A maximum likelihood method for the incidental parameter problem

TL;DR

This paper introduces a maximum likelihood approach based on invariance principles to address the incidental parameter problem, providing consistent and efficient estimators in certain models with fixed effects.

Contribution

It develops the maximal invariant likelihood estimator (MILE) using invariance, extending maximum likelihood methods to models with incidental parameters.

Findings

MILE is consistent when likelihood factorizes into marginal likelihoods.

MILE is asymptotically normal and efficient in models with Wishart distributions.

Application to IV and dynamic panel data models demonstrates practical utility.

Abstract

This paper uses the invariance principle to solve the incidental parameter problem of [Econometrica 16 (1948) 1--32]. We seek group actions that preserve the structural parameter and yield a maximal invariant in the parameter space with fixed dimension. M-estimation from the likelihood of the maximal invariant statistic yields the maximum invariant likelihood estimator (MILE). Consistency of MILE for cases in which the likelihood of the maximal invariant is the product of marginal likelihoods is straightforward. We illustrate this result with a stationary autoregressive model with fixed effects and an agent-specific monotonic transformation model. Asymptotic properties of MILE, when the likelihood of the maximal invariant does not factorize, remain an open question. We are able to provide consistent, asymptotically normal and efficient results of MILE when invariance yields Wishart distributions. Two examples are an instrumental variable (IV) model and a dynamic panel data model with fixed effects.