Model averaging in a multiplicative heteroscedastic model

Abstract

In recent years, the body of literature on frequentist model averaging in econometrics has grown significantly. Most of this work focuses on models with different mean structures but leaves out the variance consideration. In this article, we consider a regression model with multiplicative heteroscedasticity and develop a model averaging method that combines maximum likelihood estimators of unknown parameters in both the mean and variance functions of the model. Our weight choice criterion is based on a minimization of a plug-in estimator of the model average estimator’s squared prediction risk. We prove that the new estimator possesses an asymptotic optimality property. Our investigation of finite-sample performance by simulations demonstrates that the new estimator frequently exhibits very favorable properties compared with some existing heteroscedasticity-robust model average estimators. The model averaging method hedges against the selection of very bad models and serves as a remedy to variance function mis-specification, which often discourages practitioners from modeling heteroscedasticity altogether. The proposed model average estimator is applied to the analysis of two data sets on housing and economic growth.

Keywords:

• Heteroscedasticity-robust
• model averaging
• multiplicative heteroscedasticity
• plug-in
• squared prediction risk

JEL CLASSIFICATION:

• C52
• C53

Notes

1 The working model in (2) may be generalized to $y_i={\mathbf{x}}_{s,i}^{\mathrm{T}}{\mathit{\beta }}_s+f^{\ast }({\tilde{\mathbf{x}}}_{s,i},{\mathit{\gamma }}_s){ϵ}_i,$ where $f^{\ast }(·,·)$ is any other known smooth function, and all our results still hold because ML estimation remains valid under a different variance structure provided that $f^{\ast }(·,·)$ is smooth.

Additional information

Funding

Zhang’s work was supported by the National Natural Science Foundation of China (Grant nos. 71925007, 71631008 and 11871294). Wan’s work was supported by a strategic grant from the City University of Hong Kong (Grant No. 7004985). Zhao’s work was supported by a grant from the Ministry of Education in China (17YJC910011). We thank the editor, associate editor, two referees, Daniel Henderson, Regina Liu, Esfandiar Maasoumi, Jan Magnus, and Yuying Sun for helpful comments and suggestions. The article has also benefited from comments from participants at the EcoSta Conference, Hong Kong, June 2017; the IMS Pacific Rim Conference, Singapore, July 2018; and the Royal Statistical Society’s Annual Conference, Belfast, September 2019. All remaining errors are ours.