stock market index during quarter t; TREND 5 Nonlinear trend, lnt; OCT8789 5 Dummy variable equal to 1 in fourth quarter of 19871989; 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 Model