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*_{i}Δlog(*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