Observational equivalence. (Sargent, 1976.) Suppose that the money supply is determined by mt = c z_t−1 + e_t, where c and z are vectors and et is an i.i.d. disturbance uncorrelated with z_t−1. et is unpredictable and unobservable. Thus the expected component of mt is c z_{t −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 affect y.

 

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 z_t−1+β_{mt +νt}. In each specification, the disturbance is i.i.d. and uncorrelated with z_t−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, suppose mt = c z_t−1 + γw_t−1 + et and that models (i) and (ii) are as before (with their disturbances now uncorrelated with wt −1 as well as with z_t−1 and et). In this case, is it possible to distinguish between the two theories? Explain.