# Log-normalization of predictors

I have the following dependent and independent variables for my linear regression model. Since they are all in different scales (some of the are % others continuous variables), I was suggested to take the log and normalize my variables before running the regression.

    Y    X2        X3 (%)       X1 (%)
Mean 2.9 24.6   0.009517    230.992248
std  2.3 32.2   0.077092    230.992248
Min  0   1      0           0
Max  8   539    1           1


I have the following Qs:

Why should I take the log and then normalize them - rather than using just one of the two data transformations?

Should I log and normalize also my Y variable?

How would interpret my coefficient at the end of the exercise? and how can I make them humanly intelligible for a business audience?

Any easy doc reference is very appreciated!

• I don't know why you would do that, doesn't really make sense to me, and good luck interpreting that. Was there a reason given for this recommendation? Some domain knowledge behind this statement? – user2974951 Jan 11 at 7:23
• Good point, not sure there was. I think the suggestion came from two factors: X1 is not normally distributed (but is not a necessary condition for OLS, right?) so log normalization would have helped to normalize it. Second point, to ensure comparability between regressors - but that it is not really necessary for me. I was wondering whether it was incorrect to leave the data as it is.. – Filippo Sebastio Jan 11 at 7:28
• Forgive me if I misunderstood but I have to warn you that normalization is not the process of making a variable normally distributed. – gunes Jan 11 at 7:55
• Apologies, it was not clear from my message, what I meant is that my colleague suggested taking the log (not to normalize) of the continuous variable to make it a normal distribution (currently it is not). – Filippo Sebastio Jan 11 at 8:04

First of all, taking only the logarithm of predictors creates the regression equation $$y=a+\sum w_i\log x_i$$, which assumes a relationship in the form $$e^y=A\prod{x_i^{w_i}}$$. Taking log of both IVs and DV creates the problem $$\log y=a+\sum{w_i\log x_i}$$, which assumes a relation in the form $$y=A\prod{x_i^{w_i}}$$. So, the underlying assumption changes. It's not that you must take the logarithm of $$y$$, it's your underlying assumption. And if you're unsure which relation is actually meaningful, you need to do model selection and test both.
In my opinion, standardization is a common step in regression analysis. Once you set your predictors, i.e. $$\hat{x_i}=\log x_i$$, you'll solve for the regression problem $$y=\sum{w_i\hat{x_i}}+b$$. The thing is, when you actually try to solve for $$y=\sum{w_ix_i}+b$$, you'd probably apply standardization without questioning. U think, doing the same for $$\hat{x_i}$$ here would also do no harm. The purpose of standardization is to ensure that each term contributes to MSE similar amounts. And you don't need to apply standardization to $$y$$, however applying doesn't also harm.
Note: some of your variables contains $$0$$'s (or numbers close to $$0$$) I guess. Be careful when you take the logarithm.