In this paper, we demonstrate that standard central limit theorem (CLT) results do not hold for means of non‐parametric, conditional efficiency estimators, and we provide new CLTs that permit applied researchers to make valid inference about mean conditional efficiency or to compare mean efficiency across groups of producers. The new CLTs are used to develop a test of the restrictive ‘separability’ condition that is necessary for second‐stage regressions of efficiency estimates on environmental variables. We show that if this condition is violated, not only are second‐stage regressions difficult to interpret and perhaps meaningless, but also first‐stage, unconditional efficiency estimates are misleading. As such, the test developed here is of fundamental importance to applied researchers using non‐parametric methods for efficiency estimation. The test is shown to be consistent and its local power is examined. Our simulation results indicate that our tests perform well both in terms of size and power. We provide a real‐world empirical example by re‐examining the paper by Aly et al. (1990, Review of Economics and Statistics 72, 211–18) and rejecting the separability assumption implicitly assumed by Aly et al., calling into question results that appear in hundreds of papers that have been published in recent years.

中文翻译：

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非效率两阶段生产模型中条件效率测度和“可分离性”条件检验的中心极限定理

在本文中，我们证明了标准中心极限定理（CLT）的结果不适用于非参数的条件效率估计器，并且我们提供了新的CLT，允许应用研究人员对平均条件效率进行有效推断或比较均值生产者群体之间的效率。新的CLT用于开发限制性“可分离性”条件的测试，该条件对于环境变量的效率估算的第二阶段回归是必需的。我们证明，如果违反此条件，不仅第二阶段的回归难以解释且可能毫无意义，而且第一阶段的无条件效率估计也会产生误导。因此，这里开发的测试对于使用非参数方法进行效率估算的应用研究人员至关重要。该测试表明是一致的，并检查了其本地电源。我们的仿真结果表明，我们的测试在尺寸和功耗方面均表现出色。通过重新审查Aly等人的论文，我们提供了一个真实的经验示例。（1990，经济与统计评论72，211–18），并拒绝了Aly等人隐含的可分离性假设，对近年来发表的数百篇论文中出现的结果提出了质疑。