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 https://doi.org/10.15017/1522395 出版情報 : 經濟學研究. 82 (1), pp.47-68, 2015-06-30. 九州大学経済学会バージョン :published 権利関係 :
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-
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-
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]. 6 13 468., 468.AR,.,,., AR, AR,.,,. AR -49-
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-
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-
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-
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-
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-
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-
経済学研究 第 82 巻第 1 号 4 4.1 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-
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-
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,, 1970 1 2014 4. 1. GDP, CPI GDP (%) CPI (%) Rate federal fund rate, Unemploy Monetary St. Louis Adjusted Monetary Base(billion dollar), ( ) -58-
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, 10000. GDP, 10.,, 2008,., GDP, CPI,. 1970 2014, VAR,., VAR, 4-59-
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-
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-
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-
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-
経済学研究 第 82 巻第 1 号.,,. (3),,,,,.1990 2012, 19.4% 10.7%., 74% 83.5%.,,,.,,.,,.,,,,,,.,, GDP. 5 VAR,. VAR, 1,. VAR,,..,,,., VAR,., (B)23310104.. [1] C. A. Sims, Econometric and reality, Econometrica, vol.48, pp.1 48, 1980. [2] T. Amemiya, Multivariate regression and simultaneous equation models when dependent variables are truncated normal, Econometrica, vol.42, pp.999 1012, 1974. -64-
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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.414 423, 2007. [19], 3 edge snapping,,, vol.6, no.2, pp.36 52, 2013. [20],,,, vol.j97-a, no.7, pp.503 518, 2014. [21],,,, vol.81, no.4, pp.99 122, 2015. [22] The White House, Fact sheet: The president s plan to make America a magnet for jobs by investing in manufacturing, 2011. [23] Economist, Special report: Manufacturing and innovation, April 21, 2012, and Special report: Outsourcing and Offshoring, January 19, 2013. [24] J. Hatzius, The US manufacturing renascence: Fact of fiction, The Goldman Sachs Group Inc., US Economics Analyst, no.13/12, March 22, 2013. [25] G. P. Pisano and W. C. Shih, Does America really need manufacturing?, Harvard Business Review, vol.90, no.3, pp.94 102, 2012. [26] U. S. Department of Labor, U. S. Bureau of Labor Statistics, Employment Outlook 2010-2020, Monthly Labor Review, 2012. -66-
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 1 +... (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 1 +... +Φ h 1 Φ T h 1 (24), mb m y m,., -67-
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-