Extend the rate argument to other Bartlett-identity tests

Develop a full treatment of likelihood-based specification tests other than the second-Bartlett-identity panel information matrix test, establishing whether the rate argument and resulting redundancy of higher-order derivative corrections extend to other Bartlett-identity tests in panels with fixed effects.

Background

The paper’s main theoretical results concern the panel information matrix test based on the second Bartlett identity. The authors observe that the key argument depends more generally on an indicator formed by averaging within-individual likelihood moments whose variance remains of constant order after square-root-of-n scaling. Under those circumstances, plug-in estimation effects and incidental-parameter bias may become asymptotically negligible or enter only through a controlled term of order square-root-of-n divided by T.

The authors explicitly leave unresolved whether these conclusions apply to other likelihood specification diagnostics based on Bartlett identities. A complete treatment would need to derive the corresponding indicators, limiting distributions, variance estimators, and incidental-parameter effects for those alternative tests.