Inference after Model Averaging in Linear Regression Models

Abstract

This paper considers the problem of inference for nested least squares averaging estimators. We study the asymptotic behavior of the Mallows model averaging estimator (MMA; Hansen, 2007) and the jackknife model averaging estimator (JMA; Hansen and Racine, 2012) under the standard asymptotics with fixed parameters setup. We find that both MMA and JMA estimators asymptotically assign zero weight to the under-fitted models, and MMA and JMA weights of just-fitted and over-fitted models are asymptotically random. Building on the asymptotic behavior of model weights, we derive the asymptotic distributions of MMA and JMA estimators and propose a simulation-based confidence interval for the least squares averaging estimator. Monte Carlo simulations show that the coverage probabilities of proposed confidence intervals achieve the nominal level.

Weighted-Average Least Squares (WALS): Confidence and Prediction Intervals

“…Table 1 lists the nine estimators of β = (β 1 , β 2 ) that we consider in this paper. Except for LS-R and WALS, all other estimators also appear in Zhang and Liu (2019). In the remainder of this section, we describe briefly the various estimators with some emphasis on WALS.…”

“…The IC-A and IC-B postselection estimators are the LS estimators in the models with the smallest AIC and BIC respectively. As implemented in Zhang and Liu (2019), these estimators require preordering and the assumption of homoskedastic errors. There is no model averaging here, only model selection.…”

“…We assess its finite-sample performance by an extensive Monte Carlo experiment. To facilitate comparisons with the simulation study by Zhang and Liu (2019), we stay close to their framework and consider a finite model space that contains the true data-generating process (M-closed environment) but has little additional structure. Unlike Zhang and Liu (2019), who restrict attention to inference about a single auxiliary parameter, we consider inference about a single focus parameter, interpreted as the causal effect of a policy or intervention in the presence of a potentially large number of auxiliary parameters.…”

Focused information criterion for locally misspecified vector autoregressive models

“…A time series application to stationary autoregressions of in nite order is Zhang et al (2013). Zhang and Liu (2017) report results on the distribution of Mallow's and Jackknife-based model averaging weights in linear regressions with irrelevant variables.…”

Vector autoregressions

“…A time series application to stationary autoregressions of infinite order is Zhang et al (2013). Zhang and Liu (2017) report results on the distribution of Mallow's and Jackknife-based model averaging weights in linear regressions with irrelevant variables.…”