DECISION THEORY APPLIED TO A LINEAR PANEL DATA MODEL
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 dis-
tributed with a normal distribution. There is a parameter of interest γ and a nuisance
parameter τ,a N ×K matrix, where N is the cross-section sample size. The focus is on
dealing with the incidental parameters problemcreated 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 τ to (δρω),where δ is a J ×K
matrix of coefficients from the least-squares projection of τ on a N × J matrix x of
strictly exogenous variables, ρ is a K × K symmetric, positive semidefinite matrix ob-
tained from the residual sums of squares and cross-products in the projection of τ on x,
and ω is a (N −J)×K 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 invari-
ant 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.
Decision rules based on the invariant distribution for ω have a minimax property.
Given a loss function that does not depend upon ω and given a prior distribution for
(γδρ), we show how to minimize the average—with respect to the prior distribution
for (γδρ)—of the maximum risk, where the maximum is with respect to ω.
There is a family of prior distributions for (δρ) that leads to a simple closed formfor
the integrated likelihood function. This integrated likelihood function coincides with
the likelihood function for a normal, correlated random-effects model. Under random
sampling, the corresponding quasi maximum likelihood estimator is consistent for γ as
N →∞, with a standard limiting distribution. The limit results do not require normality
or homoskedasticity (conditional on x) assumptions.
KEYWORDS: Autoregression, fixed effects, incidental parameters, invariance, mini-
max, correlated random effects.