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The simple regression model is

$y=\beta_0+\beta_1 x + u$,

where $Cov(x,u) \neq 0$. To obtain consistent estimators of $\beta_0$ and $\beta_1$ when $x$ and $u$ are correlated, we can use $z$ as an instrumental variable for $x$. The instrument has to satisfy the following two assumptions:

$Cov(z,u)=0$

and

$Cov(z,x)\neq 0$

Wooldridge now writes: "instrument exogeneity means that $z$ should have no partial effect on $y$ (after x and ommited variables have been controlled for), and $z$ should be uncorrelated with the omitted variables."

I don't understand how the first assumption, $Cov(z,u)=0$, implies that $z$ has no partial effect on $y$, and why this is necessary. I understand that if $z$ were correlated with $u$ then $z$ would be endogenous and we couldn't obtain consistent estimators of $\beta_0$ and $\beta_1$. I just don't get why $z$ should have no partial effect on $y$ to be a valid instrument. Can someone explain this to me?

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If you understand that correlation between $z$ and $u$ would be problematic, the covariance assumption follows quite naturally. Correlation is just covariance divided by the product of the standard deviations, which are both positive numbers. If correlation is zero, the covariance must be zero as well. If the covariance is not zero, the correlation will inherit that.

The intuition is that you need $z$ to wiggle around $x$ (relevance), but without also altering $u$ in the process (exogeneity). That partial variation is what identifies $\beta_1$ consistently. If $z$ was also moving around $u$, you would be back near where you started. IV only uses part of the variation in $x$, the part that's uncorrelated with $u$ and is induced by $z$.

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  • $\begingroup$ Thanks for your quick answer. But why is $z$ not allowed to be correlated with $y$? $\endgroup$ – S. Ming Sep 20 '17 at 1:35
  • $\begingroup$ z can be correlated with y only through x, but not with u. $\endgroup$ – Dimitriy V. Masterov Sep 20 '17 at 2:26
  • $\begingroup$ If $z$ would be directly correlated with $y$, then we couldn't distinguish between the effect that $z$ has directly on $y$ and the effect that $x$ has indirectly on $y$ (via $z$). Is this the reason why $z$ is only allowed to affect $y$ indirectly? $\endgroup$ – S. Ming Jan 23 '18 at 19:00
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    $\begingroup$ @S.Ming I think this requirement is also about bias. IV throws away some of the variation in $x$, and only uses the good part of the variation that is induced by the instrument $z$ to estimate the effect of $x$ on $y$. If the instrument was correlated with something in $u$, then this discarding cannot happen. That is one way to $z$ cannot alter $y$. The instrument cannot also alter $y$ directly (say through a non-zero $\gamma z$ term in the $y$ equation), for the reasons you stated. $\endgroup$ – Dimitriy V. Masterov Jan 23 '18 at 19:13

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