# Time invariant instruments in fixed effects models

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