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57 TABLE 5.8 Drift exponents for FX rates. Drift exponents with standard errors found for the USD against DEM, JPY, CHF, and GBP and XAU against USD for two different powers and for the interquartile range. The sampling period extends from January 1, 1987, to December 31,1995. Currency  E[\r\]  f , 11/2 £ 2]  Interquartile   0.575 ±0.006  0.501 ±0.003  0.725 ±0.017   0.570 ±0.005  0.480 ±0.035  0.691 ±0.012   0.578 ±0.004  0.514 ±0.003  0.718±0.011   0.574 ±0.005  0.500 ±0.002  0.737 ±0.015   0.576 ±0.005  0.491 ±0.002  0.754 ±0.012 
These numbers are somewhat lower than those published by Miiller et al. (1990), but this earlier study only covered 3 years of data. Tt seems that over the years the drift exponent for p = 1 has slightly decreased from typically 0.59 to 0.57. The lower the weight of tails in the statistics, the more the empirically determined drift exponent deviates from the Gaussian random walk value of 0.5. This behavior is a consequence of the changing form of the distribution under aggregation and can also be seen as a sign of multifractality, as mentioned earlier. We repeated our analysis with only studying either negative returns or positive ones. The results show no significant asymmetry of positive and negative changes in accordance to the studies of Section 5.4.1. All results indicate a very general scaling law that applies to different currencies as well as to commodities such as gold and silver (which was additionally tested with a smaller sample). This phenomenological law becomes more important as the return distributions are unstable and the scaling law cannot be explained as a trivial consequence of a stable random process. This point will be discussed further in Section 5.5.2. Besides the evidence presented in Section 5.4, we find further evidence here for an unstable distribution because the drift exponent changes for the different measures of volatility. We find lower exponents D % 0.5 for {£[r2]}1/2 and higher exponents D 0.7 for the interquartile ranges, and these can only be explained by varying distribution forms for different time intervals. Similar scaling laws have been found by Ballocchi et al. (1999b) in a study of Eurofutures contracts20 on the London International Financial Futures Exchange (LIFFE) and the Chicago Mercantile Exchane (CME) and in stock indices by Mantegna and Stanley (2000). We report here the results for the drift exponent for p = 1 for various contracts in Table 5.9. Here again the scaling law displays a drift exponent significantly larger than that expected for a Gaussian random walk and very close to the values obtained for foreign exchange rates. The table For a full description of these data and how they are treated for such a study, see Section 2.4.
table 5.9 Drift exponents for Eurofutures. Drift exponents with standard errors found for Eurofutures contracts. The drift exponent is for E[\r\] (p = 1). All values are significantly larger than 0.5. Expiry  Eurodollar  Euromark  Sterling  March 1995  0.60 ± 0.02  0.60 ±0.01  0.61 ± 0.02  June 1995  0.66 ± 0.02  0.65 ± 0.01  0.62 ± 0.02  September 1995  0.68 ± 0.02  0.66 ± 0.01  0.62 ± 0.02  December 1995  0.64 ± 0.02  0.66 ± 0.01  0.64 ± 0.02  March 1996  0.57 ± 0.03  0.66 ± 0.01  0.63 ± 0.02  June 1996  0.70 ± 0.01  0.62 ±0.01  0.62 ± 0.02  September 1996  0.70 ±0.01  0.65 ± 0.01  0.62 ± 0.01  December 1996  0.69 ±0.01  0.63 ± 0.02  0.60 ± 0.02  March 1997  0.66 ± 0.02  0.62 ± 0.02  0.63 ± 0.02 
presents quite a dispersion of the exponents because the sample for each contract is relatively short. As a second step, we have repeated the scaling law analysis on an average of contracts. We averaged the mean absolute values of the returns (associated with each time interval) on the number of contracts. When the analysis referred to single Eurofutures, the average was computed on 9 contracts; when it referred to all Eurofutures and all contracts together, the average was computed on 36 contracts. Then we performed a linear regression for the logarithm of the computed averages against the corresponding logarithm of time intervals, taking the following time intervals: 1 day, 2 days, 1 week, 2 weeks, 4 weeks, 8 weeks, and half a year. The resulting drift exponent is 0.599 ± 0.007, remarkably close to the FX results. Ballocchi (1996) performed such studies for interbank shortterm cash rates and computed the drift exponent of the mean absolute return averaged over 72 rates (12 currencies and 6 maturities from 1 month to 1 year), and the result is again 0.569 ± 0.007, close to the numbers listed in Table 5.8. To summarize, we find drift exponents of the mean absolute return of around 0.57 for FX rates and for cash interest rates, and 0.6 for Eurofutures. 5.5.2 Distributions and Scaling Laws In this section we discuss how the distributions relate to the scaling law. There are remarkably few theoretical results on the relation between the drift exponents and the distribution aside from the trivial result for Gaussian distributions where all the exponents are 0.5. We only know of two recent papers that deal with this problem, Groenendijk et al (1996) and BarndorffNielsen and Prause (1999). The lattergives E\\r\] as a function of the aggregation and the parameters of the Normal Inverse Gaussian (NIG) Levy process. The data generated from this process do not exhibit a scaling behavior, but the relationship is very close to a straight line when
expressed in a doublelogarithmic scale. The authors show that, with a particular choice of parameters, they can reproduce what they call the "apparent scaling" behavior of the USDDEM data. There are other processes that present "apparent scaling" behavior. An example of such a process is given by LeBaron (1999a). In the literature, most scaling law results are of an empirical nature and are stylized facts directly obtained from the actual data. These results do not assume any particular datagenerating process. Therefore, formal statistical tests are needed to test whetherthe empirically observed scaling laws are consistent with a particular type of null distribution. Although the findings of these statistical tests would not change the presence of empirical scaling laws, they would provide guidance for modeling return and volatility dynamics with those distributions consistent with the observed data dynamics. We shall use here the approach of Groenendijk et al. (1996) to present a theorem they prove, which gives a good understanding of the relation between the drift exponent and other distributional properties for i.i.d. distributions. If we assume a simple random walk model x{t) = x (t  At) + s(t) (5.11) where x is the usual logarithmic price and the innovations s(t) are i.i.d., then the period return r[n At](t) has the form r[nAt](t) = x(t)  x(t  nAt) = £(tiAt) (5.12) where we have used Equation 5.11. In particular, if the e,s follow a normal distribution with mean zero and variance ct2, the variance of r[n At](t) is equal to . Groenendijk et al. (1996) consider the following quantity: 1 (£[ ( )"])  In (E[\r (At) \P]) (5.13) where At is the smallest time interval and {r (n At)} is the sequence of returns aggregated times. This quantity is directly related to the lefthand term of the scaling law shown in Equation 5.10. The theorem they prove is related to the notion of stable distributions. Let us briefly state what this notion means. This class of distributions has the following attractive property. Let {an } , denote a sequence of increasing numbers such that a~] YTi=\ £i has a nondegenerate limiting distribution, then this limiting distribution must belong to the class of stable distributions (Tbragimov and Linnik, 1971, p. 37). Ibragimov and Linnik (1971) show that the numbers a„, which satisfy this requirement, are of the form an = nl/as(n), where s(n) is a slowly varying function, that is, hm  = 1 noc s(n)
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