When a sample is drawn from or coincides with a finite population, the uncertainty of the coefficient estimators is often reported assuming the population is effectively infinite. The recent literature on finite-population inference instead derives an alternative asymptotic variance of the ordinary least squares estimator. Here, I extend the results to the more general setting of M-estimators and also find that the usual robust ‘sandwich’ estimator is conservative. The proposed asymptotic variance of M-estimators accounts for two sources of variation. In addition to the usual sampling-based uncertainty arising from (possibly) not observing the entire population, there is also design-based uncertainty, which is usually ignored in the common inference method, resulting from lack of knowledge of the counterfactuals. Under this alternative framework, we can obtain smaller standard errors of M-estimators when the population is treated as finite.

中文翻译：

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M估计量的潜在结果和有限人口推断

当从有限总体中抽取样本或与之吻合时，通常会假设总体是无限的，从而报告系数估计量的不确定性。相反，有关有限人口推断的最新文献推论出了普通最小二乘估计量的另一种渐近方差。在这里，我将结果扩展到M估计量的更一般的设置，并且还发现通常的鲁棒“三明治”估计量是保守的。拟议的M估计量的渐近方差解释了两个变化源。除了（可能）不观察整个种群而引起的通常基于采样的不确定性之外，还存在基于设计的不确定性，由于缺乏对事实的了解，通常在通用推理方法中将其忽略。在这个替代框架下，