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I'm using a cross-sectional dataset ( just one point in time) and want to use a regression. In literature, mostly an OLS regression or a regression with fixed or random effects were used. However, I cannot find if fixed effects can also be used in cross sectional data ( with just one point in time) or what would be the meaning of it....

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No, it cannot be done, because fixed effects amounts to fitting one unit-specific intercept per unit.

Now, if you only have one observation per unit, you will already fit $n$ intercepts for your $n$ cross-sectional observations. But you will also want to fit $K$ additional "real" regressors, so your regressor matrix $X$ will be of dimension $n\times (n+K)$, so that $X'X$ will be of dimension $(n+K)\times (n+K)$ but be of (at most) rank $n$, so that $(X'X)^{-1}$ will not exist. Hence, $\hat\beta=(X'X)^{-1}X'y$ cannot be computed.

Possibly, some shrinkage approach could be feasible, but I haven't seen that for this particular purpose of doing "cross-sectional fixed effects".

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