Asymptotic power of the sphericity test under weak and strong factors in a fixed effects panel data model

This paper studies the asymptotic power for the sphericity test in a fixed effect panel data model proposed by Baltagi et al. (2011), (J_BFK). This is done under the alternative hypotheses of weak and strong factors. By weak factors, we mean that the Euclidean norm of the vector of the factor loadings is O(1). By strong factors, we mean that the Euclidean norm of the vector of factor loadings is O(√n), where n is the number of individuals in the panel. To derive the limiting distribution of J_BFK under the alternative, we first derive the limiting distribution of its raw data counterpart. Our results show that, when the factor is strong, the test statistic diverges in probability to infinity as fast as O_p(nT). However, when the factor is weak, its limiting distribution is a rightward mean shift of the limit distribution under the null. Second, we derive the asymptotic behavior of the difference between J_BFK and its raw data counterpart. Our results show that when the factor is strong, this difference is as large as O_p(n). In contrast, when the factor is weak, this difference converges in probability to a constant. Taken together, these results imply that when the factor is strong, J_BFK is consistent, but when the factor is weak, J_BFK is inconsistent even though its asymptotic power is nontrivial.