研究シリーズ第40号
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- つねたけ かやぬま
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1 165
2 PEN WPI CPI WAGE IIP Feige and Pearce 166
3 167
4 168
5 169
6 Vector Autoregression n (z) z z p p p zt = φ1zt 1 + φ2zt φ pzt p + t Cov( 0 ε t, ε t j )= Σ for for j 0 j = 0 Cov( ε t, zt j ) = 0 j = > 1 ε t ε t z t n n φ1, φ P B 170 ε
7 2 P φ( B) zt = εt, φ( B) = I φ1b φ2b φ P B l B z t z, φ(b) = t l MA z t 1 = φ ( B) ε = ϕ ( B) ε = Σ ϕ ε t t j= 0 j t j 1 ϕ ( B) = φ ( B) ϕ n n ϕ0 = I ϕ φ Wold Deterministic component Invertible z z t Σφ i izt i ε t z t p 1 Sargent and Sims Index Index 1 Litterman Bayesian 171
8 Tiao and Box z t l z t ρ l z t z t l ρ l l z t zt 1, zt 2, zt 3 AR A 172
9 173
10 174
11 A 3.1 PEN WPI PEN CPI 175
12 PEN WAGE PEN IIP 176
13 WPI CPI WPI WAGE 177
14 WPI IIP GPI WAGE 178
15 CPI IIP WAGE IIP 179
16 PEN WPI WPIt 2 PENt PEN CPI PEN WAGE WAGE PEN IIP WPI CPI IIP CPI IIP WAGE IIP Tiao and Box T l ˆ ρij ( l) T 1 ˆ ( l) ˆ ρ ( l) = ij Σ( z z )( z { Σ( z z ) Σ( z z ) } 2 it it i i jt +l jt z z i z t i MA 2 2T 1 2 2T 1 2 2T 1 2 2T 1 ˆ ρ ( l) ij ) 180 j j ρ ij
17 LAGS 1 THROUGH 6 LAGS 7 THROUGH 12 LAGS 13 THROUGH 18 LAGS 19 THROUGH
18 P(l) P(l) P(l) l = 1,2, AR( p) zt = φ1zt 1 + φ2zt φpz t p + at T ˆ φ, ˆ, 1 φ 2 p P(l) () l AR l = 1,2, p P ˆ ( l) = φˆ l AR(l) ˆ φ, ˆ 1 φ 2, φˆ l z t 1, z t P P ˆ( l ) AR(l) = 0 φ 0 S l T ( = t t = l+ 1 1 t l) ( z ˆ φ z ˆ φ z ) 182 l t l ( z ˆ t φ1zt 1 ˆ φ ) lzt l AR(l) S ( 0) = z t zt l = 1,2, p φ l
19 x 2 U = S( l) / S( l 1) Bartlett 1 M ( l ) = ( N l n) l nu 2 n 2 x 2 n M (l) AR Σ 183
20 M (l) Σ AR Sims Litterman Fischer Gordon AR (3.3) MA ϕ 0, ϕ 1, ϕ 2, ϕ 0= I zt = ϕ 0 εt + ϕ 1εt 1 + ϕ 2εt 2 + = ϕ ( B) ε t ε t z ϕ l i, j j l i p AR z1, z P ε ( 0) = 0,0, j φ z ( B) z 1 +, = 0,1,2, ε t + l = P+ l l l l j ε i σ i ~ ϕ, ~, ~ 0 ϕ 1 ϕ 2, ~ϕ 0 = σ i ~ ϕ ~ ~ i= ϕ iϕ 0 184
21 Variance Decomposition MA i Li = n i l Et zi( t + l) = zi( t + l) Liε ( t + l) Liϕ 1ε ( t + l 1) Liϕ l ε 1 ( t + 1) i l E { z E z 2 } i( t+ l ) t i( t+ l) 2 k 2 j 2 σ + l ϕ σ 1 Σ k = 1 j i i l k= l 1 k= 0 l 1 Σ ϕ Σ j 2 k ϕ σ 2 k 2 j σ j 2 Σ = E ε ε ' Σ ε t Σ = S S Choleski S Ut = Sε t U t t t 185
22 EUt Ut = E Sε tε t S = SΣS = Ω Ω U t ε t zt + l Et zt + l = IS Ut + l + ϕ 1S Ut + l ϕ l 1S Ut 1 l E ( z )( ) t + l Et zt + l zt + l Et zt + l = Ω + Ω 1 1 S S ϕ 1S S ϕ ϕ l 1S ΩS ϕl 1 = Σ + ϕ Σ 1 ϕ ϕ l 1Σϕl 1 1 H = S k ϕ k i l j H k Ω jj k= l 1 2 H Ω k= 0 l 1 Σ Σ j 2 k jj PEN WPI CPI WAGE IIP VAR F R 2 186
23 F PEN R 2 = S = WPI R 2 = S =.8273 CPI R 2 = S =.8654 WAGE R 2 = S = IIP R 2 = S = PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN, WPI, CPI, WAGE, IIP t l P t ( t + l) t A(t) T l 187
24 A 188
25 189
26 k L l T l l = Σ P t ( t + l) A( t + l) / Ll Mean Error t ε Tl l = Σ P t ( t + l) A( t + l) / L Mean Absolute Error t ε Tl = Σ + ε l 2 { } 2 Root Mean Square Error 190 P( ) ( ) 2 t t + l A t l / Ll t T 2 P ( ) ( ) t t + l A t + l tεt l l Theil U = 2 Σ A( t) A( t + l) tεtl Σ RMS RMS l l 1 2 1
27 Theil U RMS Theil U RMS M E M A E 2 R M S E Theil U L l PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP RMS Theil U
28 PEN WPI CPI WAGE IIP RMS Theil U PEN WAGE RMS Theil U PEN PEN PEN PEN 192
29 193
30 WPI 194
31 CPI 195
32 196
33 IIP 197
34 PEN WPI CPI WAGE IIP WPI CPI WPI CPI WPI CPI IIP WPI IIP WPI CPI WAGE WPI CPI WAGE CPI CPI WPI CPI CPI WAGE 198
35 IIP P, C, W, Y P C W Y PEN WPI CPI WAGE 199
36 200
37 . 201
38 WPI 202
39 WPI 203
40 CPI 204
41 CPI 205
42 206
43 207
44 IIP 208
45 IIP 209
46 WPI WPI CPI WAGE IIP WPI WPI CPI WAGE WPI CPI WAGE PEN Autonomous WPI WPI CPI WAGE WPI CPI WPI 210
47 Orthogonal Decomposition of Variance z t 2 3 WPI WPI WPI CPI CPI WPI CPI WPI WAGE CPI WPI WAGE IIP 211
48 PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI CPI WAGE IIP PEN WPI 212
49 PEN WPI CPI WAGE IIP 213 PEN WPI CPI WAGE IIP PEN WPI
50 WPI CPI WPI CPI IIP WPI CPI WPI CPI WPI CPI CPI j 1 zt+ j = Σ ϕlε t+ j l + Σϕlε t+ j l l= 0 l= j t + 1 t + j z t + j t ε n 214 z t + j n + 1 t ε n z t + j PEN PEN
51 215
52 216
53 217
54 218
55 O P C W Y 219
56 WPI PEN WPI WPI CPI IIP IIP WPI Chamberlain and Leamer prior 220
57 T P Y = X β + ε T 1 T P P 1 T 1 k 1 = ( X X + kr R) ( X Y + kr r) β ε (0 σ 2 I) 2 R β = r + ν ν N(0, λ I) q P P 1 q 1 q 1 k = σ 2 / λ 2 R = I, r = 0, R = X r = 0 Shiller prior P P prior 221
58 prior prior λ j 0 λ / j prior A prior 3 222
59 223
60 PEN WPI CPI WAGE IIP RMS Theil U RMS Theil U RMS Theil U prior 224
61 P W C Y 225
62 PEN WPI CPI WAGE IIP RMS Theil U
63 n Z1t, Z nt, t = 0, ± 1, ± 2, Z t = ( Z 1t, Znt ) n n Z Yule Slutsky Bartlett Kendoll Walker Wold Yaglom n Yule-Slutsky ARMA φ ( B ) Z = θ ( B) a P t q t P φp( B) = I φ1b φ P B q θq( B) = I θ1b θ B q B φ θ n n z t = Z t η η η a t Σ φ p( B) θq( B) φ p( B) z t θ ( B) q z t + = zˆ t( ) + et ( ) l l l t z l 1 ( l) t( l) = Σ π j zt + 1 j, et ( l) = Σϕiat + l i j = 1 i= 0 227
64 Box-Jenkins π n n φ ( B) ϕ ( B) = θ ( B) P ( ) = ϕ + ϕ B + B 0 1 q ϕ ϕ = I θ ( B) π ( B) = φ ( B). q π ( B) = I π B π B ( l) ( l 1) j = π j+ 1 + ϕl 1 1 P j (1) j π π, π = π ˆ l z t( ) z t + l ˆt ( l) = t + l t, t 1 z E( z z z, e = z + zˆ ), l 1,2, t t( l) t l t( l = zˆ ( l) ˆt = 1 t t + l 1) A.5 ( l) E ( z + t z φ φ E z ) + P t ( t + l P θ1e t ( a t + l 1) θ qet ( at + l q) E ( z t E ( a t t + j t + j zt ) = zt ( j) + j, j 1, 0, j 1 ) = at + j, e (l) 0 t j ( e l =Σ 1 t( l) ) i= 0 ϕ Σϕ i i MA p = 0, q = 1 z = ( I θ B) a n = 2 θ = t z1 t = at1 11a1( t 1) 12a2( t 1) z t θ θ θ θ 2t = a2t 21a1( t 1) 22a2( t 1) 228 θ θ θ θ 12 22
65 z jt a jt a t 1 p = 1, q = 0 ( I B) z = a, n = 2 φ t t φ = φ φ φ φ φ φ z1 t = 11z1( t 1) + 12z2( t 1) + a1 t z2 t = φ 21z1( t 1) + φ22z2( t 1) + a2t z 2t 1( t 1) z2( t 1 z 1t z ) z t z t 1 ( j = 1,2, φ θ π j n = 2 MA(1) θ z1 t (1 θ11) B 0 a1t = z2t θ21b (1 θ22) B a2t a2t = β a1 t + εt ε t a t z1 t 1 11B) a1 t z = ( θ ω ω B 0 1 2t = z1 t + (1 1 θ11b θ B) ε 22 ω = β t 0 ω 1 = βθ 22 + θ21 z t η l l = 0, ± 1, ± 2, Γ( l) = E( z l z ) = γ ( l) t t { i j } i, j = 1, n 229
66 ( l) { ( l) } ρ = ρ i j ρ = γ l) /[ γ (0) γ (0)] Γ ( l) = Γ( l) i j ρ ( l) = ρ ( l) i i ( j j i j z φ ( B) θ ( B) a = ϕ ( B) a t = 1 P q t Σ j = 0 E( z tat j ) = ϕ jσ j 1 0 j < 0 z t t ( z t z l t 1 1 t P P t l t t 1 1 t g q φ LL z φ ) = z ( a a θ LL a θ ) Γ( l) = l 1 Σ j= l r r Σ j= 1 Γ( j) φ l j r l Σ j= 0 ϕ Σθ j+ l Γ( l j) φ, l > r j i 230, l = 0, LL, r 1 ϕ ( B) = φ ( B) θ ( B) = ( I + ϕ B +LL) ϕ θ = I, r = max( p, ) (i) p < q 0 q φp+ 1 = L L = φr = 0, (ii) q < p θ g + 1 = LL = θr = 0 p = 0 MA(q) (A.15) Γ g l θ jσθ j l l = 0, LL, q j =Σ 0 l > q + ( l) = 0 l > q P(z) z t AR(l) P( l) = φl l = 1,2, LL j 1 θ = 0 P(l) j
67 Γ(l) P l) 1 Γ (0) Γ(1) l = Γ(0) b ( l) l Al bl Γ bl Al Cl l > 1 ( = 1 Γ(0) LLLL Γ ( l 2) A = l M M, bl Γ( l 2) LLL Γ(0) Γ ( l 1) =, C Γ (1) z t AP( p) P ( l) = 0 l > p l Γ(1) = Γ( l 1) Akaike H. A New Look at the Statistical Model Identification I.E.E.E. Transactions on Automatic Control AC Chamberlain G. and Leamer E.E. Matrix Weighted Averages and Posterior Bounds. Journal of the Royal Statistical Society. Ser.B Feige E.L. and Pearce D.K. The Casuel Causal Relationship between Money and Income Some Caveats for Time Series Analysis The Review of Economics and Statistics. vol Litterman R.B. A Bayesian Procedure for Forecasting with Vector Antoregressions.D.P.MIT Sargent T.J. and Sims C.A. Business Cycle Modelling Without Pretending to Have Too Much A Priori Economic Theory. New Methods in Business Cycle Research Proceedings from Conference Federal Raseave Bank of Minneapolis. Sims, C.A. Macroeconomics and Reality. Econometrica vol Sims, C.A. Comparison of Interwar and Postwar Business Cycles Monetarism Reconsidered, American Economic Review Tiao, G.C.and Box.G.E.P. Modelling Multiple Time Series with Applications. Journal of American Statistical Association.vol
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Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue 2012-06 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/23085 Right Hitotsubashi University Repository
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38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t
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