This might be a silly question, but consider a model \begin{equation} Y_{it}=\rho Y_{it-1}+\delta_i+\epsilon_{it} \end{equation} Assume that $Cov(\epsilon_{it},Y_{is})=0$ for all $s\leq t-1$ (sequential exogeneity). How can we show that the fix effect estimator cannot recover $\rho$ consistently?
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What exactly do you want to know: Why the standard conditions for consistency are violated, or do you want a counterexample for consistency, or do you really want a proof that it always cannot be a consistent estimator? – frank Oct 10 '22 at 06:00
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Related: https://stats.stackexchange.com/questions/196578/difference-of-dynamic-panel-nickell-bias-and-the-incidental-parameter-probl/196674#196674 – Christoph Hanck Oct 10 '22 at 09:20