Econ226_IIIAD

Econ226_IIIAD - III. Linear state-space models A....

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1 III. Linear state-space models A. State-space representation of a dynamic system Consider following model State equation: r ± 1 8 t ² 1 ³ r ± r F r ± 1 8 t ² r ± 1 v t ² 1 Observation equation: n ± 1 y t ³ n ± k A U k ± 1 x t ² n ± r H U r ± 1 8 t ² n ± 1 w t Observed variables: y t , x t Unobserved variables: 8 t , v t , w t Matrices of parameters: F , A , H v t w t ~ i.i.d. N 0 0 , Q0 0R Q ³ r ± r R ³ n ± n
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2 Example 1: 8 t ± 1 ² C 1 C 2 C C r " 1 C r 10 C 00 01 C BBC B B C 8 t ± / t ± 1 0 0 B 0 8 j , t ± 1 ² L j " 1 8 1 t for j ² 2,3,. .., r 8 1, t ± 1 ² C 1 8 1 t ± C 2 L 1 8 1 t ± C 3 L 2 8 1 t ± C ± C p L p " 1 8 1 t ±/ t ± 1 C ± L   8 1, t ± 1 ²/ t ± 1 Observation equation: y t ² 6 ± 1 2 1 2 2 C 2 r " 1 8 t y t " 6 ² 2 ± L   8 1 t put together with state equation: C ± L   8 1 t t C ± L  ± y t " 6   ² 2 ± L   / t Conclusion: any ARMA process can be written as a state-space model.
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3 Example 2: C t ± state of business cycle D it ± idiosyncratic component for sector i C t , D it unobserved y it ± growth in sector i (observed) 8 t ± ± C t , D 1 t , D 2 t ,..., D nt   U 8 t ² 1 ± F 8 t ² v t ² 1 F ± C C 00 C 0 0 C 1 0 C 0 C 2 C 0 BBC BB 000 C C r Observation equation: y 1 t y 2 t B y nt ± 6 1 6 2 B 6 n ² + 1 10 C 0 + 2 01 C 0 BBBC B + n C 1 8 t
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4 Purpose of state-space representation: state vector 8 t contains all information about system dynamics and forecasting. 8 t ± 1 ² F 8 t ± v t ± 1 y t ² A U x t ± H U 8 t ± w t E ± y t ± j | 8 t , 8 t " 1 ,..., 8 1 , y t , y t " 1 , ..., y 1 , x t ± j , x t ± j " 1 x 1   ² A U x t ± j ± H U F j 8 t III. Linear state-space models A. State-space representation of a dynamic system B. Kalman filter Purpose of Kalman filter: calculate distribution of 8 t conditional on ( t ² £ y t , y t " 1 y 1 , x t , x t " 1 x 1 ¤ 8 t | ( t ~ N ± § 8 t | t , P t | t  
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5 8 t ± 1 ² F 8 t ± v t ± 1 y t ² A U x t ± H U 8 t ± w t v t w t ~ i.i.d. N 0 0 , Q0 0R Begin with the prior: 8 0 ~ N ± § 8 0|0 , P 0|0   § 8 0|0 ² prior best guess as to value of 8 0 P 0|0 ² uncertainty about this guess (much uncertainty ² large diagonal elements of P 0|0 ) 8 1 ² F 8 0 ± v 1 8 1 ~ N ± § 8 1|0 , P 1|0   § 8 1|0 ² F § 8 0|0 P 1|0 ² FP 0|0 F U ± Q
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6 Useful result: suppose that y 1 | x y 2 | x L N 6 1 6 2 , % 11 % 12 % 21 % 22 where 6 i and % ij may depend on x .Th en y 2 | y 1 , x L N ± m ' , M '   m ' ± 6 2 ² % 21 % 11 " 1 ± y 1 " 6 1   M ' ± % 22 " % 21 % 11 " 1 % 12 Here y 1 | x 1 , ( 0 8 1 | x 1 , ( 0 L N 6 1 6 2 , % 11 % 12 % 21 % 22 6 2 ± § 8 1|0 % 22 ± P 1|0 6 1 ± A U x 1 ² H U § 8 1|0 % 11 ± H U P 1|0 H ² R % 21 ± P 1|0 H Hence 8 1 | y 1 , x 1 , ( 0 ± 8 1 | ( 1 L N ± § 8 1|1 , P 1|1   § 8 1|1 ± § 8 1|0 ² P 1|0 H ± H U P 1|0 H ² R   " 1 ³ y 1 " A U x 1 " H U § 8 1|0 P 1|1 ± P 1|0 " P 1|0 H ± H U P 1|0 H ² R   " 1 H U P 1|0
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7 Identical calculations: if 8 t | ( t ~ N ± 8 t | t , P t | t   , then 8 t ± 1 | ( t ± 1 ~ N ± § 8 t ± 1| t ± 1 , P t ± 1| t ± 1   P t ± 1| t ² FP t | t F U ± Q P t ± 1| t ± 1 ² P t ± 1| t " P t ± 1| t H ± H U P t ± 1| t H ± R   " 1 H U P t ± 1| t § 8 t ± 1| t ² F § 8 t | t § / t ± 1| t ² y t ± 1 " A U x t ± 1 " H U § 8 t ± 1| t § 8 t ± 1| t ± 1 ² § 8 t ± 1| t ± P t ± 1| t H ± H U P t ± 1| t H ± R
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Econ226_IIIAD - III. Linear state-space models A....

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