Learning in a Misspecified VAR Model

I introduce a method to transform a T-map when agents form expectations using a misspecified learning mechanism inconsistent with a structural equation of a multivariate economic model. By transforming the perceived law of motion (PLM) into a the form of a Seemingly Unrelated Regression (SUR) the PLM and Actual Law of Motion (ALM) can be expressed in the same form. In this form, one can project the ALM into the form of the PLM. The projected T-map follows from this linear projection showing the mapping from the restricted PLM to the projected ALM. This T-map can then be used to determine whether the "Restricted Perceptions Equilibria" (RPE) of such a model are learnable under adaptive learning. I present the New Keynesian Monetary Model with inertia in both the IS and AS curves under several Taylor Rules as an example. It turns out that determinacy does not guarantee existence or uniqueness of RPE. I show a relationship between determinacy, learnability, and inertia where determinacy only changes with inertia if the REE and RPE are not stable under learning. Under restricted perceptions, an increase in inertia will always eventually lead to an explosive RPE if the RPE is not learnable. The stability properties, under learning, do not change under a simple restricted forecasting rule. Therefore, as in Bullard and Mitra (2002), the preferred policy rule is adjusting interest rates more than one for one (yet not too aggressively) with a change in inflation. The best policy to accomplish this, under the policy rules studied, is by following a Taylor rule with expected values of contemporaneous variables