Another chance for bonus points. Suppose that (Xi, Yi, Zi, $i) are independent four-tuples of scalar random variables for i = 1,...,n, with a common jointly normal distribution. All means are 0 and n is large.

Suppose further that Yi = Xiβ + $i. The variables Xi, Yi, Zi are observable, and every pair of them has a positive correlation which is less than 1. However, $i is not observable, and β is an unknown constant.

Is the correlation between Zi and $i identifiable? Can Z be used as an instrument for estimating β? Explain briefly.