# How to decompose the dependent variable in a log log model

I am considering the following log log model :

$$log(y_t) = log(\beta_0) + \sum_{i=1}^{K}\beta_ilog(x_{i,t}) + \sum_{j=K+1}^{L}\beta_jx_{j,t} + log(\epsilon_t)$$

to explain the sales of a company given several factors. After estimation of the model, I would like to "decompose" the dependent variable (the sales) by computing the contribution of each explanatory variable. Ideally, what I have in mind would be a percentage contribution, the sum of which would be 1, but I don't know how to go about it since I have some variables in log and others not, I don't even know if it's possible. So I would like to have your help on this please.

Thanks a lot!

• This doesn't make sense even without logs unless the explanatory variables are orthogonal. Mathematically, you are asking to compute the "percentage contribution" to one side of a triangle from each of the other two sides. When that side is the hypotenuse of a right triangle you can do that with the Pythagorean Theorem (provided you are willing to use squared values to represent the "contributions"), but otherwise not. This is fundamental to understanding multiple regression, because it implies the meaning of any variable depends on all the other variables in the model.
– whuber
Commented Apr 14, 2023 at 12:28
• @whuber Thank you for this comment. You are saying that trying to have a contribution (which implicitly depends on the others explanatory variables) does not make sense since we don't have any "link / relation" between the variables (i.e any link between the side of the triangle) except in a particular case, for the triangle it is clear with Pythagorean theorem but when talking about the orthogonality in regression I don't see. Could you develop this please ? Don't hesitate to develop the math if you have this in mind I will be happy to learn about this. Commented Apr 14, 2023 at 14:01
• @whuber Putting aside the idea of having a percetange of contribution, in classical linear model I often see contribution as $\beta_i\times x_{i,t}$ but does this make sense ? It implies to fix the value of $x_{i,t}$ ? Commented Apr 14, 2023 at 14:45
• See stats.stackexchange.com/a/113207/919 for one explanation.
– whuber
Commented Apr 14, 2023 at 15:31
• @whuber Thank you for this link it is well explained and I think I see what you had in mind by talking about orthogonality between explanatory variables which is a strong assumption indeed which is not satisfied in my case.. Commented Apr 17, 2023 at 6:56