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.

Question

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.

Quizplus · Accepted Answer

The Answer of 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.

The Two Conditions for Instrument Validity Are corr⁡(Zi,Xi)≠0 and corr⁡(Zi,ui)=0\operatorname { corr } \left( Z _ { i } , X _ { i } \right) \neq 0 \text { and } \operatorname { corr } \left( Z _ { i } , u _ { i } \right) = 0corr(Zi​,Xi​)=0 and corr(Zi​,ui​)=0

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