The Solution of Singular Linear Difference Systems under Rational Expectations
Robert G. King, Mark W. Watson · International Economic Review · 1998
Many linear 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. For this singular linear dierence system under rational expectations, we show there is a unique stable solution under two requirements: (i) the determinental polynomial jAz Bj is not zero for some value of z, and (ii) a rank condition is satisfied which is a direct generalization of Blanchard and Kahn's (1980) requirement for the nonsingular system. The unique solution is characterized using a familiar approach: a canonical variables transformation which separates the dynamics associated with stable and unstable eigenvalues. In singular models, however, there are also canonical variables associated with infinite eigenvalues. These new canonical variables arise from nonexpectational behavioral relations or dynamic identities present in the singular linear dierence system.