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 m t is c ′ zt −1 , and the unexpected component is et. In setting the money supply, the Federal Reserve responds only to vari- ables that matter for real activity; that is, the variables in z directly affect y .

Now consider the following two models: (i ) Only unexpected money mat- ters, so yt = a′zt−1+bet+vt;(ii)all money matters, so yt = α′zt−1+βmt +νt. 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, suppose mt =c′zt−1+γ′wt−1+et and that models (i) and (ii) are as before (with their distubances now uncorrelated with wt −1 as well as with zt −1 and et). In this case, is it possible to distinguish between the two theories? Explain.

(Romer 311)

Romer. Advanced Macroeconomics, 4th Edition. McGraw-Hill Learning Solutions, 2012. VitalBook file.