Fixed Effects model - Variable that only changes across time and not individuals

[Linus Seb]

I have a fixed effects model where the dependent variable is house prices, and I want to include a variable for interest rates, that varies over time (t) but not individuals (i). One potential issue is that I have also included time fixed effects, which could potentially open up some endogeneity issues. Is it an isseu if I include this variable?
Can I interact a variable that changes across time and not individuals with a variable that changes across individuals but not time? The only issue I see with this is the interpretation, but this is not of interest as this approach is used as a robustness check.
Thanks for any help.

[Andrew Musau]

This variable will be collinear with the time-effects. The way to think about it is that the time effects capture the effects of all such variables (invariant across units, varying across time), so once you have included them in the model, you don't have to worry about not explicitly including such variables. If you explicitly want to study the effects of interest rates, then you have a problem as its coefficient is not identified with the inclusion of time effects. If it is a control variable, you are fine.

[Clyde Schechter]

One potential issue is that I have also included time fixed effects, which could potentially open up some endogeneity issues. Is it an isseu if I include this variable?

If you have a variable that is invariant across panels (individuals, in your case) and varies only with time, it will be colinear with the time fixed effects. Either it will be dropped or one of the time indicators will be dropped in order to resolve this. (Which gets dropped is a bit idiosyncratic and hard to predict, but it doesn't really matter which it is.) The implication is that your time indicators will no longer be valid estimators of time-specific shocks and your interest rate variable, if it isn't omitted, will not be a valid estimator of the interest rate effect on house prices. So if estimating the effect of interest rates is important for answering your research question, you cannot use time fixed effects. If, on the other hand, you were introducing it only as a covariate ("control variable"), just leave it out--the time fixed effects will automatically adjust for it anyway.

Can I interact a variable that changes across time and not individuals with a variable that changes across individuals but not time?

Yes, you can do this. The "main effects" of the interaction will be omitted due to colinearity with the panel and time fixed effects. (Well, the individual-invariant one may be preserved and one of the time indicators dropped instead.) But the interaction term is not colinear (provided you do not use individual#time fixed effects, just individual fixed effects and time fixed effects.) The interpretation of the interaction term is the same as it would be in any other case. Remember, however, that the time indicators are no longer valid estimators of time shocks and the coefficient of the firm_invariant "main" effect, if it is not omitted, is also uninterpretable.