# Why are time-invariant variables perfectly collinear with fixed effects?

Say I want to measure weight as a function of gender and individual over time.

Something I dont quite understand is why a fixed effect model for individuals always block out gender?

What is the proof that it will be perfectly collinear?

From my understanding most fixed effects models introduce individual dummies, but I dont see how this will generate perfect collinearity.

A fixed effects model can be regarded as a regression with a dummy variable for each group. This dummy variable is time invariant. If you have another variable which is time invariant for a group it is a multiple of the dummy for that group and is thus perfectly colinear with that dummu.

This is a good question, and is important to better understand fixed effects. Let's review the concept of Perfect multicollinearity first. Perfect multicollinearity occurs when two or more independent variables in a regression model exhibit a deterministic (perfectly predictable or containing no randomness) linear relationship. Suppose you have a model that $$Y_i=\beta_o+\beta_1X_{i1}+\beta_2X_{i2}+\epsilon_i$$ and $$X_{i2}=\alpha_o+\alpha_1X_{i1}$$ By doing easy substitution, you then get $$Y_i=\beta_o+\beta_2\alpha_o+(\beta_1+\beta_2\alpha_1)X_{i1}+\epsilon_i$$

You can find that now the equation only contains $$X_{i1}$$. Obtaining individual regression coefficients (for example, $$X_{i2}$$) for every variable is impossible if you have perfect multicollinearity.

There are 2 cases in econometrics that could have collinearity: among each other, or with the fixed-effects. The above example is for the former case. Your questions is on the latter case.

Why the above example have collinearity problem? Because in the equation, some independent variable(s) can perfectly predictable other independent variable(s). Once I know $$X_{i1}$$, I can definitely know $$X_{i2}$$. That's what I say "perfectly predictable". Once you include fixed effects, then such "perfectly predictable" could happen.

In your example, you include individual fixed effects, which is equivalent to you have a set of dummy variables to denote every single person in your model. Thus, you can imagine there is one variable, say, $$X_{i1}$$ means that whether this unit is first person. Once you know it is the first person, then you know the gender. Because gender is fixed within individual level. This is exactly what I call "perfectly predictable". But what if people can change their gender? Then no problem at all. Even you know this is the first person (because you include individual fixed effect), you cannot judge the gender for sure, because gender is not fixed. Of course, we assume that people cannot change their gender. But if this variable is school grade, then there will be no problem at all (since people can change their school grade). The take-away is Anything that does not vary within the levels (could be individual, time, etc.) is taken care of and cannot be included.

• So how does one handle perfect multicollinearity - dropping the intercept or omitting variables? Commented Jan 26, 2022 at 5:29