This paper examines the asymptotic risk of nested least-squares averaging estimators
when the averaging weights are selected to minimize a penalized leastsquares
criterion. We find conditions under which the asymptotic risk of the
averaging estimator is globally smaller than the unrestricted least-squares estimator.
For theMallows averaging estimator under homoskedastic errors, the condition
takes the simple form that the regressors have been grouped into sets of
four or larger. This condition is a direct extension of the classic theory of James–
Stein shrinkage. This discovery suggests the practical rule that implementation of
averaging estimators be restricted to models in which the regressors have been
grouped in this manner. Our simulations show that this new recommendation results
in substantial reduction in mean-squared error relative to averaging over all
nested submodels.We illustrate the method with an application to the regression
estimates of Fryer and Levitt (2013).
Keywords. Shrinkage, efficient estimation, averaging, risk.
JEL classification. C13.