Some Asymptotic Properties Of Least Squares Model Averaging

Model averaging has drawn a lot of attentions from researchers in the past decades as a powerful tool of forecasting in many areas such as econometrics,social sciences and medical studies.Although a single model can be effective in some cases,it still has the defects of losing information and high prediction risk.Model averaging combines several candidate models by assigning larger weights to the better models,by which,it often provides more precise predictions than a single model does.So far,a series of model averaging methods,such as linear model,mixed linear model,generalized linear model and generalized mixed linear model,have been developed in this field.For model averaging,asymptotic optimality of weights selection has been commonly researched on.In recent years,some scholars began to investigate the large sample properties of parameter estimations,but these results are not complete.For the most basic linear model averaging,people usually consider two different cases: one is all nested candidate models are underfitted and the other is the nested candidate models contain the true model.In case one,there exist the results of asymptotic optimality on weights selection,but without those of large sample properties on parameter estimations.In case two,there exist the results of large sample properties on parameter estimations,but without those of asymptotical optimality on weights selection.Aiming at this problem,this paper systematically studies and proves the unconcluded parts under the condition of candidate models with fixed dimensions,and comprehensively summarizes the asymptotic optimality of weights selection and large sample properties of parameter estimations in these two cases.In conclusion,we show that the weights selection will asymptotically put all the weights to the largest model when all the nested candidate models are underfitted.The asymptotical distribution of the model averaging coefficients is the same as that of the least squares estimations of the largest model.When the candidate models contain the true model,if the penalty coefficient tends to infinity and is an infinitesimal quantity of the square root of the sample size,the weights selection is asymptotically optimal.If the penalty coefficient is equal to 2,the weights selection is not asymptotically optimal.The results of this paper,together with the results in the existing literature,give the complete theoretical properties of the linear model averaging when the dimensions of candidate models are fixed.Finally,numerical experiments strongly favor the theoretical results.