自然利子率について:理論整理と計測
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1 * ** No.03-J * **
2 No. 03-J * 997 E52, E58 * obuyuki.oda@boj.or.jp ju.muraaga@boj.or.jp
3 ( (2 2 i = r π φ π π π φ ( y y (- ( y Blider (998 2 Bomfim (997, 200Laubach ad Williams (2003Neiss ad Nelso (200 Plaier ad Scrimgeour (2002 Bak for Ieraioal Selemes (2003
4 y φ π φ y r
5 ( (-3 (
6 4 5 ( Laubach ad Williams (
7 HP 5
8 6 (2-6 Summers (983 6
9 > ( (2-2 = aural rae of oupu aural rae of ieres 7 (2-2 7
10 8 4 (equilibrium real ieres rae (eural rae of ieres 9 ( (2 0 IS AS 8 (200, Wicksell (936 Woodford (2003 Woodford (2003 8
11 IS (2-3 9
12 0
13 A L g A F (K, A L ( K (2 A L F (K, A L = A L F (k, A L f (k (3- k K /A L r r = F K (K, A L = f k (3-2 ( F K F (K, A L K F (K, A L = K (A L -
14 c u ( c = u ( c ( r /( ρ (3-3 2 u ( u c ( c u ( c ( - 3,4 (3-3 ( c c c = σ( r ρ (3-4 5 ( r 2 2 (3-5 r = - g A (3-F (3-3 ( (3-4 [ - ( - -/2]E 0 [(c -c /c ] 2 [ - ( - -/2]Var[(c -c /c ] Var (3-7(3-0 2
15 (2 (3 - ( (2(3 (3-4 balaced growh pah (3-4 r r (3-2 k K /A L K g A A L g A (3- g A g A (3-4 g A r r = - g A (3-5 (-2 6 g A (3-5 r - g A (3-6 (3-5(3-8 Chadha ad Dimsdale (
16 7 - r g A (3-7 g A 0 (3-7 r r (3-8 (
17 3 AS Woodford (2003 IS AS 5
18 g A y c z ẑ zˆ log( z / z z z (4- z z z z 0 exp( g A r i (4-2 r rˆ log (4-3 r i iˆ log (4-4 i r A σ g i r π π π ρ logπ log( P / P 8 (4-5 (4-6 (
19 IS IS (4-7 ( (2 IS u (c ; c 9 i ( c ; ξ /( ρ c i = E Π uc ( c ; ξ u (
20 c y 20 Dixi-Sigliz C Y P Dixi-Sigliz c y C Y (4-8 (4-9 ( Y ; ξ /( ρ c i = E Π uc ( Y ; ξ u (4-9 Y C i (4-0 (4- Y y 0 ( i θ θ di θ θ (4-0 C c 0 ( i θ θ di θ θ (4- dy ( i y ( i θ = dp ( i p ( i (4-9 (4-9 9 Y ˆ E Yˆ σ ( iˆ E rˆ (4-2 = π ˆ r σ ( g E g (4-3 uc ( y;0 σ (4-4 y u ( y;0 cc 20 Woodford (2003 8
21 g uc ξ ( y;0 σ ξ u ( y;0 c u = u cξ cc ( y;0 ξ ( y ;0 y (4-5 (4-2 IS (4-30 Yˆ x x Yˆ (4-6 (4-2 IS x ˆ σ i E rˆ (4-7 = Ex ( π ˆ [( ˆ ( ˆ σ g Y E g YT ] r (4-8 (4-7 IS 2 (4-8 (4-7 iˆ π rˆ ( E AS 2 IS IS ( (2 (3 9
22 4-32 (4-33 AS (4-34 (4-36 ((3 ( i y (i = A f (h (i (4-9 A i h (i i u (C ; v (h (i; 22 (2 real margial cos i h (i w (i w ( i v h( h ( i ; ξ = uc ( C ; ξ P f - ( w ( i v h( f ( y ( i / A ; ξ = uc ( Y ; ξ (4-20 P i y (i Y w (i/p i w (i f - (y (i/a y (i i s (i 22 v ~ v ( y ( i ; ξ v ( f ( y ( i / A ; ξ = v ( h ( i ; ξ (4-F 20
23 (4-20 v h( f ( y ( i / A ; ξ s ( i = Ψ ( y ( i / A u ( Y ; ξ A c (4-2 Ψ ( x / f ( f ( x Aˆ log( A / A A A exp( g 0 A s (i  s ( ( (, ;, ˆ i = s y i Y ξ A (4-22 p (ip 23 p ( i = µ s ( y (, ;, ˆ i Y ξ A (4-23 P / ( > ( (4-22 (4-23 (3 (3-23 (
24 (3-2 2 (3- aural rae of oupu 25 Y i p (i = P y (i = Y Y (4-23 s( Y, Y ; ξ, Aˆ = µ ( Y Y A 27 (4-25 Y Y, Aˆ ( ξ ( ( AS Y e efficie rae of oupu Y e e e e s( Y, Y ; ξ, Aˆ = Y Y (4-48 (
25 AS AS 0A ˆ = 0 Y s ( Y, Y ;0,0 = µ (4-27 Y (4-27 AS (4-2 (4-2 y (i Y logy (4-25 sˆ( i ω yˆ ( i σ Yˆ ( ω σ Yˆ (4-28 yˆ ( i log( y ( i / Y (4-29 Yˆ log( Y / Y (4-30 sˆ ( i log( µ s ( i (4-3 (4-4 yˆ ( i = ˆ (4-28 Y sˆ = ( ω σ ( Yˆ Yˆ (
26 (3-2 saggered price adjusme Calvo ( (4-32 Calvo (983 0 << 29 Calvo (983 π γ ω κ γ = E ( Yˆ Yˆ π κ σ ωθ (4-F2 (4-F3 24
27 π = βe π sˆ (4-33 ζ α αβ β ζ > 0 ρ α ωθ Calvo (983 (4-33 (4-32 π = βe π ( Yˆ Yˆ κ (4-34 κ ( ω σ ζ ( = α ( α αβ ω σ ωθ > 0 (4-35 x Yˆ π = βe π κx (4-36 (4-34 (4-36 AS U ( u (C ; (2 v (h (i; u (C ; u (Y ; (4-9 v (h (i; v (f -- (y (i/a ; (4-37 Y y (i u (Y ; v (f - (y (i/a ; U β U = Ω β L = 0 = 0 Ξ (4-38 Ω Ξ L (
28 30 L = π λ( x x 2 * 2 (4-39 x * e log( Y / Y λ κ/ θ s ( Y, Y ;0,0 = µ > e e s( Y, Y ;0,0 = 0 (4-40 (4-4 ( 4-27 (4-42 (4-4 (4-35 x * (4-39 x * (4.39 x * (4-38 (4-7 (4-36 =0 β L (4-43 (4-7 iˆ (4-36 (4-7 x ( (4-F2 2 * 2 L = ( π E π λ( x x (4-F4 (4-38 L (4-38 (4-F4 (4-35 (4-4 (4-F3 (4-4 26
29 x (4-36 (4-43 λ E 0 [ π ( x x ] = 0 κ (4-44 (4-36 x x s s x s = s = 0 (4-45 IS AS IS AS x iˆ = Ex σ ( Eπ rˆ ( 4-7 π = βe π κx ( 4-36 (4-7 rˆ x (
30 28 x = 0 ( AS (4-7 ] ˆ ( ˆ [( ˆ Y g E Y g r σ ( 4-8 ( log (log ( ˆ ˆ ( ( ( ˆ ˆ ( ( ˆ ( = = A A A g E g Y Y E g E g Y E Y g g E g Y E Y g r g r σ σ σ σ σ σ σ ρ σ ρ σ ( ( log (log g E g Y Y E r σ ( ( (4-47 ( (4-48 (4-46 ( ( ( ( A q g L E g r = ω σ ω σ ρ σ (4-F5
31 σ g E g ( (4-46 (4-48 (4-47 ( (4-2 y (i Y log Y (4-27 Yˆ σ = σ g ωq ω (4-48 Yˆ g q 33 (4-48 g q L (
32 g uc ξ ( Y ;0 ξ ( 4-5 u ( Y ;0 Y cc q v~ v~ yξ yy ( Y ;0 ξ ( Y ;0 Y (4-49 g 34 Gˆ u C q 35 Â v h g = Gˆ ( s C G (4-50 q = ( ω Aˆ ω νh (4-5 s G (4-48 Yˆ (4-46 Yˆ (4-46 (4-50 (4-5 AR( ( (4-49 Â 36 AR( 30
33 r = ( σ g ( σ ω ( ω A ρ [( ρ ( ρ A G Aˆ Gˆ ( ω υ( s G ρ ( ρ h h ] C C (4-52 G C A h 4 Gˆ C Â h AR( - g A (2-3 AR( (4-46 (4-46 (4-47 I(0 3
34 ((3 37 ( Bomfim (200 (2 MIT-Pe-SSRC (MPS Bomfim (997 (3 (2 ( (2 37 Bomfim (200 32
35 2 Neiss ad Nelso (200 DGEM FRB Laubach ad Williams (2003 (3 Laubach ad Williams (2003 (5-(5-6 33
36 IS y y = a ( y a 3 y a ( y [( r r ( r 2 r 2 ] 2 ε y (5- AS π = b π π i π i b3( y y 3 i = 2 4 i = 5 b 4 π m b b π 5 m ε 2 ( b b (5-2 y = y g ε3 (5-3 = g (5-4 g ε4 r = c g z (5-5 = d (5-6 z z ε5 y y r r π π m g z ε 2 ε ε 3 ε 4 ε 5 IS AS IS y y r AS r IS AS AS 34
37 4 IS AS 4 g (5-3 ( r g c z (5-5 (4-46 AR( 4 40 Laubach ad Williams (2003 ε 4 Clark (987 Kuer (994 Hamilo (989Kim ad Piger ( Laubach ad Williams (2003 z AR(2 35
38 ε i (i =,2,,5 42 ε ε 2 y r z uobserved compoes model GDP CPI IS AR(3 z AR( I(0 I( 2 I(0 I( z z AR( z AR( d κ d = γ γ γ= 0.95 d κ 42 ε 5 ε4 (5-(5-6 ε4 0 pile-up problem Laubach ad Williams (2003 Sock ad Waso (998 media ubiased esimaor ε4 ε 5 σ 4 / σ 5 43 uobserved compoes model Harvey (989Hamilo (994, Chaper 3 36
39 IS AS a 2 b (5-6 Laubach ad Williams ( GDP ( HP GDP 37
40 ( ( (2 GDP 00 TFP 6 (3 GDP (2 (2 TFP 6 (3 (200 (2(3 ( g z a( T ( <T a( T a( Harvey (989, Hamilo (994, Chaper 3 38
41 IS AS Krugma (998 39
42 40
43 ( (2 (3HP Hodrick- Presco Taylor (993 i = r π φ π π π φ ( y y (6- ( y 50 i π y r π y y r y r 48 HP 49 (2 AS ( r 2.0% y 2.2% φ π = φ y =0.5 Taylor (993 4
44 42 y y x r y y r i φ π π φ π φ π π φ π π π = = ( ( ( (6-2 ( y y x = IS π y π y IS AS (4-46 (4-47 r r ~ ( log log ( ~ y y E Y Y E r = σ σ (6-3 = σ HP HP y y HP HP HP r y y E y y E r ~ ~ (6-4 r HP y y (6-2 ( ( ~ HP Y HP y y r i = φ π π φ π π (6-5 (6- (6-5 (6-5
45 Laubach ad Williams ( Laubach ad Williams (
46 y r = c g z 2 y c g 3 HP =,600 HP y r ~ HP ,54 L HP 54 44
47 Var[π] Var[x ] L = α Var[ x ] ( α Var[ π] (6-6 α 0 α α α φ π φ y α 45
48 HP 3 α α HP HP 5 5 HP HP 46
49 Laubach ad Willliams (2003 HP 47
50 5 48
51 GDP GDP Workig Paper NAIRU Workig Paper Bak for Ieraioal Selemes, Moeary Policy i a Chagig Evirome, BIS Coferece Papers, forhcomig, Blider, Ala S., Ceral Bakig i Theory ad Pracice, Cambridge: MIT Press, Bomfim, Aulio, The Equilibrium Fed Fuds Rae ad he Idicaor Properies of Term-Srucure Spreads, Ecoomic Iquiry, Vol.35, No.4, pp , 997., Measurig Equilibrium Real Ieres Raes: Wha Ca We Lear from Yields o Idexed Bod?, Fiace ad Ecoomics Discussio Series , 200. Calvo, Guillermo A., Saggered Prices i a Uiliy-Maximizig Framework, Joural of Moeary Ecoomics, Vol.2, pp , 983. Chadha, Jagji S. ad Nicholas H. Dimsdale, A Log View of Real Raes, Oxford Review of Ecoomic Policy, Vol.5, No.2, pp.7-45, 999. Clark, Peer K., The Cyclical Compoe of U.S. Ecoomic Aciviy, Quarerly Joural of Ecoomics, Vol.02, No.4, pp , 987. Hamilo, James D., A New Approach o he Ecoomic Aalysis of Nosaioary Time Series ad he Busiess Cycle, Ecoomerica, Vol.57, No.2, pp , 989. Hamilo, James D., Time Series Aalysis, Priceo Uiversiy Press, 994. Harvey, Adrew C., Forecasig, Srucural Time Series Models ad he Kalma Filer, Cambridge Uiversiy Press, 989. Kim, Chag-Ji ad Jeremy Piger, Commo Sochasic Treds, Commo Cycles, ad Asymmery i Ecoomic Flucuaios, Joural of Moeary Ecoomics, Vol.49, pp.89-2, Krugma, Paul R., I s Baaack: Japa s Slump ad he Reur of he Liquidiy Trap, Brookigs Paper o Ecoomic Aciviy, 2, pp ,
52 Kuer, Keeh N., Esimaig Poeial Oupu as a Lae Variable, Joural of Busiess ad Ecoomic Saisics, Vol.2, No.3, pp , 994. Laubach, Thomas ad Joh C. Williams, Measurig he Naural Rae of Ieres, Review of Ecoomics ad Saisics, forhcomig, Neiss, Kaharie S. ad Edward Nelso, The Real Ieres Rae Gap as a Iflaio Idicaor, Bak of Eglad Workig Paper No.30, 200. Plaier, L. Chrisopher ad Dea Scrimgeour, Esimaig a Taylor Rule for New Zealad wih a Time-varyig Neural Real Rae, Discussio Paper Series, DP2002/06, Reserve Bak of New Zealad, Sock, James H. ad Mark W. Waso, Asympoically Media Ubiased Esimaio of Coefficie Variace i a Time Varyig Parameer Model, Techical Workig Paper 20, Naioal Bureau of Ecoomic Research, 996. Summers, Lawrece H., The No-adjusme of Nomial Ieres Raes: A Sudy of he Fisher Effec, i James Tobi (ed., Macro-Ecoomics, Prices ad Quaiies, Blackwell, 983. Taylor, Joh B., Discreio Versus Policy Rules i Pracice, Caregie-Rocheser Coferece Series o Public Policy, Vol.39, pp.95-24, 993. Wicksell, Ku, Ieres ad Prices: A Sudy of he Causes Regulaig he Value of Moey, Eglish Traslaio, Lodo: Macmilla, 936. Woodford, Michael, Ieres ad Prices: Foudaios of a Theory of Moeary Policy, Priceo: Priceo Uiversiy Press,
53 g Yˆ v ~ q v ~
54
55 [( r r ( r 2 r 2 ] 3 y y = a ( y y a2 ( y 2 y 2 ε b b b π m m = b π π i π i b3( y y b4 π b5 π ε2 3 i= 2 4 i= 5 y = y g ε3 g g = ε4 a 2 (5- (5-2 (5-3 (5-4 r = c g z (5-5 = d (5-6 z z ε5 a a a b b b b b c d σ ( = Var( ε σ 2 ( = Var( ε σ 3( = Var( ε σ 4 ( = Var( ε σ 5 ( = Var( ε ( σ4 Sock ad Waso (996media ubiased esimaor σ3 λ = σ σ g 3 4 λg = 0. 2
56
57
58
59
60
61 Var[x ] Var[] L L Var[x ]( Var[]
62 Var[x ] Var[] L L Var[x ]( Var[]
63
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72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
More informationI ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
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59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
More informationm dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
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More information7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
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.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
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