Observational equivalence. (Sargent, 1976.) Suppose that the money supply is determined by mt = c'zt-1 + et, where c and z are vectors and et is an i.i.d. disturbance uncorrelated with zt-1. et is unpredictable and unobservable. Thus the expected component of mt is c'zt-1, and the unexpected component is et. In setting the money supply, the Federal Reserve responds only to variables that matter for real activity; that is, the variables in z directly affecty.
Now consider the following two models: (i) Only unexpected money matters, so yt = a'zt-1 + bet + vt;
(ii) All money matters, so yt = α'zt-1 + βmt + vt. In each specification, the disturbance is i.i.d. and uncorrelated with zt-1 and et.
(a) Is it possible to distinguish between these two theories? That is, given a candidate set of parameter values under, say, model (i), are there parameter values under model (ii) that have the same predictions? Explain.
(b) Suppose that the Federal Reserve also responds to some variables that do not directly affect output; that is, supposemt = c'zt-1 + y'wt-1 + et and that models (i) and (ii) are as before (with their distubances now uncorrelated with zt-1 as well as withzt-1 and et).
In this case, is it possible to distinguish between the two theories? Explain.