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I am running a fixed effects model of this kind $$y_{i,t}=\beta_0+\beta_1x_{i,t}+\gamma_i+\lambda_t+\varepsilon_{i,t}$$ Where $y_{i,t}$ is my outcome variable of interest, $x_{i,t}$ is an endogenous variable and $\gamma_i$ and $\lambda_t$ are unit and time fixed effects.

To address the endogeneity of $x_{i,t}$ I want to use an instrumental variable approach. As an instrument, I want to use $Z_i$, which unfortunately is time-invariant. My question is: is it possible to run a 2sls model with fixed effects using a time-invariant instrument? My concern is that in the first stage equation ($x_{i,t}=\alpha_0+\phi Z_i+\gamma_i+\lambda_t+\varepsilon_{i,t}$) the coefficient on the instrument $\phi$ would be unidentified given that it would be absorbed by the fixed effects $\gamma_i$. While I am aware that one should include all of the regressors of the second stage equation in the first stage, I was wondering whether there are some approaches allowing to use time-invariant instruments in fixed effects models

Thanks a lot in advance for your help

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You cannot use an IV that is time-invariant in a fixed effects approach. Citing Wooldridge:

It's clear that a time-constant IV cannot be used in fixed effects, so one shouldn't try. As Sebastian noted, Stata will drop collinear variables, but not always the one that it should. Whenever one does fixed effects manually, this can happen.

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