Forecasting the Correlation Structure

coefficient is negative 17 times, and is significant four times. This negative coefficient is reinforced in the pooled analysis of Table 3. This result is consistent with the observations of Erb et al. 1994, suggesting greater comovement across international equity markets when world markets are declining. Finally, during the global crash of October 1987 international market correlations increased dramatically, leading us to expect a positive coefficient for the October 1987 dummy variable. Instead, this dummy coefficient is consistently negative across all equations, indicating that the fitted model over-estimates the correlation structure during this period. These high-fitted values are attributable to the strong positive association between world market volatility and the correlation for all 21 pairs of countries, combined with the extraordinarily large value for world volatility experienced in the fourth quarter of 1987. 14 When the model is re-estimated without the October 1987 dummy, the coefficient on world volatility declines by 25 in the pooled regression, and is reduced in most equations of the SUR model. In this light, the October 1987 dummy should be viewed as a tool that allows the model to reveal the influence of world volatility and the remaining 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 model in light of the forecasting goals we address next. In summary, the estimated pooled and SUR models indicate the following. For correlations estimated with both US dollar and home currency returns: 1 world market volatility WLDVOL ij is positively associated with the r ij ; 2 a positive trend TREND in the r ij 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 XRSD ij has a dampening effect on the r ij . In addition, to a lesser extent, for US dollar returns: 4 term structure differentials LOSH i 2 LOSH j are negatively related to r ij ; while for home currency returns: 5 real interest differentials INT i 2 INT j are negatively related to r ij ; and 6 the world market return is negatively associated with the r ij .

V. Forecasting the Correlation Structure

The above analysis improves our understanding of how and why the correlation structure changes over time, and thus contributes to the dialogue on global integration. A stringent test of the validity of any such economic model is its out-of-sample forecasting ability. In this light, we compare the forecasting ability of our economic model with that of four atheoretical forecasting models. 15 14 The mean correlation across the twenty-one pairs of national equity markets during the fourth quarter of 1987 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 world market 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. Determinants of International Correlation 459 Forecasting Models Consider first the SUR economic model specified in Equation 2. Assuming that the relation in Equation 2 also holds in period t 1 1, and taking conditional expectations, we obtain the one-step-ahead forecast: Er ijt1 1 uI t 5 b 1 b 1 E t uIND i 2 IND j u t1 1 uI t 1 b 2 E t uINFL i 2 INFL j u t1 1 uI t 1 b 3 E t [ uINT i 2 INT j u t1 1 uI t ] 1 b 4 E t uLOSH i 2 LOSH j u t1 1 uI t 1 b 5 E t [ uSIZE i 2 SIZE j u t1 1 uI t ] 1 b 6 E t GAP ijt1 1 uI t 1 b 7 E t [TRADE ijt1 1 uI t ] 1 b 8 E t uXRCH ij u t1 1 uI t 1 b 9 E t XRSD ijt1 1 uI t 1 b 10 E t [WLDVOL t1 1 uI t ] 1 b 11 E t WLDMKT t1 1 uI 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 t e ijt1 1 uI t ; 3 where E t [X ,t1 1 uI t ] represents the expectation in period t of variable X , in period t 1 1, conditional on information available in period t, and E t [e ijt1 1 uI t ] 5 0. Observe that all variables in Equation 2 with time subscripts appear in the forecast, Equation 3, as conditional expectations. If we replace the conditional expectation of each such right- hand-side variable in Equation 3 with its actual value in period t, we obtain a forecast of the correlation generated from this regression model. This approach is taken here to generate forecasts with the economic model. 16 Initially the SUR economic model in Equation 2 is re-estimated over the first 16 years of the 22-year sample period 1972–1987. The resulting 21 fitted equations in this model are then employed in Equation 3 to generate out-of-sample one-step-ahead forecasts of the correlation 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 is generated 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 approach would be appropriate, for example, if each right-hand-side macroeconomic variable followed a random walk. An alternative approach would be to use other forecasting methodology to generate one-step-ahead quarterly forecasts of each conditional expectation on the right-hand side of Equation 3 to employ in forecasting r ijt1 1 . 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 across different pairs of countries. This result implies that different economic variables are important to differing degrees for different pairs of countries. By employing the SUR model to forecast the correlations rather than the pooled model, we allow different sets of economic variables to be more important in forecasting the correlations for different pairs of countries. In addition, the economic forecasting model is fitted using Fisher transformations of the correlation as the dependent variable [z 5 .5ln1 1 r ijt 1 2 r ijt ] to ensure that all forecasted correlations range between 21 and 11. Estimation results are generally robust with respect to those presented in Tables 4 and 5, and are available upon request. Finally, we wish to address the use of a long 16-year estimation period for the economic and ARIMA forecasting models. First, the model parameters are generally robust when we split the sample into two eleven-year sample periods 1972–1982 and 1983–1993. Hence, the choice of sample period has little bearing on the estimation results. Second, due to the large number of variables specified in the economic model, a relatively long estimation period is desirable to have sufficient degrees of freedom to appeal to the asymptotic properties of the estimation technique. For example, it would be unfeasible to employ annual periods for estimation and forecasting, due to the reduction in degrees of freedom associated with the use of such longer periods. 460 K. Bracker and P. D. Koch The forecasting performance of the economic model is then compared with that of four alternative models. We discuss these alternative models in order, from the simplest to the more complex. The first model represents a one-step-ahead forecast of no change from the previous quarter’s observation on the correlation matrix. While this naive model may seem like a weak challenge, McNees 1992 has shown it to perform well in predicting many economic variables. A second alternative model, similar to the first, sets the one-step-ahead forecast of each bilateral correlation equal to the historical average over the past eight quarterly observations. The third challenger model employs an “empirical Bayes” approach similar to the top performing model for Kaplanis 1988. This model effectively generates a correlation forecast for every pair of markets each quarter that regresses toward the global mean across all 21 correlation forecasts see Kaplanis, 1988, for details. These first three models offer interesting alternatives, since most of the mean-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 and Sarnat 1970; Odier and Solnik 1993. The final alternative model is an ARIMA model for each time series of pairwise correlations. While the structure of each ARIMA forecasting model proves to be stable across the estimation and holdout periods, the ARIMA model parameters are re-estimated for each quarter throughout the holdout sample with updated data. 18 Forecasting Results The performance of these five models is evaluated on the basis of minimizing the Mean Squared Error MSE across the set of 21 forecasts every quarter. Each model is applied to 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 each quarter of the 6-year holdout period, as well as for this entire 6-year period and three 2-year subperiods. In addition, due to the influence of the October 1987 international crash on the forecasts for the first quarter of 1988, we provide results for the subperiod covering the second quarter of 1988 through 1993. Results for the first quarter of 1988 reveal that, while the forecast performance of the historical average model and ARIMA model is not greatly affected by the October 1987 crash, 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’s correlation matrix to forecast the current quarter’s matrix. Similarly, the economic model forecast is also heavily influenced by the value of world market volatility in the previous quarter. This behavior leads to forecasts from these three models that dramatically over-estimate the correlation matrix for the first quarter of 1988. In contrast, the historical average model and ARIMA model are not as greatly affected by the aberrant behavior during 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’s RiskMetrics 1996 package to forecast variances and covariances is a special case of the general class of ARIMA models. Our application of this approach fits the unique ARIMA model that is appropriate for each pairwise time series of correlations and employs it to forecast. Details on the fitted ARIMA models employed in this application, along with their diagnostics, are available upon request. Of the twenty-one ARIMA models estimated in this study, 14 models have a statistically significant AR1 term. When a lagged r ijt2 1 term is added to the economic model, regression results are not significantly altered. Determinants of International Correlation 461 portion of the information set employed by these two models to forecast the first quarter of 1988. The aberrant behavior of the forecasts for the first quarter of 1988 leads us to focus on the 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 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, we generate 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. No Change Historical Average Empirical Bayes Approach ARIMA Model Economic Model 1988: Q1 0.23582 0.05622 0.23255 0.05966 0.34584 Q2 0.04593 0.01039 0.03563 0.01523 0.01705 Q3 0.01656 0.02020 0.01684 0.01275 0.01520 Q4 0.02621 0.02219 0.02334 0.01590 0.02274 1989: Q1 0.02742 0.01169 0.03019 0.01845 0.01234 Q2 0.02828 0.02915 0.02753 0.02540 0.02960 Q3 0.01825 0.02046 0.01341 0.02479 0.02258 Q4 0.10241 0.05733 0.10197 0.06236 0.05325 1990: Q1 0.03070 0.01098 0.02533 0.01093 0.01358 Q2 0.03904 0.02867 0.02539 0.02031 0.02637 Q3 0.02732 0.03096 0.02013 0.03326 0.02371 Q4 0.02643 0.02412 0.02206 0.01970 0.01950 1991: Q1 0.02423 0.03718 0.02301 0.04201 0.01917 Q2 0.02754 0.02911 0.03018 0.02598 0.03092 Q3 0.08582 0.07538 0.07409 0.08491 0.05707 Q4 0.12171 0.01972 0.12367 0.03298 0.03729 1992: Q1 0.02047 0.02133 0.02505 0.02366 0.01518 Q2 0.04106 0.01474 0.03215 0.01751 0.02139 Q3 0.09475 0.07905 0.09166 0.04455 0.09985 Q4 0.05275 0.04678 0.03858 0.01891 0.02324 1993: Q1 0.02023 0.04538 0.01818 0.03098 0.02411 Q2 0.05255 0.05408 0.04891 0.04602 0.03560 Q3 0.03758 0.04817 0.03152 0.03565 0.05326 Q4 0.03979 0.01312 0.03195 0.01456 0.01693 Avg MSE 1988–1989 0.06261 0.02845 0.06018 0.02932 0.06482 Avg MSE 1990–1991 0.04785 0.03201 0.04298 0.03376 0.02845 Avg MSE 1992–1993 0.04490 0.04033 0.03975 0.02898 0.03620 Avg MSE 1988–1993 0.05179 0.03360 0.04764 0.03069 0.04316 Avg MSE 1988Q2–1993 0.04378 0.03262 0.03960 0.02943 0.03000 462 K. Bracker and P. D. Koch performance in terms of MSE is given by the naive model of no change from the previous quarter, for both US dollar and home currency returns. The empirical Bayes approach does not fare much better, ranking as the fourth best model for US dollar returns and the third best 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, we generate 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. No Change Historical Average Empirical Bayes Approach ARIMA Model Economic Model 1988: Q1 0.18207 0.04718 0.17842 0.05782 0.14023 Q2 0.03717 0.03166 0.02877 0.07376 0.05535 Q3 0.01447 0.02179 0.01400 0.04268 0.02686 Q4 0.03869 0.02325 0.03457 0.02115 0.02596 1989: Q1 0.02746 0.01462 0.02643 0.02236 0.01900 Q2 0.04085 0.04554 0.03917 0.04323 0.02241 Q3 0.02063 0.02607 0.01358 0.01493 0.01664 Q4 0.12879 0.07718 0.12431 0.08159 0.09777 1990: Q1 0.04058 0.01172 0.03416 0.02967 0.02957 Q2 0.03356 0.02441 0.02341 0.04069 0.01615 Q3 0.05459 0.06773 0.04112 0.08063 0.05529 Q4 0.03111 0.02706 0.03252 0.04661 0.01734 1991: Q1 0.01729 0.04554 0.01297 0.07746 0.03167 Q2 0.04325 0.02921 0.04360 0.04297 0.03477 Q3 0.07832 0.06299 0.06410 0.12861 0.07363 Q4 0.07186 0.01206 0.07740 0.01590 0.01796 1992: Q1 0.02660 0.03601 0.02509 0.01858 0.01979 Q2 0.06685 0.02081 0.05317 0.03585 0.04029 Q3 0.04976 0.02867 0.03596 0.02324 0.02587 Q4 0.04929 0.04835 0.04821 0.02709 0.02726 1993: Q1 0.02118 0.06143 0.01728 0.03878 0.02317 Q2 0.03718 0.09656 0.02644 0.05821 0.03868 Q3 0.04247 0.04261 0.03431 0.03072 0.03205 Q4 0.05033 0.01918 0.04074 0.02017 0.02869 Avg MSE 1988–1989 0.06127 0.03591 0.05741 0.04469 0.05053 Avg MSE 1990–1991 0.04632 0.03509 0.04116 0.05782 0.03455 Avg MSE 1992–1993 0.04296 0.04420 0.03515 0.03158 0.02948 Avg MSE 1988–1993 0.05018 0.03840 0.04457 0.04470 0.03818 Avg MSE 1988Q2–1993 0.04445 0.03802 0.03875 0.04413 0.03375 Determinants of International Correlation 463 returns ranking first, but does considerably worse fourth for returns on a home currency basis. This outcome suggests that time series regularities may be more stable and predictable for correlations across exchange rates than across equity index returns. The historical average over the previous eight quarters consistently offers reasonably accurate forecasts that are worthy of note, ranking third for US dollar returns and second for home currency 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 are not necessarily overwhelming, the economic model does dominate in its ability to forecast during this holdout period. Similar results over the shorter 2-year holdout samples solidify the relative dominance of the economic model, especially for home currency forecasts. After documenting that the economic model dominates the other forecasting models in consistently 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 U m , regression U r , and disturbance U d . Besides displaying a forecast profile with the minimum MSE, the optimal forecasting model should ideally yield values of U m and U r that approach zero while U d approaches one. Such a forecast profile would typify forecast errors that fluctuate randomly about zero, and thus display little tendency for the model to systematically over- or under-estimate the actual values. These Theil decompositions are presented in Tables 8 and 9 for US dollar returns and home currency returns, respectively. Again we focus on the forecast period covering the holdout 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 U m 5 bias component; ii U r 5 regression component; and iii U d 5 disturbance component. For the optimal forecasting model, the bias and regression components should approach zero while 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 the entire holdout sample period and over various subperiods from the holdout sample. 1988–1993 1988–1989 1990–1991 1992–1993 1988Q2–1993 No Change U m 0.0033640 0.0309081 0.0013259 0.0002840 0.0000814 U r 0.2459002 0.2576696 0.3095426 0.2124636 0.2119127 U d 0.7507358 0.7114224 0.6891315 0.7872524 0.7880059 Historical Average U m 0.0023420 0.0201028 0.1173250 0.3099923 0.0024740 U r 0.0158330 0.0036388 0.0463054 0.0057332 0.0187329 U d 0.9818250 0.9762583 0.8363696 0.6842745 0.9787931 Empirical Bayes U m 0.0086228 0.0436621 0.0000000 0.0022806 0.0006563 U r 0.1434385 0.1704913 0.2125032 0.0802900 0.0996870 U d 0.8479386 0.7858466 0.7874968 0.9174294 0.8996567 ARIMA Model U m 0.0127502 0.0060997 0.1749821 0.0329274 0.0181984 U r 0.0111766 0.0000253 0.0489162 0.0241418 0.0119014 U d 0.9760732 0.9938750 0.7761018 0.9429309 0.9699002 Economic Model U m 0.0265206 0.0124100 0.0005429 0.1322994 0.0048421 U r 0.1225076 0.2622355 0.0764341 0.0555171 0.0391367 U d 0.8509718 0.7253545 0.9230230 0.8121835 0.9560212 464 K. Bracker and P. D. Koch Tables 8 and 9. Table 8 indicates that for US dollar returns the economic, historical average, and ARIMA models offer similarly desirable forecast profiles in terms of their Theil decomposition. However, the performance of the ARIMA model drops somewhat for home currency returns in Table 9, leaving the economic and historical average models as the dominant performers in terms of Theil decomposition.

V. Summary and Conclusions