ALLS1TehnAppx-808.ex Opimal Moneary Poliy in an Operaional Medium-Sized DSGE Model: Tehnial Appendix Malin Adolfson Sveriges Riksbank Sefan Laséen Sveriges Riksbank Jesper Lindé Sveriges Riksbank Lars E.O. Svensson Sveriges Riksbank Firs draf: Oober 2006 This version: Augus 2008 Absra Tehnial appendix o Opimal Moneary Poliy in an Operaional Medium-Sized DSGE Model JEL Classifiaion: E52, E58 Keywords: Opimal moneary poliy, foreass, judgmen This ehnial appendix of Adolfson, Laséen, Lindé, and Svensson [1] onains he deailed speifiaion of he maries A, B, C, andh in he model, as well as he deailed speifiaion of he measuremen equaion and he relaed maries D 0, D, and Ds. 1. Definiion of he maries A, B, C, D, andh and he insrumen rule [f X f x ] We wan o idenify he submaries in he model, X +1 = A 11 X + A 12 x + B 1 i + Cε +1, Hx +1 = A 22 x + A 21 X + B 2 i. We speify he predeermined variables X =(X ex0 ) 0, he forward-looking variables x,he poliy insrumen i, and he shok veor ε as follows (he number of he elemens in eah veor,x pd0 All remaining errors are ours. The views expressed in his paper are solely he responsibiliy of he auhors and should no be inerpreed as refleing he views of Sveriges Riksbank.
is also lised): X ex b 1 b 1 2 ˆ 3 ˆ, 1 4 ˆν 5 ˆζ 6 ˆζ h 7 ˆζ q 8 ˆλ f 9 ˆλ m 10 ˆλ mi 11 b φ 12 ˆΥ 13 b z 14 b z 1 15 ˆλ x 16 ˆε R 17 b π 18 b π 1 19 ˆτ k 20 ε ˆτ y 21 ˆτ 22 ˆτ w 23 ĝ 24 ˆτ k 1 25 ˆτ y 1 26 ˆτ 1 27 ˆτ w 1 28 ĝ 1 29 ˆπ 30 ŷ 31 ˆR 32 ˆπ 1 33 ŷ 1 34 ˆR 1 35 ˆπ 2 36 ŷ 2 37 ˆR 2 38 ˆπ 3 39 ŷ 3 40 ˆR 3 41 x ˆπ d 1 ˆπ m 2 ˆπ mi 3 ŷ 4 ˆπ x 5 b w 6 ĉ 7 bĩ 8 ˆψ z 9 ˆP k 0 10 Ŝ 11 Ĥ 12, X pd ˆk 13 ˆq 14 ˆμ 15 â 16 17 18 ˆγ md ˆγ mid ˆγ x 19 b x 20 m x 21 b k +1 22 b m +1 23 b k 1 42 b m 2 43 ˆR 1 3 44 ˆπ d 1 4 45 ˆπ m 1 5 46 ˆπ mi 1 6 47 ŷ 1 7 48 ˆπ x 1 8 49 b w 1 9 50 ĉ 1 10 51 bĩ 1 11 52 ˆψ z, 1 12 53 ˆP k 0 1 13 54 Ŝ 1 14 55 Ĥ 1 15 56, i ˆR, ˆk 1 16 57 ˆq 1 17 58 ˆμ 1 18 59 â 1 19 60 ˆγ md 1 20 61 ˆγ mid 1 21 62 ˆγ x ε ε z ε ν ε ζ ε ζ h ε ζ q ε λ ε λ m ε λ mi... 1 2 3 4 5 6 7 8 9... ε φ ε Υ ε z ε λx ε εr ε π ε 0 τ ε 0 x 10 11 12 13 14 15 16 : 20 21 : 23 1 22 63 b x 1 23 64 m x 1 24 65 b k 1 25 66 û 1 26 67 ˆπ d 2 27 68 ˆπ m 2 28 69 ˆπ d 3 29 70 ˆπ m 3 30 71 0., 2
1.1. Predeermined variables Firs we speify he exogenous variables and he shoks, hen he endogenous predeermined variables. Row 1 - Saionary ehnology shok ˆ +1 = ρ ˆ + σ ε ε,+1. ˆ : A 11 (1, 1) = ρ, ε,+1 : C (1, 1) = σ ε. The noaion means ha he value of he persisene of he ehnology shok, ρ,isplaedinhe firs row and firs olumn of he marix A 11. The noaion ˆ : means ha he parameer value perains o ha speifi variable. Row 2 - Lagged saionary ehnology shok Row 3 - Permanen ehnology shok ˆ =ˆ. ˆ : A 11 (2, 1) = 1. ˆ,+1 = ρ μz ˆ + σ εz ε z,+1. ˆ : A 11 (3, 3) = ρ, ε z,+1 : C (3, 2) = σ ε. Row 4 - Lagged permanen ehnology shok ˆ =ˆ. ˆ : A 11 (4, 3) = 1. Row 5 - Firm money demand shok ˆν +1 = ρ νˆν + σ εν ε ν,+1. Row 6 - Consumpion preferene shok ˆν : A 11 (5, 5) = ρ ν, ε ν,+1 : C (5, 3) = σ εν. ˆζ +1 = ρ ζ ˆζ + σ εζ ε ζ,+1. ˆζ : A 11 (6, 6) = ρ ζ, ε ζ,+1 : C (6, 4) = σ εζ. 3
Row 7 - Labor supply shok ˆζ h +1 = ρ ζ hˆζ h + σ εζ h ε ζ h,+1. Row 8 - Household money demand shok ˆζ h : A 11 (7, 7) = ρ ζ h, ε ζ h,+1 : C (7, 5) = σ. εζ h ˆζ q +1 = ρ ζ qˆζh + σ εζ q ε ζ q,+1. Row 9 - Markup shok - domesi firms ˆζ q : A 11 (8, 8) = ρ ζ q, ε ζ q,+1 : C (8, 6) = σ εζ q. ˆλ d +1 = ρ λˆλd + σ ελ ε λ,+1. ˆλ d : A 11 (9, 9) = ρ λ, ε λ,+1 : C (9, 7) = σ ελ. Row 10 - Shok o subsiuion elasiiy beween domesi and foreign onsumpion goods ˆλ m +1 = ρ λ m ˆλm + σ ελ m ε λ m,+1. ˆλ m : A 11 (10, 10) = ρ η m, ε λ m,+1 : C (10, 8) = σ ελ m. Row 11 - Shok o subsiuion elasiiy beween domesi and foreign invesmen goods ˆλ mi mi +1 = ρ λ mi ˆλ + σ ε ελ mi λ mi,+1. Row 12 - Risk premium shok ˆλ mi : A 11 (11, 11) = ρ η mi, ε λ mi,+1 : C (11, 9) = σ. εη mi b φ +1 = ρ φ b φ + σ ε φε φ,+1. b φ : A 11 (12, 12) = ρ φ, ε φ,+1 : C (12, 10) = σ ε φ. 4
Row 13 - Invesmen speifi ehnology shok ˆΥ +1 = ρ Υ ˆΥ + σ ευ ε Υ,+1. ˆΥ : A 11 (13, 13) = ρ Υ, ε Υ,+1 : C (13, 11) = σ ευ. Row 14 - Relaive (asymmeri) ehnology shok (beween foreign and domesi) b z +1 = ρ z b z + σ ε z ε z,+1. b z : A 11 (14, 14) = ρ z, ε z,+1 : C (14, 12) = σ ε z. Row 15 - Lagged relaive (asymmeri) ehnology shok (beween foreign and domesi) b z = b z. b z : A 11 (15, 14) = 1. Row 16 - Markup shok - exporing firms ˆλ x,+1 = ρ λx ˆλx + σ ελx ε λx,+1. ˆλ x : A 11 (16, 16) = ρ λx, ε λx,+1 : C (16, 13) = σ ελx. Row 17 - Moneary poliy shok R,+1 = ρ εr R + σ εεr ε εr,+1. R : A 11 (17, 17) = ρ εr, ε εr,+1 : C (17, 14) = σ εεr. Row 18 - Inflaion arge b π +1 = ρ π b π + σ ε π ε π,+1. b π : A 11 (18, 18) = ρ π, ε π,+1 : C (18, 15) = σ ε π. Row 19 - Lagged inflaion arge b π = b π. b π : A 11 (19, 18) = 1. 5
Row 20 29 - Fisal VAR(2) τ +1 = Θ 1 τ + Θ 2 τ 1 + e τ,+1, (1.1) where Θ 1 = Θ 1 0 Θ 1, Θ 2 = Θ 1 0 Θ 2 and e τ = Θ 1 0 S τ ε τ N (0, Σ τ ). Rewrie (1.1) as VAR(1): ˆτ +1 Θ1 eτ,+1 ˆτ Row 30 41 - Foreign VAR(4) = Θ2 I 0 ˆτ ˆτ 1 + ˆτ : A 11 (20 : 24, 20 : 24) = Θ 1, ˆτ 1 : A 11 (20 : 24, 25 : 29) = Θ 2, ˆτ : A 11 (25:29, 20 : 24) = I, ε τ,+1 : C (20 : 24, 16 : 20) = hol Θ 1 0 S τ Θ 1 0 0 0 S τ. X +1 = Φ 1 X + Φ 2 X 1 + Φ 3 X 2 + Φ 4 X 3 + e X,+1,. (1.2) where Φ 1 = Φ 1 0 Φ 1, Φ2 = Φ 1 0 Φ 2, e., and e X = Φ 1 0 S x ε x N (0, Σ x ). Rewrie (1.2) as VAR(1): X +1 Φ 1 Φ2 Φ3 Φ4 X X e X X 1 = I 0 0 0 X,+1 1 0 I 0 0 X + 0 X 2 2 0 0 0 I 0 X 3 0 X : A 11 (30 : 32, 30 : 32) = Φ 1, X 1 : A 11 (30 : 32, 33 : 35) = Φ 2, X 2 : A 11 (30 : 32, 36 : 38) = Φ 3, X 3 : A 11 (30 : 32, 39 : 41) = Φ 4, X : A 11 (33 : 35, 30 : 32) = I, X 1 : A 11 (36 : 38, 33 : 35) = I, X 2 : A 11 (39 : 41, 36 : 38) = I, ε x,+1 : C (30 : 32, 21 : 23) = hol Φ 1 0 S x Φ 1 0 S 0 0 x. 0 Row 42 - Physial apial sok (noe ha b k+1 is predeermined variable no. 42 in period +1and forward-looking variable no. 22 in period ): b k+1 = b k+1, b k : A 12 (42, 22) = 1. Row 43 - Finanial asses (noe ha b m +1 is predeermined variable no. 43 in period +1 and forward-looking variable no. 23 in period ): b m +1 = b m +1, b m +1 : A 12 (43, 23) = 1. 6
Row 44 66 - Lagged forward-looking variables ˆR : B 1 (44, 1) = 1 ˆπ d : A 12 (45, 1) = 1 ˆπ m : A 12 (46, 2) = 1 ˆπ mi : A 12 (47, 3) = 1 ŷ : A 12 (48, 4) = 1 ˆπ x : A 12 (49, 5) = 1 b w : A 12 (50, 6) = 1 ĉ : A 12 (51, 7) = 1 bĩ : A 12 (52, 8) = 1 Row 66 - Lagged physial apial sok ˆψ z : A 12 (53, 9) = 1 ˆP k 0 : A 12 (54, 10) = 1 Ŝ : A 12 (55, 11) = 1 Ĥ : A 12 (56, 12) = 1 û : A 12 (57, 13) = 1 ˆq : A 12 (58, 14) = 1 ˆμ : A 12 (59, 15) = 1 â : A 12 (60, 16) = 1 ˆγ md : A 12 (61, 17) = 1 ˆγ mid : A 12 (62, 18) = 1 ˆγ x : A 12 (63, 19) = 1 b x : A 12 (64, 20) = 1 m x : A 12 (65, 21) = 1 b k 1 : A 11 (66, 42) = 1 Row 67 - Lagged apaiy uilizaion û 1 ˆk 1 b k 1 ˆk 1 : A 11 (67, 57) = 1 b k 1 : A 11 (67, 66) = 1 7
Rows 68-71 - Lagged inflaion ˆπ d 1 : A 11 (68, 45) = 1 ˆπ m 1 : A 11 (69, 46) = 1 ˆπ d 2 : A 11 (70, 68) = 1 ˆπ m 2 : A 11 (71, 69) = 1 1.2. Forward-looking variables Row 1 - Domesi inflaion ˆπ d b π = β 1+κ d β κ dβ (1 ρ π ) 1+κ d β m d α ˆR f = m d αĥ αˆk + b w + Rewrie he above equaions as: ˆπ d +1 ρ π b π + κ d ˆπ d 1 b π 1+κ d β b π + (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) ˆ + Ĥ ˆk + b w + ˆR f ˆε, νr v(r 1) + 1 ˆR ν(r 1) 1 + v(r 1) + 1 ˆν β 1+κ d β ˆπd +1 = ˆπ d + κ d 1+κ d β ˆπd 1 + m d + ˆλ d, νr v(r 1) + 1 ˆR ν(r 1) 1 + v(r 1) + 1 ˆν ˆε + αˆ + (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) + (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) + (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) (1 κ d)(βρ π 1) b π 1+κ d β. αĥ (1 ξ d)(1 βξ d ) αˆk ξ d (1 + κ d β) b w + (1 ξ d)(1 βξ d ) νr ξ d (1 + κ d β) v(r 1) + 1 ˆR 1 + ν(r 1) v(r 1) + 1 ˆν + (1 ξ d)(1 βξ d ) αˆμ ξ d (1 + κ d β) z ˆε + (1 ξ d)(1 βξ d ) ˆλ d ξ d (1 + κ d β) 8
ˆπ d +1 : H (1, 1) = β 1+κ d β ˆπ d : A 22 (1, 1) = 1 ˆπ d 1 : A 21 (1, 45) = κ d 1+κ d β b π β : A 21 (1, 18) = 1 1+κ d β ρ π κ d 1+κ d β κ dβ (1 ρ π ) 1+κ d β ˆλ d : A 21 (1, 9) = (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) ˆλ d : A 21 (1, 9) = 1 (resaled) Marginal os : b w : A 22 (1, 6) = (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) Ĥ : A 22 (1, 12) = (1 ξ d)(1 βξ d ) α ξ d (1 + κ d β) ˆk : A 22 (1, 13) = (1 ξ d)(1 βξ d ) α ξ d (1 + κ d β) ˆε : A 21 (1, 1) = (1 ξ d)(1 βξ d ) ξ d (1 + κ d β) ˆ : A 21 (1, 3) = (1 ξ d)(1 βξ d ) α ξ d (1 + κ d β) ˆν : A 21 (1, 5) = (1 ξ d)(1 βξ d ) ν(r 1) ξ d (1 + κ d β) v(r 1) + 1 ˆR 1 : A 21 (1, 44) = (1 ξ d)(1 βξ d ) νr ξ d (1 + κ d β) v(r 1) + 1 Row 2 - Impored onsumpion inflaion ˆπ m b π = β ˆπ m +1 1+κ m β ρ π b π + κ m m ˆπ 1 b π 1+κ m β κ mβ (1 ρ π ) b π + (1 ξ m)(1 βξ m ) 1+κ m β ξ m (1 + κ m β) m m m x ˆγ x ˆγ md m m + m ˆλ, 9
β +1 : H (2, 2) = 1+κ m β ˆπ m : A 22 (2, 2) = 1 ˆπ m 1 : A 21 (2, 46) = κ m 1+κ m β ˆπ m b π : A 21 (2, 18) = 1 β 1+κ m β ρ π κ m ˆλ m : A 21 (2, 10) = (1 ξ m)(1 βξ m ) ξ m (1 + κ m β) ˆλ m : A 21 (2, 10) = 1 (resaled) Marginal os : m x : A 22 (2, 21) = (1 ξ m)(1 βξ m ) ξ m (1 + κ m β) ˆγ x : A 22 (2, 19) = (1 ξ m)(1 βξ m ) ξ m (1 + κ m β) ˆγ md : A 22 (2, 17) = (1 ξ m)(1 βξ m ) ξ m (1 + κ m β) Row 3 - Impored invesmen inflaion ˆπ mi b π = 1+κ m β κ mβ (1 ρ π ) 1+κ m β β ˆπ mi +1 1+κ mi β ρ π b π + κ mi mi ˆπ 1 b π 1+κ mi β κ miβ (1 ρ π ) b π + (1 ξ mi)(1 βξ mi ) 1+κ mi β ξ mi (1 + κ mi β) m mi m x ˆγ x ˆγ mi,d m mi + mi ˆλ, ˆπ mi β +1 : H (3, 3) = 1+κ mi β ˆπ mi : A 22 (3, 3) = 1 ˆπ mi 1 : A 21 (3, 47) = κ mi 1+κ mi β Marginal os : ˆγ mid : A 22 (3, 18) = (1 ξ mi)(1 βξ mi ) ξ mi (1 + κ mi β) ˆγ x : A 22 (3, 19) = (1 ξ mi)(1 βξ mi ) ξ mi (1 + κ mi β) m x : A 22 (3, 21) = (1 ξ mi)(1 βξ mi ) ξ mi (1 + κ mi β) ˆλ mi : A 21 (3, 11) = (1 ξ mi)(1 βξ mi ) ξ mi (1 + κ mi β) ˆλ mi : A 21 (3, 11) = 1 (resaled) b π : β A 21 (3, 18) = 1 1+κ mi β ρ π κ mi 1+κ mi β κ miβ (1 ρ π ) 1+κ mi β 10
Row 4 - Oupu gap 0= ŷ + λ fˆ + λ f αˆk λ f αˆ + λ f (1 α) Ĥ ŷ : A 22 (4, 4) = 1 Ĥ : A 22 (4, 12) = λ f (1 α) ˆk : A 22 (4, 13) = λ f α Row 5 - Expor prie inflaion ˆπ x b π = ˆ : A 21 (4, 1) = λ f ˆ : A 21 (4, 3) = λ f α β ˆπ x +1 1+κ x β ρ π b π + κ x x ˆπ 1+κ x β 1 b π κ xβ (1 ρ π ) b π + (1 ξ x)(1 βξ x ) 1+κ x β ξ x (1 + κ x β) m x + ˆλ x, ˆπ x +1 : H (5, 5) = β 1+κ x β ˆπ x : A 22 (5, 5) = 1 ˆπ x 1 : A 21 (5, 49) = κ x 1+κ x β Marginal os : m x : A 22 (5, 21) = (1 ξ x)(1 βξ x ) ξ x (1 + κ x β) ˆλ x : A 21 (5, 16) = (1 ξ x)(1 βξ x ) ξ x (1 + κ x β) ˆλ x : A 21 (5, 16) = 1 (resaled) b π : β A 21 (5, 18) = 1 1+κ x β ρ π κ x 1+κ x β κ xβ (1 ρ π ) 1+κ x β Row 6 - Real wage d η 0 ŵ 1 + η 1 ŵ + η 2 ŵ +1 + η 3 ˆπ b π + d η4 ˆπ +1 ρ b π π 0 = E +η 5 ˆπ 1 b π + η6 ˆπ ρ π b π +η 7 ˆψτ z + η 8 Ĥ + η 9ˆτ y + η 10ˆτ w + η 11ˆζh bπ = (1 ω ) γ d (1 η ) bπ d + ω (γ m ) (1 η ) bπ m 11
where η b w ξ w σ L λ w b w 1+βξ 2 w b w βξ w b w ξ w b w βξ w b w ξ w κ w b w βξ w κ w 1 λ w (1 λ w )σ L (1 λ w ) τ y (1 τ y ) (1 λ w ) τ w (1+τ w ) (1 λ w ) η 0 η 1 η 2 η 3 η 4 η 5 η 6 η 7 η 8 η 9 η 10 η 11. (1.3) Colle erms and use he definiion of bπ : µ η 2 ŵ +1 + η 4ˆπ d +1 = η 0 ŵ 1 η 1 ŵ η 3 + η 6 (1 ω ) γ d (1 η ) bπ d η 6 ω (γ m ) (1 η ) bπ m η 5 µ(1 ω ) γ d (1 η ) bπ d 1 η 5 (ω ) )(γ m ) (1 η bπ m 1 η 7 ˆψτ z η 8 Ĥ η 9ˆτ y η 10ˆτ w η 11ˆζh +[η 3 + ρ π η 4 + η 5 + ρ π η 6 ] b π 12
ˆπ d +1 : H (6, 1) = η 4 ŵ +1 : H (6, 6) = η 2 µ bπ d : A 22 (6, 1) = η 3 + η 6 (1 ω ) γ d (1 η ) bπ m : A 22 (6, 2) = η 6 ω (γ m ) (1 η ) ŵ : A 22 (6, 6) = η 1 ˆψ τ z : A 22 (6, 9) = η 7 Ĥ : A 22 (6, 12) = η 8 Resaling he labor supply shok so is oeffiien is of he same magniude as ŵ i hˆζh : A 21 (6, 7) = η 11 > 0 A 21 (6, 7) = η ˆζ h 1 : if η 11 < 0 A 21 (6, 7) = η 1 =0 A 21 (6, 7) = η 11 b π : A 21 (6, 18) = (η 3 + ρ π η 4 + η 5 + ρ π η 6 ) ˆτ y : A 21 (6, 21) = η 9 ˆτ w : A 21 (6, 23) = η 10 bπ d 1 : A 21 (6, 45) = η 5 (1 ω ) γ d (1 η ) bπ m 1 : A 21 (6, 46) = η 5 ω (γ m ) (1 η ) ŵ 1 : A 21 (6, 50) = η 0 Row 7 - Consumpion bβ ĉ +1 + μ 2 z + b 2 β ĉ b ĉ 1 + b ˆμz βˆ,+1 + E +( bβ)( b) ˆψ z + τ 1+τ (μ z bβ)( b)ˆτ +( bβ)( b)ˆγ d ( b) ˆζ bβˆζ =0 +1 bγ d = ω (γ m ) (1 η ) bγ md ˆ,+1 = ρ μz ˆ ˆζ +1 = ρ ζ ˆζ bβ ĉ +1 = μ 2 z + b 2 β ĉ b ĉ 1 + b 1 βρ μz ˆ +( bβ)( b) ˆψ z + τ 1+τ ( bβ)( b)ˆτ +( bβ)( b) ω (γ m ) (1 η ) bγ md ( b) bβρ ζ ˆζ 13
ĉ +1 : H (7, 7) = bβ ĉ : A 22 (7, 7) = μ 2 z + b 2 β ĉ 1 : A 21 (7, 51) = b ˆψ z : A 22 (7, 9) = ( bβ)( b) bγ md : A 22 (7, 17) = ( bβ)( b) ω (γ m ) (1 η ) ˆ : A 21 (7, 3) = b 1 βρ μz Resaling of ˆζ o give i he same oeffiien as ĉ hˆζ : A 21 (7, 6) = ( b) i bβρ ζ ˆζ : A 21 (7, 6) = μ 2 z + b 2 β ˆτ : A 21 (7, 22) = τ 1+τ ( bβ)( b) Row 8 - Invesmen n E ˆPk + ˆΥ h io ˆγ i,d μ 2 zs 00 ( ) b ĩ bĩ 1 β b ĩ +1 bĩ +ˆ βˆ,+1 =0 bγ i,d = ω i γ i,mi (1 η i ) bγ mi,d ˆ,+1 = ρ μz ˆ μ 2 zs 00 ( ) βbĩ +1 = μ 2 zs 00 ( )(1+β)bĩ μ 2 zs 00 ( )bĩ 1 ˆP k +ω i γ i,mi (1 η i ) bγ mi,d + μ 2 zs 00 ( ) 1 βρ μz ˆ ˆΥ bĩ +1 : H (8, 8) = μ 2 zs 00 ( ) β bĩ : A 22 (8, 8) = μ 2 zs 00 ( )(1+β) bĩ 1 : A 21 (8, 52) = μ 2 zs 00 ( ) ˆP k : A 22 (8, 10) = 1 bγ mi,d : A 22 (8, 18) = ω i γ i,mi (1 η i ) ˆ : A 21 (8, 3) = μ 2 zs 00 ( ) 1 βρ μz Resaling of ˆΥ o give i he same oeffiien as bĩ ˆΥ : A 21 (8, 13) = μ 2 zs 00 ( )(1+β) 14
Row 9 - Lagrange muliplier (firs order ondiion wih respe o m +1 ) E μˆψ z + μˆψ z,+1 μˆπ d +1 μˆ,+1 + μ k βτ ˆR + τ k 1 τ k (β μ)ˆτ k +1 = 0 ˆ,+1 = ρ μz ˆ μˆψ z,+1 μˆπ d +1 = μˆψ z μ k βτ ˆR + μˆ,+1 τ k 1 τ k (β μ)ˆτ k +1 ˆ,+1 = ρ μz ˆ 0 = μˆψ z + μˆψ z,+1 μˆ,+1 + μ k βτ ˆR μˆπ d +1 + τ k 1 τ k (β μ)ˆτ k +1 ˆ,+1 = ρ μz ˆ ˆτ k +1 = Θ 1 (1, 1) ˆτ k + Θ 1 (1, 2) ˆτ y + Θ 1 (1, 3) ˆτ + Θ 1 (1, 4) ˆτ w + Θ 1 (1, 5) ĝ + Θ 2 (1, 1) ˆτ k 1 + Θ 2 (1, 2) ˆτ y 1 + Θ 2 (1, 3) ˆτ 1 + Θ 2 (1, 4) ˆτ w 1 + Θ 2 (1, 5) ĝ 1 k μˆψ z,+1 μˆπ d +1 = μˆψ z + μρ μz ˆ μ βτ ˆR τ k 1 τ k (β μ) Θ 1 (1, 1) ˆτ k τ k 1 τ k (β μ) Θ 1 (1, 2) ˆτ y τ k 1 τ k (β μ) Θ 1 (1, 3) ˆτ τ k 1 τ k (β μ) Θ 1 (1, 4) ˆτ w τ k 1 τ k (β μ) Θ 1 (1, 5) ĝ τ k 1 τ k (β μ) Θ 2 (1, 1) ˆτ k 1 τ k 1 τ k (β μ) Θ 2 (1, 2) ˆτ y 1 τ k 1 τ k (β μ) Θ 2 (1, 3) ˆτ 1 τ k 1 τ k (β μ) Θ 2 (1, 4) ˆτ w 1 τ k 1 τ k (β μ) Θ 2 (1, 5) ĝ 1 ˆπ d +1 : H (9, 1) = μ ˆψ z,+1 : H (9, 9) = μ ˆψ z : A 22 (9, 9) = μ ˆR : B 2 (9, 1) = μ k βτ 15
Row 10 - Prie of apial ˆ : A 21 (9, 3) = μρ μz ˆτ k : A 21 (9, 20) = τ k 1 τ k (β μ) Θ 1 (1, 1) ˆτ y : A 21 (9, 21) = τ k 1 τ k (β μ) Θ 1 (1, 2) ˆτ : A 21 (9, 22) = τ k 1 τ k (β μ) Θ 1 (1, 3) ˆτ w : A 21 (9, 23) = τ k 1 τ k (β μ) Θ 1 (1, 4) ĝ : A 21 (9, 24) = τ k 1 τ k (β μ) Θ 1 (1, 5) ˆτ k 1 : A 21 (9, 25) = τ k 1 τ k (β μ) Θ 2 (1, 1) ˆτ y 1 : A 21 (9, 26) = τ k 1 τ k (β μ) Θ 2 (1, 2) ˆτ 1 : A 21 (9, 27) = τ k 1 τ k (β μ) Θ 2 (1, 3) ˆτ w 1 : A 21 (9, 28) = τ k 1 τ k (β μ) Θ 2 (1, 4) ĝ 1 : A 21 (9, 29) = τ k 0 = ˆψ z +ˆ,+1 ˆψ β(1 δ) z,+1 + τ k β(1 δ) 1 τ k ˆτ k +1 μ, z ˆr k +1 = ρ μz ˆ + b w +1 + 1 τ k (β μ) Θ 2 (1, 5) ˆPk,+1 + ˆP k β(1 δ) ˆr +1 k νr v(r 1) + 1 ˆR ν(r 1) + v(r 1) + 1 ρ νˆν + Ĥ+1 ˆk +1 ˆτ k +1 = Θ 1 (1, 1) ˆτ k + Θ 1 (1, 2) ˆτ y + Θ 1 (1, 3) ˆτ + Θ 1 (1, 4) ˆτ w + Θ 1 (1, 5) ĝ + Θ 2 (1, 1) ˆτ k 1 + Θ 2 (1, 2) ˆτ y 1 + Θ 2 (1, 3) ˆτ 1 + Θ 2 (1, 4) ˆτ w 1 + Θ 2 (1, 5) ĝ 1 16
β(1 δ) ˆψ z,+1 + β(1 δ) ˆk+1 ˆPk,+1 + β(1 δ) " = ˆψ z + ρ μz β(1 δ) # ρ μ μz z β(1 δ) ν(r 1) v(r 1) + 1 ρ νˆν b w +1 + β(1 δ) Ĥ μ +1 z ˆ + ˆP k β(1 δ) νr v(r 1) + 1 ˆR τ k β(1 δ) 1 τ k Θ1 (1, 1) ˆτ k + τ k β(1 δ) 1 τ k Θ1 (1, 2) ˆτ y + τ k β(1 δ) 1 τ k Θ1 (1, 3) ˆτ + τ k β(1 δ) 1 τ k Θ1 (1, 4) ˆτ w + τ k β(1 δ) 1 τ k Θ1 (1, 5) ĝ + τ k β(1 δ) 1 τ k Θ2 (1, 1) ˆτ k 1 + τ k β(1 δ) 1 τ k Θ2 (1, 2) ˆτ y 1 μ + τ k β(1 δ) z 1 τ k Θ2 (1, 3) ˆτ 1 + τ k β(1 δ) 1 τ k Θ2 (1, 4) ˆτ w 1 + τ k β(1 δ) 1 τ k Θ2 (1, 5) ĝ 1 b w +1 : H (10, 6) = β(1 δ) ˆψ z,+1 : H (10, 9) = 1 β(1 δ) ˆP k,+1 : H (10, 10) = Ĥ +1 : H (10, 12) = β(1 δ) û +1 : H (10, 13) = β(1 δ) ˆψ z : A 22 (10, 9) = 1 ˆψ z : A 22 (10, 9) = 1 ˆP k : A 22 (10, 10) = 1 17
ˆR : B 2 (10, 1) = β(1 δ) νr v(r 1) + 1 ˆ : A 21 (10, 3) = ρ μz β(1 δ) ρ μ μz z ˆν : A 21 (10, 5) = β(1 δ) ν(r 1) v(r 1) + 1 ρ ν ˆτ k : A 21 (10, 20) = τ k ˆτ y : A 21 (10, 21) = τ k ˆτ : A 21 (10, 22) = τ k ˆτ w : A 21 (10, 23) = τ k ĝ : A 21 (10, 24) = τ k ˆτ k 1 : A 21 (10, 25) = τ k ˆτ y 1 : A 21 (10, 26) = τ k ˆτ 1 : A 21 (10, 27) = τ k ˆτ w 1 : A 21 (10, 28) = τ k ĝ 1 : A 21 (10, 29) = τ k 1 τ k β(1 δ) Θ1 (1, 1) 1 τ k β(1 δ) Θ1 (1, 2) 1 τ k β(1 δ) Θ1 (1, 3) 1 τ k β(1 δ) Θ1 (1, 4) 1 τ k β(1 δ) Θ1 (1, 5) 1 τ k β(1 δ) Θ2 (1, 1) 1 τ k β(1 δ) Θ2 (1, 2) 1 τ k β(1 δ) Θ2 (1, 3) 1 τ k β(1 δ) Θ2 (1, 4) 1 τ k β(1 δ) Θ2 (1, 5) Row 11 - Change in nominal exhange rae (UIP) 1 s φ E Ŝ+1 = φ s Ŝ + ˆR ˆR + φ a â b φ Ŝ+1 : H (11, 11) = 1 s φ Ŝ : A 22 (11, 11) = φ s â : A 22 (11, 16) = φ a ˆR : B 2 (11, 1) = 1 b φ : A 21 (11, 12) = 1 ˆR : A 21 (11, 32) = 1 18
Row 12 - Hours worked (aggregae resoure onsrain) (1 ω ) γ d η ĉ + η y ˆγ d +(1 ω i ) γ id η i ĩ b ĩ + η y iˆγ id + g y ĝ + y y i = λ d hˆε + α ˆk ˆ +(1 α) Ĥ 1 k τ r k k y ĉ + η y ˆγ d i λ d hˆε + α ˆk ˆ +(1 α) Ĥ 0 = (1 ω ) γ d η 0 = (1 ω ) γ d η +(1 ω i ) γ id η i ĩ +(1 ω i ) γ id η i ĩ y + 1 k τ r k k y bγ d = ω (γ m ) (1 η ) bγ md bγ i,d = ω i γ i,mi (1 η i ) bγ mi,d y ĉ +(1 ω ) γ d η y η iω i γ i,mi (1 η i ) bγ mi,d λ d α λ dˆε + λ d αˆ λ d (1 α) Ĥ 1 ˆk b k b ĩ + η iˆγ id + g y ĝ + y y 1 ˆk b k ŷ η f ˆγ x + b z ŷ η f ˆγ x + b z y η ω (γ m ) (1 η ) bγ md +(1 ω i ) γ id η i ĩ y b ĩ y ĉ : A 22 (12, 7) = (1 ω ) γ d η y η f ˆγ x + g y ĝ + y y ŷ + y y b z 1 ˆk 1 τ k r k k y bĩ : A 22 (12, 8) = (1 ω i ) γ id η i ĩ y Ĥ : A 22 (12, 12) = λ d (1 α) ˆk : A 22 (12, 13) = λ d α 1 k τ r k k 1 y μ z γ d η bγ md : A 22 (12, 17) = (1 ω ) bγ mi,d : A 22 (12, 18) = (1 ω i ) ˆγ x : A 22 (12, 19) = y y η f ˆε : A 21 (12, 1) = λ d ˆ : A 21 (12, 3) = λ d α b z : A 21 (12, 14) = y y ĝ : A 21 (12, 24) = g y ŷ : A 21 (12, 31) = y y b k : A 21 (12, 42) = 1 k τ r k k 1 y y y η ω (γ m ) (1 η ) γ id η i ĩ y η iω i γ i,mi (1 η i ) 1 τ k r k k y 1 b k 19
Row 13 - Capial-servies flow û = 1 ˆr k 1 τ k σ a σ a (1 τ k ) ˆτ k, û ˆk b k, ˆr k = ˆ + b w + ˆR f + Ĥ ˆk. 0= (1 + σ a ) ˆk + σ a b k +ˆ + b w + ˆR f + Ĥ 0= (1 + σ a ) ˆk + σ a b k + b w + Ĥ + νr v(r 1) + 1 ˆR 1 +ˆ + b w : A 22 (13, 6) = 1 Ĥ : A 22 (13, 12) = 1 ˆk : A 22 (13, 13) = (1 + σ a ) τ k (1 τ k ) ˆτ k ν(r 1) v(r 1) + 1 ˆν τ k (1 τ k ) ˆτ k ˆ : A 21 (13, 3) = 1 ˆν : ν(r 1) A 21 (13, 5) = v(r 1) + 1 ˆτ k : A 21 (13, 20) = τ k (1 τ k ) b k : A 21 (13, 42) = σ a ˆR =1 : νr A 21 (13, 44) = v(r 1) + 1 Row 14 - Real money balanes ˆq = 1 ˆζ q + τ k σ q 1 τ k ˆτ k ˆψ z R R 1 ˆR 1, 0= σ q ˆq ˆψ z R R 1 ˆR 1 + ˆζ q + τ k 1 τ k ˆτ k, ˆψ z : A 22 (14, 9) = 1 ˆq : A 22 (14, 14) = σ q ˆζ q : A 21 (14, 8) = 1 ˆτ k : A 21 (14, 20) = τ k 1 τ k ˆR 1 : A 21 (14, 44) = R R 1 20
Row 15 - Loan marke learing ν wh ˆν + b w + Ĥ = μ m π d ˆμ + b m ˆπ d ˆ qˆq, 0= ν whˆν ν wh b w ν whĥ + μ m π d ˆμ + μ m π d b m μ m π d ˆπ d μ m π d ˆ qˆq, Row 16 - Ne foreign asses ˆπ d : A 22 (15, 1) = μ m π d b w : A 22 (15, 6) = ν wh Ĥ : A 22 (15, 12) = ν wh ˆq : A 22 (15, 14) = q ˆμ : A 22 (15, 15) = μ m π d ˆν : A 21 (15, 5) = ν wh ˆ : A 21 (15, 3) = μ m π d b m : A 21 (15, 43) = μ m π d â = y m x η f y ˆγ x + y ŷ + y b z +( m +ĩ m )ˆγ f m ĉ + m η (1 ω ) γ d (1 η ) ˆγ md ĩ m bĩ +ĩ m η i (1 ω i ) γ id (1 η i ) ˆγ mid + R â 1, π ˆγ f = m x +ˆγ x, 0 = â [ y ( m +ĩ m )] m x η f y ( m +ĩ m ) ˆγ x, + y ŷ + y b z m ĉ + m η (1 ω ) γ d (1 η ) ˆγ md ĩ m bĩ +ĩ m η i (1 ω i ) γ id (1 η i ) ˆγ mid + R â 1, π 21
ĉ : A 22 (16, 7) = m bĩ : A 22 (16, 8) = ĩ m â : A 22 (16, 16) = 1 ˆγ md : A 22 (16, 17) = m η (1 ω ) γ d (1 η ) ˆγ mid : A 22 (16, 18) = ĩ m η i (1 ω i ) γ id (1 η i ) ˆγ x : A 22 (16, 19) = η f y +( m +ĩ m ) m x : A 22 (16, 21) = [ y +( m +ĩ m )] b z : A 21 (16, 14) = y ŷ : A 21 (16, 31) = y â 1 : A 21 (16, 60) = R π Row 17 - Relaive prie - impored ons vs. domesi 0= ˆγ md +ˆγ md 1 +ˆπ m ˆπ d ˆπ d : A 22 (17, 1) = 1 ˆπ m : A 22 (17, 2) = 1 ˆγ md : A 22 (17, 17) = 1 ˆγ md 1 : A 21 (17, 61) = 1 Row 18 - Relaive prie - impored invesmen vs. domesi 0= ˆγ mid +ˆγ mid 1 +ˆπ mi ˆπ d ˆπ d : A 22 (18, 1) = 1 ˆπ mi : A 22 (18, 3) = 1 ˆγ mid : A 22 (18, 18) = 1 ˆγ mid 1 : A 21 (18, 62) = 1 Row 19 - Relaive prie - expor vs. foreign 0= ˆγ x +ˆγ x 1 +ˆπ x ˆπ ˆπ x : A 22 (19, 5) = 1 ˆγ x : A 22 (19, 19) = 1 ˆπ : A 21 (19, 30) = 1 ˆγ x 1 : A 21 (19, 63) = 1 22
Row 20 - Real exhange rae b x = ω (γ m ) (1 η md )ˆγ ˆγ x, m x 0 = b x ω (γ m ) (1 η md )ˆγ ˆγ x, m x m x = m x 1 + bπ d bπ x S b ˆγ x = ˆγ x 1 +ˆπ x ˆπ ˆγ md = ˆγ md 1 +ˆπ m ˆπ d Row 21 - Marginal os expor ˆγ md : A 22 (20, 17) = ω (γ m ) (1 η ) ˆγ x, : A 22 (20, 19) = 1 b x : A 22 (20, 20) = 1 m x : A 22 (20, 21) = 1 0= m x + m x 1 + bπ d bπ x b S bπ d : A 22 (21, 1) = 1 bπ x : A 22 (21, 5) = 1 S b : A 22 (21, 11) = 1 m x : A 22 (21, 21) = 1 m x 1 : A 21 (21, 65) = 1 Row 22 - Physial apial sok (noe ha b k+1 is predeermined variable no. 42 in period +1and forward-looking variable no. 22 in period ): 0= b k+1 +(1 δ) 1 b k (1 μ δ) 1 ˆμ z + 1 (1 δ) 1μz ˆΥ + 1 (1 δ) 1μz bĩ, z b k+1 : A 22 (22, 22) = 1, b k : A 21 (22, 42) = (1 δ) 1, ˆ : A 21 (22, 3) = (1 δ) 1, ˆΥ : A 21 (22, 13) = 1 (1 δ) 1μz, ˆΥ : A 21 (22, 13) = S 00 μ 2 z (1 + β) µ1 (1 δ) 1μz (resaled shok!) bĩ : A 22 (22, 8) = 1 (1 δ) 1. 23
Row 23 - Finanial asses (noe ha b m +1 is predeermined variable no. 43 in period +1 and forward-looking variable no. 23 in period ) 1.3. Speifying D 0= b m +1 +ˆμ ˆ ˆπ d + b m b m +1 : A 22 (23, 23) = 1 ˆ : A 21 (23, 3) = 1 b m : A 21 (23, 43) = 1 ˆπ d : A 22 (23, 1) = 1 ˆμ : A 22 (23, 15) = 1 We also speify he arge variables, for onveniene inluding some variables of ineres whih may have zero weigh in he loss funion. They are he quarerly and four-quarerly CPI inflaion gaps relaive o seady-sae inflaion, whih equals he offiial inflaion arge; he oupu gap relaive o seady-sae oupu (we may also use he gap relaive o poenial oupu, meaning he flexprie and flexwage oupu level); he quarerly ineres rae; he quarerly ineres-rae differenial; and he real exhange rae. The inflaion and ineres raes are all measured a an annual rae. Then he veor of arge variables, Y, and he orresponding marix D saisfy where Y [4ˆπ pi Quarerly CPI inflaion gap: 4ˆπ pi, ˆπ pi, ŷ, 4 ˆR, 4( ˆR ˆR 1 ), b x ] 0 D ˆπ pi =ˆπ pi +ˆπ pi 1 +ˆπpi 2 +ˆπpi 3. =4[(1 ω )(γ d ) (1 η ) bπ d + ω (γ m ) (1 η ) bπ m ], (1.4) X x i, bπ d : D(1,n X +1)=4(1 ω )(γ d ) (1 η ), bπ m : D(1,n X +2)=4ω (γ m ) (1 η ). Four-quarer CPI inflaion gap: ˆπ pi =ˆπ pi +ˆπ pi 1 +ˆπpi 2 +ˆπpi 3. (1.5) ˆπ d : D(2,n X +1)=(1 ω )(γ d ) (1 η ), ˆπ d 1 : D(2, 45) = (1 ω )(γ d ) (1 η ), ˆπ d 2 : D(2, 68) = (1 ω )(γ d ) (1 η ), ˆπ d 3 : D(2, 70) = (1 ω )(γ d ) (1 η ), bπ m : D(2,n X +2)=ω (γ m ) (1 η ), bπ m 1 : D(2, 46) = ω (γ m ) (1 η ), bπ m 2 : D(2, 69) = ω (γ m ) (1 η ), bπ m 3 : D(2, 71) = ω (γ m ) (1 η ). 24
Alernaively, we ould inorporae he exogenous ime-varying inflaion arge in he inflaion gap, 1 Y [4(ˆπ pi b π ), ˆπ pi 4b π, ŷ, 4 ˆR, 4( ˆR ˆR X 1 ), b x ] 0 D x i, in whih ase b π : D(1, 18) = 4, b π : D(2, 18) = 4. Oupu gap, Ineres rae, ŷ : D(3,n X +4)=1. ˆR : D(4,n X + n x + n i )=4. Ineres-rae differenial, ˆR : D(5,n X + n x + n i )=4, ˆR 1 : D(5, 44) = 4. Real exhange rae, b x : D(6,n X + 20) = 1. 1.4. Defining f X and f x - Ineres rae rule i = f X X + f x x h i ˆR = ρ R ˆR 1 +(1 ρ R ) b π + r π ˆπ 1 b π + ry ŷ 1 + r x b x 1 +r π ˆπ ˆπ 1 + r y (ŷ ŷ 1 )+ε R ˆR = ρ R ˆR 1 +(1 ρ R ) b π +(1 ρ R ) r πˆπ 1 (1 ρ R ) r π b π +(1 ρ R ) r y ŷ 1 +(1 ρ R ) r x b x 1 + r πˆπ r πˆπ 1 + r y ŷ r y ŷ 1 + ε R = ρ R ˆR 1 +(1 ρ R )[1 r π ] b π + r π (1 ω )(γ d ) (1 η ) bπ d +r π (ω )(γ m ) (1 η ) bπ m +[(1 ρ R ) r π r π ](1 ω )(γ d ) (1 η ) bπ d 1 +[(1 ρ R ) r π r π ] ω (γ m ) (1 η ) bπ m 1 +r y ŷ +[(1 ρ R ) r y r y ]ŷ 1 +(1 ρ R ) r x b x 1 + ε R 1 Noe ha one ould alernaive replae 4 π by π + π 1 + π 2 + π 3 and onsider he laer he ime-varying four-quarer inflaion arge. 25
ˆR 1 : f X (1, 44) = ρ R b π : f X (1, 18) = (1 ρ R )[1 r π ] bπ d : f x (1, 1) = r π (1 ω )(γ d ) (1 η ) bπ m : f x (1, 2) = r π ω (γ m ) (1 η ) bπ d 1 : f X (1, 45) = [(1 ρ R ) r π r π ](1 ω )(γ d ) (1 η ) bπ m 1 : f X (1, 46) = [(1 ρ R ) r π r π ](ω )(γ m ) (1 η ) ŷ : f x (1, 4) = r y ŷ flex : f x (1,n x +4)= r y ŷ 1 : f X (1, 48) = (1 ρ R ) r y r y ŷ flex 1 : f X (1,n X +48)= ((1 ρ R ) r y r y ) b x 1 : f X (1, 64) = (1 ρ R ) r x ε R : f X (1, 17) = 1 2. Poenial oupu under flexible pries and wages - expanding he sae The flexible prie and wage model is parameerized under i) ξ d = ξ m = ξ mi = ξ x = ξ w =0,ii) seing he four markup as well as he fisal shoks o zero (σ ελ = σ ελ m = σ = σ ελ mi ε λ x =0, and Θ 1 0 S τ =0 5x5, respeively), and iii) we solve he sysem under he assumpion ha ˆπ pi =0.The flexprie model an be wrien where, and X +1 H f x +1 H f This implies he following soluion where x flex = " x flex i flex # H 0 1 nx = A f X x i, A f + C 0 nx n ε 0 ni n ε A 11 A 12 B 1 A 21 A 22 B 2 0 1 nx e 1 0 bπ d : e 1 (1, 1) = (1 ω )(γ d ) (1 η ), bπ m : e 1 (1, 2) = ω (γ m ) (1 η ). X flex = M flex X flex 1 + Cflex ε, x flex = F flex Xflex, = F flex M flex X flex 1 + F flex C flex ε, ε +1, (2.1) and F flex =[ Fx flex F flex i ] 0. We expand he sae of he disored eonomy wih he predeermined sae variables in he flexprie model so ha Ẋ =(X 0,X flex0 ) 0, whih implies Ẋ +1 = A 11 Ẋ + Ȧ12x + Ḃ1i + Ċε +1, 26,
Ḣx +1 = Ȧ21Ẋ + Ȧ22x + Ḃ2i, A where Ȧ11 = 11 0 nx nx A 0 nx nx M flex, Ȧ12 = 12, 0 nx nx Ȧ 21 = B A 21 0 nx nx, Ȧ22 = A 22, Ḃ 1 = 1, Ḃ 0 2 = B 2, nx ni C Ċ = C flex, and Ḣ = H. We also need o hange he arge variable marix o inlude he flexprie oupu gap X Y [4ˆπ pi, ˆπ pi, ˆπ d, (y y flex ), 4 ˆR, 4( ˆR ˆR 1 ), b x ] 0 Ḋ X flex x i. Ḋ = D X D X flex D x 0 D i, where D above has been pariioned suh ha D = D X D x 0 D i and D X flex is defined so ha ŷ flex : D X flex(4,n X +4)= F flex x,4. 3. The measuremen equaion and he definiion of he maries D 0, D, and D s Le s (X, 0 Ξ 0 1,x0,i 0 ) 0 and all s he sae of he eonomy (alhough i inludes nonpredeermined variables). The soluion o he model under opimal poliy under ommimen an be wrien in Klein form as s +1 X +1 x +1 i +1 = M 0 0 f x M 0 0 f i M 0 0 s + I f x f i Cε+1. As noed in Adolfson e al. [1, Appendix C], i an also be wrien in AIM form as s +1 = Bs + Cε +1, (3.1) where C ΦΨ and he innovaion Cε has he disribuion N(0,Q) wih Q E[ Cε +1 ( Cε +1 ) 0 ]= C C 0.Consider he sae unobservable, le Z denoe he n Z -veor of observable variables, and le he measuremen equaion be Z = D 0 + D s s + η, (3.2) where η is an n Z -veor of measuremen errors wih disribuion N(0, Σ η ) (he veor η here should no be onfused wih veor η of oeffiiens speified in (1.3)). We assume ha he marix R is diagonal wih very small elemens. 2 Equaions (3.1) and (3.2) orrespond o he sae-spae form for he derivaion of he Kalman filer in Hamilon [3]. Noe ha he marix D s is relaed o he marix D [ D X Dx Di ] (pariioned onformably wih X, x,andi )as D s =[ D X 0 D x Di ], sine s inludes he veor of predeermined variables Ξ 1. 2 The role of he measuremen errors in he esimaion of he model and he sae of he eonomy is furher disussed in Adolfson e al. [2]. 27
3.1. Differene speifiaion The veor of observable variables is π d ln(w /P ) ln C ln I b x R Ĥ... Z = 1 2 3 4 5 6 7... y ln X ln M π pi π def,i ln Y π R 8 9 10 11 12 13 14 15 and orresponds o he daa used in esimaing Ramses. Nex, we need speify how he daa orresponds o he model variables: 0 (3.3) π d = 400(π 1)π +4πˆπ d, ln(w /P ) = 100ln + b w +ˆ, ln C = 100ln + ĉ + d + m η d m h (γ m ) 1 (γ d ) 1i bγ md +ˆμ z, ĩ ln I = 100ln + bĩ + ĩ d +ĩ m η ĩ d ĩ m γ i,mi 1 i γ i,d 1 bγ mi,d ĩ ĩ + 1 k τ r k k 1 û +ˆμ, z b x = b x, R = 400(R 1)R +4R ˆR, Ĥ = Ĥ, y = 100ln + ŷ +ˆ, ln X = 100ln + h ŷ η f bγ x, + b z ln M = 100ln + m m +ĩ m ĉ + ĩm m +ĩ m bĩ m i +ˆ, m +ĩ m η (1 ω )(γ d ) (1 η ) bγ md γ i,d (1 η i ) bγ mi,d +ˆ, ĩm m +ĩ m η i(1 ω i ) π pi 400(π 1)π +4πˆπ pi +4π τ 1+τ ˆτ (3.4) = 400(π 1)π +4π(1 ω )(γ d ) (1 η ) bπ d +4πω (γ m ) (1 η ) bπ m +4π τ 1+τ ˆτ, π def,i = 400(π 1)π +4πĩ π d ˆπ d +4πĩ π m ˆπ mi ĩ +4π π d Id η I d +I m i ω i γ i,mi (1 η i ) ĩ π m Im η I d +I m i (1 ω i ) γ i,d (1 η i ) ln Y = 100ln + ŷ + b z +ˆ, π = 400(π 1)π +4πˆπ, R = 400(R 1)R +4R ˆR. bγ mi,d, Domesi inflaion π d = 400(π 1)π +4πˆπ d, 28
D 0 (1, 1) = 400(π 1)π D s 1,n X +1 = 4π Change in real wage ln(w /P ) = 100 ln + b w b w 1 +ˆ, Change in real onsumpion ln C = 100 ln +ĉ ĉ 1 + d + m η d m D 0 (2, 1) = 100 ln μ z b w : Ds 2,n X +6 =1 b w 1 : Ds (2, 50) = 1 ˆ : Ds (2, 3) = 1 d + m η d m h (γ m ) 1 (γ d ) 1i bγ md h (γ m ) 1 (γ d ) 1i bγ md 1 +ˆ, D 0 (3, 1) = 100 ln ĉ : Ds 3,n X +7 =1 ĉ 1 : Ds (3, 51) = 1 bγ md : Ds 3,n X +17 = d + m η d m h(γ m ) 1 (γ d ) 1i bγ md 1 : D s (3, 61) = d + m η d m h(γ m ) 1 (γ d ) 1i ˆ : Ds (3, 3) = 1 Change in real invesmen ĩ ln I = 100 ln +bĩ bĩ 1 + ĩ d +ĩ m η ĩ d ĩ m γ i,mi 1 i γ i,d 1 ĩ ĩ ĩ ĩ d +ĩ m η ĩ d ĩ m γ i,mi 1 i γ i,d 1 bγ mi,d 1 ĩ ĩ + 1 k τ r k k 1 ˆk 1 k τ r k k 1 b k û = ˆk b k bγ mi,d 1 τ k r k k 1 û 1 +ˆ, 29
D 0 (4, 1) = 100 ln μ z bĩ : Ds 4,n X +8 =1 bĩ 1 : Ds (4, 52) = 1 bγ mi,d : Ds 4,n X +18 = bγ mi,d 1 : Ds (4, 62) = ĩ ˆk : Ds 4,n X +13 = b k : Ds (4, 42) = 1 k τ r k k 1 ĩ ĩ d +ĩ m η ĩ d ĩ m γ i,mi 1 i γ i,d 1 ĩ ĩ ĩ d +ĩ m η ĩ d ĩ m γ i,mi 1 i γ i,d 1 ĩ ĩ 1 k τ r k k 1 û 1 : Ds (4, 67) = 1 k τ r k k 1 ˆ : Ds (4, 3) = 1 Real exhange rae b x = b x b x : D s 5,n X +20 =1 Ineres rae R = 400(R 1)R +4R ˆR, D 0 (6, 1) = 400(R 1)R ˆR : Ds 6,n X + n x + n i =4R Hours worked Ĥ = Ĥ, Ĥ : Ds 7,n X +12 =1 Change in oupu (GDP) y = 100 ln +ŷ ŷ 1 +ˆ, D 0 (8, 1) = 100 ln ŷ : Ds 8,n X +4 =1 ŷ 1 : Ds (8, 48) = 1 ˆ : Ds (8, 3) = 1 30
Change in real expors ln X = 100 ln +ŷ ŷ 1 η f bγ x, + η f bγ x, 1 + b z b z 1 +ˆ, Change in real impors ln M = 100 ln + m m +ĩ m ĉ + m D 0 (9, 1) = 100 ln ŷ : Ds (9, 31) = 1 ŷ 1 : Ds (9, 34) = 1 bγ x, : Ds 9,n X +19 = η f bγ x, 1 : Ds (9, 63) = η f b z : Ds (9, 14) = 1 b z 1 : Ds (9, 15) = 1 ˆ : Ds (9, 3) = 1 m m +ĩ m ĉ 1 1 + m +ĩ m η (1 ω )(γ d ) (1 η ) bγ md ĩm m +ĩ m η i(1 ω i ) m m +ĩ m η (1 ω )(γ d ) (1 η ) bγ md ĩm m +ĩ mb ĩ ĩm m +ĩ mb ĩ 1 γ i,d (1 η i ) bγ mi,d + ĩm m +ĩ m η i(1 ω i ) γ i,d (1 η i ) bγ mi,d 1 +ˆ, CPI inflaion D 0 (10, 1) = 100 ln ĉ : Ds 10,n X +7 = m m +ĩ m ĉ 1 : Ds (10, 51) = m m +ĩ m bγ md : D s 10,n X +17 = m m +ĩ m η (1 ω )(γ d ) (1 η ) bγ md 1 : Ds (10, 61) = m m +ĩ m η (1 ω )(γ d ) (1 η ) bĩ : Ds 10,n X +8 = ĩm m +ĩ m bĩ 1 : Ds (10, 52) = ĩm m +ĩ m bγ mi,d : Ds 10,n X +18 = ĩm m +ĩ m η i(1 ω i ) bγ mi,d 1 : Ds (10, 62) = ĩm m +ĩ m η i(1 ω i ) ˆ : Ds (10, 3) = 1 γ i,d (1 η i ) π pi = 400(π 1)π +4π(1 ω )bπ d +4πω bπ m +4π τ 1+τ ˆτ 4π τ 1+τ ˆτ 1, 31 γ i,d (1 η i )