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101 FIGURE 10.4 Linear correlation coefficients calculated using increasingly small subintervals, T/N = (365 days, 128 days, 32 days, and 7 days), for the FX return pair USDDEM  USDITL. The dashed lines above and below zero correlation are 95% confidence ranges assuming normally distributed returns. between 365 and 504, so the statistical significance of correlation values was always comparable. The correlation between some financial instruments can be described as reasonably stable. However, brief but large breaks can be observed in almost all cases, and the additional statistics provided by time series of higher frequency are essential to detect such occurrences. In addition, we observed decreasing absolute values all the correlations we examined when going to higher data frequencies. This will be further discussed further in Section 10.5. 10.4.2 The Exponential Memory of Return Correlations The linear correlation values shown in Figures 10.2 through 10.7 can be seen as time series in their own right. A certain stability of correlation levels seems to indicate that markets have a memory for these levels. This memory can be investigated by considering the autocorrelation of the correlation time series. We
DJIA  AMEX 1990 1992 1994 Time (year) DJIAAMEX 1996 1990 1992 1994 Time (year) 1996 DJIAAMEX 1990 1992 1994 Time (year) DJIA  AMEX 1996 1990 1992 1994 Time (year) 1996 FIGURE 10.5 Linear correlation coefficients calculated using increasingly small subintervals, T/N = (365 days, 128 days, 32 days, and 7 days), for the stock index pair DJIA AMEX (both expressed in USD). The dashed lines above and below zero correlation are 95% confidence ranges assuming normally distributed returns. focus on the weekly correlation measurements displayed in the lower right plots of Figures 10.2 through 10.7. These weekly correlations are computed from 20min returns. The autocorrelation analysis was performed for different lags ( ), using Equation 10.8. Results of these calculations are shown in Figure 10.8. Shown along with each autocorrelation curve are the 95% confidence limits for a normally distributed random process. The differences in the behaviors of the six correlation pairs are striking. For foreign exchange rate correlations, we observe a significantly positive autocorrelation extending to long lags up to 50 to 100 weeks. Correlation structures have a long memory. For correlations between implied forward interest rates, we find a positive autocorrelation above the significance limit for lags up to 3 or 4 months, which means a reduced but still long memory. The correlation of the stock index pair (Down Jones and AMEX Stock Index) behaves differently as it dives below significance already at the first lag of 1 week. The market has no consistent
FIGURE 10.6 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 USD 36 months  DEM 36 months. The dashed lines above and below zero correlation are 95% confidence intervals assuming 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 USD 36 months  DEM 36 months. The dashed lines above and below zero correlation are 95% confidence ranges assuming normally distributed returns. memory of the level, but this may be due to the strong Epps effect of the 20min returns, explained in Section 10.5. Figure 10.8 also shows that autocorrelation values for each of the six instrument pairs decline roughly exponentially but with markedly different attenuation rates. To better gauge the difference in autocorrelation attenuation for these correlation pairs, the autocorrelations shown in Figure 10.8 were modeled by a simple exponential function: Y = Ae~xx (10.12)
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