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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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