Log models (2/2)

This post will be a mathematical elaboration of the previous.

In the previous post we established that a log-log model is where we have a logged dependent variable and we are regressing that on a bunch of logged independent variables.

 Before we begin, we need to remind ourselves of some log rules:

 
Given our log-log model, let's say that we are interested in finding out what Y is.
How can we do this?

We need to anti-log our model:

 

This result is another reason why log models are attractive. This rises from the fact that we have assumed non-linearity in our independent variables. Specifically, we are trying to estimate the degree of this non-linearity. 
We want to estimate β1 and β2 in order to discover the non-linear effects of X1 and X2 on Y.

For all intents and purposes, this is a much more realistic assumption than the wholly linear model.