Non-Classical Measurement Error I have a question similar to this thread.  Although there was one answer, it did not capture the essence of the poster's question.
In essence, I am trying to estimate $y = a + \beta x^* + \epsilon$.  Unfortunately, I do not observe $x^*$, but rather a proxy $x = x^* + \nu$.  I know that in such cases, the model suffers from endogeneity (classical error-in-variables).
In my case, $\nu \ge 0$ and so $\mathbf{E}[\nu]\ne0$.
The question is can I still use traditional instrumental variable techniques, or why not?  If not, what options do I have?  (Bonus: How can I implement this in Stata.)
 A: If you have a model for you measurement error (e.g. you can assume a distribution/family of distributions for $\nu$), you can build a fully Bayesian model, that will give you the exact answer - such models are now feasible for reasonably sized datasets (I think the limit is around $N < 10^6$) with Stan language (StataStan interface in your case). Fully Bayesian in this context just means that you specify the full process that you think generates your data - if you can simulate your data, you can write a Stan model. As a bonus, you also get estimates for $x^*$. Take a look at the "Measurement error" section in the Stan manual for more guidance. 
The R package brms (built on top of Stan) supports normally distributed measurement error out of the box, for other types of error, you would have to write your own model (you can however use the code generated by brms as a start).
If you want to stay in the classical statistics, you have to assume $\nu$ is normally distributed. This answer has some clues, and I think there are other ways, but cannot find them now.
