Distribution Theory of the Least Squares Averaging Estimator

Abstract

This paper derives the limiting distributions of least squares averaging estimators for linear regression models in a local asymptotic framework. We show that the averaging estimators with fixed weights are asymptotically normal and then develop a plug-in averaging estimator that minimizes the sample analog of the asymptotic mean squared error. We investigate the focused information criterion (Claeskens and Hjort, 2003), the plug-in averaging estimator, the Mallows model averaging estimator (Hansen, 2007), and the jackknife model averaging estimator (Hansen and Racine, 2012). We find that the asymptotic distributions of averaging estimators with data-dependent weights are nonstandard and cannot be approximated by simulation. To address this issue, we propose a simple procedure to construct valid confidence intervals with improved coverage probability. Monte Carlo simulations show that the plug-in averaging estimator generally has smaller expected squared error than other existing model averaging methods, and the coverage probability of proposed confidence intervals achieves the nominal level. As an empirical illustration, the proposed methodology is applied to cross-country growth regressions.

On uniform asymptotic risk of averaging GMM estimators

“…Hence, cannot be consistently estimated. Our solution to this problem follows the popular approach in the literature which replaces d by an estimator whose asymptotic distribution is centered at d ; see Liu () and Charkhi, Claeskens, and Hansen () for similar estimators in the least square estimation and maximum likelihood estimation problems, respectively.…”

Model averaging in predictive regressions

“…The risk of these models tends to infinity with the sample size, and hence the asymptotic approximations break down. To obtain a useful approximation, we follow Hjort and Claeskens (), Hansen () and Liu (), and use a local‐to‐zero asymptotic framework to approximate the in‐sample MSE. More precisely, the parameters γ are modelled as being in a local neighbourhood of zero.…”

“…The optimal weights, however, are infeasible, as they depend on the unknown parameter ψ. Similar to Liu (), we propose a plug‐in estimator to estimate the optimal weights for the forecasting model. We first estimate the asymptotic risk by plugging in an asymptotically unbiased estimator.…”

“…The proposed forecast combination method is the prediction counterpart to the plug‐in averaging estimator proposed in Liu (). Similar to Liu (), we employ a drifting asymptotic framework and use the asymptotic risk to approximate the finite sample MSE. However, we focus attention on the global fit of the model instead of a scalar function of parameters.…”

“…Following Hjort and Claeskens (), Hansen () and Liu (), we examine the asymptotic risk of least‐squares averaging estimators in a local asymptotic framework where the regression coefficients of potentially relevant predictors are in a local neighbourhood of zero. This local‐to‐zero framework ensures the consistency of the averaging estimator while, in general, it presents an asymptotic bias.…”