Time and region fixed effects (When is it appropriate to use c. and i. for the time trends?)

[Francisco]

I did come to realize that when controlling for time effects and region effects in a regular regression, it is pretty normal and understandable to use the prefix .i for both time and region variables.

For example, reg y x i.year i.region, cluster(region)

However, when the interaction is also required, there are some posts around here and other sites suggesting the use of the prefix c. for the time variable instead of the i. prefix.

For example, reg y x c.year##i.region, cluster(region)

Of course, the use of i.year##i.region will control for a) year fixed effects, b) region specific effects, and c) effects happening each year in each region. However, the use of c.year##i.region seems to be multiplying the dummy variables for each region by the year variable. The latter would suggest it is a sort of region-trend control variable.


QUESTIONS:

1. Is my understanding correct?
2. Is the use of either c. or i. contingent on what one wants to estimate?
3. What is the best way to proceed in a difference-in-difference estimation? In particular, in a diff-in-diff approach with a staggered rollout at the region-year level.

[Jeff Wooldridge]

What is the unit of observation? I take it you have pooled cross sections.

[Francisco]

Hello Professor Wooldridge, I really appreciate your engagement in this post.

I am having the same issue in two research projects. One is at the commuting_zone-year level and the other is at the county-year level.

[Jeff Wooldridge]

I see. Panel data but where the unit is sub-regional. If the key variables vary at the commuting zone or county level, you get the most robustness by putting the the full set of interactions. This allows each region to have its own unrestricted “trend.” If you use c.year, you’re assuming a linear trend for each region. So it’s always up or always down.

Using i.year can remove a lot of variation, but it’s preferred if it works out.

[Francisco]

Thank you so much! Your explanation about the unrestricted.vs.linear (i. vs c.) is really neat.

Indeed, the key variables vary at the commuting zone or country level, respectively. Therefore, for my purposes, using i. would be preferred.

Do you have any preference between areg, xt-fe, xi: reg, or any other command, to deal with this big number of interactions?

[Jeff Wooldridge]

I actually think just using -reg- as you did above may be the best. I haven't used reghdfe, which allows for multiple fixed effects but I'm not sure about interactions.

One would of caution: if you use

reg y x i.region##i.year, vce(cluster region)

you must ignore the standard errors on all of the region-year interactions. They are meaningless. However, the standard errors you get on the b^ are fine, provided you have a reasonable number of regions.

[Francisco]

Great! Thank you so much!

I will proceed as suggested.

[Kien Hoang]

I actually think just using -reg- as you did above may be the best. I haven't used reghdfe, which allows for multiple fixed effects but I'm not sure about interactions.

One would of caution: if you use

reg y x i.region##i.year, vce(cluster region)

you must ignore the standard errors on all of the region-year interactions. They are meaningless. However, the standard errors you get on the b^ are fine, provided you have a reasonable number of regions.

Hi Professor Jeff Wooldridge. I came across this post as I have a similar problem. Can you please elaborate on why the se of region-year interactions are meaningless?

Thank you so much.