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 [\emph{Econometrica} \textbf{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 a nested model setting commonly adopted in the area of MA. 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. Several specific methods for constructing the candidate model sets are proposed. Implications on minimax adaptivity are given as well. The results from simulations back up our theoretical findings.

翻译：过去几十年来，模型平均（MA）作为模型选择（MS）统计方法的替代工具引起了广泛关注。Hansen [《计量经济学》**75** (2007) 1175–1189] 提出了一种Mallows模型平均（MMA）方法，通过最小化Mallows的$C_p$准则来选择模型权重。MMA的主要理论依据是渐近最优性（AOP），即所得到的MA估计量的风险/损失在渐近意义上等价于最优但不可实现平均模型的风险/损失。现有文献中，MMA的AOP要么通过将权重约束在特殊离散权重集中得到证明，要么通过限制候选模型数量来证明。然而，本文首先证明，在这些限制条件下，MA的最优风险成为不可达到的目标，且MMA的收敛速度可能慢于MS。在此背景下，一个尚未解决的基础性问题是：当考虑适当广泛的候选模型集且模型权重未受有害约束时，MMA估计量能否渐近地达到候选模型最优凸组合的表现？我们在模型平均领域常用的嵌套模型设定下回答了这一问题。我们给出了MMA风险的有穷样本不等式，并证明在不对候选模型施加非自然限制的条件下，MMA的AOP在一般连续权重集中得以在特定温和条件下成立。提出了若干构建候选模型集的具体方法，并讨论了其对极小极大适应性的含义。模拟结果进一步支持了我们的理论发现。