Group invariance, likelihood ratio tests, and the incidental parameter problem in a high-dimensional linear model
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Date
2013-01-23
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Abstract
This paper considers a linear panel data model with reduced rank regressors and interactive fixed eff ects.
The leading example is a factor model where some of the factors are observed, some others not. Invariance
considerations yield a maximal invariant statistic whose density does not depend on incidental parameters. It
is natural to consider a likelihood ratio test based on the maximal invariant statistic. Its density can be found
by using as a prior the unique invariant distribution for the incidental parameters. That invariant distribution
is least favorable and leads to minimax optimality properties. Combining the invariant distribution with
a prior for the remaining parameters gives a class of admissible tests. A particular choice of distribution
yields the spiked covariance model of Johnstone (2001). Numerical simulations suggest that the maximal
invariant likelihood ratio test outperforms the standard likelihood ratio test. Tests which are not invariant
to data transformations (i) are uniquely represented as randomized tests of the maximal invariant statistic
and (ii) do not solve the incidental parameter problem.
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Keywords
factor models, incidental parameters, integrated likelihood, invariance, likelihood ratio test, minimax, panel data models