  We consider benchmarked empirical Bayes (EB) estimators under the basic
area-level model of Fay and Herriot while requiring the standard benchmarking
constraint. In this paper we determine the excess mean squared error (MSE) from
constraining the estimates through benchmarking. We show that the increase due
to benchmarking is O(m^{-1}), where m is the number of small areas.
Furthermore, we find an asymptotically unbiased estimator of this MSE and
compare it to the second-order approximation of the MSE of the EB estimator or,
equivalently, of the MSE of the empirical best linear unbiased predictor
(EBLUP), that was derived by Prasad and Rao (1990). Morever, using methods
similar to those of Butar and Lahiri (2003), we compute a parametric bootstrap
estimator of the MSE of the benchmarked EB estimator under the Fay-Herriot
model and compare it to the MSE of the benchmarked EB estimator found by a
second-order approximation. Finally, we illustrate our methods using SAIPE data
from the U.S. Census Bureau, and in a simulation study.
