Nelson-Siegel Nelson-Siegel 1992 2007 15 1 Nelson and Siegel(1987) 2 FF VAR 1996 FF B)
1 Nelson-Siegel 15 90 1 Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel 3 4 5 1 Litterman and Scheinkman(1991) Frye(1997) 2 Nelson-Siegel Diebold and Li(2006) Diebold, Rudebusch and Aruoba(2006) 2
2 Nelson-Siegel Nelson-Siegel 3 Nelson-Siegel Nelson-Siegel 2.1 Nelson-Siegel m f(m) f(m) = L + Se mλ + Cmλe mλ (1) L S C λ 4 y(m) m y(m) = 1 m m 0 f(u)du (2) ( 1 e mλ ) ( 1 e mλ y(m) = L + S + C e mλ) (3) mλ mλ Nelson-Siegel (3) 5 ( 1 e mλ t ) ( 1 e mλ t ) y t (m) = L t + S t + C t e mλt (4) mλ t mλ t 2.2 Nelson-Siegel B 3 Christensen, Diebold and Rudebusch (2007) Duffie and Kan (1996) (3) 4 Nelson and Siegel(1987) 5 3
6 1992 1 2007 5 100 300 7 1 20 8 1 X Y Z 1992 1980 1980 10 1990 9 20 1990 10 m 1 m s s t L S C λ L t S t C t λ t t s ( 1 e m j λ t ) ( 1 e m j λ t )] 2 [y t (m j ) L t S t C t e m jλ t (5) m j λ t m j λ t j=1 L t S t C t λ t λ t λ t λ (5) L t S t C t (1, 1 e m j λ m j λ, 1 e m j λ m j λ e mjλ ), (j = 1,,s) {y t (m j )}(j = 1,,s) λ ˆL t (λ), Ŝt(λ), Ĉt(λ) U t (λ) = s j=1 [ y t (m j ) ˆL t (λ) Ŝt(λ) ( 1 e m j λ m j λ ) ( 1 e m Ĉt(λ) j λ )] 2 e m jλ m j λ (6) λ t U t(λ) ˆλ 6 90 7 1998 11 8 3.1 1 9 1987 20 1996 30 1999 10 3.3 96 4
λ 0.0327 11 L t S t C t ˆL t (ˆλ), Ŝt(ˆλ), Ĉt(ˆλ) 3.3 8 1: 1992 1 2007 5 2.3 (3) 2 3 S t,c t S(m) = 1 e mλ mλ C(m) = 1 e mλ mλ (7) e mλ (8) λ 2 2 λ 0.0327 C(m) 48 60 C t 4 5 λ C(m) S(m) 0 <m S t t y t (m) =L t + S t S(m)+C t C(m) S t,c t λ Nelson-Siegel 11 Diebold and Li (2006) 1985 1 2000 12 192 30 λ =0.0609 5
L t 12 012 013 014 015 016 017 018 019 012 0103 010438 0105 0103 010438 0105 023 47 70 95209243267290305339 2: λ (S(m)) (C(m)) e mλ m S t S(m) C t C(m) λ λ Nelson-Siegel λ L, S, C L, S, C λ 3 4 2 5 10 15 S t S(m) C t C(m) 24 (S t S(m)) 60 C (C t C(m)) 1999 1999 2001 2003 2005 2001 2003 2004 2005 12 0 <m 0 <L t + S ts(m)+c tc(m) 3.3.1 6
07237 0123 0423 0523 0623 51 7 657 67 8905 8904 8901 8903 890 890 890 890 89077 89076 89075 89074 89071 89073 8907 8907 3: S S t S(m) 2,5,10,15 (S ts(m)) 534 0534 0234 01 5 2 1 5 215 25 678091 67809 67809 678094 67809 67809 67809 678099 678055 678052 678051 67805 67805 678054 67805 67805 4: C C t C(m) 2,5,10,15 (C tc(m)) 3 L t S t C t 3 x t 7
3.1 13 FF 14 Nelson-Siegel 3 Nelson-Siegel 3.2 FF L S C 3 VAR 2 3.3 L t S t C t x t =(L t,s t,c t ) y t H y t = Hx t 1 e 1 m 1 λ 1 e m 1 λ m 1 λ m 1 λ e m 1λ H =. (9) e msλ 1 1 e msλ m sλ 1 e msλ m sλ x t t 1 M t 1 1 x t 1 F G y t = Hx t + w t, w t N(0,R) (10) x t = µ + F x t 1 + GM t 1 + v t, v t N(0,Q) (11) t y t M t 1 13 2006 8 14 GDP 8
Ang and Piazzesi(2003) VAR Diebold, Rudebusch and Aruoba(2006)( DRA(2006)) Nelson-Siegel 3 15 3.2 3.2.1 L t,s t,c t Z t L t,s t,c t M t a 16 1) Z t = µ + a M t + ξz t 1 + v t, v t N(0,σ 2 ) 2) Z t = µ + a M t + u t, u t = φu t 1 + v t, v t N(0,σ 2 ) 1992 1 2007 5 AIC CPID t : CPI CPI IIPD t : 2000 BDR t : TOPIXD t : ( 15 (2007) 10 1995 2005 1 3 5 2 16 9
US10 t : 10 17 FF t : FF 18 TDUMMY t 99 2 0 1999 2 t L t CPID t BDR t L t 1 u t L t = 0.231 + 0.074 BDR t + 5.740 CPID t + 0.918 L t 1 + v t (3.164) (1.940) (1.457) (40.776) S t (US10 t FF t ) TOPIXD u t v t S t = 2.971 0.151 (US10 t FF t ) 0.007 TOPIXD t + u t ( 6.820) ( 1.986) ( 2.303) u t = 0.952 (41.510) u t 1 + v t C t FF t IIPD t IIPD t TDUMMY 99 2 S t C t = 4.104 + 0.352 FF t + 4.916 (IIPD t TDUMMY t )+u t ( 5.278) (2.021) (2.542) u t = 0.892 (23.942) u t 1 + v t = 1 1: L S C 0.980 0.906 0.847 = 1.741 2.130 2.035 17 18 effective federal funds rate( ) 10
L t S t C t TOPIX TOPIX 1999 FF 24 60 120 5 35 72 132 35 72 132 90301213456 78 904 905 903 906 904 907 905 908 906 90 907 90 908 9022 90 9021 90 9023 9022 9024 9021 9025 9023 9026 9024 9027 9025 9028 9026 9027 5: 3.2.2 t t + k y t+k y t L t,s t,c t 1) Z t+k Z t = k 1 k 1 k 1 ξ j a M t+k j +( ξ j µ +(ξ k 1)Z t )+ ξ j v t+k j j=0 k 1 2) Z t+k Z t = a (M t+k M t )+(φ k 1)u t + φ j v t+k j Z 1 1999 5 2000 5 6 2 j=0 j=0 j=0 11
7 7 FF 412 312 0120 3 4 5 02 0002 34 56 60 78 3079 354 326 370 408 447 6: 99/05-00/05 1999 5 2000 5 01241 124 123 125 126 127 128 129 410 012343 58 81 6 41 453 478 41 316 33 7: 1999 5 2000 5 12
3.3 Q 1996 1 2007 5 3.3.1 3.1 (10) (11) y t = Hx t + w t, w t N(0,R) x t = µ + F x t 1 + GM t 1 + v t, v t N(0,Q) y t y t =(y (m 1) t,,y (ms) t ) s t 12 240 12 y t =(y (12) t,y (24) t,,y (240) t ) x t x t =(L t,s t,c t ) M t µ (3 1) (10) (11) G 19 F, G R, Q λ f u ( ) 19 DRA(2006) " # " #" # " # H 0 x t w t = + ( ) 0 I M t 0 " # " # " #" # " # = 0 F 0 F 1 x t 1 v t + + ( ) S y t M t x t M t 1 F 2 (10) (11) y t x t M t DRA(2006) FF F 1,F 2 L t,s t,c t M t 1 z t 13
z z u f u (z) = z exp ( z u u ) z<u (12) u y t = f u (Hx t + w t ) (13) fu 1 (y t ) u R 1 σ 2 σ 2 I Q 3.3.2 y t = f u (Hx t + w t ), w t N(0,σ 2 I) (14) x t = µ + F x t 1 + v t, v t N(0,Q) (15) 2 4 f u ( ) u u =exp(c) c λ 0.036 2 0.0327 2.2 AIC 7676.42 3.3.3 CPID t :CPI CPI( ) IIPD t : 2000 BDR t : TOPIXD t : US10 t : 10 FF t : FF US10 t FF t : 14
2: F, µ F Const. L t S t C t µ L t+1 0.875* -0.052 0.020 0.280* (0.087) (0.079) (0.016) (0.092) S t+1 0.014 0.928* -0.021-0.285* (0.107) (0.098) (0.020) (0.114) C t+1 0.029 0.087 0.929* -0.069 (0.184) (0.168) (0.035) (0.193) * 5% 3: Q Q L t S t C t L t 0.038 (0.196) S t 0.059 (1.516) C t 0.177 (0.276) 4: σ 2,λ,c σ 2 λ c u = exp(c) 0.002 0.036-7.354* 0.001 (0.041) (0.195) (1.000) (0.001) 15
AIC CPID t IIP t TOPIXD t BDR t FF t AIC 7681.01 5 7 2.2 L t,s t,c t 034321 8 01 02 567081 567089 56708 567088 567044 56704 567043 56704 567042 56704 567041 567049 8: L t,s t,c t (10) (11) L t,s t,c t 2.2 * 2.2 F 1 x t (I F ) 1 µ DRA(2006) FF FF 1996 1992 IIPD t S t+1 TOPIX S t+1 TOPIXD t TOPIX C t+1 TOPIXD t TOPIXD t 4 5 16
5: F, µ,g F Const. G L t S t C t µ CPID t IIPD t TOPIXD t BDR t FF t L t+1 0.763* -0.135 0.017 0.378* 1.653-0.930* 0.001 0.272-0.019 (0.143) (0.123) (0.020) (0.103) (0.983) (0.348) (0.001) (0.172) (0.018) S t+1-0.170 0.765* -0.036-0.331* -0.124 1.282* -0.003* -0.153 0.037 (0.173) (0.149) (0.025) (0.126) (0.988) (0.416) (0.001) (0.210) (0.022) C t+1 0.619* 0.635* 0.889* -0.462* 0.522 0.708 0.008* -0.337 0.020 (0.290) (0.251) (0.044) (0.215) (0.996) (0.629) (0.002) (0.351) (0.038) 6: Q Q L t S t C t L t 0.036 (0.038) S t 0.054 (0.091) C t 0.161 (0.508) 7: σ 2,λ,c σ 2 λ c u = exp(c) 0.002 0.036-16.942* 0.000 (0.001) (0.021) (1.000) (0.000) 17
u 0 12 240 f u ( ) 3.4 FF µ =(I F ) 1 µ F x t µ = F k (x t k µ)+ = j=1 k k 1 F j 1 GM t j + F i v t i j=1 F j 1 GM t j + F i v t i x t p x (p) t M t q M (q) t M (q) t k 1 x(p) t Φ pq (k) i=0 Φ pq (k) = M (q) x (p) t =(F k 1 G) pq (16) t k (F k 1 G) pq F k 1 G (p, q) t k M (q) t k 1 t k x (p) t Ψ (k) pq = k r=1 i=0 Φ (r) pq (17) q t k x t Ψ q (k) HΨ q (k) =(Ψ (k) 1q, Ψ(k) 2q, Ψ(k) 3q ) FF HΨ (k) q 9 k 1 3 6 36 FF 1 FF 9 36 10 3 0.4 3 1.2 FF 9 6 7 18
k 36 01231 123 124 125 1267 723 724 725 726 7 95 7 9 95 73 83 793 73 9: FF 1 FF 4 1980 90 2000 20 04 3 06 5 10 3 2000 2 1996 2003 6 VaR 2006 6 1 2 21 20 Ahrend, Catte and Price 2006 21 95 9 98 19
4.1 µ =(I F ) 1 µ x t = x t µ (11) x t = F x t 1 + GM t 1 + v t k>0 x t+k = F (F x t+k 2 + GM t+k 2 + v t+k 1 )+GM t+k 1 + v t+k = k k 1 F k x t + F j 1 GM t+k j + F i v t+k i j=1 i=0 k k 1 x t+k x t =(F k I) x t + F j 1 GM t+k j + F i v t+k i j=1 i=0 t t + k x x t 3 (10) y t+k y t = H(x t+k x t )+(w t+k w t ) = H( x t+k x t )+(w t+k w t ) = k k 1 H(F k I) x t + H F j 1 GM t+k j + H F i v t+k i +(w t+k w t ) j=1 i=0 2 VAR ˆx t H ˆx t 10 13 16 H ˆµ Mean 22 H ˆµ 9 0.5% 1999 2 99 4 2000 8 2001 3 2006 3 CPI 2001 0% 22 (14) f u(h ˆx t) u H ˆx t 20
4.2 1996 2007 1996 2003 6 VaR 2003 6 06 6 01231 1234 4235 5236 6237 723 814 1618 45 68 81 97419465438491517559 10: (96 1 03 6 ) 96 1 03 6 11 VAR 1996 VAR 05235 523 0423 0123 41 67 75 89458461437485159118 11: (96 1 03 6 ) 21
12 TOPIX IIPD CPID FF VAR 14 15 TOPIXD FF IIP 0123 0124 0125 0126 01271 126 125 76 58 81 94719756738791614669 523 12: (96 1 03 6 ) 4235 1617 1234 1717 01231 45 67 71 89418465437481519558 13: (03 6 06 6 ) 03 6 06 6 TOPIXD 2000 22
IIPD CPID 723 01231 124 125 1267 73 85 51 64716783795761314336 14: (03 6 06 6 ) 0123 0124 01251 124 123 126 75 83 31 64716785793761514556 15: (03 6 06 6 ) 23
4.3 2006 6 07 5 523 2006 6 07 5 4235 1717 17 18 1234 113 01231 45 67 71 89418465437481519558 16: (06 6 07 5 ) 06 6 07 5 VAR FF 0234 0235 0236 02372 236 235 234 0117 85 52 46124187195142726774 17: (06 6 07 5 ) 24
1267 0123 0124 01251 124 123 75 8331 64716785793761514556 18: (06 6 07 5 ) 5 Nelson-Siegel 1992 2007 15 1 1992 2007 5 3 2 5 2 TOPIX 10 FF FF 1999 3 90 1996 25
VAR 1996 2003 26
McColluch(1971,1975) Steeley(1991) B B 2 Steeley(1991) 3 B t 1 τ Pt 0 (τ)(0 <τ) P 0 t (τ) = n 1 i= 3 λ i B i (τ) B i (τ),i = 3, 2,,n 1 [ξ 0,ξ n ] ξ 3 < <ξ 0 < <ξ n < <ξ n+3 n +7 3 B λ i (i = 3,,n 1) t L Pt l (l =1,,L) a l t(l =1,,L) t + τ j (j =1,,M) C lj v m t,c,b,λ P t 1 + a 1 t c 11 c 1M v m t =., C =..... Pt L + a L t c L1 c LM B = B 3 (τ 1 ) B n 1 (τ 1 )..... B 3 (τ M ) B n 1 (τ M ), λ = λ 3. λ n 1 v m t v t t v t t = CBλ λ v m t v t t 0 1 Pt 0 (0) = 1 λ λ { λ =argmin (v m t CBλ) (v m t CBλ) w λ =1 } λ 27
w =(B 3 (0),,B n 1 (0)) λ = α + 1 µw α w Kw Kw K = { (CB) CB } 1 α = K(CB) v m t λ τ P 0 t (τ) P 0 t (τ) = n 1 i= 3 λ i B i (τ) λ =( λ 3,, λ n 1 ) τ Ỹt(τ) Ỹ t (τ) = 1 ( ) τ log P 0 t (τ) B ξ 3 < <ξ 0 < <ξ n < <ξ 3 t t 2 t L t L t λ γ { ˆλ =argmin (v m CBλ) (v m CBλ)+γλ Zλ w λ =1 } (18) λ Z (i, j) Z i,j Z i,j = ξn ξ 0 ξn ξ 0 ( d k )( d k ) dv 2 B i(v) dv 2 B j(v) dv (19) { dk dv 2 P 0 t (v)}2 dv = λ Zλ (20) 28
λ Zλ Pt 0 (v) λ Zλ γ γ γ Steeley(1991) λ ˆλ = ˆα + 1 w ˆα w ˆKw ˆKw (21) ˆK = { (CB) CB + γz } 1 (22) ˆα = ˆK(CB) v m (23) [1] Ahrend, R., P. Catte, and R. Price, Factors Behind Low Long-Term Interest Rates, OECD Economic Department Working Papers, No. 490, 2006. [2] Ang, A., and M. Piazzesi, A no-arbitrage vector autoregression of term structure dynamics with macroeconomic and latent variables, Journal of Monetary Economics, Vol. 50, 745 787, 2003. [3] Christensen, J. H. E., F. X. Diebold, and G. D. Rudebusch, The Affine Arbitrage-Free Class of Nelson-Siegel Term Structure Models, NBER Working Paper Series, No. 13611, 2007. [4] Diebold, F. X., and C. Li, Forecasting the term structuring of government bond yields, Journal of Econometrics, Vol. 130, 337 364, 2006. [5] Diebold, F. X., M. Piazzesi, and G. D. Rudebusch, Modeling Bond Yields in Finance and Macroeconomics, The American Economic Review, Vol. 95, 415 420, 2005. [6] Diebold, F. X., G. D. Rudebusch, and S. B. Aruoba, The macroeconomy and the yield curve: a dynamic latent factor approach, Journal of Econometrics, Vol. 131, 309 338, 2006. 29
[7] Duffie, D., and R. Kan, A Yield-Factor Model of Interest Rates, Mathematical Finance, Vol. 6, No. 4, 379 406, 1996. [8] Frye, J., Principals of Risk: VAR through Factor-Based Interest Rate Scenarios, VAR : understanding and applying value-at-risk, 275 287, 1997. [9] Litterman, R., and J. Scheinkman, Common factors affecting bond returns, Journal of Fixed Income, Vol. 1, 54 61, 1991. [10] McCulloch, J. H., Measuring the term structure of interest rates, The Journal of Business, Vol. 44, 19 31, 1971. [11] McCulloch, J. H., The tax-adjusted yield curve, The Journal of Finance, Vol. 30, 811 830, 1975. [12] Nelson, C. R.and A. F. Siegel, Parsimonious Modeling of Yield Curves, The Journal of Business, Vol. 60, No. 4. 473-489, 1987. [13] Piazzesi, M., Bond Yields and the Federal Reserve, Journal of Political Economy, Vol. 113, no.2, 311 344, 2005. [14] Steeley, J. M., Estimating the gilt-edged term structure: basis splines and confidence intervals, Journal of Business Finance and Accounting, Vol. 18, 513 529, 1991. [15] 2007 2007. 30