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


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$


$ 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!

Best Answer

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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