Model Averaging OLS and 2SLS: An Application of the WALS Procedure

More commonly, applied and theoretical researchers are examining model averaging as a tool when considering estimation of regression models. Weighted-average least squares (WALS), originally proposed by Magnus and Durbin (1999, Econometrica) within the framework of estimating some of the parameters of a linear regression model when other coefficients are of no interest, is one such model averaging method with their proposed approach being a Bayesian combination of frequentist ordinary least squares and restricted least squares estimators. We generalize their work, along with that of other researchers, to consider averaging ordinary least squares (OLS) and two stage least squares (2SLS) estimators when possibly one or more regressors are endogenous. We derive asymptotic properties of our weighted OLS and 2SLS estimator under a local misspecification framework, showing that results from the existing WALS literature apply equally well to our case. In particular, determining the optimal weight function reduces to the problem of estimating the mean of a normally distributed random variate, which is unrelated to the details specific to the regression model of interest, including the extent of correlation between the explanatory variable(s) and the error term. We illustrate our findings with two examples. The first example, from a commonly adopted econometrics textbook, considers returns to schooling, and the second case is a growth regression application, which examines whether religion assists in explaining disparities in cross-country economic growth.