Decision theory applied to a linear panel data model

Abstract

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 gamma and a nuisance parameter tau, a N x K matrix, where N 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 tau to (delta, rho, omega), where delta is a J x K matrix of coefficients from the least-squares projection of tau on a N x J matrix x of strictly exogenous variables, rho is a K x K symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross-products in the projection of tau on x, and omega is a (N - J) x 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 invariant statistic does not depend upon omega. There is a unique invariant distribution for omega. 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 omega 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 omega have a minimax property. Given a loss function that does not depend upon omega and given a prior distribution for (gamma, delta, rho), we show how to minimize the average-with respect to the prior distribution for (gamma, delta, rho)-of the maximum risk, where the maximum is with respect to omega. There is a family of prior distributions for (delta, rho) that leads to a simple closed form for 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 gamma as N -> infinity, with a standard limiting distribution. The limit results do not require normality or homoskedasticity (conditional on x) assumptions.