On Asymptotic Optimality of Least Squares Model Averaging When True Model Is Included

TL;DR

This paper investigates the asymptotic properties of least squares model averaging when the true model is included, revealing conditions under which asymptotic risk optimality holds despite loss of loss optimality.

Contribution

It overcomes previous technical limitations by analyzing the asymptotic behavior with diverging penalty factors and fixed true model dimension, contrasting different weight sets.

Findings

Asymptotic risk optimality holds under diverging penalty factors.

Loss optimality does not hold in the same setting.

Different weight sets can lead to opposite asymptotic properties.

Abstract

Asymptotic optimality is a key theoretical property in model averaging. Due to technical difficulties, existing studies rely on restricted weight sets or the assumption that there is no true model with fixed dimensions in the candidate set. The focus of this paper is to overcome these difficulties. Surprisingly, we discover that when the penalty factor in the weight selection criterion diverges with a certain order and the true model dimension is fixed, asymptotic loss optimality does not hold, but asymptotic risk optimality does. This result differs from the corresponding result of Fang et al. (2023, Econometric Theory 39, 412-441) and reveals that using the discrete weight set of Hansen (2007, Econometrica 75, 1175-1189) can yield opposite asymptotic properties compared to using the usual weight set. Simulation studies illustrate the theoretical findings in a variety of settings.