  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 smoothing
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
involves only some tuning to trim boundary observations, 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.
