The Davies Problem: A New Test for Random Slope in the Hierarchical Linear Model

__Abstract__ Crucial inference for the hierarchical linear model concerns the null hypothesis of no random slope. We argue that the usually applied statistical test suffers from the so-called Davies problem, that is, a nuisance parameter is only identified under the alternative. We propose an easy-to-implement methodology that exploits this property. We provide the relevant critical values and demonstrate through simulations that our new methodology has better power properties.