*Bounty: 50*

*Bounty: 50*

Consider the following linear regression model:

$$

y_{i}=x_{i}beta+epsilon_{i}

$$

In the above, the error term is not conditionally mean independent

of $x_{i},$ that is $mathbb{E}left(epsilon_{i}|X_{i}right)neq0$.

Consider now, an instrument $z_{i}$ that can be used to identify

$beta$ by exploiting exogenous variation in $x_{i}$. That is

$$

x_{i}=gamma z_{i}+v_{i}

$$

is used to obtain fitted values of $x_{i}$, which are then used

in place of $x_{i}$ in the original regression. The instrument is

plausibly exogenous and is also relevant (i.e. it helps predicts $x_{i}$).

In my study, however, it seems that the instrument systematically

underpredicts $x_{i}$ for higher values of $y_{i.}.$ In other words,

the residual term

$$

x_{i}-hat{gamma}z_{i}=tilde{v_{i}}

$$

is systematically increasing with $y_{i}$ such that $covleft(y_{i},tilde{v_{i}}right)>0.$

Does this then imply that this instrument is invalid? Can anything

be done to solve the issue if any?