A solution to log of dependent variables with negative observations

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.