  In the past decades, model averaging (MA) has attracted much attention as it
has emerged as an alternative tool to the model selection (MS) statistical
approach. Hansen [Econometrica 75 (2007) 1175--1189] introduced a Mallows model
averaging (MMA) method with model weights selected by minimizing a Mallows'
$C_p$ criterion. The main theoretical justification for MMA is an asymptotic
optimality (AOP), which states that the risk/loss of the resulting MA estimator
is asymptotically equivalent to that of the best but infeasible averaged model.
MMA's AOP is proved in the literature by either constraining weights in a
special discrete weight set or limiting the number of candidate models. In this
work, it is first shown that under these restrictions, however, the optimal
risk of MA becomes an unreachable target, and MMA may converge more slowly than
MS. In this background, a foundational issue that has not been addressed is:
When a suitably large set of candidate models is considered, and the model
weights are not harmfully constrained, can the MMA estimator perform
asymptotically as well as the optimal convex combination of the candidate
models? We answer this question in both nested and non-nested settings. In the
nested setting, we provide finite sample inequalities for the risk of MMA and
show that without unnatural restrictions on the candidate models, MMA's AOP
holds in a general continuous weight set under certain mild conditions. In the
non-nested setting, a sufficient condition and a negative result are
established for the achievability of the optimal MA risk. Implications on
minimax adaptivity are given as well. The results from simulations back up our
theoretical findings.
