T a 2 1 prior for 2 a gamma with mean 2 standard

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  ± T ´ A 2   " 1 prior for 2 : A Gamma with mean 2 standard deviation 0.5 E 1 Gamma with mean 1.5 standard deviation 0.25
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13 prior for all parameters: f 2   f ( " 1 | 2   f + | ( , 2   Have analytical expressions for f Y | + , (   from likelihood (same as f Y | + , ( , 2   f + , ( | 2   from prior f + , ( | 2 , Y   from posterior From these we could calculate the value of f Y | 2   ± f Y | + , ( , 2   f + , ( | 2   f + , ( | Y , 2  
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14 Using this and prior f 2   , we can use numerical methods (Metropolis- Hastings or importance sampling) to sample from f 2 | Y   - f Y | 2   f 2   to get inference about true structural parameters 90% posterior confidence A ² 1.3,2.8   E 1 ² 1.0,1.6   By sampling 2 m   from f 2 | Y   and then ( m   from f ( | 2 m   , Y   and + m   from f + | ( m   , 2 m   , Y   , we can generate posterior distribution of VAR parameters
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15 Identifying impulse-response function using structural model u gt u zt u ft ± @ g v gt @ z v zt @ f v ft E v t v t U   ± I 3 A 0 y t ± a ² A 1 y t " 1 ² A 2 y t " 2 ² C ² A p y t " p ² v t Can calculate from structural model A 0 2   " 1 ± ± y t ± v t U One option: ± y t ² s ± v t U ± ± y t ² s ± / t U ± / t ± v t U ± ' s m   A 0 2 m     " 1
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16 Problem: ( m   p A 0 2 m     " 1 ¡ A 0 2 m     " 1 ¢ U ' m   A 0 2 m     " 1 OK for impulse- response, but variance shares won’t add up Another option: QR factorization Proposition: Any n k   matrix C can be written as n k   C ± n n   Q n k   R where Q U Q ± I n and R is upper triangular How to find: qr command in Gauss or Matlab (1) Find QR factorization of ¡ A 0 2 m     " 1 ¢ U ± Q 2 m     R 2 m     A 0 2 m     " 1 ± ¡ R 2 m    ¢ U ¡ Q 2 m    ¢ U ¡ R 2 m    ¢ U is thus lower triangular
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17 (2) Find Cholesky factorization of ( m   ± P m   P m   U so P m   is lower triangular Assumption: P m   S ¡ R 2 m    ¢ U (3) Use / t ± P m   ¡ Q 2 m    ¢ U v t E / t / t U   ± P m   ¡ Q 2 m    ¢ U Q 2 m     P m   ± P m   I n P m   U ± ( m   In other words, ± y t ² s ± v t U ± ' s m   P m   ¡ Q 2 m    ¢ U
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18 II. Vector autoregressions A. Introduction B. Normal-Wishart priors for VAR’s C. Bayesian analysis of structural VAR’s D. Identification using inequality constraints E. Integrating VARs with dynamic general equilibrium models F. Selecting priors for DSGEs Traditional approach: set priors for each parameter independently just augment with another independent prior when a new parameter is introduced
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19 Model 1: y t ± c ² / t / t L N 0, @ / 2   Prior 1: c L N m , 5 2   implies: y t L N m , @ / 2 ² 5 2   Model 2: y t ± c ² by t " 1 ² / t Suppose we keep specification of / t and prior for c
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