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What are the consequences for the OLS estimator when weak exogeneity holds but strict exogeneity does not? Thank you.

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Keeping aside other assumptions. If strict exogeneity hold the OLS parameter are unbiased and consistent. If weak form hold, but not strict, the OLS parameters are consistent but not unbiased. However pay attention about the fact that exogeneity is a causal concept. About the last point read here: Strict exogeneity and lagged variables

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