Empirical Bayes Forecasts of One Time Series Using Many Predictors

Appendix677 KB


We consider both frequentist and empirical Bayes forecasts of a single time series using a linear
model with T observations and K orthonormal predictors. The frequentist formulation considers
estimators that are equivariant under permutations (reorderings) 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 is proportional to T 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 class of nonGaussian 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 case,
where the "prior" is the weak limit of the empirical cdf of the true parameter values. Monte Carlo
results are encouraging. The new estimators are used to forecast monthly postwar U.S.
macroeconomic time series using the first 151 principal components from a large panel of


Appendix: http://www.economics.harvard.edu/faculty/stock/files/eb3_app.pdf
Last updated on 08/09/2012