Log Transformation of a Quadratic variable

[Deepika Deshpande]

Hello,

I have a basic question on a model I am running on STATA. My question pertains more to the model than to STATA, however, I thought I would ask this group for any suggestions.

My model is of the following form: Y= a+bx+cx^2

I expect to get an inverted U curve. The problem I am facing is that my independent variable x is highly skewed and hence I plan to log transform it. However, after log transforming it, I believe I cannot get a curvature. Does anyone have any suggestions on how I should proceed?

[Maarten Buis]

I see no reason why that would be a problem y = a + b ln(x) + c ln(x)^2 is still linear in the parameters, so that all works. This of course assumes that x is strictly positive, otherwise taking a logarithm makes no sense.

[Joro Kolev]

I do not see any problem with having a skewed regressor. And if you start with these elaborate nonlinear transformations you will have problems with interpreting your results after that.

If I were you I would just estimate the model as it is, without taking logs of x.

[Deepika Deshpande]

Thank you, Maarten and Joro.

[Deepika Deshpande]

Hi Joro, Wrt your response #3, my understanding is that fixed effects panel regression does not assume the regressors to be normally distributed - it only requires the residuals to be normally distributed. Would you agree with that understanding. (If my understanding is correct, I will not do the log transformation, although I am rather confused why many statistical analyses perform such transformations.)

[Maarten Buis]

Your statement that the distribution of the explanatory variables is irrelevant is correct. The reason why people still use logarithmic transformations is to allow for a specific type of non-linearity, or they like elasticities. You can combine all kinds of transformations to allow for ever more elaborate non-linearity, including combining a quadratic and a logarithm. However, once you get that far, you could consider other transformations like a linear spline (see help mkspline). This often gives a better balance between a very free shape and interpretable coefficients.

[Deepika Deshpande]

Thank you, Maarten. Apologies for asking a few questions to reconfirm my understanding regarding the requirements for fe regression:
1. The explanatory variables do not need to be normally distributed (- you already confirmed this above)
2. However, the explanatory variables do need to be i.i.ds (- is there any way for me to test this?)
3. The dependent variable does not need to be normally distributed either.
4. Extreme values in independent and dependent variables should be avoided. This can be rectified through winsorization.