Econ226_IIIE

Econ226_IIIE - III. Linear state-space models A....

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1 III. Linear state-space models A. State-space representation of a dynamic system B. Kalman filter C. Using the Kalman filter D. Bayesian analysis of linear state-space models E. Solutions to linear rational expectations models 1. Problem statement A E t y t ± 1 ² By t ± Cx t x t ± 1 ² o x t ± / t ± 1 x t exogenous e.g., if z t ² ± k ± 1   z t ² C 1 z t " 1 ± C 2 z t " 2 ± C ± C p z t " p ± v t
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2 ± kp ± 1   x t ± z t z t " 1 B z t ² p ² 1 o ± C 1 C 2 C C p " 1 C p I k 0 C 00 BB CB B C I k 0 y t ± ± n k ± 1   k t ± n d ± 1   d t k t ± predetermined (chosen by agents at t " 1 ) k t ± h ± x t " 1 , x t " 2 ,... x 1 , k 0   d t ± m ± x t , x t " 1 ,..., x 1 , k 0   goal of solution method: find h ± .   and m ± .   E. Solutions to linear rational expectations models 1. Problem statement 2. Blanchard-Kahn solution method
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3 A E t y t ± 1 ² By t ± Cx t This method assumes A is nonsingular. Drawback: if original system involves E t z t ± 2 or z t " 2 , can be written in canonical form using companion form, but at cost of making A singular. A E t y t ± 1 ² By t ± Cx t Find Jordan form of A " 1 B : A " 1 B ² V " 1 JV J ² J 1 0 C 0 0J 2 C 0 BB C B 00 C J s J i ² 5 i 10 C 0 0 5 i 1 C 0 5 i C 0 BBB C B 000 C 5 i
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4 Order eigenvalues 5 i such that first n s are less than or equal to unity in modulus and next n u are greater than unity in modulus. Assumption for unique stationary solution: n s ± n k J ± J s 0 0J u diagonal elements of J s are all t 1 in modulus diagonal elements of J u are all ² 1 in modulus
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Econ226_IIIE - III. Linear state-space models A....

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