Interpretation of an interaction term when coefficient of one of the constitutive term is missing

[Sahawal Alidou]

Greetings,
I am having a debate with a friend about some regression results. Say two independent variables x and y and their interaction term x*y. After a fixed effect regression, coefficient on x is missing but that of y and x*y are displayed. Question is : can we still give an interpretation to the coefficient on x*y ?
Thanks in advance.

[Phil Bromiley]

You didn't get a quick answer. You'll increase your chances of a helpful answer by following the FAQ on asking questions - provide Stata code in code delimiters, readable Stata output, and sample data using dataex.

If a coefficient on a variable is "missing" then I would think you would want to look more carefully at your model and data to see what is going on. If it happened to be very small, that wouldn't be a problem (and the parameters on main effects depend on where you put the zero points on the interacting variables which is often arbitrary). But any time you don't get a parameter estimate at all would suggest to me you need to look harder at data construction etc.

It is certainly possible to have a case where you can't estimate a main effect but can estimate the interaction. If, for example, you have a fixed effect model, you can't get parameter estimates on any variable that doesn't vary within panels, but you can use these in interactions.

[Sahawal Alidou]

Thank you Phil.
Best.

[Jeff Wooldridge]

This is a fairly common situation when using fixed effects. I trust x does not vary over time but y does. So you cannot estimate the effect of x overall; you can only see whether the effect depends on y. For y, things are better: I can estimate the effect of y for any value of x. Using margins I can estimate the average partial effect.

For example, suppose y is time spent in job training and x is a time constant attribute — highest grade completed. I can estimate the effect of job training on wages and see if it varies by education. I cannot estimate the effects of schooling unless I use some other estimation method.