# #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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