Title

Empirical Bayes Regression With Many Regressors

Author(s)

Thomas Knox Thomas Knox (University of Chicago)

James H. Stock James Stock (Harvard University)

Mark W. Watson Mark Watson (Princeton University)

Abstract

We consider frequentist and empirical Bayes estimation of linear regression coefficients with T observations and K orthonormal regressors. The frequentist formulation considers estimators that are equivariant under permutations of the regressors. The empirical Bayes formulation (both parametric and nonparametric) treats the coefficients as i.i.d. and estimates their prior. Asymptotically; when K =ρΤδ for 0 < ρ < 1 and 0 < δ ≤ 1; the empirical Bayes estimator is shown to be: (i) optimal in Robbins' (1955; 1964) sense; (ii) the minimum risk equivariant estimator; and (iii) minimax in both the frequentist and Bayesian problems over a wide class of error distributions. Also; the asymptotic frequentist risk of the minimum risk equivariant estimator is shown to equal the Bayes risk of the (infeasible subjectivist) Bayes estimator in the Gaussian model with a "prior" that is the weak limit of the empirical c.d.f. of the true parameter values.

Creation Date

2004-01

Section URL ID

Paper Number

2004-1

URL

http://www.princeton.edu/~mwatson/papers/knox_stock_watson_empb_AS_1.pdf

File Function

Jel

C13, C20

Keyword(s)

Large model regression, equivariant estimation, minimax estimation, shrinkage estimation

Suppress

false

Series

13