Econ226_IIIF

# C 1 l µ t c 1 ² o ss costs monetary policy taylor

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C 1 " )   L µ t C ± 1 ² o s.s. costs monetary policy (Taylor Rule) (10) R µ t ± > R µ t " 1 ² 1 " >  £ = t ² r = = µ t " 1 " = t " 1   ² r Y § Y t " § Y t P  ¤ ² r = = µ t " = µ t " 1   ² r Y § Y t " § Y t p " § Y t " 1 " § Y t " 1 p   ² 1 t R = t ± inflation target = t ± > = = t " 1 ² 1 t = § Y t p ± output level if prices perfectly flexible

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6 y t ± C t , C t " 1 , R µ t , R µ t " 1 , K µ t , K µ t " 1 , Î t , Î t " 1 , Q µ t , w t , w t " 1 , L µ t , = µ t , = µ t " 1 , § Y t , r µ K , t   U x t ± â t b , â t I , 1 t Q , â t L , 1 t w , â t a , 1 t p , â t G , = t , 1 t R   U equations (1)-(10) (along with lag definitions) can be written as A E t y t ² 1 ± By t ² Cx t while shocks satisfy x t ² 1 ± o x t ² / t ² 1 (note also E t x t ² 1 ± o x t ) Observed data: OLS regression of log real consumption on constant and time trend residual ± z 1 t ± C t Same for log of investment yields Î t Other data: GDP, real wages, GDP deflator, nominal interest rate Treat Q µ t , r µ K , t , â t a as unobservable
7 state equation: 8 t ² 1 ± F 8 t ² v t ² 1 (came from solution to DSGE model) observation equation: z t ± H U 8 t Let 2 ± parameters of structural model Use Kalman filter to evaluate likelihood function p z 1 ,..., z T | 2   prior p 2   posterior p 2 | Z   - p 2   p Z | 2   prior p 2   posterior p 2 | Z   - p 2   p Z | 2   we can sample from posterior using method such as Metropolis- Hastings 5 2   ± log p 2   ² log p Z | 2

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8 5 2   ± log p 2   ² log p Z | 2   (1) Find mode of posterior distribution using numerical optimization 2 ' ± arg max 5 2   (2) Find Hessian of posterior distribution H 2 '   ± " ± 2 5 2   ± 2 ± 2 U 2 ± 2 ' (3) Set 2 j   ±
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