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102 DEM 36M  DEM 912M DEM 36M  DEM 912M 1990 1992 1994 1996 Time (year) DEM 36  DEM 912M ilil f 1990 1992 1994 Time (year) 1996 0 " >. Q 1990 1992 1994 1996 Time (year) DEM 36M  DEM 912M 1990 1992 1994 Time (year) 1996 FIGURE 10.7 Linear correlation coefficients calculated using increasingly small subintervals, T/N = (365 days, 128 days, 32 days, and 7 days), for the implied forward interest rate pair DEM 36 months  DEM 912 months. The dashed lines above and below zero correlation are 95% confidence ranges assuming normally distributed returns. where is an exponential attenuation length and is a simple weight. These parameters were fitted to the data in Figure 10.8, starting with the first lag (neglecting the zeroth lag, which is equal to one by definition) and only to the point where autocorrelation data fell below the 68% of the upper confidence limit, thus focusing on the initial decay of autocorrelation. The autocorrelations of USDDEM USDGBP correlations are shown in Figure 10.9. The results for the autocorrelations of correlation data shown in Figure 10.8 are also reported in Table 10.4. The DJIAAMEX pair is excluded here due to the lack of correlation in the 20min returns, as mentioned above. The goodness of fit can be judged by the x2 value, divided by the degree of freedom m in fitting. This is also shown in Table 10.4. Each autocorrelation value was assumed to have a stochastic error of 1/>J~N, where N is the total number of correlation observations considered when calculating an individual autocorrelation value. All x2/m values are below 1 or just slightly above 1, indicating
100 200 Lag (weeks) usd/dem  usd/itl 100 200 Lag (weeks) djiaamex 100 200 Lag (weeks) 100 200 300 Lag (weeks) dem 3M  dem 9M 100 200 300 Lag (weeks) usd 3M  dem 3M 100 200 Lag (weeks) FIGURE 10.8 Autocorrelations of weekly correlation coefficients of returns, as plotted in Figures 10.2 through 10.7. The 95% confidence ranges corresponding to normally distributed random distributions are shown as dotted curves, where the curvature is caused by the decreasing size of the sample with growing lags. The total sampling period T is from January 7, 1990, to January 5, 1997. usd/dem  usd/nlg usd/dem  usd/gbp
ffl g.5c I ; .2 . ; 3 0.23 ! ~ , . ig.oo 2 . 30.00 4 . 5 .00 60.00 Lag (weeks) FIGURE 10.9 Fit of an exponential function to the autocorrelation of USDDEM USDGBP weekly correlation coefficients. that Equation 10.12 describes the data rather well. Adding a second exponential function to Equation 10.12 did not significantly improve the goodness of fit in all cases unless the data point at zero lag (defined as being equal to one) was added to the data set. Table 10.4 shows considerable values of the amplitude A (which cannot exceed 1) and a long memory of the correlation level. The pair USDDEM  USDNLG has the longest memory with an exponential attenuation length of more than 80 weeks, which is just over one and a half years. The autocorrelation model of Equation 10.12 with the parameters of Table 10.4 can be the basis of a correlation forecast. This forecast could be remarkably longterm due to the long memory in correlation, depending on the instrument pair. 10.5 CORRELATION BEHAVIOR AT HIGH DATA FREQUENCIES Previous authors have observed a dramatic decrease in correlation as the time intervals of the returns enter the intrahour level, for both stock (Epps, 1979) and FX returns (Guillaume et al., 1997; Low et al., 1996). We follow the suggestion of Low et al. (1996) by referring to this phenomenon as the Epps (1979) effect after the first identifiable author to thoroughly document it. In this discussion, the Epps
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