AN ASYMPTOTIC THEORY FOR LEAST SQUARES MODEL AVERAGING WITH NESTED MODELS
嵌套模型下最小二乘模型平均的渐近理论AI 解读 ↓
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嵌套模型下最小二乘模型平均的渐近理论
- 研究问题
- 研究嵌套候选模型下最小二乘模型平均的权重、极限分布与最优性，区分所有模型均欠拟合和候选集合包含真实模型的情形。
- 方法与思路
- 作者分析包含 Mallows 模型平均在内的一类权重选择方法，分别考察两种候选集合，并研究权重空间和惩罚因子增长的影响。
- 材料陈述的贡献
- 摘要称，全欠拟合时权重趋向最大模型；包含真实模型时特定惩罚与权重条件可实现最优性，而固定维数的 Mallows 方法不满足该性质。
- 阅读关注点
- 应核对渐近最优性的损失定义、特殊权重空间和惩罚增长阶数，以及模型维数与候选集合限制如何影响权重集中和正态性结果。
- 阅读边界
- 仅据所给 Crossref 摘要，无法重建具体限制；对固定维数 Mallows 方法的结论不能推广为所有模型平均方法均不最优。
原文摘要
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.