  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.
