Table 1 Simple Wavelet Analysis for stocks in the S&P 500 Index as of December 31 st 1998 ^ Shapiro- GMM Normality 6 0.9664 0.00281 11.36 4.14 55 7 0.9790 0.00300 56.58 31.69 45 8 0.9689 0.00319 403.49 233.02 205 9 0.9746 0.00316 2393.66 1906.18 280 10 0.9712 0.00324 16876.25 15267.88 353 11 0.9664 0.00335 133227.62 122728.61 360 12 0.9638 0.00338 1020755.86 983949.49 360 ^ 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 6 0.0015 0.01027 0.9071 22.53* 7 0.0003 0.00797 0.9761 6.89* 8-0.0007 0.01026 0.9013 1403.81* 9 0.0013 0.01025 0.9441 1670.33* 10 0.0005 0.01020 0.9349 10009.33* 11 0.0003 0.01024 0.9391 12140.37* 12-0.0003 0.00941 0.9459 25032.07* * 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 6 0.0034 0.01098 0.9595 4.15* 7 0.0008 0.00882 0.9783 6.15* 8-0.0003 0.00835 0.9620 90.77* 9 0.0005 0.00866 0.9434 1157.58* 10 0.0004 0.00783 0.9443 5582.16* 11 0.0002 0.0074 0.9149 18645.23* 12-0.0003 0.00648 0.8882 194321.92* * The null hypothesis of Normality for the corresponding wavelet time series is rejected at the 1 % significance level. 1
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 6 0.9835 0.00034 1.503 0.078 12 7 0.9906 0.00017 1.642 0.086 12 8 0.9945 0.00011 2.132 0.126 32 9 0.9970 0.00005 2.039 0.082 25 10 0.9974 0.00102 1.893 0.066 15 11 0.9992 0.00001 1.988 0.068 11 12 0.9996 0.00001 1.944 0.063 9 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 6 0.9650 0.00070 5.70 0.304 168 7 0.9739 0.00046 10.27 0.541 317 8 0.9791 0.00032 20.17 0.882 590 9 0.9829 0.00021 35.52 1.055 902 10 0.9843 0.00103 64.43 1.195 998 11 0.9873 0.00009 117.15 1.518 1000 12 0.9880 0.00006 236.25 2.293 1000 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
Table 6 Variance Ratio Analysis for stocks in the S&P 500 Index as of December 31 st 1998 ^ Variance Adjusted Z q 2 0.9693 0.00348-0.6207 0.06178 8 4 0.9178 0.00699-0.6803 0.05374 5 8 0.9634 0.01175-0.1842 0.04675 1 16 1.0677 0.03040 0.0478 0.05862 5 ^ 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* 2 0.9557-0.7949 4 0.8572-1.1975 8 1.2249 0.9780 16 1.6026 1.4097 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
Table 8 Variance Ratio Analysis for the S&P 100 Index as of December 31 st 1998 Variance Heteroscedasticity Adjusted Z q* 2 1.1949 2.9867* 4 1.2789 1.9609^ 8 1.6114 2.1036^ 16 1.4645 0.9607 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 2 0.9693 0.00348-0.6207 0.06178 8 4 0.9178 0.00699-0.6803 0.05374 5 8 0.9634 0.01175-0.1842 0.04675 1 16 1.0677 0.03040 0.0478 0.05862 5 Independent Normal variates N n are obtained using a Box Muller algorithm coded in C++. 4
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 2 1.0017 0.00136 0.0226 0.03133 3 4 0.9904 0.00328-0.1159 0.02863 4 8 0.9386 0.00708-0.3217 0.02784 2 16 0.8203 0.01115-0.4597 0.02387 5 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