研究シリーズ第40号

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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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