A solution to log of dependent variables with negative observations

Abstract

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.

Appendices

Appendix A: Simulations

In the simulations, we assume the following data generating process (DGP):

$${\mathrm{ln}}\,y=PI+{\beta }_{0}+{\beta }_{1}{\mathrm{ln}}\,x+v,$$

where β₀ = β₁ = 1; $x \sim U\left(0,1\right)$ for positive valued y and $x \sim \gamma U\left(0,1\right)$ for negative valued y observations; $v \sim U\left(0,{\sigma }_{v}^{2}\right)$; and PI is either iπ or 0. Here, iπ represents negative profits and 0 represents positive profits. We picked different distributions for x depending on the sign of y because losses may have different magnitudes compared to profits. In our benchmark setting, we set γ = 0.2. We also repeated the experiment for γ = 0.5. We considered σ_v = 0.1 and σ_v = 0.2 cases. The sample size is n = 1000 and 500 observations have negative profit and 500 observations have positive profit. We ran the Monte Carlo experiment for 500, 000 repetitions. We announce the means of parameter estimates; means of confidence intervals; empirical sizes for 5% significance level; and correlations of predicted y with true values of y. In the first column (Complex), we announce the results for our estimator; in the second column (Drop), we present results obtained when we drop observations with y < 0; in the third column (Bos-Koetter), we present results for Bos-Koetter method; and in the fourth column (Constant) we present results for the model which adds constant to y to assure that transformed y has positive values. Since the Constant method transforms y variable in a different way (i.e., ${\mathrm{ln}}\,\left(y+C\right)={\beta }_{0}+{\beta }_{1}{\mathrm{ln}}\,x+v$), in that scenario we only examine the correlation of y by its prediction, which is given as follows:

$$\hat{y}=\exp ({\hat{\beta }}_{0}+{\hat{\beta }}_{1}{\mathrm{ln}}\,x)-C,$$

where $C=\min (\left|y\right|)+c$ is the adjusting constant and c = 0.01 is a small number to assure that the logarithm is evaluated at positive values. We present the simulation results in Table 6.

Table 6 Simulation Results for σ_v = 0.1 and γ = 0.2

Based on our Monte Carlo experiment results, we see that the Complex and Drop approaches outperform the Bos-Koetter approach in terms of parameter bias and validity of empirical sizes (which must be 0.05). Similarly, these two approaches outperform the Constant approach in terms of predicting y. When γ is large so that there are observations with relatively small negative values (i.e., large C), the Constant approach seems to be much less reliable in terms of predicting y. Since the Complex approach uses all data, this approach has an advantage in terms of getting smaller confidence intervals. This is useful especially when the percentage of negative observations is large. Overall, the Complex approach may not only gives good Monte Carlo experiment results but also it has the ability to make predictions related to negative profits, which makes it a reasonable choice for empirical applications.

Appendix B: Alternative negative profit modeling

An alternative way to model negative profit is given as follows:³⁶

$$y=\exp \left({\beta }_{0}+{x}^{\prime}\beta \right)v,$$

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where $y\in {\mathbb{R}}$ is the profit; $x\in {{\mathbb{R}}}^{k}$ is a vector of explanatory variables that may or may not include interaction terms and the constant term; $\beta \in {{\mathbb{R}}}^{k}$ is a vector parameters excluding the constant term and β₀ is the constant term; and $v\in {\mathbb{R}}$ is an i.i.d. error term with $E\left[v\right]=0$. To differentiate the models, we call this model Alternative Complex model. Hence, depending on the sign of v, the profit can be negative or positive. When we take the logarithm of this equation, we have:

$$\begin{array}{*{20}{l}}{\mathrm{ln}}\,y={\beta }_{0}+{x}^{\prime}\beta +{\mathrm{ln}}\,v\Rightarrow \\ {\mathrm{ln}}\,\left(\left|y\right|\right)+i\pi D={\beta }_{0}+{x}^{\prime}\beta +{\mathrm{ln}}\,\left(\left|v\right|\right)+i\pi D\Rightarrow \\ {\mathrm{ln}}\,\left(\left|y\right|\right)={\beta }_{0}+{x}^{\prime}\beta +{\mathrm{ln}}\,\left(\left|v\right|\right)\Rightarrow \\ {\mathrm{ln}}\,\left(\left|y\right|\right)={\beta }_{0}+E\left[{\mathrm{ln}}\,\left|v\right|\right]+{x}^{\prime}\beta +\omega \Rightarrow \\ {\mathrm{ln}}\,\left(\left|y\right|\right)={\beta }_{00}+{x}^{\prime}\beta +\omega ,\end{array}$$

where $\omega =\mathrm{ln}\,\left(\left|v\right|\right)-E\left[\mathrm{ln}\,\left|v\right|\right]$ is an error term with mean zero and ${\beta }_{00}={\beta }_{0}+E\left[\mathrm{ln}\,\left|v\right|\right]$ is a constant term.

One important difference between Complex model and Alternative Complex model is that the constant term in the Alternative Complex model includes β₀ and $E\left[\mathrm{ln}\,\left|v\right|\right]$. Thus, interpretation of the constant term depends on which one of these models we have in our mind. Another difference is that the Alternative Complex model does not have a dummy variable for negative profits. In practice, we may be agnostic about whether the Complex model or Alternative Complex model is the true model and include the negative profit dummy variable in the model. In this case, the constant term’s interpretation depends on which one of these models is the true model. If the value of the constant term is not important, this approach would be more robust than any one of these two models.