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23 *At* «mAt► 1111 to *3 % tc, 1,,1,,1.,1 ll n t\o 1,.1.,1,.1 t2 5 t% t\\ FIGURE 3.3 An overlapping scheme of time intervals to compute a homogeneous time series of overlapping returns. All returns are measured over a time interval of size mAi. The intervals on the top time axis alone are not overlapping; an overlapping scheme arises when the phaseshifted intervals on the two lower time axes are added. In this example, the overlap factor is m = 3. where the r are independent. The investigated returns are sums of smallinterval returns: T,ri (3.18) The conclusion of Miiller (1993) is that the method of overlapping leads to a distinct but not overwhelming increase of precision and significance in most applications. Some main derivations and results of that study are presented here. There is one special case where overlapping does not help, which is the empirical mean of returns. Assume a large sample with n nonoverlapping intervals, covering the mn small intervals of size At with returns from r, to rmn. The same information can be used to compute the m(n  1) + 1 overlapping returns from rmto rmn. The mean of these overlapping returns is , mn . mn m E" = „fan + i EErfw = (319) m (=) m (n  1) + 1 t1 m (n  1) + 1 m  1 E+wi,)! + m , E r> , ml m{n 1) + 1 1 v,. , , , .. m / ( 1) + 1 m"+1" «(,11) + ] l = \ 1 ~m The corresponding nonoverlapping mean is . n . mn  Yrmi = Tn (3.20) n n (=1 (=1 by using Equation 3.18. A comparison to the last term of Equation 3.19 shows that both means are essentially equal for 3> 1; they are based on the sum of the smallinterval returns r. The remaining difference vanishes with 1 in the largesample limit, whereas the error of both means is proportional to 05 in the same limit.8 Overlapping remains a valid method, but it is unable to reduce the error of the empirical mean. Miiller (1993) has more details of this analysis.
Applications other than the computation of mean returns are more interesting. The most important example is realized volatility. A special version of Equation 3.8 is used: N+j I (3.21) 7 = 1 where the returns r, are defined by Equation 3.16 and the returns are overlapping if m > 1. An analytical exploration of this realized volatility is possible under the assumption of Equation 3.17 and the special choice p = 2. By substituting Equation 3.18, we obtain N / ;=i \/=i Nm+j+j (3.22) The expectation value of vf can be computed by expanding this expression where the cross products of r vanish due to the iid assumption. The result is E[v2] (3.23) This is the theoretical expectation of the squared return of the Brownian motion over an interval of size m . Equation 3.21 is thus an unbiased estimator, at least for p = 2. The realized volatility vj of Equation 3.21 has an error whose variance is E{\vfm 2]2} = E[vf]m2o4 = E I E riNm+j+j  m2 rx4 (3.24) where Equations 3.22 and 3.23 have been used. The further computation of this expression is somewhat tedious because of the two squares. A long sum is obtained where each term is a constant times four factors of the type .. The expectation values of these terms follow from the Gaussian i.i.d. assumption: (1) rf has an expectation of 4 (the fourth moment of the Gaussian random variable), (2) r i r has an expectation of a4 (if i j), and (3) each term with a factor x, to an odd power has the expectation zero. The resulting variance of the error is E{[v2  m a2]2} 2 m (2m2 + \)oA 3N m (m  1) 2 (2 m2 + 1) N. (3.25) for N >m.
This has to be compared to the corresponding error of an overlapfree computation from the same sample. There are only n nonoverlapping return observations (with N = mn). The variance of the error is 2 m2 ct4 2 m3 a4 n ~ N (3.26) This has been computed as a special case of Equation 3.25 (case m = 1 with only n observations, but with a variance ma1 where m is the original overlap factor). Equations 3.25 and 3.26 are now compared. The ratio of the two error variances is E{u? ma2]2} 3 + 3~m1 for N » m (3.27) This ratio is < 1, so overlapping is indeed a means to reduce the stochastic error of realized volatility to a certain extent. In the limit of a very high overlapping factor m, the error variance is reduced to a minimum of twothirds of the value without overlapping. An overlap factor of only m = 2 already reduces the error variance to 75%. This finding can also be formulated by defining the effective number of observations, neff. This is the number of nonoverlapping observations that would be needed to reach the same error variance as that based on overlapping observations. From Equations 3.25 and 3.26, we obtain "eft = 3 m N 2m2 + l 3 m2 2m2+l for n » 1 (3.28) This can be expressed as a rule of thumb. Using the method of overlapping enhances the significance of realized volatility like adding up to 50% of independent observations to a nonoverlapping sample. In Miiller (1993), an analogous study was made for "realized covariance," the empirically measured covariance between two time series, based on simultaneous overlapping returns of both series. The result is similar. The estimator based on overlapping returns is unbiased and has a reduced error variance. The effective number of observations is again given by Equation 3.28. In two cases, we have found an error reduction due to overlapping intervals. This provides a motivation for a general use of overlapping returns, also in other cases such as realized volatility with another exponent p (see Equation 3.8) and in other studies such as the analysis of the distribution of returns. The user has to be aware of and to account for the serial dependence due to overlapping and its possible effects on the results, as we have done in the derivations presented earlier. The increase in significance can roughly be expected to be as given by Equation 3.28, with some deviations due to the nonBrownian nature of the raw
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