Log models (1/2)

In the previous post, we established that the case for a wholly linear model is:

β1 is the marginal effect of X1 on the dependent variable Y.
If X1 changes by one unit, assuming that all other variables are constant, what happens to Y?

This is where the log model is of some use to us:






In this model, β1 captures the corresponding increase in lnY due to an increase of 1 unit in lnX1.
However, this is a very cumbersome description and it doesn't really tell us much.
Assuming that all variables are held constant and the only thing we vary is X1, we can differentiate the above:

 With some minor adjustments, the above can be restated as:

The above describes the percentage change in Y over the percentage change in X.
Anyone familiar with economics should immediately recognise this result.
β1  represents the concept of elasticity within this log model.

We can generalise this specific case in terms of our log model. We can say that the beta coefficients show us the partial elasticity of our dependent variable with respect to a particular independent variable.

Now, we are ready to adjust our "cumbersome" description:

Assuming that all the other variables are held constant, β1 shows us the effect of a percentage increase of 1% in X1 on the dependent variable.

The above example will happily hold for when both our dependent and independent variables are logged, but what happens if we only log our dependent variable?

We can still think about β1 in percentage terms, it shows what the percentage increase in Y is from a 1 unit increase in X1.

If I have a non log dependent variable and a logged independent variable:

β1 in this context represents the increase in Y that is associated with a 1% increase in X1.



In sum, the reason to have a log-log model is that all of our variables are on the same scale. Therefore, we don't need to worry about units which allows for a sharper comparison between coefficients.