Theoretical results of frequentist model averaging mainly focus on asymptotic optimality and asymptotic distribution of the model averaging estimator. However, even for basic least squares model averaging, many theoretical problems have not been well addressed yet. This article discusses asymptotic properties of a class of least squares model averaging methods with nested candidate models that includes the Mallows model averaging (MMA) of Hansen (2007, Econometrica 75, 1175–1189) as a special case. Two scenarios are considered: (i) all candidate models are under-fitted; and (ii) the true model is included in the candidate models. We find that in the first scenario, the least squares model averaging method asymptotically assigns weight one to the largest candidate model and the resulting model averaging estimator is asymptotically normal. In the second scenario with a slightly special weight space, if the penalty factor in the weight selection criterion is diverging with certain order, the model averaging estimator is asymptotically optimal by putting weight one to the true model. However, MMA with fixed model dimensions is not asymptotically optimal since it puts nonnegligible weights to over-fitted models. The theoretical results are clearly summarized with their restrictions, and some critical implications are discussed. Monte Carlo simulations confirm our theoretical results.

频率论模型平均的理论结果主要集中在模型平均估计量的渐近最优性和渐近分布上。然而，即使对于基本的最小二乘模型平均，许多理论问题也没有得到很好的解决。本文讨论了一类具有嵌套候选模型的最小二乘模型平均方法的渐近特性，其中包括 Hansen 的 Mallows 模型平均 (MMA)（2007 年，Econometrica75, 1175–1189) 作为特例。考虑两种情况：（i）所有候选模型都欠拟合；(ii) 真实模型包含在候选模型中。我们发现，在第一种情况下，最小二乘模型平均法将权重 1 渐近地分配给最大的候选模型，并且所得模型平均估计量是渐近正态的。在具有稍微特殊的权重空间的第二种情况下，如果权重选择标准中的惩罚因子以一定顺序发散，则模型平均估计器通过将权重 1 赋给真实模型来渐近最优。然而，具有固定模型维度的 MMA 并不是渐近最优的，因为它给过度拟合的模型赋予了不可忽略的权重。理论结果清楚地总结了它们的局限性，并讨论了一些重要的影响。蒙特卡罗模拟证实了我们的理论结果。