Negative observations pose a problem in econometric models that apply log-transformation to the data. We propose a simple yet effective solution to this problem by extending the domain of numbers to the set of complex numbers. In particular, this approach suggests that we can replace the negative values with their absolute values and estimate this transformed model with conventional estimation methods. Moreover, we extended this approach to logs of independent variables with negative observations as well. Using our method, we estimated the profit efficiencies of the US banks and illustrated that different treatments for observations with negative profits (loss) may lead to substantially different efficiency estimates. We also showed the importance of controlling bank heterogeneity when estimating efficiency.

中文翻译：

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带有负观测值的因变量对数的解决方案

负观测值在将对数转换应用于数据的计量经济学模型中造成了问题。通过将数字的域扩展到复数的集合，我们提出了一个简单而有效的解决方案。尤其是，这种方法建议我们可以将负值替换为其绝对值，并使用常规估计方法估计此转换模型。此外，我们还将这种方法扩展到具有负观测值的自变量的对数。使用我们的方法，我们估算了美国银行的利润效率，并说明对负利润（亏损）的观察结果采用不同的处理方法可能会导致效率估算值大不相同。我们还显示了在评估效率时控制银行异质性的重要性。