AN ASYMPTOTIC THEORY FOR LEAST SQUARES MODEL AVERAGING WITH NESTED MODELS
研究了嵌套候选模型下最小二乘模型平均方法的渐近性质，发现当所有模型欠拟合时，该方法渐近地将权重分配给最大模型；当真实模型包含在内时，若权重选择准则的惩罚因子以特定阶数发散，则模型平均估计量渐近最优。
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.