U.S. Stock Returns, the Berry-Esseen Theorem, and Statistical Testing

Neither existing theory nor prior empirical work can tell us the impact of non-normality on required sample sizes for Student-t tests of the mean in U.S. stock returns. Prior empirical work and bounds from a modified Berry-Esseen theorem do suggest, however, that the answer should vary with market capitalization, driven by third moments. For two-tailed nominally 5%-sized one-sample tests, we find that at least 100 observations are needed for large-capitalization stocks, and at least 200 observations are needed for small-capitalization stocks. Larger sample sizes are required for significance levels below 5%, or if one-tailed tests are used with skewed data.