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.
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 $\beta\%$.
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%?
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$
$ 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!