7 DSGE 2013 3 7 1 118 1.1............................ 118 1.2................................... 123 1.3.............................. 125 1.4..................... 127 1.5...................... 128 1.6.............. 130 2 MCMC 131 2.1.............................. 132 2.2............................... 132 2.3 M-H......... 134 2.4 MCMC.......................... 136 3 DSGE 137 3.1.............................. 138 3.2............................ 138 3.3 DSGE.............. 140 4 142 4.1......................... 142 4.2............................. 145 117
DSGE Dynamic Stochastic General Equilibrium Model DSGE 5 2 DSGE DSGE ω 0 < ω < 1 1 DSGE Blanchard and Kahn VAR 3 MCMC 2 5 4 1 1 1.1 1. 2. 118
3. 4. 5. A P (A) [0, 1] A P (A) P (A) = A A A A P (A ) P (A ) = 365 A A 1 = 1 = 0 P (A ) 1 0 [0, 1] 0.3 (1) P (A) A P (1) P (B A) = P (A B) P (A) (2) 119
(2) P (A B) A B A P (A) P (B A) A B A B A {,,, } B {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. P ({ } {1}) 1/52 P ({1} { }) 1/13 (2) P ({1} { }) = P ({ } {1}) P ({ }) = 1 52 1 4 = 1 13 (3) (2) A B P (A B) = P (A B) P (B) (4) P (A B) = P (B A)P (A) (2) P (A B) = P (B A)P (A) P (B) (5) (5) 1.4 *1 ω ω 0 < ω < 1 ω = 0, 1 *1 1 120
ω 0 1 0 1 4. E P 1 ω 1 2 ω 2 Ω = {ω 1, ω 2,..., ω 52 } 1 E 1 = {ω 1 } 1 1 1 1 E 2 = {ω 1, ω 14, ω 27, ω 40 } E 3 = {ω 1, ω 2,..., ω 13 } 2 P P P (Ω) = 1 1/52 P (E 1 ) = 1/52 P (E 2 ) = 4/52 = 1/13 P (E 3 ) = 13/52 = 1/4 1 121
"! #! $! %! & ' ( ) * +, ++ +-! +.! / "!/ #! 0!! / &#! 1 Ω ω i! )*+,-./01 23!"#$%& 4 (1!"#$%&'!"#$%& ' ( 2 2 2 Ω F *2 Ω F (Ω, F, P ) P P : F [0, 1] (Ω, F, P ) X Ω R X : Ω R P P (E) X X(ω) 1 y a *2 122
ay 2ay 3ay 4ay 4. 5. 1.4 1.2 p(x) X(ω) X(ω) z X(ω) z E X z P (E X z ) = z p(x)dx (6) *3 R B P 1 3 *4 *3 *4 P 0 θ 1 P (x = k; θ) = θ k (1 θ) 1 k, k 0, 1 (7) 0 θ 1 P (x = k; θ) = n! = n i=1 i n! k!(n k)! θk (1 θ) n k, k = 0, 1, 2,..., n (8) 123
p(x) N(µ, σ 2 ) < X < Ga(a, s) X 0 Be(a, b) 0 X 1 Unif(a, b) a X b, a, b R 1 2πσ 2 exp ( (x µ)2 1 σ 2 ) µ σ 2 s a Γ(a) xa 1 exp( x ) as as2 s Γ(a+b) Γ(a)Γ(b) xa 1 (1 x) b 1 a ab a+b 1 b a b a 2 (a+b) 2 (a+b+1) (b a) 2 12 1 Γ( ) 0.0 0.1 0.2 0.3 0.4 N(µ = 0, σ 2 = 1 ) 0.0 0.2 0.4 0.6 0.8 1.0 Ga(s = 1, a = 1 ) 0.0 0.5 1.0 1.5 4 2 0 2 4 Be(a = 2, b = 2 ) 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Unif(a = 1, b = 1 ) 1.0 0.5 0.0 0.5 1.0 3 p(x) X E(X) = xp(x)dx (9) µ Var(X) = (x µ) 2 p(x)dx (10) X σ 2 P ({ X µ kσ}) 1 k 2, k > 1 (11) 124
1.3 p(a, b) A, B A B p(b a) p(a b) = p(a, b) p(a) (12) p(a) A p(a) = p(a, b)db p(a b) = p(b a)p(a) p(b) (13) *5 (13) *5 p(a) p(b) p A (a) p B (b) p(a b) p(b a) 125
1. (13) p(a b) a y b θ p y θ (y θ) 2. p θ (θ) (13) p(b) 3. y obs p y θ (y θ) y p y θ (y θ) l(θ; y obs ) 4. p θ y (θ y obs ) p θ y (θ y obs ) = l(θ; yobs )p θ (θ) p y (y obs ) (14) p y (y obs ) θ p y (y obs ) = l(θ; y obs )p(θ)dθ θ p θ y (θ y obs ) l(θ; y obs )p(θ) (15) 5. p θ y (θ y obs ) 5. θ θp θ y(θ y obs )dθ MCMC 2 p y (y obs ) y obs 1.5 *6 θ < θ < 0 < θ < 0 θ 1 a θ b *6 126
*7 DSGE *8 2 p y θ (y θ) 1.4 DSGE 3 DSGE 3 1.4 *9 N 1 0 0 1 k = {k i } N i=1, k i {0, 1} *7 *8 Jeffreys Jeffreys improper ( ) *9 127
θ N f(θ; k) = θ k i (1 θ) 1 k i (16) i=1 θ [0, 1] Be(a, b) π(θ) θ a 1 (1 θ) b 1 (17) a, b * 10 π(θ k) θ K+a 1 (1 θ) N K+b 1, K = k i (18) θ Be(K + a, N K + b) 2.1 N 4 θ 1 θ = 0.6 a, b a = 2, b = 2 4 0.5 N = 5, 20, 100, 500 N N N θ N θ 2 MCMC 1.5 p y (y obs ) y obs m *10 Be(a, b) a a+b ab (a+b) 2 (a+b+1) 128
N = 5 N = 20 0 2 4 6 8 10 0 1 2 3 4 5 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 N = 100 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 N = 1000 0.0 0.2 0.4 0.6 0.8 1.0 4 θ M 1, M 2,..., M m M i θ i (14) M i M i θ i p(θ i y, M i ) = p(y θ i, M i )p(θ i M i ) p(y M i ) (19) * 11 M i y (13) a y b M i M i p(m i y) = p(y M i)p(m i ) p(y) (21) p(y M i ) (19) M i p(m i ) *11 p(b a, c) = p(a b, c)p(b c) p(a c) (20) 129
* 12 p(y) p(y) = m p(y M i )p(m i ) (22) i=1 i j p(m i y) p(m j y) p(m i y) p(m j y) = p(y M i)p(m i ) p(y M j )p(m j ) (23) *12 p(m i ) = p(m j ) = 1/m p(m i y) p(m j y) = p(y M i) p(y M j ) (24) p(y M i )/p(y M j ) * 13 p(y M i ) Bayesian Econometrics Koop[2003] 1.6 * 14 GMM AIC BIC * 15 *12 p(m i ) = 1/m *13 2 ln (p(y M i )) BIC *14 *15 MCMC Geweke 2.4 130
* 16 MCMC * 17 2 MCMC Markov Chain Monte Carlo MCMC 1 MCMC MCMC 2 DSGE MCMC *16 *17 DSGE-VAR 131
M-H 2.1 * 18 [0, 1] * 19 R 2.2 1.4 1 0 i k i (k i {0, 1}) k i {k 1, k 2,... } W t {, } 0.9 0.1 0.5 0.5 {W 1, W 2,... } [ ] 0.9 0.5 Q = 0.1 0.5 (25) π + π = 1 [ ] [ ] [ ] 0.9 0.5 π π = 0.1 0.5 π π (26) π = 5/6, π = 1/6 *18 *19 M-H 132
a t 4 2 0 2 4 0 50 100 150 200 5 AR(1) AR(1) ε t σ 2 1 < ρ < 1 a t a 0 = ā a t+1 = ρa t + ε t+1 (27) a 0 = ā T a t, t T (27) AR(1) a t ( σ 2 ) N 0, 1 ρ 2 q(a t, a t+1 ) N(ρa t, σ 2 ) MCMC 5 σ 2 = 1, ρ = 0.5 0, 100, 100 3 a t N(0, 4/3) (28) 133
2.3 M-H MCMC AR(1) MCMC MCMC MCMC 1 M-H Metropolis-Hastings * 20 M-H q f(θ) * 21 M-H θ (0) i i 1 step 1. θ q(θ (i 1), θ) step 2. { } α(θ (i 1), θ) f( θ)q( θ, θ (i 1) ) = min f(θ (i 1) )q(θ (i 1), θ), 1 (29) step 3. α(θ (i 1), θ) θ θ (i) = θ θ θ (i) = θ (i 1) f(θ) M-H q(ϕ, θ) = q(θ, ϕ) q *20 MCMC M-H *21 q step 1. M-H 134
α(θ (i 1), θ) { } α(θ (i 1), θ) f( θ) = min f(θ (i 1) ), 1 (30) random walk chain * 22 θ M-H θ 0.2 0.5 * 23 q random walk f(θ) θ θ (0) i i 1 step 1. θ θ (i 1) + ϵ, ϵ N(0, σ 2 ) step 2. { } α(θ (i 1), θ) f( θ) = min f(θ (i 1) ), 1 (31) step 3. α(θ (i 1), θ) θ θ (i) = θ θ θ (i) = θ (i 1) M-H Ga(s = 5, a = 1) * 24 σ 2 0.1, 20, 5000 3 1,200 200 1,000 6 *24 Ga(s = 5, r = 1) *22 Metropolis M-H *23 Bayesian Econometrics Koop[2003] 98 *24 135
step 3. 0.944, 0.492, 0.045 6 3 σ 2 = 20 Ga(s = 5, r = 1) σ 2 f Ga(s, r) s/r 2 σ 2 = 5 σ 2 σ σ σ 6 θ step 1. MN(0, Σ) 3.3 DSGE 2.4 MCMC MCMC 136
6 5 Geweke 10% 50% θ A, θ B Z = θ A θ B Sθ A(0)/N A + Sθ B(0)/N B (32) 1 * 25 t Z 2 * 26 Z 2 10% 50% 3 DSGE 2 DSGE VAR η t [ˆxt+1 ŝ t+1 ] ] [ˆxt = D + Rη t+1 (33) ŝ t MCMC 3.2 *25 x t, t = 1, 2,..., N x t Var(x t )/N Var(x t ) x t MCMC x t x t x t S x (0)/N S x (0) x t 0 *26 137
3.1 t = 1, 2,... y t = Zα t + ϵ t, ϵ t MN m (0, H), (34) α t+1 = Dα t + Rη t+1, η t MN r (0, Q), (35) α 1 MN r (a 1, P 1 ) (36) * 27 (34) y t m α t r Z (m, r) y t α t (35) D, R (r, r) * 28 ϵ t, η t α 1 y t, α t 3.2 ϑ = {Z, H, D, R, Q, a 1, P 1 } Y T = {y i } T i=1 p(y T ) = p(y 1, y 2,..., y T ) = p(y T Y T 1 )p(y T 1 Y T 2 ) p(y 2 y 1 )p(y 1 ) (37) ϑ p(y T ) p y ϑ (y ϑ) p(y t Y t 1 ) E(y t Y t 1 ) Var(y t Y t 1 ) Y T = Y obs T l(ϑ; YT obs) * 29 2 *30 *27 *28 (36) *36 α 1 α 1 a 1, P 1 *29 138
a t = E(α t Y t 1 ), P t = Var(α t Y t 1 ) a t+1 = E(α t+1 Y t ) = DE(α t Y t ) (38) P t+1 = Var(α t+1 Y t ) = DVar(α t Y t )D + RQR (39) ν t = y t E(y t Y t 1 ) = y t Za t (40) F t = Var(y t Y t 1 ) = Var(ν t Y t 1 ) (41) M t = Cov(α t, ν t Y t 1 ) (42) M t α t ν t [ ] ([ ] [ ]) αt at Pt M Y t 1 MN r+m, t 0 ν t M t F t {ν t, Y t 1 } Y t (43) E(α t Y t ) = E(α t ν t, Y t 1 ) = a t + M t F 1 t ν t (44) Var(α t Y t ) = Var(α t ν t, Y t 1 ) = P t M t F 1 t M t (45) M t = E ( (α t a t )νt ) Y t 1 = E ( (α t a t )(y t Za t ) ) Y t 1 = E ( (α t a t )(Zα t + ϵ t Za t ) ) Y t 1 = E ( (α t a t )(α t a t ) Z ) Y t 1 = P t Z F t = Var(Zα t + ϵ t Za t Y t 1 ) = Var(Zα t + ϵ t Y t 1 ) = ZP t Z + H (46) (47) α t ϵ t a t, P t t = 1, 2,... T a 1, P 1 ν t = y t Za t F t = ZP t Z + H K t = DP t Z Ft 1 L t = D K t Z a t+1 = Da t + K t ν t (48) (49) (50) (51) (52) P t+1 = DP t L t + RQR (53) 139
E(y t Y t 1 ) = Za t, Var(y t Y t 1 ) = F t = ZP t Z + H Y T = YT obs ϑ l(ϑ; YT obs) k y MN(µ, Σ) y, µ, Σ y = µ = Σ = [ y(1) y (2) ] [ µ(1) µ (2) ] [ ] Σ(11) Σ (12) Σ (21) Σ (22) (54) (55) (56) y (1) k 1 y (2) k 2 k = k 1 + k 2 µ, Σ y (2) = y (2) y (1) MN(µ (1 2), Σ (1 2) ) µ (1 2) = µ (1) + Σ (12) Σ 1 (22) (y (2) µ (2)) (57) Σ (1 2) = Σ (11) Σ (12) Σ 1 (22) Σ (12) (58) * 30 3.3 DSGE DSGE DSGE VAR Y T = Y obs T ] [ˆxt α t = ŝ t Z H (33) D Q R θ H(θ), D(θ), Q(θ), R(θ) (59) *30 MN(µ, Σ) p(y) exp [ 1 2 (y µ) Σ(y µ) ] 140
2.3 θ p θ (θ) y t = Zα t + ϵ t, ϵ t MN m (0, H(θ)), α t+1 = D(θ)α t + R(θ)η t+1, η t MN r (0, Q(θ)), α 1 MN r (a 1, P 1 ) (60) l(θ; Y T ) (31) f(θ) l(θ; Y T )p θ (θ) θ p θ y (θ Y T ) l(θ; Y T )p θ (θ) θ DSGE θ (0) i i 1 step 1. θ MN(θ (i 1), Σ) step 2. Blanchard and Kahn D(θ), R(θ) step 3. l( θ; YT obs) step 4. p θ ( θ) [ ] l( p = min θ;y)p θ ( θ) l(θ (i 1) ;y)p θ (θ (i 1) ), 1 step 5. p θ θ (i) = θ θ θ (i) = θ (i 1) step. 1 Σ 2.4 Blanchard and Kahn R(θ) Sims R θ 141
4 5 1980 1999 4.1 6 π t = βπ t+1 + κˆx t (61) ˆx t = ˆx t+1 (î t π t+1 ) + ν t (62) î t = ϕ π π t + ϕ y ˆx t + v t (63) v t+1 = ρ v v t + u t+1 (64) â t+1 = ρ A â t + ε t+1 (65) ν t = â t+1 â t (66) * 31 (65) (66) ν t = â t+1 â t = (ρ A 1)â t + ε t+1 t t + 1 (62) ν t (ρ A 1)â t ϕ π > 1, ϕ y > 0 ϕ π ϕ π = ϕ π 1 β β = 0.99 5 π t = βπ t+1 + κˆx t (67) ˆx t = ˆx t+1 (î t π t+1 ) + (ρ A 1)â t (68) î t = (1 + ϕ π )π t + ϕ y ˆx t + v t (69) v t+1 = ρ v v t + u t+1 (70) â t+1 = ρ A â t + ε t+1 (71) u t, ε t σ u, σ ε θ = *31 κ = (1 ω)(1 ωβ)(γ+1) ω 142
{γ, ω, ϕ π, ϕ y, ρ A, ρ v, σ u, σ ε } * 32 ˆx t π t α t = î t v t, η t = â t [ ut ε t 0 0 ] 0 0, R = 0 0 1 0 0 1 (72) θ Blanchard and Kahn α t+1 = D(θ )α t + Rη t+1, η t MN(0, Q(θ )) (73) GDP GDP 3 7 x obs t, πt obs, i obs t 1980 2 1999 1 76 * 33 x obs t x obs = ˆx t (74) π obs t π obs = π t + ϵ π,t (75) (i obs t ī obs )/4 = î t * 34 x obs, π obs, ī obs ϵ π,t 3 3 1 ϵ π,t θ = {γ, ω, ϕ π, ϕ y, ρ A, ρ v, σ u, σ ε, σ ϵπ } (76) 1/4 (76) *32 *33 OECD Economic Outlook No. 90 *34 (60) Z Z = 1 0 0 0 0 0 1 0 0 0 (77) 0 0 1 0 0 143
7 θ 2 * 35 γ ϕ π, ϕ y ω 0 < ω < 1 *35 µ η 144
γ 1.0 0.5 ω 0.8 0.1 ϕ π 0.5 0.25 ϕ y 0.5 0.25 GDP ρ A 0.8 0.05 AR(1) ρ v 0.8 0.1 AR(1) σ ε 0.5 0.5 ε t σ u 0.5 0.5 u t σ ϵπ 0.5 0.5 ϵ π,t 2 AR(1) 1 σ ε, σ u, σ ϵπ * 36 4.2 3.3 2 125,000 25,000 200,000 M-H 0.304, 0.306 Geweke 3 Z 1, Z 2 1 2 Geweke 1 Z 2 Z γ *36 Dynare 4.3.1 Dynare Octave MATLAB DSGE Dynare Blanchard and Kahn (73) Dynare VAR 145
Z 1 Z 2 γ 2.271 0.511 2.509 1.004 ω 0.348 0.091 0.222 0.369 ϕ π 0.430 0.037 0.195 0.834 ϕ y 0.875 0.455 3.268 0.043 ρ A 0.926 0.013 3.648 0.344 ρ v 0.394 0.191 0.776 0.391 σ u 0.229 0.095 0.679 0.452 σ ε 0.958 0.010 0.411-0.381 σ ϵπ 0.879 0.016 0.186 0.415 3 Geweke ω ϕ π GDP ϕ y GDP AR(1) ρ A ρ v (73) t 1 α t = D t 1 α 1 + D t i 1 Rη i+1 (78) i=1 θ α t α 1 η t y t = Zα t + ϵ t y t α 1 η t ϵ t (74) (76) 8 * 37 GDP 90 97 GDP GDP *37 θ (i) 146
6 4 2 $%& '()*+,"-./+,"- GDP!"# GDP!"# % 0 2 4 6 1980 81 82 83 84 85 86 87 88 89 1990 91 92 93 94 95 96 97 98 99+ % % 3 2.5 2 1.5 1 0.5 0 0.5 1 GDP!"#!"!#$%&'()* +%, -./0 12345678!"#$% GDP&'()*) 1.5 198081 82 83 84 85 86 87 88 89 1990 91 92 93 94 95 96 97 98 99 2.5 2 1.5 1 0.5 0 0.5 1!"# $!% "&'()*+, -.)*+,!"# 1.5 1980 81 82 83 84 85 86 87 88 89 1990 91 92 93 94 95 96 97 98 99$ 8 80 147