#StackBounty: #instrumental-variables Instrument systematically underpredicting endogenous variable based on values of outcome variable

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?


Get this bounty!!!

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