Nuisance parameters, modified profile likelihood and Jacobian prior

In a model with nuisance parameters, the maximum likelihood estimators (MLE) of the parameters of interest can be biased. One can reduce the bias due to the presence of the nuisance parameters by removing the O(1) bias of the profile likelihood score. To achieve this, we propose the Jacobian integrated likelihood (JIL) obtained by using a prior consisting of the Jacobian determinant of the new nuisance parameters, which are functions of the original nuisance parameters and are independent of the dependent variable. Our JIL is closely related to the modified profile likelihood (MPL) in Barndorff-Nielsen and Cox (1994). We propose the adjusted MPL, which is easier to compute and can also remove the O(1) bias of the profile likelihood score. For panel fixed effects models, both the JIL and the adjusted MPL can remove the bias of order O(T−1) in the MLE as the cross-sectional size (N) increases. We give the conditions when the estimators from the adjusted MPL and the JIL are the same and consistent with T = o(N). Although the adjusted MPL and the JIL do not always exist, one can use their first-order conditions to obtain bias-reduced estimators. The theoretical results are demonstrated by panel binary choice models and dynamic panel linear models with exogenous and predetermined regressors.