時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 時 永 祥 三 松 野 成 悟 1 Sims( ) (Vector Auto-Regressive: VAR),, [1]-[8]. VAR, (Auto- Regressive: AR) VAR ( V
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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 時永, 祥三九州大学 : 名誉教授 松野, 成悟宇部工業高等専門学校経営情報学科 : 教授 出版情報 : 經濟學研究. 82 (1), pp.47-68, 九州大学経済学会バージョン :published 権利関係 :
2 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 時 永 祥 三 松 野 成 悟 1 Sims( ) (Vector Auto-Regressive: VAR),, [1]-[8]. VAR, (Auto- Regressive: AR) VAR ( VAR ) [9]-[13]., VAR,, [9]-[13]. VAR, 3, 1) AR [9][10], 2) AR, [11]-[13]. 3, 3) [14]-[16](,, )., VAR, ( ), AR.,., 2 VAR,.,, 1,.,,, [17]-[21]., AR,,.,.,,,. -47-
3 経済学研究 第 82 巻第 1 号.,. AR,.,. 1,,.,,,. 2 VAR 2.1 VAR,.,., VAR, [1].,., ( ), GDP,,,, 6. 4, 24. VAR,,.,,,,.,,., 2, 2 y 1,t y 2,t. y 1,t = π 10 + π (1) 11 y 1,t 1 + π (1) 12 y 1,t 2 + π (1) 21 y 2,t 1 + π (1) 22 y 2,t 2 + v 1,t (1) y 2,t = π 20 + π (2) 11 y 1,t 1 + π (2) 12 y 1,t 2 + π (2) 21 y 2,t 1 + π (2) 22 y 2,t 2 + v 2,t (2) v 1,t,v 2,t,.,,,,., 2 VAR(2). (1), (2) n, L, Y t = AX t + B + V t,y t =[y 1,t,y 2,t,..., y n,t ] T,V t =[v 1,t,v 2,t,..., v n,t ] T (3) -48-
4 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 X t =[y 1,t 1,y 1,t 2,..., y 1,t L,y 2,t 1,y 2,t 2,..., y 2,t L,..., y n,t 1,y n,t 2,..., y n,t L ] T (4) π (1) 11 π (1) 12...π (1) 1L π (1) 21 π (1) 22...π (1) 2L... π (1) n1 π (1) n2...π (1) nl A = π (2) 11 π (2) 12...π (2) 1L π (2) 21 π (2) 22...π (2) 2L... π (2) n1 π (2) n2...π (2) nl... (5) π (n) 11 π (n) 12...π (n) 1L π (n) 21 π (n) 22...π (n) 2L... π (n) n1 π (n) n2...π (n) nl B =[π 10,π 20,..., π n0 ] T (6),,. VAR,,..,. 2.2 AR VAR VAR,,,.,, AR VAR [9]-[13]. VAR, AR.,, Volcker, Burns, Greenspan [10].,, (1) Time-Varying Parameter (TVP)-VAR, (2) Markov Switching(MS)-VAR 2. 2,,,. TVP-VAR, AR, [9][10]. n,, Y t =[y 1,t,y 2,t,..., y n,t ] T. (3) (6), AR,, A(t). AR,. AR,, [9][10] , 468.AR,.,,., AR, AR,.,,. AR -49-
5 経済学研究 第 82 巻第 1 号,., AR,,. 2 MS-VAR, AR [11]-[13].,, AR,,., AR,,., a t, σt 2 h t = ln σ t.,. h t = h t 1 + I 2 η t,η t N(0,c h ) (7) a t = a t 1 + I 3 ξ t,ξ t N(0,c a ) (8), N(0,c h ),N(0,c a ), c h,a h., I 2,I 3,, ( ).,,., AR,., VAR,,.,. AR,.,,,,.,,,,,,., AR,,,,. AR,,, VAR,.,,. 2.3 : VAR, AR., VAR., AR,, -50-
6 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用., VAR., VAR,. 3., 1), AR, 2),, 3), VAR 2 AR. 1),, VAR,. 2),,.,,.,,.,.3), VAR,., VAR, AR., AR,,., VAR,. Y t,. t =1, 2,..., N: Y t t Y t : t K: Y t I k : Y t k, t t I k, Y t VAR,. θ, A,B. L(Y θ) exp[ 1 2 K k=1 t I k (Y t AX t B) T (Y t AX t B)] (9) 3 VAR 3.1 ( ), [14]-[21]., 1) [14]-[16], 2) [17]-[21] -51-
7 経済学研究 第 82 巻第 1 号. 1,.,.,,., 2,, [18]-[21].,,,., [18] (photon),,.,,.,. t =1, 2,..., N: t j =1, 2,..., J: ( J ) y j,t :j t K j :j I j,k :j k, t t I j,k λ j,k :j, k r j,t :j t 1, ; r j,t = (10) 0, otherwise; λ j =(λ j,1,..., λ j,kj ), Λ = (λ 1,..., λ J ), R t =(r 1,t,..., r J,t ) T, R =(R 1,R 2,..., R N ), θ =(R, Λ)., R t t,, R, R t,. J,. f(y θ) = J Kj j=1 k=1 t I j,k λ yj,t j,k exp( λ j,k) y j,t! (11) λ j,k,,., R =(R 1,R 2,..., R N ),, ( ),.,., j t r j,t j,. r j,t j -52-
8 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用,,., j r j,t, B. 3, r j,t j,. b i B,B = (000, 001,..., 111) (12), P 000,P 001,..., P 111, z i., z 1 = P 000,z 2 = P 001,..., z 8 = P 111 (13)., z 1 = P 000,z 2 = P 001,..., z 8 = P 111, P.,., 2 θ =(R, Λ), z i,z i (P 000,P 001,..., P 111 ) P.,,,,.,. (1),.,., R R t ( ) B b T j. R R i.,. f(r) Rt =b T f(r i Y ) (14) j, b j, S j., R t, [17],,.,,,. (2) Λ λ j,k, 2 ν, γ G(ν, γ)., λ j,k G(ν, γ)., R, Φ( Φ=(P, γ) ), j k, s j,k n j,k,. λ j,k R, γ, Y G(ν + s j,k,γ+ n j,k ) (15) (3) -53-
9 経済学研究 第 82 巻第 1 号 θ, Φ, z j α j, S j. ( ) B, α j B. (4) γ α j α j + S j (16) 2 γ,. 1 ν,. γ R, Λ G(ν K J J j K j, λ j,k ) (17) j=1 j=1 k=1,, θ, Φ,,, Λ.. j, 1 y j,t, VAR Y t., Y t, Y j,t j.,,., 4,,, (11), (9),.,. f(y R) =exp[ 1 2 J K j j=1 k=1 t I j,k (Y j,t A (j) X j,t B (j) ) T (Y j,t A (j) X j,t B (j) )] (18), θ =(R, Λ), Λ VAR, R., (18) A, B(AR ), j,, (j). A (j),b (j). 3.2 VAR, Y j,t., R, P f(y R),f(R Φ),,., Φ=(P, γ). f(r Y )= f(r, Φ Y ) f(y R)f(R Φ)f(Φ)dΦ (19) -54-
10 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用,, Gibbs sampling,. (19), R,. R,, ( ) r j,t,.,,., Gibbs sampling. Gibbs sampling,,.,,., Gibbs sampling.,,, (19),,.,.,. Gibbs sampling,.,,. (1) R, P. (2), R,.,, Λ,., R, P., f(r) Ri =b j f(r i Y ), b j S j., α j α j + S j, P α j. (3), (18).,,.,. -55-
11 経済学研究 第 82 巻第 1 号 VAR,, VAR.,,,.,. (1),.,.,., J j, j, K j ( ),.,. (2), (3) (6) A, B, (3) v j,t, VAR.,, VAR,, AR., 2, (v j,t ). (3),,., 1), 2), 3) VAR, 2.,,.,,,., AR,,.,. : N = 240 : J =2 : K 1 =4,K 2 =2 α j = (100, 2, 3, 2), (3) (6) AR A, B v t,.,,,. -56-
12 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 1: ( : y 1,t, : f(r j,t = 1)) 1, VAR y 1,t ( ).2 (No.1, No.2) 1 ( 1 y 1,t ). 1,. j, t (r j,t =1 ) f(r j,t = 1).,. 2 P,, P 00,P 01,P 10,P 11,. 3,., K j = N t=1 r j,t, K.K j K j = m, m =1, 2,... f(k j = m)..,,. 4.2,,.,.,,. 1, -57-
13 経済学研究 第 82 巻第 1 号 2: P 00,P 01,P 10,P 11.,,,., VAR.,,, GDP (CPI), VAR.,,, VAR., GDP, CPI,,, 1. 4,, GDP, CPI GDP (%) CPI (%) Rate federal fund rate, Unemploy Monetary St. Louis Adjusted Monetary Base(billion dollar), ( ) -58-
14 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 3: K 1,K 2,. j = 1, j =2. 4, 5 ( ) GDP,,. 4, 5 (GDP),. j, t (r j,t =1 ) f(r j,t = 1). 6,., K j = N t=1 r j,t, K.K j K j = m, m =1, 2,... f(k j = m)..,, 3, 2.,, 6, 2 5,, 3., GDP CPI. GDP, CPI, 7,.,, GDP CPI,,., CPI ( ), 10, GDP, 10.,, 2008,., GDP, CPI, , VAR,., VAR, 4-59-
15 経済学研究 第 82 巻第 1 号 4: GDP ( : y 1,t, : f(r 1,t = 1)),,. VAR,,,,., GDP CPI,,.,, GDP CPI,. 10, 10 ( ) ( ).,, 8, 9 4., U1, U2, U3, U4 4,.,, J1, J2, J3, J4., 2. (1), GDP CPI -60-
16 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 5: GDP ( : y 1,t, : f(r 2,t = 1)).4, 4 U4,.,.,,,., 8, GDP CPI, U4, (U1, U2, U3).,, 10 ( )., 3 4, GDP CPI.,, GDP CPI. (2),,.,,.,, -61-
17 経済学研究 第 82 巻第 1 号 6: K 1,K 2 7: GDP, CPI ( :, : ).,., J4,,,,. 9, J4,,.,,,,, GDP,.,. 4.3, GDP.,,.,. -62-
18 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 8: ( GDP ) 9: ( CPI ),., 3 [22]-[26]. (1), 2,,.,., GDP 2000.,,.,,,. (2),., 1), 2),, 3),.,,, -63-
19 経済学研究 第 82 巻第 1 号.,,. (3),,,,, , 19.4% 10.7%., 74% 83.5%.,,,.,,.,,.,,,,,,.,, GDP. 5 VAR,. VAR, 1,. VAR,,..,,,., VAR,., (B) [1] C. A. Sims, Econometric and reality, Econometrica, vol.48, pp.1 48, [2] T. Amemiya, Multivariate regression and simultaneous equation models when dependent variables are truncated normal, Econometrica, vol.42, pp ,
20 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 [3] R. B. Litterman, Forecasting with Bayesian vector autoregressions-five years of experience, Journal of Business, and Economic Statistics, vol.4, pp.25 38, [4] C. W. J. Granger, Investigating causal relations by econometric models and cross-spectral methods, Econometrica, vol.37, pp , [5] A. S. Dexter, D. Levi and B. R. Nault, Sticky prices : The impact of regulation, Journal of Monetary Economics, vol.49, pp , [6] W. Oh and K. Lee, Causal relationship between energy consumption and GDP revisited : The case of Korea , Energy Economics, vol.26, pp.51 59, [7],,,, 229, [8],, VAR,, vol.81, no.1, pp.21 43, [9] C. A. Sims and T. Zha, Bayesian methods for dynamic multivariate models, International Economic Review, vol.39, no.4, pp , [10] C. A. Sims and T. Zha, Where there regime switches in U. S. monetary policy?, The American Economic Review, vol.96, no.1, pp.54 81, [11] G. Koop, R. Leon-Gonzalet and R. W. Strachan, On the evolution of the monetary policy transmission mechanism, Journal of Economic Dynamics & Control, vol.33, pp , [12] G. E. Primiceri, Time varying structural vector autoregressions and monetary policy, Review of Economic Studies, vol.72, pp , [13] Y. Liu and J. Morley, Structural evolution of the postwar U. S. economy, Journal of Economic Dynamics & Control, vol.42, pp.50 68, [14] A. Andreiu, A. Doucet, S. S. Singh and B. Tadic, Particle methods for change detection, system identification and control, Proc. IEEE, vol.93, pp , [15] F. Caron, M. Davy, A. Doucet, E. Duflos and P. Vanheeghe, Bayesian inference for linear dynamic models with Dirichlet process mixtures, IEEE Transactions on Signal Processing, vol.56, no.1, pp.71 84, [16] E. Fox, E. B. Sudderth, M. I. Jordan and A. S. Willsky Bayesian nonparametric inference of switching dynamic linear models, IEEE Transactions on Signal Processing, vol.59, no.4, pp , [17] E. Punskaya, C. Andrieu, A. Doucet and W. J. Fizgerald, Bayesian curve fitting using MCMC with applications to signal segmentation, IEEE Transactions on Signal Processing, vol.50, no.3, pp ,
21 経済学研究 第 82 巻第 1 号 [18] N. Dobigeon, J. Y. Tourneret and J. D. Scargle, Joint segmentation of multivariate astronomical time series: Bayesian sampling with a hierarchical model, IEEE Transactions on Signal Processing, vol.55, no.2, pp , [19], 3 edge snapping,,, vol.6, no.2, pp.36 52, [20],,,, vol.j97-a, no.7, pp , [21],,,, vol.81, no.4, pp , [22] The White House, Fact sheet: The president s plan to make America a magnet for jobs by investing in manufacturing, [23] Economist, Special report: Manufacturing and innovation, April 21, 2012, and Special report: Outsourcing and Offshoring, January 19, [24] J. Hatzius, The US manufacturing renascence: Fact of fiction, The Goldman Sachs Group Inc., US Economics Analyst, no.13/12, March 22, [25] G. P. Pisano and W. C. Shih, Does America really need manufacturing?, Harvard Business Review, vol.90, no.3, pp , [26] U. S. Department of Labor, U. S. Bureau of Labor Statistics, Employment Outlook , Monthly Labor Review,
22 時系列区分化手法による時間変化多変量ベクトル自己回帰モデルの推定と金融政策ショック分析への応用 A. VAR ( ) VAR 1, ( ).,, 1,., AR MA,., (1), (2) VAR.,, (1),, y t., v t =(v 1t,v 2t,..., v Mt ),, Σ v = E[v t vt T ].,., MA. y t = μ + v t + M 1 v t (20), μ Θ i. μ = E[y t ]=(I Θ 1 Θ 2... Θ p ) 1 ν (21) M i kj, i j k.,. VAR(p), P Σ v P T = I.,. y t = μ + Φ i w t i, Φ i = M i P 1,w t =(w 1t,w 2t,..., w Mt ) T = Pv t (22) i=0 w t, 1,., Φ i 1, w t y t.,, (, ),. h. (22), h,. Σ(h) =Σ v + M 1 Σ v M1 T M h 1 Σ v Mh 1 T (23), P Σ v P T = I,. Σ(h) =Φ 0 Φ T 0 +Φ 1 Φ T Φ h 1 Φ T h 1 (24), mb m y m,., -67-
23 経済学研究 第 82 巻第 1 号 h,, ( )., 1. B. ( ), (Dhirichle Process: DP). DP.,., 1, 2,..., K z 1,z 2,..., z K.. P (Z) = 1 B(α) ΠK i=1z αi 1 i, K z i = 1 (25) B(α) = ΠK i=1 Γ(α i) Γ( K i=1 α (26) i) α i, Γ(.).,. i=1 α i α i + m i (27) m i z i ( i).,. (13) z i = P 000,P 001,..., P 111 (, (13) 3 ), z i,. m i, P 000,P 001,..., P 111,. 時永祥三 九州大学名誉教授 松野 成悟 宇部工業高等専門学校経営情報学科教授 -68-
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