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104 TABLE 10.5 Correlation results characterizing the Epps effect. Minima, maxima, and mean values (averaged over time intervals between I and 2 days) of correlation coefficients. These correlations are functions of the time interval of return measurement, as plotted in Figures 10.10 and 10.1 I. Also given are the time intervals at which the maxima occurred and the time intervals where the coefficient reaches 90% of the mean value. This latter time interval in the last column is called the stabilization intervalthe threshold after which the Epps effect no longer affects correlation results. The sampling period was from January 9, 1990, to January 7, 1997. Instrument Min. Max. Interval Mean 90% Stabtliz. pair corr. corr. of max. corr. of interval (days) (12 days) mean (min) t SDDEM  USDGBP 0.55 0.86 7.2 0.79 0.71 10 ISDDEM USDNLG 0.78 1.00 14.0 0.99 0.89 15 I SDFRF USDITL 0.49 0.86 12.0 0.79 0.71 25 USDNLG  USDFRF 0.69 0.99 16.9 0.97 0.87 25 USDFRF  USDGBP 0.48 0.86 7.2 0.80 0.72 30 USDJPY  DEMJPY 0.34 0.62 19.4 0.48 0.43 30 DF.MGBPUSDGBP 0.23 0.75 17.0 0.45 0.41 170 DJIAAMEX 0.00 0.86 13.3 0.77 0.69 320 DEM 36MDEM 912M 0.40 0.90 19.2 0.82 0.74 340 to give a more accurate reference level of the correlation drop due to the Epps effect, Table 10.5 also reports the arithmetic mean of all correlation values based on time intervals between 1 and 2 days. A total of 224 correlation values (i.e., 224 aggregated time series) belong to this range of time intervals. Although there is no best choice of time interval for the general case, this mean value is considered as a reasonable reference level of correlation in its stable region. When moving to shorter time intervals, the Epps effect makes the correlation values decline. As a threshold value of the Epps effect, Table 10.5 also shows the time interval at which correlations drop to 90% of the reference level. This estimation of the Epps threshold or, seen from the other side, the correlation stabilization interval has the advantage that it can be uniformly applied to all cases, and it does not deliver obviously misleading results based on the maximum value of correlation. We find that even currency pairs that are highly correlated in the long term become much less correlated in the intrahour data frequency range. Miiller et al. (1997a) propose a hypothesis of heterogeneous markets where the market agents differ in their perception of the market, have differing risk profiles, and operate under different institutional constraints (see also Section 7.4). If the financial markets are indeed composed of heterogeneous agents with different time horizons of interest, then the Epps effect in correlation estimations can be interpreted as a cutoff between groups of agents. Shortterm traders focus on the rapid movements of individual rates rather than multivariate sets of assets. For these shortterm agents, correlations between instruments play a secondary role. Other, less rapid agents reestablish "correct" correlations after market shocks, but this takes some time.
Stabilization vs. Tick Activity 2 4 6 8 1/sqrt(A1 * A2) (min/tick) FIGURE 10.12 Correlation stabilization intervals (= stabilization points, in min) as a function of the inverse square root of the product of the tick frequencies. The same instrument pairs and stabilization intervals as in Table 10.5 are plotted. When considering the stabilization interval in the last column of Table 10.5, we find that the Epps effect already vanishes at return measurement intervals of 10 min for the correlation of the most frequently quoted financial instruments. For pairs of less frequently quoted instruments, the Epps effect may last for hours, up to 6 hr in Table 10.5. This indicates a relationship between tick frequencies (and perhaps liquidity) of instruments and the duration of the Epps effect. Two correlation studies were made to probe this relationship: (1) standard correlation between the stabilization interval and the tick frequency of the more frequently quoted instrument of the pair and (2) standard correlation between the stabilization interval and the tick frequency of the less frequently quoted instrument. Mean tick frequencies per business day were taken, as listed in Table 2.2. The greater of the two tick frequencies is estimated to have a standard correlation of 0.59 to the stabilization interval. The corresponding standard correlation of the lower tick frequency is 0.65. These values are significant to 92% and 95% confidence levels, respectively, assuming a normal random distribution. Therefore we conclude that, to a reasonable level of confidence, both of the tick frequencies substantially affect the stabilization interval after which the Epps effect of correlations vanishes. Tick frequency and stabilization interval are inversely related. This can be seen graphically in Figure 10.12 where the stabilization interval is plotted versus the inverse geometric mean of the two tick frequencies. When two tick frequencies are very high, as on the lefthand side of Figure 10.12, the stabilization interval becomes small. On the other side, at very low data frequencies, a plateau in the correlation stabilization interval appears to exist at a data interval of 300 to 400 min. This would indicate that the Epps effect does not play a substantial role in attenuating correlation values beyond 6hr return measurement intervals,
oven if the instruments involved are very inactive (< 100 data updates per business day). It should be stated that these are indicative and preliminary results and an enhanced study with more instruments and statistics is called for. 10.6 CONCLUSIONS The problems associated with estimation of correlation at higher data frequencies have been discussed and illustrated using examples. An easytouse covolatility adjusted correlation estimator, which correctly accounts for missing or nonexistent data, has been proposed. The effect of this new formulation is to estimate correlation when data exist, and to make no update to the correlation calculation when data do not exist. The input of the method is homogeneous time series linearly interpolated between the ticks. At times when tick intervals are longer than the return measurement intervals, the weight of the return observations tends to / . Because the estimator is adjusted by covolatility, some of the information from the more frequent of the two time scries involved will not be fully utilized, and statistical significance can be degraded. With growing data frequency, this degradation is inevitable but tolerable because the statistical significance based on highfrequency sampling is high by nature. Covolatility adjusted correlation is an estimator complementary to other estimators. Its fluctuating weighting of observations is an alternative to time scale transformations such as the #time discussed in Chapter 6. Empirically estimated linear correlation coefficients of returns vary over time. The return correlations of some financial instrument pairs widely fluctuate from week to week, whereas other correlations are very stable over periods of many years. It was observed that longterm historical stability is not a guarantee of future correlation stability. This was evidenced through the examination of USDITL and USDGBP return correlations with the USDDEM rate. The crisis of 1992, when the involvement of ITL and GBP in the European Monetary System (EMS) was suspended, was reflected by a dramatic and rapid change in their correlations with other currencies thereafter. Correlation calculated over a long data sample (years) has an averaging effect, and increased structure in correlations is observed when correlations are calculated over smaller periods (weeks). Depending on the time horizon of interest, there is pertinent information to be gained when calculations are performed over smaller periods using highfrequency data. The selfmemories of return correlations have been modeled as exponential attenuations through estimation of the autocorrelation of linear correlation coefficients. Correlation values have memories for their past values that differ between instument pairs and often extend over years rather than only weeks. The understanding of this correlation memory is a first step toward correlation forecasting. The behavior of the correlation coefficient as a function of the time interval of return measurement has been investigated. Nonzero correlations of returns are dramatically attenuated when this interval decreases and enters the intrahour region. This behavior is called the Epps effect and depends on the pair of investi
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