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Instrumental variable analysis for non-linear endogenous variable
If $Cov(x_1, \varepsilon_i)\ne 0$, it is probably the case that $Cov(x_1^2, \varepsilon)\ne 0$, and you would need an instrument for $x_1^2$.
If $z$ is your instrument for $x_1$, then as Oda said, it ...
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