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In the classical errors-in-variables problem, we are looking at the effect of $x^*_i$ on $y$, but $x^*_i$ is misreported. We have observations $x_i = x^*_i +u_i$, where $u_i$ has zero mean mean, variance $\sigma^2_u$, and is independent from $y_i$ and $x^*_i$. In this case, OLS produce biased and inconsistent estimates of the coefficients.

A common solution is to obtain a second measurement $z_i = x^*_i +v_i$, where $v_i$ has zero mean, variance $\sigma^2_v$ and is independent from $y_i, x^*_i, $ and $u_i$ and then use $z_i$ as an instrumental variable for $x_i$ to obtain consistent estimates of the coefficients.

My question is, what if there is a biased measurement error; for example what if $x_i = \gamma x^*_i +u_i$ where $\gamma>0$ and we still have $z_i = x^*_i +v_i$. Is there an estimation procedure that would give a consistent estimator of the coefficients? What about bias?

Thanks.

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