Multicollinearity with fixed effects regression and dummy variable

Question

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

Answer 1

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.