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99 small deviations that exist are due to the statistical error (~ 2%) of these tests. The third column of Table 10.1, by contrast, shows standard correlation values severely affected by the interpolationfiHed data gaps. This simple example illustrates one of the original design goals of the covolatility adjusted linear correlation measure: correlation is measured where data exist, and the calculation is not updated where data do not exist. Tests with foreign exchange data were performed to exemplify the effect of the covolatility adjusted correlation measure on time series with fluctuating data frequency and volatility. Homogeneous time series of USDDEM prices were generated according to Section 3.2.2, equally spaced by 3min intervals, once in physical time, once in ??time as explained in Chapter 6. USDDEM has a high data frequency (see Table 2.2), but is also characterized by large intraday and intraweek fluctuations of both data frequency and volatility as shown in Section 5.6.2 and Figure 5.12. Absolute value of USDDEM returns were used because they are known to have autocorrelations of greater magnitude than actual returns. Three autocorrelation functions are investigated: (1) standard autocorrelation (Equation 10.8) of 18min returns in physical time, (2) standard autocorrelation of 18min returns in iztime (see Chapter 6), and (3) autocorrelation measured by the covolatility adjusted correlation coefficients (Equation 10.5), analyzing 18min in physical time. The covolatility computation was done in 3min intervals and with m  6 (Equation 10.3), resulting in a covolatility value every 18 min. Results of these measurements are shown in Figure 10.1. A total data period of 6 months was used, ranging from January 1, 1996, to July 1, 1996. The covolatility adjusted autocorrelation values (bullets in Figure 10.1) are significantly lower than the corresponding standard autocorrelation values, but close to the standard autocorrelation of the series equally spaced in iztime. We ascribe the high level of standard autocorrelation in physical time to the weekly seasonality of the data. The high absolute returns during working days and the low values on weekends are responsible for part of the high standard autocorrelation at lags up to about 1 day. As described in Chapter 6, iztime eliminates seasonality and thus the part of the autocorrelation caused by seasonality. The covolatility behaves similarly in the following respect. Weekends with their data gaps have extremely low covolatility values, so they are practically eliminated from the statistics. Weekly seasonality no longer affects the statistics. At lags around 24 hr, the picture is different. The covolatility adjusted autocorrelation approaches the value of the standard autocorrelation in a clear peak which indicates daily seasonality. Unlike iztime, which deforms time to eliminate seasonality, the covolatility adjusted correlation measure was designed to give a high weight to the most active periods, with no intention to hide all the seasonalities. Removing seasonality is not always desirable, so we find the covolatility adjusted correlation estimation to be a suitable method for many applications. In addition, the simplicity of this methodology lends itself to wider use.
1 1 8 CM < LU E5° co Z) • Volatility Weighted AC (physical time) A Standard AC (physical time) + Standard AC (theta time) Lag (hours) FIGURE 10.1 Autocorrelation of the absolute values of USDDEM returns as a function of the time lag. The triangles ( ) refer to standard autocorrelation of absolute returns, equally spaced in physical time. Bullets (•) refer to the covolatility adjusted autocorrelation of the same absolute returns. Crosses (+) refer to standard autocorrelation of absolute returns, equally spaced in ?time. Sampling period: January I, 1996, to July I, 1996. 10.4 STABILITY OF RETURN CORRELATIONS When correlation is calculated between two time series, the assumption is that this quantity does not vary over time. For the case of financial time series this is seldom occurs, although time variance of the correlation coefficient over time can sometimes be small. This issue is critical for portfolio pricing and risk management where hedging techniques can become worthless when they are most needed, during periods known as correlation "breakdown," or relatively rapid change. Boyer et al. (1997) have also demonstrated that a detection of correlation breakdown or other structural breaks by splitting a return distribution into a number of quantiles can yield misleading results. We use highfrequency data to estimate correlations literally as a function of time for a number of different financial time series in an effort to better understand the level of change that can occur. Highfrequency correlation estimations are contrasted with lowerfrequency estimates for the same sample periods. The "memory" that correlation coefficients have for their past values is also estimated for a number of examples using a simple
TABLE 10.2 Data sampling for correlation as function of time. Four different sampling schemes are selected to divide the total sampling period of size T from January 7, 1990, to January 5, 1997. Correlation  Data frequency  Number of returns  95% confidence  calculation  (number of returns  per correlation  band  period  per day)  calculation  1.96/7   f = nN/T    365 days    0.10  128 days    0.10  32 days    0.10  7 days    0.09 
and appropriate parameterization. Such estimations can be applied to longterm correlation forecasting, which is required, for example, to price or hedge financial options involving multiple assets, Gibson and Boyer (1997). 10.4.1 Correlation Variations over Time The general stability of correlation coefficients was examined using various correlation calculation intervals and data frequencies. This involved examination of a fixed historical time series over a time period T, from January 7,1990, to January 5,1997. The time series of returns ( ( )) was then divided into N subsets of equal duration (T/N) from which correlation coefficients were computed according to Equation 10.1. Four values of N were selected, while the total period T always remained constant. A homogeneous series of n returns was then chosen inside each period of size T/N via linear interpolation, so each correlation coefficient is based on n observations. Similar numbers n were selected for all the four values of N in order to maintain nearly uniform statistics, as shown in Table 10.2. In this table, the number of return observations per day, / = n N/ T, is also given. Results from these calculations are shown in Figures 10.2 to 10.7, where correlations versus time are displayed, and dashed lines above and below zero correlation are 95% confidence ranges assuming normally distributed random distributions. The confidence limits are slightly nonuniform due to small variations in statistics. Some correlations were computed with fewer observations than n because of missing observations. Whenever a weight from Equation 10.3 was equal to zero, the corresponding observation was ignored. The weights were not used for any other purpose, and the correlations remain standard linear correlations defined by Equation 10.1. Correlation coefficient mean values and variances are given for each pair of financial instruments and for each of the four calculation frequencies in Table 10.3. Having virtually the same statistical significance for all correlation calculations shown in Figures 10.2 through 10.7, we can make a number of observations about
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