Combining Significance of Correlated Statistics with Application to Panel Data

The inverse normal method, which is used to combine P-values from a series of statistical tests, requires independence of single test statistics in order to obtain asymptotic normality of the joint test statistic. The paper discusses the modification by Hartung (1999, Biometrical Journal, Vol. 41, pp. 849-855), which is designed to allow for a certain correlation matrix of the transformed P-values. First, the modified inverse normal method is shown here to be valid with more general correlation matrices. Secondly, a necessary and sufficient condition for (asymptotic) normality is provided, using the copula approach. Thirdly, applications to panels of cross-correlated time series, stationary as well as integrated, are considered. The behaviour of the modified inverse normal method is quantified by means of Monte Carlo experiments