This article aims at achieving two distinct goals. The first is to extend the existing LM test of overdispersion to the situation where the alternative hypothesis is characterized by the correlated random effects model. We obtain a result that the test against the random effects model has a certain max-min type optimality property. We will call such a test the LM test of overdispersion. The second goal of the article is to draw a connection between panel data analysis and the analysis of multiplicity of equilibrium in games. Because such multiplicity can be viewed as a particular form of neglected heterogeneity, we propose an intuitive specification test for a class of two-step game estimators.