Econometrica: Jan, 2009, Volume 77, Issue 1
Decision Theory Applied to a Linear Panel Data Model
Gary Chamberlain, Marcelo J. Moreira
This paper applies some general concepts in decision theory to a linear panel data model. A simple version of the model is an autoregression with a separate intercept for each unit in the cross section, with errors that are independent and identically distributed with a normal distribution. There is a parameter of interest γ and a nuisance parameter τ, a × matrix, where is the cross‐section sample size. The focus is on dealing with the incidental parameters problem created by a potentially high‐dimension nuisance parameter. We adopt a “fixed‐effects” approach that seeks to protect against any sequence of incidental parameters. We transform to (δ, ρ, ω), where δ is a × matrix of coefficients from the least‐squares projection of on a × matrix of strictly exogenous variables, ρ is a × symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross‐products in the projection of τ on , and is a (−) × matrix whose columns are orthogonal and have unit length. The model is invariant under the actions of a group on the sample space and the parameter space, and we find a maximal invariant statistic. The distribution of the maximal invariant statistic does not depend upon ω. There is a unique invariant distribution for ω. We use this invariant distribution as a prior distribution to obtain an integrated likelihood function. It depends upon the observation only through the maximal invariant statistic. We use the maximal invariant statistic to construct a marginal likelihood function, so we can eliminate ω by integration with respect to the invariant prior distribution or by working with the marginal likelihood function. The two approaches coincide.