What's the difference in growth of Y in a linear regression model when using a log-lin model or a lin-log model

Question

Description

I'm currently studying a chapter on linear regression analysis. I have come to a section where we study the interpretation of the coefficients with logarithmically transformed variables. I would like to know what happens to Y when an absolute change in the value of X, or a relative change in the X occurs.

Linear regression

The formula presented here is the basic linear representation of my dataset.

$ Y = \alpha + \beta X + \epsilon$

When you take the first derivative of the formula you get:

$dY = \beta * dX $

$\beta $ is the slope of the formula, thus when X's value increases with an increment of 1 $ Y $ increases with a value of $\beta $. On the other hand when X increases with 1%, Y increases with $\beta\%$.

Log-Lin model

What I don't understand is how much Y grows when we transform the original formula logarithmically.

$\log(Y) = \alpha + \beta X + \epsilon$

When I take the first derivative of this formula I become:

$\ dY/Y = \beta*dX $

How do I interpret this formula? Does this mean that when $ dX = 1$ that $Y $ grows with $ \beta \% $ ? What happens when to Y when X grows with 1%?

Lin-Log model

Converserly, when I apply the same reasoning to the following transformation, is my conclusion still valid?

The first derivative of:

$ Y = \alpha + \beta*log(X) + \epsilon$

is:

$ dY = \beta * (dX/X) $

So that, when there is an in increase in X of 1%, Y increases with $\beta$ What if X increases its value with 1, how much does Y increase?

This was my first question. If the format or the content of the question can be improved please let me know.

Many thanks in advance!

Answer 1

So in the top model, $Y=\alpha+\beta X+u$ a 1 unit change in X relates to a 1$\beta$ unit change in Y. So whatever units you are using, its unit change in both.

With $\ln(Y)=\alpha+\beta X+u$ then we have that a 1 unit change in X relates to a $\beta*100%$ percent changes in Y. That is because the LHS in the derivative is the growth rate in Y.

With $Y=\alpha+\beta \ln X+u$ , we now have the opposite, so a 1 percent change in X relates to a $\beta/100$ unit change in Y.

The reason why we multiply by 100 in the log-lin case is because as X changes by 1% the $\beta$ needs to be converted from percent to units. The opposite in the lin-log case.

Edit: I should add that in the log-lin model, that interpretation is only an approximation that works for small $\beta$. The exact percentage difference is:

$100*[exp(\beta*\Delta X)-1]$ This is easily seen when you do a percentage change between the model for 2 different values of X.