GLS under monotone heteroskedasticity

The generalized least square (GLS) is one of the most basic tools in regression analyses. A major issue in implementing the GLS is estimation of the conditional variance function of the error term, which typically requires a restrictive functional form assumption for parametric estimation or tuning parameters for nonparametric estimation. In this paper, we propose an alternative approach to estimate the conditional variance function under nonparametric monotonicity constraints by utilizing the isotonic regression method. Our GLS estimator is shown to be asymptotically equivalent to the infeasible GLS estimator with knowledge of the conditional error variance, and is free from tuning parameters, not only for point estimation but also for interval estimation or hypothesis testing. Our analysis extends the scope of the isotonic regression method by showing that the isotonic estimates, possibly with generated variables, can be employed as first stage estimates to be plugged in for semiparametric objects. Simulation studies illustrate excellent finite sample performances of the proposed method. As an empirical example, we revisit Acemoglu and Restrepo's (2017) study on the relationship between an aging population and economic growth to illustrate how our GLS estimator effectively reduces estimation errors.