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103 TABLE 10.4 Autocorrelation study. The autocorrelations of weekly correlation data shown in Figure 10.8 are fitted to the parametrization given in Equation 10.12, for five financial instrument pairs. A large value of A. indicates a long, exponentially decaying memory of the correlation level. The variable X2/m indicates the goodness of fit (good fits have a value around I or preferably below); m is the degree of freedom of fitting. Instrument    (weeks)  X2/m  USDDEM   USDNLG  0.35 ±0.01  80.9 ±7.1  0.65  USDDEM   USDGBP  0.62 ± 0.02  31.5 ± 1.8  0.15  USDDEM   USDITL  0.61 ±0.02  59.8 ±3.0  0.34  DEM 36m  DEM 912m  0.27 ± 0.04  10.0 ±2.6  1.30  USD 36m   DEM 36m  0.23 ± 0.03  21.8 ±5.4  0.41 
effect is characterized and investigated for a number of foreign exchange rates, stock indices, and implied forward interest rate pairs through the examination of the same 7 years of highfrequency return values as used in the previous sections. The basis of our exploration was a set of homogeneous time series of returns, equally spaced by 5min intervals, for several financial instruments. Linear interpolation was used in the sense of Equtions 3.2 and 3.6. For each time series of 5min returns, we obtained 499 additional time series through aggregation: 10min returns, 15min returns,... , 2500min returns. To cover longer time intervals, 877 more time series were obtained by further aggregation in coarser steps: 2530min returns, 2560min returns, ... , 28810min returns, where a 28810min interval roughly corresponds to 20 days. Various calculations were performed with these many times series. The most interesting results are the correlation coefficients for returns of different financial instruments, using the same time interval size. This can be done for all interval sizes of the aggregated time series. The resulting correlation coefficients are plotted as a function of the time interval size in Figure 10.10. When returns are computed with high frequency, over intervals distinctly shorter than 1 day, the correlation levels diminish in Figure 10.10. The same effect is better viewed in Figure 10.11, where the same data are shown with a logarithmic scale of interval sizes and where the data point farthest to the left (highest data frequency) corresponds to the correlation calculated using linearly interpolated, homogeneous time series of 5min returns. Table 10.5 gives the minimum and maximum values for the linear correlation coefficient data shown in Figure 10.10. Also given are the time intervals at which maxima occurred. We noted several problems when taking the maximum of correlation as a function of the time interval. The correlation graphs reach a more or less stable maximum level at time intervals exceeding one day, but this stability is not perfect for any correlation graph. Moreover, the maximum of correlation is affected by increasing stochastic noise for large time intervals. In an attempt
USD/DEM  USD/NLG 0 5 10 15 Data Interval (days) USD/DEM  USD/GBP 0 5 10 15 Data Interval (days) DJIA  AMEX USD/JPY  JPY/DEM 0 5 10 15 Data Interval (days) USD/FRF  USD/GBP 0 5 10 15 20 Data Interval (days) DEM 36 monthDEM 912 month 5 10 15 20 0 5 10 15 Data Interval (days) Data Interval (days) FIGURE 10.10 Linear correlation coefficients calculated for six pairs of financial instruments as a function of the size of the time interval of returns. For all calculations the total sampling period remained constant (from January 9, 1990, to January 7, 1997,) causing the 95% confidence ranges to be narrow at high data frequencies and wider as the time interval increases. Rapid declines in correlation at very high data frequencies are noted in all cases.
s § USD/DEM  USD/NLG 0.1 1.0 10.0 Data Interval (days) USD/DEM  USD/GBP £ d USD/JPY  JPY/DEM 0.1 1.0 10.0 Data Interval (days) USD/FRF  USD/GBP 0.1 1.0 10.0 Data Interval (days) DJIA  AMEX 0.1 1.0 10.0 Data Interval (days) DEM 36 monthDEM 912 montr £ 0.1 1.0 10.0 Data Interval (days) 0.1 1.0 10.0 Data Interval (days) FIGURE 10.1 I Linear correlation coefficients calculated for six pairs of financial instruments as a function of the size of the time interval of returns. The same data as ir Figure 10.10 are shown with a logarithmic horizontal axis. Rapid declines in correlation very high, intraday data frequencies are noted in all cases.
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