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!
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.
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