A chance for bonus points. Three investigators are studying the following model: Yi = Xi + $i

A chance for bonus points. Three investigators are studying the following model: Yi = Xiβ + $i for i = 1,...,n. The random variables are all scalar, as is the unknown parameter β. The unobservable $i are IID with mean 0 and finite variance, but X is endogenous. Fortunately, the investigators also have an n×1 vector Z, which is exogenous and not orthogonal to X. Investigator #1 wishes to fit the model by OLS.
Investigator #2 wants to regress Y on X and Z; the coefficient of X in this multiple regression would be the estimator for β. Investigator #3 suggests βˆ = Z
Y/Z
X. Which of the three estimators would you recommend? Why? What are the asymptotics? To focus the discussion, assume that (Xi, Yi, Zi, $i) are IID four-tuples, jointly normal, mean 0, and var(Xi) = var(Zi) = 1. Assume too that n is large. As a matter of notation, Yi is the ith component of the n×1 vector Y ; similarly for X.