I would never apply a log transform to a variable having negative values and surely not apply it to an absolute value. It is meaningless. Well, sometimes negative values can appear in business data, like sales of rarely sold products reflecting returned items not compensated by new sales. But in such a case I would agglomerate several individual products to avoid the phenomenon. With sometimes zero values, adding a constant, e.g., 1 before taking the log will help but it often appears that the results are sensitive to the constant being added.

Can't you interpret negative values ​​another way? For example, instead of recording a profit of -$1 million, you record a loss of $1 million. I think it has the same effect as your equation, which considers the absolute value of x, and doesn't need to be moved.

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Dear Professor Guy Mélard

I am currently working on an empirical analysis to estimate the relationship between financial development and renewable energy consumption in Morocco. The model I am using is specified as follows:

RENC=α0+α1FDB+α2FDM+α3GDP+α4ROIL+α5TO+α6FDI+ϵ

The indices for financial development (FDB and FDM) contain negative values. These indices were constructed from various financial development variables. The remaining variables (GDP, ROIL, TO, FDI) do not have negative values, so a standard logarithmic transformation could be applied to them if necessary.

I encountered your suggestion regarding the addition of a constant before taking the logarithm. This approach, while potentially useful, raises a few specific questions in my context:

- Should the constant be added only to the variables containing negative values (FDB and FDM), or should it be applied to all variables in the model to maintain consistency? _Could you please explain in detail how the constant addition would affect the interpretation of the results? Specifically, how does this approach help in maintaining the economic significance and comparability of the transformed variables? Thank you for your time and assistance.

Are you mixing stocks and flows? It sounds as if your financial development variables are changes in some financial variables - additions to or subtractions from the financial variables. If you aggregate these to give stocks of the variables that will always be positive.

I agree with Guy Mélard . I have never encountered a case where it made sense to add to a variable with negative values to make these values positive. the resulting system will not make sense.

On another matter, I would doubt the validity of ARDL in this case as there will be feedback from the renewable energy variable to the oil variable.

Dear Professor John C Frain ,

Thank you for your feedback. I would like to clarify that the financial development variables (FDB and FDM) are composite indices constructed using PCA. The standardization process resulted in values between -3 and 3, introducing negative values not from incremental changes but from making the data comparable.

This standardization causes difficulties with logarithmic transformation. I aim to understand better how to manage these negative values for robust and meaningful economic analysis.

Yes, you are right, Professor. The use of the ARDL model in this context can be questioned due to the possibility of feedback between renewable energy consumption and real oil prices. Indeed, several studies highlight that the bidirectional interaction between these variables could bias the results of an ARDL model.

Thank you again for your valuable advice and assistance.

A log transform seems to have no reason in your case. Why do you want to use it?

Professor Guy Mélard ,

I understand that the logarithmic transformation may not seem necessary at first glance. However, after conducting several analyses, I found reasons justifying its use in this particular case.

Some variables exhibit significant skewness and extreme values. The skewness test revealed significant skewness for FDM (1.084) and FDI (1.512).

The coefficients of the log-transformed variables can be interpreted as percentages, thus facilitating the economic interpretation of the results.

Nobody (I hope) has said that explanatory variables should be symmetrically distributed. I am curious on how you can interpret percentages of a variable taking negative values. Anyway, don't expect to be published with your model. I will stop here this discussion.

Guy Mélard Thank you very much for your valuable feedback and assistance.

I have added a constant to the data to ensure all variables are positive before applying the logarithmic transformation.

Your insights have been very helpful, and I appreciate your guidance

This is nonsense

I would like to clarify another methodology I used. In constructing the financial development composite index, I applied the logarithmic transformation (ln) to the initial variables, which were all positive. Following this, I standardized the transformed variables to ensure comparability, a crucial step for conducting Principal Component Analysis (PCA). This process resulted in the creation of the composite index, which included some negative values due to standardization. This approach is based on a methodology I found in an academic article, "Dynamics between Financial Development, Renewable Energy Consumption, and Economic Growth: Some International Evidence" by Kassi Diby Francois Noel Koffi. The standardization step is necessary to build a meaningful composite index from various financial variables. I hope this explanation provides a clearer understanding of my approach. I appreciate your assistance and look forward to any further guidance you can offer.

To log-transform a series with negative values, you can use a log transformation after shifting the data to ensure all values are positive. For instance, apply a transformation like log(x+C)\text{log}(x + C)log(x+C), where CCC is a constant greater than the absolute value of the minimum data point.

Chuck A Arize In the equation

log( y)= a + b log (x)

b is an elasticity. How do you interpret b in the equation?

log(y) = a + b log (x+c).