Instrumental variable solution to an extended Frisch problem

In signal processing and time series analysis applications we often encounter cases in which a number of (noise-free) variables are linearly related and we want to make inferences on the number and the form of the linear relations among those variables from noisy observations of them. The Frisch problem is concerned with the aforementioned inferences under the assumption that the components of the observation noise vector are mutually uncorrelated. In this paper we extend the Frisch problem by allowing the noise vector components to be correlated in an arbitrary (and unknown) way. The Extended FRIsch problem of this paper is called EXFRI for short. To make EXFRI solvable we basically assume that the observation noise is temporally white whereas the noise-free signals are temporally correlated. We show that, under the assumptions made, the EXFRI problem has a computationally simple and statistically elegant Instrumental Variable (IV) solution, which is essentially based on a canonical correlation decomposition procedure.