# RDP 2011-02: Long-term Interest Rates, Risk Premia and Unconventional Monetary Policy Appendix C: Anticipated Structural Changes Under Rational Expectations

## Appendix C: Anticipated Structural Changes Under Rational Expectations

Following Cagliarini and Kulish (2008), write the model in matrix form as follows

where the state vector is defined by

and where y1,t is an (n1 × 1) vector of exogenous and some endogenous variables, and y2,t is an (n2 × 1) vector with those endogenous variables for which conditional expectations appear, zt+1, (k × 1), contains leads of y2,t; in the model above, however, zt+1 = y2,t+1 and k = n2. The dimension of yt is n × 1, where n = n1 + n2 + k. Also, we assume εt to be an l ×1 vector of serially uncorrelated processes, and are (n1 + n2) × n matrices, is (n1 + n2) × 1 and is (n1 + n2) × l.

ηt is the vector of expectations revisions given by,

where Augment the system defined by Equation (C1) with the k equations from Equation (C2) to obtain Equation (8) reproduced below

A unique rational expectations solution takes the form:

Consider that at the beginning of forecast horizon, the monetary authority announces how the policy parameters will vary in the future. An announcement of this form entails a form of structural change, from the perspective of the standard solution for rational expectations models. This induces a sequence of structures of the form, Therefore, the system evolves as follows

Under regularity conditions the solution for yt+1,...,yt+T satisfies

After t + T, the standard solution for applies.