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.

翻译：广义最小二乘法（GLS）是回归分析中最基础的工具之一。实施GLS的主要挑战在于误差项条件方差函数的估计，这通常需要对参数估计施加严格的功能形式假设，或对非参数估计使用平滑参数。本文提出了一种替代方法，通过利用同调回归法在非参数单调性约束下估计条件方差函数。我们证明，所提出的GLS估计量在渐近意义上等价于已知条件误差方差的不可实现GLS估计量，且仅需通过调整边界观测值进行微调——这不仅适用于点估计，也适用于区间估计或假设检验。我们的分析扩展了同调回归法的适用范围，表明同调估计量（可能包含生成变量）可作为第一阶段的估计量，用于代入半参数对象。模拟研究显示该方法具有优异的有限样本性能。在实证案例中，我们重新审视了Acemoglu和Restrepo（2017）关于人口老龄化与经济增长关系的研究，以说明我们的GLS估计量如何有效降低估计误差。