Testing the null hypothesis of zero serial correlation in short panel time series: a comparison of tail probabilities

It is known that the analysis of short panel time series data is very important in many practical problems. This paper calculates the exact moments up to order 4 under the null hypothesis of no serial correlation when there are many independent replications of size 3. We further calculate the tail probabilities under the null hypothesis using the Edgeworth approximation for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=3, 4$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5,$$\end{document} when the structure of the pdf (probability density function) of the test statistic is in essence different. Finally, we compare the three types of tail probabilities, namely, the Edgeworth approximation, the normal approximation and the exact probabilities through a large scale simulation study.