More on Regression Coefficient Interpretation

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(y_i) = α + βX_i + ε_i    ;  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?

Specifically,

      β = [∂log(y_i) / ∂X_i] = [∂logy_i / ∂y_i][∂y_i / ∂X_i] = [∂y_i  / ∂X_i] / y_i,

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(y_i) = α + βX_i + γ [X_iΔlog(Z_i)] + ε_i    ;  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(y_i) / ∂X_i] = [∂logy_i / ∂y_i][∂y_i / ∂X_i] = [∂y_i  / ∂X_i] / y_i.

However, this expression now equals [β + γ Δlog(Z_i)].

So, a one unit change in X leads to a percentage change in y that's equal to 100*[β + γ Δlog(Z_i)]%.

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!

© 2018, David E. Giles