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1 Table 1 Simple Wavelet Analysis for stocks in the S&P 500 Index as of December 31 st 1998 ^ Shapiro- GMM Normality ^ 360 of the 500 stocks in the Index at the end of 1998 have a complete history for the selected sample interval. Table 2 Simple Wavelet Analysis for the S&P 500 Index Wave Shapiro-Wilk GMM Normality Deviation Normality * * * * * * * * The null hypothesis of Normality for the corresponding wavelet time series is rejected at the 1 % significance level. Table 3 Simple Wavelet Analysis for the S&P 100 Index Wave Shapiro-Wilk GMM Normality Deviation Normality * * * * * * * * The null hypothesis of Normality for the corresponding wavelet time series is rejected at the 1 % significance level. 1
2 Table 4 Simple Wavelet Analysis for 1000 Simulations of a Lognormal Stock Process with an annual drift µ of 13 % and annual volatility σ of 35 %^ Shapiro- GMM Normality Independent Normal variates N n are obtained using a Box Muller algorithm coded in C++. Table 5 Simple Wavelet Analysis for 1000 Simulations of a Lognormal Stock Process with a mid-sample drift and volatility shift^ Shapiro- GMM Normality Independent Normal variates N n are obtained using a Box Muller algorithm coded in C++. Annual drift µ and volatility σ are assumed to shift from 13 % and 25 % to 5 % and 65 % respectively at mid-sample. 2
3 Table 6 Variance Ratio Analysis for stocks in the S&P 500 Index as of December 31 st 1998 ^ Variance Adjusted Z q ^ 360 of the 500 stocks in the Index at the end of 1998 have a complete history for the selected sample interval. Table 7 Variance Ratio Analysis for the S&P 500 Index as of December 31 st 1998 Variance Heteroscedasticity Adjusted Z q* The Heteroscedasticity-robust test statistic Z q is computed as Z q = (nq) 1/2 M q / (ϑ q) 1/2. Under the random walk null hypothesis, the value of the variance ratio is 1 and the test statistic Z q is asymptotically distributed as 3
4 Table 8 Variance Ratio Analysis for the S&P 100 Index as of December 31 st 1998 Variance Heteroscedasticity Adjusted Z q* * ^ ^ The Heteroscedasticity-robust test statistic Z q is computed as Z q = (nq) 1/2 M q / (ϑ q) 1/2. Under the random walk null hypothesis, the value of the variance ratio is 1 and the test statistic Z q is asymptotically distributed as ^ One-sided test rejection of the random walk null hypothesis at 5 % Table 9 Variance Ratio Analysis for 1000 Simulations of a Lognormal Stock Process with an annual drift µ of 13 % and annual volatility σ of 35 %^ Variance Adjusted Z q Independent Normal variates N n are obtained using a Box Muller algorithm coded in C++. 4
5 Table 10 Variance Ratio Analysis for 1000 Simulations of a Lognormal Stock Process with a mid-sample drift and volatility shift^ Variance Adjusted Z q Independent Normal variates N n are obtained using a Box Muller algorithm coded in C++. Annual drift µ and volatility σ are assumed to shift from 13 % and 25 % to 5 % and 65 % respectively at mid-sample. 5
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