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I am struggling with basic intuition about individual fixed effects. I have panel data with a binary treatment $D_{it}$, where some individuals never receive the treatment. I am running a regression with individual fixed effects and time dummies: $$y_{it} = \alpha_{i}+\lambda_{t}+\beta D_{it} + \epsilon_{it}$$ My understanding is that with individual fixed effects, $\beta$ is basically determined as a differences in the within-individual means when $D_{it}=1$ and $D_{it}=0$. Does this mean that I don't need to include the observations for which $D_{it}=0$ for all $t$?

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Observations for which $D_{i,t}=0$ for all $t$ will contribute to your estimation, but only through the determination of the various $\lambda_t$. It is therefore usefull to include them to estimate more precisely the coefficients associated to the $\lambda_t$.

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