# Can we include fixed effects in cross sectional data?

I'm new to econometrics and recently learned about fixed effects. If I have a cross sectional data, is it possible to include fixed effects? What seemed strange is that the number of dummy variables will be exactly the same as the number of observations in this case and there will be some covariates which means the number of parameters exceed the observations. So is it correct that it's impossible?

• I agree with your reasoning that this will imply that the $X'X$ matrix used in the OLS formula will no longer be invertible. Commented Apr 17, 2023 at 12:46
• @ChristophHanck Thanks for your reply! Could you clarify why X'X matrix will no longer be invertible? Is it because it's not a square matrix? Commented Apr 17, 2023 at 19:55
• $X$ will then be $n\times (n+K)$, $X'X$ will then be a $(n+K)\times(n+K)$ matrix, where $K$ is the number of "other" regressors next to the fixed effects. Since the rank of a matrix is the minumum of row and column rank and since the rank cannot exceed its dimension and $\operatorname{rank}(AB) \leq \operatorname{min}(\operatorname{rank}(A), \operatorname{rank}(B))$ (math.stackexchange.com/questions/978/…), we have the result. Commented Apr 18, 2023 at 5:42