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：在过去几十年中，模型平均（MA）作为模型选择（MS）统计方法的替代工具，吸引了广泛关注。Hansen [《计量经济学》**75** (2007) 1175–1189] 提出了一种Mallows模型平均（MMA）方法，该方法通过最小化Mallows的$C_p$准则来选取模型权重。MMA的主要理论依据是渐近最优性（AOP），即所得MA估计量的风险/损失渐近等价于最优但实际不可行的平均模型的风险/损失。文献中通过将权重约束在特定的离散权重集内或限制候选模型数量，证明了MMA的AOP。然而，本文首先表明，在这些限制下，MA的最优风险成为一个不可达的目标，且MMA可能比MS收敛得更慢。在此背景下，一个尚未解决的根本性问题是：当考虑适当大的候选模型集且模型权重未受到有害约束时，MMA估计量能否渐近地达到候选模型最优凸组合的性能？我们在MA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风险的有限样本不等式，并表明，在没有对候选模型施加不自然限制的条件下，MMA的AOP在一般连续权重集中成立，仅需满足某些温和条件。我们提出了几种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影响。模拟结果支持了我们的理论发现。