Expectational Stability of Resonant Frequency Sunspot Equilibria

We consider the stability under adaptive learning of the complete set of solutions to the basic linear forward looking model in which the current value of the state variable depends linearly on the (subjectively) expected value of the state next period and the coefficient of the expected state is bigger than one in absolute value. In addition to the fundamentals solution, the literature describes both finite-state Markov sunspot solutions, satisfying a resonant frequency condition, and autoregressive solutions depending on an arbitrary martingale difference sequence. We clarify the relationships between these solutions and show that the stability properties of equilibria may depend crucially on the representation used by agents in the learning process. Only the finite-state Markov sunspot solutions can be stable under learning