ON ASYMPTOTIC OPTIMALITY OF LEAST SQUARES MODEL AVERAGING WHEN TRUE MODEL IS INCLUDED

Asymptotic optimality is a key theoretical property in model averaging. Due to technical difficulties, existing studies rely on restricted weight sets or on the assumption that there is no true model with fixed dimensions in the candidate set. The focus of this article is to overcome these difficulties by studying a class of least squares model averaging methods with nested candidate models, which includes Mallows model averaging (MMA) of Hansen (2007, Econometrica 75, 1175–1189) and parsimonious model averaging (PMA) of Zhang et al. (2020, Journal of the American Statistical Association 115, 972–984) as special cases. The main results are as follows. First, MMA with fixed model dimensions is neither asymptotically loss optimal nor asymptotically risk optimal. Second, when the true model dimension diverges, MMA is asymptotically both loss and risk optimal. Third, when the true model dimension is fixed, PMA is asymptotically risk optimal but not asymptotically loss optimal, which differs from the corresponding result in Fang, Yuan, and Tian (2023, Econometric Theory 39, 412–441). This finding also reveals that the discrete weight set of Hansen (2007) and the usual weight set can lead to different asymptotic properties: asymptotic loss optimality holds under the former but fails under the latter. Finally, when the true model dimension diverges, PMA is asymptotically both loss and risk optimal. Simulation studies illustrate the theoretical findings in a variety of settings.