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Consider a simple version of the model without regressors, which is enough to highlight the problem with fixed effects: \begin{equation} y_{im}=\alpha_i+\rho y_{i,m-1}+\eta_{im}\quad(m=1,\ldots,M;\;i=1,\ldots,n) \end{equation} Further assume that $E(\alpha_i\eta_{im})=0$ and $E(y_{i0}\eta_{im})=0$ for all $m$, as well as $E(\eta_{i,m}\eta_{i,h})=0$ as well ...


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No. $R^2$ in instrumental variables regression is not useful. Since one of the explanatory variables $x$ is correlated with the error $\epsilon$ we can't decompose the variance of the outcome $y$ into $\beta^2 Var(x) + Var(\epsilon)$, so the obtained $R^2$ has neither a natural interpretation, nor can it be used for computation of F statistics. ...


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Since you have only one instrument and one treatment, you can use the Anderson-Rubin (AR) confidence interval by inverting the AR test. The AR test is uniformly most powerful unbiased in this setting, and it has correct coverage rates regardless of the strength of the instrument. This review might be useful. This and other "weak-instrument robust" methods ...


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The R package ivmodel implements the LIML estimator (link here).


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