To illustrate the impact of the correlation between sample means and sample
standard deviations on calculated test statistics, we conduct the following experiment.
We randomly draw 1,000 samples of 50 observations from a )~2 distribution
with one degree of freedom. We choose the ~,2 distribution for two reasons.
First, we know with certainty that the population mean for this distribution is
one. 1 Second, like the distribution of buy-and-hold abnormal returns, the Z 2 distribution
is positively skewed with a skewness measure of 2.83. In the 1,000
samples, we reject the null hypothesis that the sample mean is equal to one (the
population mean) at the 5% theoretical significance level in favor of the alternative
hypothesis that the sample mean is significantly less than one in 6.6% of
all samples and in favor of the alternative hypothesis that the sample mean is
significantly greater than one in no samples. We refer to this as the skewness bias.