Weighted-average least squares (WALS) : confidence and prediction intervals

We extend the results of De Luca et al. (2021) to inference for linear regression models based on weighted-average least squares (WALS), a frequentist model averaging approach with a Bayesian flavor. We concentrate on 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 representing the nuisance component of the model. In our Monte Carlo simulations we compare the performance of WALS with that of several competing estimators, including the unrestricted least-squares estimator (with all auxiliary regressors) and the restricted least-squares estimator (with no auxiliary regressors), two post-selection estimators based on alternative model selection criteria (the Akaike and Bayesian information criteria), various versions of frequentist model averaging estimators (Mallows and jackknife), and one version of a popular shrinkage estimator (the adaptive LASSO). We discuss confidence intervals for the focus parameter and prediction intervals for the outcome of interest, and conclude that the WALS approach leads to superior confidence and prediction intervals, but only if we apply a bias correction.