Fixed Effects Dropping Variables

[Kevin Morris]

Hi folks,
First time poster here. I'm running into something strange that I haven't encountered before, and am likely missing something silly.
I'm running a model looking at county-level variation while using fixed-effects for state-level variation. However, whenever I use a dummy variable that applies to each county in the state, that dummy gets dropped due to collinearity when I use xtreg, fe.
In particular, I'm looking at the impact a Republican governor has on my independent variable.
However, when I run the model using just "reg" and i.state, the variables (IE, Republican control) are no longer dropped.
Any thoughts on what I'm missing here?

[Clyde Schechter]

Since you don't show the actual outputs you are getting, I cannot be certain, but it is most likely this:

When Stata encounters a group of colinear variables in a regression, it breaks the colinearity by omitting one of the variables. The manuals do not, as far as I have seen, make any particular promise about which variable(s) will be selected for omission. But a little experimentation suggests that of the variables mentioned in varlist, Stata chooses the one (or several if need be) mentioned last. In a fixed-effects regression using the -fe- option of -xtwhatever-, it will select a variable from the varlist to omit and preserve the absorbed effects.

Now, clearly a variable like having a Republican governor will be constant for all observations in any state (assuming we are not looking at more than one point in time) and therefore that variable is colinear with the state effects. So in -xtreg, fe- it gets dropped.

When you run -regress i.republican_control perhaps_other_variables i.state-, the colinearity among i.state and republican_control will be resolved by removing a state indicator (over and above the one that is already omitted to break the colinearity among the state "dummies.") I think if you inspect your -reg- output carefully you will see that an extra state has been dropped. That is why you don't see republican_control omitted.

I should point out that from a mathematical perspective, any method of breaking the colinearity is arbitrary, and in such a model, the coefficients of all of the variables that are involved in a colinearity are artifacts of the methods and have no real-world meaning. (The coefficients of the other variables are not dependent on the choice of how the colinearity is broken, and you don't have to worry about them. Also, all predictions made, say, with -margins- are invariant to the choice of how the colinearity is broken.) Implication: even though you have found a way to get the appearance of an effect estimate for republican control, it is in fact a meaningless number. It is in principle impossible to estimate the effect of a variable that does not vary within fixed-effect units in a fixed-effects model. It's linear algebra and there is no way around it.

If your research goals require that you estimate an effect of republican control, it cannot be done in a state-fixed-effects model. For that you must either use a random-effects or between-effects model.

[Kevin Morris]

Hi Clyde,
Thanks so much for the thorough response; you were exactly right.
I think is may be the first time I've constructed a model in which fixed or random effects seemed like they needed to be controlled for and my variable of interest is a dummy of this sort, and I'm having a little trouble fleshing out the intuition behind it. I'm basically trying to say, "even after account for county level demographics and state-wide effects, Republican control* has impact X." Does switching this straight to a random-effects model allow me to do that? When I run it using random effects, by key covariates maintain their significance and I have plenty of observations.

Thanks so much for helping a random stranger on the internet!

* My actual-key variable isn't in fact republican control, but rather a factor that usually, but not always, applies to every county or no county in a given state, so I'm realizing that that variable is suffering from the same problem.

[Clyde Schechter]

To the extent that a random effects model is appropriate, it does get around this problem and allow you to estimate the effects of variables that are constant within-state.

The problem you may encounter is that whereas fixed-effects regression is consistent, random effects are subject to omitted variable bias due to other effects that are constant within state but not measured. In some disciplines, people like to use the Hausman test to determine whether a random effects model is usable. In other disciplines, this issue is just swept aside. So you might want to check the literature in your field to see what people typically do.

My actual-key variable isn't in fact republican control, but rather a factor that usually, but not always, applies to every county or no county in a given state, so I'm realizing that that variable is suffering from the same problem.

If you were to use this variable in a fixed effects model, because it is not always constant within state, it will not be omitted from the model. BUT, the states within which it is constant provide no information about its effect. So your estimate of the effect may be underpowered (and the output about number of observations and number of groups will not reveal this problem).