# Multicollinearity with fixed effects regression and dummy variable

Im doing a fixed effects regression on a panel data with $$N*T=870$$ observations where N = 290 and T = 3. The entities are districts observed over a total of 3 time periods. I want to interact my main independent variable with a dummy variable, where the dummy variable takes the value 1 if a "large" district and 0 o.w. The equation looks like this: $$y_{it}={\beta_1}a_{i}+{\beta_2}b_{t}+{\beta_3}c_{it}+{\beta_4}D_{i}*c_{it}+e_{it}$$, where y is the dependent variable, a is the fixed effects for entity i, b is the time fixed effects, c is the main independent variable for entity i and time t, and D is the dummy variable. Will this regression cause issues with multicollinearity, since the entity fixed effects ($$a_i$$) are essentially dummies for each entity?

Example data:

year, entity, indepdent_variable, D
1     1       0.1                 1
2     2       0.3                 0
3     3       0.5                 1

• Are you worried about multicollinearity, or perfect multicollinearity? In case of the former, this is a nice blog: statisticalhorizons.com/multicollinearity. In case of the latter, your statistical program will normally inform you about that. I don't immediately see a problem with your regression (but maybe I'm missing something).
– Tom
Apr 29, 2021 at 19:30

Perfect collinearity is a potential concern. Note $$D_i$$ is $$i$$-subscripted, exhibiting variation across districts but not over time. Whatever you're hoping to capture with this time-invariant dummy is already adjusted for via the inclusion of district fixed effects. A district's size is a time-constant attribute. A firm that is large in 2018 is large in 2020, absent any redistricting efforts during the years under consideration.
Software will invariably drop $$D_i$$ as it is 'collinear' with the district fixed effects. Put differently, the simple effect for district size is indistinguishable from $$a_i$$. In applied work, its common to interact time-invariant regressors with time dummies, but I suppose it would depend upon the specifics of your research question.
In short, $$\beta_4$$ is estimable. And, even though collinearity is built into the model, it is unlikely to result in estimation issues.
• Thank you Thomas. The code im using: lm(y ~ c + I(D * c) + as.factor(District) + as.factor(Year)). Note that c is not time constant and do vary across districts. To calculate clustered standard error on a district level I use coeftest(model, vcov=vcovHC(model,type="HC0",cluster="District")). Would you say this is correct? Apr 30, 2021 at 6:25
• Yes. The code you supplied clusters on your districts and produces heteroskedastic-consistent variance estimates. Also, I don't recommend creating the interaction manually using I(.), though the code you supplied should produce equivalent results. Let software do the heavy lifting for you. In some instances, you may forget to include one of the constituent terms and end up with a misspecified model. The good news is that coeftest() will spit out your coefficients and adjusted standard errors, absent the collinear variable. Again, the function should exclude D for you. Apr 30, 2021 at 20:08