Suppose I regress (with OLS) $y$ on $x_1$ and $x_2$. Suppose I have i.i.d. sample of size n, and that $x_1$ is observed with error but $y$ and $x_2$ are observed without error. What is the probability limit of the estimated coefficient on $x_1$?
Let us suppose for tractability that the measurement error of $x_1$ is "classical". That is the measurement error is normally distributed with mean 0 and is uncorrelated with $x_2$ or the error term.