Testing with Vectors of Statistics: Revisiting Combined Hypothesis Tests with an Application to Specification Testing

We review tests of null hypotheses that consist of many subsidiary null hypotheses, including tests that have not received much attention in the econometrics literature. We study test performance in the context of specification testing for linear regressions based on a Monte Carlo study. Overall, parametric tests that use (transformed) P-values corresponding to all subsidiary null hypotheses outperform the well-known minimum P-value test and a recently proposed test that relies on the non-parametric estimation of the joint density of all subsidiary test statistics.