DEM 3-6M - DEM 9-12M

DEM 3-6M - DEM 9-12M

1990 1992 1994 1996 Time (year)

DEM 3-6 - DEM 9-12M

ilil- f

1990

1992 1994 Time (year)

1996

0 "

>. Q

1990 1992 1994 1996 Time (year)

DEM 3-6M - DEM 9-12M

1990

1992 1994 Time (year)

1996

FIGURE 10.7 Linear correlation coefficients calculated using increasingly small sub-intervals, T/N = (365 days, 128 days, 32 days, and 7 days), for the implied forward interest rate pair DEM 3-6 months - DEM 9-12 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 USD-DEM -USD-GBP 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 DJIA-AMEX pair is excluded here due to the lack of correlation in the 20-min 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)

djia-amex

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 USD-DEM USD-GBP 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 USD-DEM - USD-NLG 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 long-term 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