I get a lot of direct email requests from people wanting help/guidance/advice of various sorts about some aspect of econometrics or other. I like being able to help when I can, but these requests can lead to some pitfalls - for both of us.
More on that in a moment. Meantime, today I got a question from a Ph.D student, "J", which was essentially the following:
" Suppose I have the following regression model
log(yi) = α + βXi + εi ; i = 1, 2, ...., n .
How do interpret the (estimated) value of β?"
I think most of you will know that the answer is:
"If X changes by one unit, then y changes by (100*β)%".
If you didn't know this, then some trivial partial differentiation will confirm it. And after all, isn't partial differentiation something that grad. students in ECON should be good at?
β = [∂log(yi) / ∂Xi] = [∂logyi / ∂yi][∂yi / ∂Xi] = [∂yi / ∂Xi] / yi,
which is the proportional change in y for a unit change in X. Multiplying by 100 puts the answer into percentage terms.
So, I responded to "J" accordingly.
So far, so good.
But then I got a response:
"Actually, my model includes an interaction term, and really it looks like this:
log(yi) = α + βXi + γ [XiΔlog(Zi)] + εi ; i = 1, 2, ...., n.
How do I interpret β?"
Whoa! That's not the question that was first asked - and now my previous answer (given in good faith) is totally wrong!
Let's do some partial differentiation again, with this full model. We still have:
[∂log(yi) / ∂Xi] = [∂logyi / ∂yi][∂yi / ∂Xi] = [∂yi / ∂Xi] / yi.
However, this expression now equals [β + γ Δlog(Zi)].
So, a one unit change in X leads to a percentage change in y that's equal to 100*[β + γ Δlog(Zi)]%.
This percentage change is no longer constant - it varies as Z takes on different sample values. If you wanted to report a single value you could evaluate the expression using the estimates for β and γ, and either the sample average, or sample median, value for Δlog(Z).
This illustrates one of the difficulties that I face sometimes. I try to respond to a question, but I really don't know if the question being asked is the appropriate one; or if it's been taken out of context; or if the information I'm given is complete or not.
If you're a grad. student, then discussing your question in person with your supervisor should be your first step!