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Could anyone elaborate on why fixed effects (or within estimator) will not work in the probit setting? Thanks in advance.

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A model with "fixed effects" has individual intercepts, say $\alpha_{i}$, for each individual $i$ in your sample. This means the number of parameters you are trying to estimate grows just as quickly as your sample size $n$ does. This is called the incidental parameters problem and generally causes inconsistency of the maximum likelihood estimator.

In a linear model this is not a problem because the nuisance parameters $\alpha_{i}$ are eliminated by first-differencing. But in a non-linear model, like probit, this is not possible.

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