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?
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1$\begingroup$ I agree with your reasoning that this will imply that the $X'X$ matrix used in the OLS formula will no longer be invertible. $\endgroup$– Christoph HanckCommented Apr 17, 2023 at 12:46
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$\begingroup$ @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? $\endgroup$– chunguc1004Commented Apr 17, 2023 at 19:55
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$\begingroup$ $X$ 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. $\endgroup$– Christoph HanckCommented Apr 18, 2023 at 5:42
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If I have a cross sectional data, is it possible to include fixed effects?
A fixed effects analysis is attempting to exploit variation within a unit, usually over time. Without serial observations for units, there is no within-unit variation to exploit. If you're going to estimate fixed effects with dummy variables for each cross-sectional unit, then these are singletons. In simple terms, the dummies index each unit at a single point in time. Your model would be estimating just as much parameters as there are individual measurements.
So is it correct that it's impossible?
Correct.
Even absent the inclusion of additional covariates, you would not have enough residual degrees of freedom to estimate a model.
answered Apr 17, 2023 at 23:07