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

ACTA WASAENSIA 127Table 15. Probability values of falsely rejecting H 0 : ρ ≥ 1 (and const.= 0) in DickeyFuller tests of the form ∆p t =(ρ−1)p t−1 +(const.)+ t over the full sampleof 20,000 observations.∆p t =(ρ − 1)p t−1 +const.+ t : ∆p t =(ρ − 1)p t−1 + t :p-values: ρ const. ρp 1 1.95 · 10 −13 0.7709 2.03 · 10 −13p 2 2.63 · 10 −11 0.5691 3.08 · 10 −11in table 15, no matter whether an (insignificant) constant is included into the Dickey-Fuller regressions or not. In that respect our price series look still as unsatisfactory asthose of Lux & Marchesi (2000), but with the hindsight of the simulation study by Lux& Marchesi (1999) it appears likely that the failure of the simulations to produce integratedprices is again just due to the simplifying assumption of constant fundamentalvalues. Therefore, as in Lux & Marchesi (2000), figure 13 should be mainly regardedas a visualization of the behaviourally explained difference between trading prices andfundamental values rather than trading prices as such.The two upper panels of figure 14 contain the logreturns of the two stocks calculatedas the difference between the simulated the logarithmic trading prices p 1 and p 2 overunit time steps asr i,t = p i,t − p i,t−1 , i =1, 2. (5.16)The third panel contains the logreturn of the equal weighted index calculated as1r EW,t =ln2 exp(r 1,t)+ 1 2 exp(r 2,t) . (5.17)Assuming a symmetric setup with equally many stocks issued by both companies, thereturns of a capitalization weighted index may be calculated asexp(p 1,t−1 )r CW,t =lnexp(p 1,t−1 )+exp(p 2,t−1 ) exp(r exp(p 2,t−1 )1,t)+exp(p 1,t−1 )+exp(p 2,t−1 ) exp(r 2,t) ,(5.18)which are plotted in the last panel of figure 14. All time series are clearly heteroscedasticwith similar intermittent outbreaks of volatility as in figures 1 and 2 of section 4.3.2,and discussed as stylized facts of real financial returns in section 2.5.Table 16 contains summary statistics for the above mentioned return series. All time