In the case of a single endogenous variable and a single instrumental variable, the IV estimator is given by

$b_{IV} = \frac{cov(z,y)}{cov(z,y)}$

It is often mentioned that "the instrument should not affect the dependent variable directly, but only through the endogenous variable". But if the instrument should not affect the dependent variable directly,

$cov(z,y) = 0$

would hold, making

$b_{IV} = 0$.

So what is happening?

In the case of a single endogenous variable and a single instrumental variable, the IV estimator is given by

$b_{IV} = \frac{cov(z,y)}{cov(z,x)}$

It is often mentioned that "the instrument should not affect the dependent variable directly, but only through the endogenous variable". But if the instrument should not affect the dependent variable directly,

$cov(z,y) = 0$

would hold, making

$b_{IV} = 0$.

So what is happening?