My understanding of the endogeneity issue is that $\mathbb{E}[Y|X]$ is inconsistently estimated. For instance if we know $\mathbb{E}[Y|X] = \theta X$, then the estimator $\hat{\theta}$ is inconsistent (or biased). My question is, provided that endogeneity issue exists, do we still have $$\mathbb{E}[Y] = \mathbb{E}[\mathbb{E}[Y|X]]? $$ My concern is if $\mathbb{E}[Y|X]$ is incorrectly estimated, then the outer expectation may be incorrect as well.

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