Friday, July 13, 2018

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(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!

© 2018, David E. Giles


  1. Dear Dave, although this seems an extremely "elementary" topic, this text seems good as an addition to the explanation you have provided,

  2. Dear Prof. Dave,
    This was some illustration in econometrics and
    ....................How a grad student should be approaching an equation............More precicesly ask the right question...
    Thanks for the post...