Description
wals computes the WALS estimates of a linear regression model with iid errors. The model contains two subsets of regressors: k1 focus regressors, which are included in the model on theoretical or substantive grounds; and k2 auxiliary regressors, whose inclusion is less certain. We assume that k2>0 and that k=k1+k2<n, where n denotes the sample size.
WALS is a frequentist model-averaging estimator that relies on a preliminary semi-orthogonal transformation of the auxiliary regressors and a Bayesian shrinkage analysis of the associated normal location model. This approach is attractive because it performs well in finite samples, offers a transparent notion of prior ignorance, and is not restricted to sequences of nested models. Equally important, it is numerically stable and computationally efficient due to its preliminary semi-orthogonal transformation of the auxiliary regressors.
In the Bayesian shrinkage step, WALS places a prior on the 'population t-ratio' of the transformed auxiliary parameters. The Bayesian framework serves to construct an estimator with desirable theoretical properties (admissibility, bounded risk, robustness, and optimality in terms of minimax regret), while the performance of this estimator is evaluated in the classical frequentist setting. In large samples, the WALS estimator is uniformly sqrt-n consistent, and its asymptotic distribution is (multivariate) normal only under certain conditions on the auxiliary parameters.
The wals command also produces plug-in estimators of the sampling moments (bias, SE, and RMSE), as well as re-centered and asymmetric simulation-based WALS confidence intervals constructed from the bias-corrected posterior mean.
Help files
After installation, you can view the estimation options by typing in Stata
help wals
and the post-estimation options by typing
help wals postestimation
Key references (in chronological order)
Magnus, J. R., Powell, O., and Prüfer, P. (2010). A comparison of two model averaging techniques with an application to growth empirics. Journal of Econometrics, 154, 139-153.
Magnus, J. R., and De Luca, G. (2016). Weighted-average least squares: A review. Journal of Economic Surveys, 30, 117-148.
De Luca, G., Magnus, J. R., and Peracchi, F. (2022). Sampling properties of the Bayesian posterior mean with an application to WALS estimation. Journal of Econometrics, 230, 299-317.
De Luca, G., Magnus, J. R., and Peracchi, F. (2023). Weighted-average least squares (WALS): Confidence and prediction intervals. Computational Economics, 61, 1637-1664.
De Luca, G., Magnus, J. R., and Peracchi, F. (2025). Bayesian estimation of the normal location model: A non-standard approach. Oxford Bulletin of Economics and Statistics, 87, 913-923.
De Luca, G., and Magnus, J. R. (2025b). Weighted-average least squares: improvements and extensions. The Stata Journal, 25(3), 587-626.