, 1), 2) (Markov-Switching Vector Autoregression, MSVAR), 3) 3, ,, , TOPIX, , explosive. 2,.,,,.,, 1
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2 , 1), 2) (Markov-Switching Vector Autoregression, MSVAR), 3) 3, ,, , TOPIX, , explosive. 2,.,,,.,, 1986 Q Q2,,. :, explosive, recursiveadf, MSVAR, 4
3 1 1.1,, , 1), 2) (Markov-Switching Vector Autoregression, MSVAR), 3) ,., 1985.,.,,,,.,, ).,,.,, ( 1 ). 90,,,.,.,,,,.,,., 1987, , 2),.,, 2000,,,, 3., Phillips et al.(2012), AR(1). AR(1),.. x t = µ x + δx t 1 + ε t,x (1.1) δ > 1,, explosivear(1)., 0 < δ < 1, AR(1), δ = 1,.,,,.,, explosive. 1) ), 2
4 1.2, (2009) Phillips et al.(2012), recursiveadf, 225 TOPIX : 225( ), TOPIX( ), ( /m 2 ). 1.2,, FRB view BIS view 2 3). FRB view.,,.,,.,,., BIS View,..,,, 3) (2008) 3
5 1.2.,,.,.,., 1), 2), 3).,,,.,,MSVAR,.,,.,,.,,.. 目的 手段 バブル期間の推定 単位根検定 バブル構造変化の推定 MSVAR バブル期間のみ株価に影響を与える変数の存在 インパルス応答関数 2:.,,,,.,.. 2,,, MSVAR. 3, 4
6 recursive ADF. 4, MSVAR,,. 5,., 6. 2, MSVAR.,. (2007),,,,,. (2000),,,, 4. (2000),,,,.,,,., 5,., (2012, 2013),, 2,,,.,, (2008) (Dynamic General Equilibrium, DGE),.,,., ,. MSVAR, Hamilton(1989)., Krolzig(1998) Ox MSVAR.,. Fujiwara(2004) MSVAR,,,,., Inoue and Okimoto(2007) MSVAR,
7 3, recursiveadf. recursiveadf,,,, MSVAR. 3.1, (2009) Phillips et al.(2012),., t D t, t P t, t E t., R( ). P t = R E t(p t+1 + D t+1 ) (3.1) Campbell and Shiller(1989),. p t = p f t + b t (3.2) p f t = κ γ 1 ρ + (1 ρ) ρ i E t p t+i (3.3) i=0 b t = lim i ρ i E t p t+i (3.4) E t (b t+1 ) = 1 ρ b t = (1 + exp(d p)b t ) (3.5) κ = log(ρ) (1 ρ) log( 1 1) (3.6) ρ, p t = log(p t ), d t = log(d t ), γ = log(1 + R)ρ = 1/(1 + exp(d p)), d p = E[log(D t /P t )]. p f t,,. b t,,. b t = 1 ρ b t 1 + ε b,t (1 + g)b t 1 + ε b,t, E t 1 (ε b,t ) = 0 (3.7) g = 1 ρ 1 = exp(d p) > 0,., ε b,t, b t.,(2) b t 0., (3.2), b t = 0,, p t = p f t = µ + (1 ρ) ρ i E t (d t+1+i ), 0 < ρ < 1 (3.8) i=0 6
8 3.2 ADF, d t p t., d t p t. d t p t = κ γ 1 ρ ρ i E t ( d t+1+i ) (3.9), d t I(0), p t d t (1, 1)., = 1 L, L., b t 0,, (3.7) b t explosive, (3.2), p t explosive., b t p t., d t I(1) I(0), p t explosive,., p t d t.,, p t explosive 4).,, explosive. i=0 3.2 ADF,, Augumented Dickey-Fuller(ADF). t x t,. x t = µ x + δx t + J ϕ j x t j + ε x,t (3.10) j=1, J. I(1), δ = 1, I(0), δ < 1, explosivear, δ > 1. δ t ADF. ADF = t δ=1 = ˆδ 1 ˆσ δ (3.11). ˆσ δ, ˆδ. H 0, δ 1, H 1, δ > 1 5). 3.3 sup ADF, ADF, ADF Evans(1991).,, sup ADF. sup ADF. sup ADF = sup ADF [nr] (3.12) r [r 0,1] 4) d t I(1),. (2009) TOPIX, I(1). 5), δ = 1, δ < 1,. 7
9 3.4 recursiveadf, [ ], ADF [nr], 1 [nr] ADF., r 0., n = 500, r 0 = 0.1, sup ADF [nr] = max{adf 50, ADF 51, ADF 52,..., ADF 500 } (3.13) r [r 0,1] [nr 0 ] = 50, ADF 50 1, ADF,. ADF,,.,, (explosive),,, I(1)., sup ADF,,., ADF, sup ADF,. 3.4 recursiveadf, recursiveadf. n 0 n 1 ADF ADF (n 0, n 1 )., 1 n 0 < n 1 n. n. inf ADF (r) = sup ADF (r) = min ADF (1, k) (3.14) k [nr],...,n max ADF (1, k) (3.15) k [nr],...,n rangeadf (r) = sup ADF (r) inf ADF (r) (3.16) inf ADF, sup ADF., rangeadf, sup ADF inf ADF,. (2009),, sup ADF, ( )., TOPIX( ), ,TOPIX, 2,. (2009) Phillips et al(2012), recursiveadf, k [nr] + 1 k ADF rollingadf.,,, recursiveadf 6). 6) Phillips et al(2012), recursiveadf,,. 8
10 4 MSVAR, MSVAR 7). recursiceadf,, MSVAR,, K MSVAR(p). y t = Φ (1) 1 (s t)y t 1 + Φ (2) 2 (s t)y t Φ (p) p (s t )y t p + ε t (4.1) s t = j j = 1,, K (4.2) y t,, s t 1 K. s t, s t s t., i j p ij, P (s t = j s t 1 = i, s t 2 = k,...) = P (s t = j s t 1 = i) = p ij., E(ε t ε t) = Ω(s t )., K P. p 11 p 21 p M1 p 12 p 22 p M2 P = p 1M p 2M p MM M p ij = 1, i, j (1, M) (4.3) P (j, i), p ij. (2, 1), 1 2., 3 3 MSVAR(2). j=1 y 1,t y 2,t = + ϕ (1) 11 ϕ (1) 12 ϕ (1) 13 ϕ (1) 21 ϕ (1) 22 ϕ (1) 23 (s t ) y 3,t ϕ (1) 31 ϕ (1) 32 ϕ (1) 33 ϕ (2) 11 ϕ (2) 12 ϕ (2) 13 ϕ (2) 21 ϕ (2) 22 ϕ (2) 23 (s t ) y 1,t 2 y 2,t 2 ϕ (2) 31 ϕ (2) 32 ϕ (2) y 3,t 2 33 p 11 p 21 p 31 s t = [1, 2, 3] P = p 12 p 22 p 32 p 13 p 23 p 33 y 1,t 1 y 2,t 1 (4.4) y 3,t 1 + ε 1 ε 2 (4.5) ε 3 3 p ij = 1 (4.6), 1. j=1 7) VAR (2010) 9
11 4.2,. P = p p 11 p 22 0 (4.7) 0 1 p 22 p 33, , p 11, p 22, p 33, P.,,, MSVAR. 4.2,.,. ξ k ij(h) = E ty t+h ε jt st = =s t+h =k (4.8), k y t, 1 (h k )h. MSVAR,,.,,.,. 4.3 MSVAR., EM (Gibbs sampler), 8)., R MSBVAR., Patrick(2015), MSVAR., MSBVAR,, Inoue and Okimoto(2007)., MSVAR. θ = [θ 1, θ 2, θ 3, θ 4] (4.9) θ 1 = [s 1, s 2,, s T ] (4.10) θ 2 = [p 11, p 12, p 21, p 22,, p MM ] (4.11) θ 3 = [vech(ω(1)), vech(ω(2)),, vech(ω(k)) ] (4.12) θ 4 = [β(1), β(2),, β(n) ] (4.13) β(j) = [vec(φ (1) 1 (j)), vec(φ (2) 2 (j)),, vec(φ (p) p (j)) ] (4.14) 8), (2007) 10
12 4.3 vech( ), vec( ) vech = ( ) (4.15) ( ) vec = (4.16) 5 6 6, p(θ ŷ T ) 9), θ (0), j = 0 N. 1) θ (j+1) 1 p(θ 1 θ (j) 2, θ(j) 3, θ(j) 4, ŷ T ) ( ), Baum-Hamilton-Lee-Kim(BHLK) filter and smoother., forward-filter-backward-sample. 2) θ (j+1) 2 p(θ 2 θ (j) 1, θ(j) 3, θ(j) 4, ŷ T ) ( ) Dirichlet. 3),,. 4) θ (j+1) 3 p(θ 3 θ (j) 1, θ(j) 2, θ(j) 4, ŷ T ) ( ). 5) θ (j+1) 4 p(θ 4 θ (j) 1, θ(j) 2, θ(j) 3, ŷ T ) ( ). 6) j + 1 = N.,. MSVAR, EM,,,. MSVAR,,,,., EM 9) ŷ T = {y p+1, y p+2,, y t }. 11
13 .,, (MCMC). MCMC,,.,,,.. 5,, ADF MSVAR. ADF,. MSVAR, ADF, ADF, 225 TOPIX., (CPI), ( 3 ). r 0 = 0.1, ADF : 225 TOPIX 225, TOPIX ADF
14 5.1 5%., , TOPIX, , ,, , 1,, AR(1) 1, explosivear(1) 10). 3, recursiveadf. supadf,, explosive /4 1990/ /5 1990/ : 225 TOPIX recursive ADF 225, TOPIX. ( ) ( ) : AR(1) (x t = µ x + δx t 1 + ε x,t ) δ ( 225, TOPIX) 10) AR(1), 2. 13
15 5.1 supadf (r) infadf (r) rangeadf (r) (r 0 = 1/10) supadf (r) infadf (r) rangeadf (r) (r 0 = 1/10) : recursiveadf ( 225, TOPIX),. sup ADF, 225 TOPIX explosive. 225 explosive, TOPIX explosive, explosive AR(1), δ 1.,.1, 225 TOPIX., , TOPIX ,, , TOPIX , 225 TOPIX. 225, 225, TOPIX., TOPIX,, TOPIX, 225., 225 TOPIX 11), TOPIX 225, TOPIX ). 2,TOPIX, (2009). (2009) explosive, , 2., (2009) ( ),. 3, (2000 IT 2007 ).,, ,, 11) 10 AIC 2 VAR(6)., (2010). 12), 225 TOPIX. 14
16 5.2 MSVAR, explosive,.,,,. 5.2 MSVAR, 4 MSVAR MSVAR,. MSVAR, (2010). (2010),, 1985, 86, 87,.,,.,, 225,,,, (CPI) 5.,, 2 13).,,,., 2 5 MSVAR (1)., 1980Q1 2014Q4.,,,,. 5, Nikkei-E-I-Oil-CPI., MSVAR, burn-in 1000, E CPI Nikkei Oil I Time Time 5:. 13) 3, 2. 15
17 5.2 MSVAR MSVAR 14). 6, 2., Q1 Q Q2 Q3., Q1 Q2, (=Q2) (=Q2),., 2008 Q2 Q3,., (=Q3), 1., , , 6994., 1, 225 explosive,,.,, 1,,., 7., , (I Nikkei).,,,.,. 2, 1 2, (Nikkei Nikkei, I I).,,, 2. 3, (Nikkei I, Nikkei CPI).,,. 4, 1, (Oil I)., 1,.,. 1,.,,,,. 14) MSVAR.B 16
18 5.2 MSVAR state1 state : (x, y ). 7: 1 ( 95% ). 17
19 5.2 MSVAR 8: 2 ( 95% ). Nikkei Nikkei E I Shock to E I Oil CPI Oil CPI Response in 9: 1,2 ( 95% ).,. 18
20 5.3 (, ).,.,.,.,. 1,, , 225.,, ). 2, IT,,., 225.,.,, 16)., 5,,.,,,, ,.,, ,. 15) MSVAR,. 16) 1990,,,. 19
21 5.3 y t = β 0 + (β 1 + β 2 D 1 (t))rate t d + ε t (5.1) y t = β 0 + (β 1 + β 3 D 2 (t))rate t d + ε t (5.2) y t = β 0 + (β 1 + β 2 D 1 (t) + β 3 D 2 (t))rate t d + ε t (5.3) y t = β 0 + (β 1 + β 2 D 1 (t))rate t d + β 4 ST t + β 5 EX t + β 6 MB t + ε t (5.4) { 0 (t S k ) D 1 (t) = t + 1 t 0 (t S k ), D 2(t) = t t = 1,., 175 (5.5) { {1986Q1,, 1990Q2} ( ) S k = (5.6) {1986Q1,, 1990Q1} (TOPIX ) y t 225 or TOPIX( ) Rate t ST t TOPIX or 225( ) EX t MB t D 1 (t) D 2 (t), 4., 225 TOPIX 2., (5.1), ( 1), (5.2), ( 2), (5.3), 1 2 ( 3), (5, 4), (5.1), ( 225, TOPIX ),, ( 4). D 1 (t), S k 1, 2, 3, 1, 0.,, 225, 1986 Q Q2, TOPIX, 1986 Q Q1., D 2 (t), 1, 2, 3, 1., 1970 Q Q4.,, β 2,. 1,,, 2,., 3 4, β 2,., 5. I) (d = 0), II) 1 (d = 1), III) 1, IV) 1, V) D j (t) = k, j = 1, 2 k k
22 5.3, I), II)1, III, IV), V), (4 ) (5 ) (2 ) (2 ) , I II, III, IV, V. Nikkei225 TOPIX / I n 1 I N n 2 I N n 3 I N n 4 I N n 1 I T n 2 I T n 3 I T n 4 I T n t 1 I N t 2 I N t 3 I N t 4 I N t 1 I T t 2 I T t 3 I T t 4 I T t II n 1 II N n 2 II N n 3 II N n 4 II N n 1 II T n 2 II T n 3 II T n 4 II T n t 1 II N t 2 II N t 3 II N t 4 II N t 1 II T t 2 II T t 3 II T t 4 II T t III n 1 III N n 2 III N n 3 III N n 4 III N n 1 III T n 2 III T n 3 III T n 4 III T n t 1 III N n 2 III N n 3 III N n 4 III N n 1 III T n 2 III T n 3 III T n 4 III T n IV n 1 IV N n 2 IV N n 3 IV N n 4 IV N n 1 IV T n 2 IV T n 3 IV T n 4 IV T n t 1 IV N t 2 IV N t 3 IV N t 4 IV N t 1 IV T t 2 IV T t 3 IV T t 4 IV T t V n 1 V N n 2 V N n 3 V N n 4 V N n 1 V T n 2 V T n 3 V T n 4 V T n t 1 V N t 2 V N t 3 V N t 4 V N t 1 V T t 2 V T t 3 V T t 4 V T t 3: (n, t TOPIX.),,,,.,,.,,,.,, 17). 17), 21
23 5.3 Excange.Rate Time.Dummy O.N.Rate Bubble.Dummy.T TOPIX Bubble.Dummy.N Nikkei Monetary.Base Time Time 10: , β 2 β 3 10%, 4., I II, II., III V 1. 4, TOPIX 1 (= ), k 2., β 3 β 2 TOPIX, 1. Nikkei225 TOPIX I n t II n t III n t IV n t V n t 4: ( 10% β 2, II β 3 ) 22
24 5.3 II n IV t 5., TOPIX 4,,.,., (, 1986 Q Q2 ),., 4,,,., 6 V n., 5 t., k k 2,., R 2, 5 6, k 2.. (1986 Q Q2), ( 225, TOPIX).,., k k 2, 18).,.,. 1, TOPIX,., (5.0% 3.0%) 19).,, ,, 1986 Q Q2, ,,,. 2,,.,.,,,.,, 18), k 3, k 2 t R 2. 19) % 4.5%, 3 10, 4.5% 4.0%, 4 21, 4.0% 3.5%, 11 1, 3.5% 3.0%,, 2 23, 3.0% 2.5%. 23
25 5.3. Nikkei225 TOPIX β ( ) (1.741) (1.237 ) (1.875 ) ( 1.108) (1.853 ) (1.478 ) (1.963 ) (1.621) β ( ) ( 0.516) ( 0.657) ( 0.975) (0.874) ( 1.108) ( 0.979) ( 1.226) ( 1.271) β ( ) ( ) ( ) ( ) ( ) ( ) (0.620) β ( ) ( 0.128) (0.828) ( 0.013) (0.733) β ( ) ( ) ( ) β ( ) ( 0.079) ( 0.700) β ( ) (2.322 ) ( 1.930) R : II n IV t ( t,,,,, 0.1%, 1%, 5%, 10% ) Nikkei225 TOPIX β ( ) (1.822) (1.237 ) (1.948 ) ( 1.038) (1.901 ) (1.478 ) (2.002 ) (1.571) β ( ) ( 0.596) ( 0.657) ( 0.995) (0.869) ( 1.193) ( 0.979) ( 1.235) ( 1.311) β ( ) ( ) ( ) ( ) ( ) ( ) (0.806) β ( ) ( 0.128) (0.799) ( 0.013) (0.692) β ( ) ( ) ( ) β ( ) ( 0.076) ( 0.689) β ( ) (2.330 ) ( ) R : V n ( t,,,,, 0.1%, 1%, 5%, 10% ) 24
26 6, 3, , TOPIX , , explosive., MSVAR,,,.,,,,.,, 1986 Q Q2,., 2. 1,..,,. 2, ( ).,,,.,.,., ,,., , , 2010,,., 20).,.,,,.,.,. 1,.,. (2000), ,,.,,,. 2, MSVAR.,, 20) , 200, ,
27 .,.,,,., 21),,.,, MSVAR 22). [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] Campbell, J. Y., and Shiller R., The Dividend-Price Ratio and Expectation of Future Dividend and Discount Factors, Review of Financial Studies 1, 1989, [14] Evans, George, W., Pitfalls in Testing for Explosive Bubbles in Asset Prices, Economic Review 81, 1991, pp [15] Fujiwara, Ippei, Evaluationg Monetary Policy When Nominal Interest Rates are Almost Zero, Research and Statistics Department, Bank of Japan, 2004, pp ) (2000) 22),. 26
28 [16] Hamilton, James, A New Approach to the Economic Analysis of Nonstationary Time Series and the Buiness Cycle, Econometrica 57(2), 1989, pp [17] Inoue, Tomoo, Okimoto, Tatsuyoshi, Were There Structural Breaks in the Effect of Japanese Monetary Policy? Re-evaluating Policy Effects of the Lost Decade, Faculty of Economics and IG555, Yokohama National University, 2007, pp [18] Krolzig, Hans, M., Econometric Modelling of Markov-Switching Vector Autoregressions using MSVAR for Ox, Institute of Economics and Statistics and Nuffield College, Oxford, 1998, pp [19] Patrick, Brandt, Package MSBVAR, CRAN, 2015, pp.1-92 [20] Phillips, Peter, C. B., Wu, Yangru and Yu, Jun, Explosive Behavior in the 1990s Nasdaq : When Did Exuberance Escalate Asset Values?, Couwles Foundation Paper NO.1349, 2012, pp
29 .A 225, TOPIX, NEEDS (CPI) = =100 IMF 85 6, 7:.B MSVAR B.1 MSVAR Naive SR Time-Series SE p p p p : MSVAR 28
30 B.2 MSVAR B.2 MSVAR Naive SR Time-Series SE Nikkei Nikkei E I Oil CPI E Nikkei E I Oil CPI I Nikkei E I Oil CPI Oil Nikkei E I Oil CPI CPI Nikkei E I Oil CPI : 1 MSVAR 29
31 B.2 MSVAR Naive SR Time-Series SE Nikkei Nikkei E I Oil CPI E Nikkei E I Oil CPI I Nikkei E I Oil CPI Oil Nikkei E I Oil CPI CPI Nikkei E I Oil CPI : 2 MSVAR 30
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