BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

24 ACTA WASAENSIAinferences about squared returns and volatility. In particular, the sample standarddeviation defined in (2.21) applied to squared returns is biased with sampling variancedecling slower than 1/n, which implies errors in the ACF-estimates of squared returns,used for example in ARCH-modelling, with wider than expected confidence bands 19 .Furthermore, the slower than n −1 decline in autocorrelations (see (2.17)) implies thatthe infinite sum of autocorrelations is no longer finite, such that there exists no characteristiccorrelation time after which the process may be approximated as Markovian 20 .Crato & de Lima (1994) find long range dependence in the daily squared returns of3 US stock indices in the time period from January 1980 to December 1990. Lobato& Savin (1996) extend this finding for absolute and squared returns of the S&P 500index and the 30 constituents of the Dow Jones Industrial Average between July 1962and December 1994. Lux (1996a) finds evidence for long memory in daily returns ofthe German share index DAX and its 30 constituents in 1959—88.Long range dependence in high frequency equity returns has been reported for the USstock market e.g. by Cizeau et al. (1997); Liu et al. (1997, 1999) and for the Italianstock market by Raberto, Scalas, Cuniberti & Riani (1999) 21 .2.7 MultiscalingWhen Ding et al. (1993) calculated the sample ACF as a function of various powersq of the absolute daily S&P 500 index returns ACF(|r| q ), 22 they found that it wasmonotonically increasing for q 1 and monotonically decreasing for q 1 independentofthetimelagconsidered. Thisfinding has been later confirmed for the same indexby Pasquini & Serva (1999). Nonlinear scaling of the sample ACF in powers of q hasalso been reported for the German Dax index by Lux (1996a), for the British FT-SE19 see Beran (1994: Chapter 1) and the discussion in Mikosch (2003b).20 see the discussion in Mantegna & Stanley (2000).21 For evidence of long memory in financial time series of assets other than equities see the referencesin the review studies by Farmer (2000); Cont (2001) and Lux & Ausloos (2002).22 see section 2.6.