The Two Conditions for Instrument Validity Are $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \text { and } \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0$

question 24

Essay

The two conditions for instrument validity are $\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \text { and } \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0$
The reason for the inconsistency of OLS is that $\operatorname { corr } \left( X _ { i } , u _ { i } \right) \neq 0$
But if X and Z are correlated, and X and u are also correlated, then how can Z and u not be correlated? Explain.

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