Mean GMM. Standard error

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Derrick Morton
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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

Tests for One Variance

Tests for One Variance Chapter 65 Introduction Occasionally, researchers are interested in the estimation of the variance (or standard deviation) rather than the mean. This module calculates the sample size and performs power