The solution of singular linear difference systems under rational expectations

Many linear rational expectations macroeconomic models can be cast in the first-order form, AE(t)y(t+1) = By(t) + CE(t)x(t), if the matrix A is permitted to be singular. We show that there is a unique stable solution under two requirements: (i) the determinantal polynomial \Az-B\ is not zero for some value of z, and (ii) a rank condition. The unique solution is characterized using a familiar approach: a canonical variables transformation separating dynamics associated with stable and unstable eigenvalues. In singular models, however, there are new canonical variables associated with infinite eigenvalues. These arise from nonexpectational behavioral relations or dynamic identities present in the singular linear difference system.