1 Nelson-Siegel Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel Litterman and Scheinkman(199
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1 Nelson-Siegel Nelson-Siegel Nelson and Siegel(1987) 2 FF VAR 1996 FF B)
2 1 Nelson-Siegel Nelson and Siegel(1987) 3 Nelson-Siegel 3 Nelson-Siegel 2 3 Nelson-Siegel 2 Nelson-Siegel Litterman and Scheinkman(1991) Frye(1997) 2 Nelson-Siegel Diebold and Li(2006) Diebold, Rudebusch and Aruoba(2006) 2
3 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
4 X Y Z 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
5 λ L t S t C t ˆL t (ˆλ), Ŝt(ˆλ), Ĉt(ˆλ) : (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 C(m) 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) λ =
6 L t : 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 S t S(m) C t C(m) 24 (S t S(m)) 60 C (C t C(m)) <m 0 <L t + S ts(m)+c tc(m)
7 : S S t S(m) 2,5,10,15 (S ts(m)) : C C t C(m) 2,5,10,15 (C tc(m)) 3 L t S t C t 3 x t 7
8 FF 14 Nelson-Siegel 3 Nelson-Siegel 3.2 FF L S C 3 VAR 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 GDP 8
9 Ang and Piazzesi(2003) VAR Diebold, Rudebusch and Aruoba(2006)( DRA(2006)) Nelson-Siegel 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 ) AIC CPID t : CPI CPI IIPD t : 2000 BDR t : TOPIXD t : ( 15 (2007)
10 US10 t : FF t : FF 18 TDUMMY t t L t CPID t BDR t L t 1 u t L t = BDR t CPID t 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 = (US10 t FF t ) TOPIXD t + u t ( 6.820) ( 1.986) ( 2.303) u t = (41.510) u t 1 + v t C t FF t IIPD t IIPD t TDUMMY 99 2 S t C t = FF t (IIPD t TDUMMY t )+u t ( 5.278) (2.021) (2.542) u t = (23.942) u t 1 + v t = 1 1: L S C = effective federal funds rate( ) 10
11 L t S t C t TOPIX TOPIX 1999 FF : 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 j=0 j=0 j=0 11
12 7 7 FF : 99/05-00/ :
13 3.3 Q (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 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
14 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 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 AIC CPID t :CPI CPI( ) IIPD t : 2000 BDR t : TOPIXD t : US10 t : 10 FF t : FF US10 t FF t : 14
15 2: F, µ F Const. L t S t C t µ L t * * (0.087) (0.079) (0.016) (0.092) S t * * (0.107) (0.098) (0.020) (0.114) C t * (0.184) (0.168) (0.035) (0.193) * 5% 3: Q Q L t S t C t L t (0.196) S t (1.516) C t (0.276) 4: σ 2,λ,c σ 2 λ c u = exp(c) * (0.041) (0.195) (1.000) (0.001) 15
16 AIC CPID t IIP t TOPIXD t BDR t FF t AIC L t,s t,c t : 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 IIPD t S t+1 TOPIX S t+1 TOPIXD t TOPIX C t+1 TOPIXD t TOPIXD t
17 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 * * * (0.143) (0.123) (0.020) (0.103) (0.983) (0.348) (0.001) (0.172) (0.018) S t * * * * (0.173) (0.149) (0.025) (0.126) (0.988) (0.416) (0.001) (0.210) (0.022) C t * 0.635* 0.889* * * (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 (0.038) S t (0.091) C t (0.508) 7: σ 2,λ,c σ 2 λ c u = exp(c) * (0.001) (0.021) (1.000) (0.000) 17
18 u 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 FF 1 FF FF
19 k : FF 1 FF VaR Ahrend, Catte and Price
20 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 H ˆµ Mean 22 H ˆµ 9 0.5% CPI % 22 (14) f u(h ˆx t) u H ˆx t 20
21 VaR : ( ) VAR 1996 VAR : ( ) 21
22 12 TOPIX IIPD CPID FF VAR TOPIXD FF IIP : ( ) : ( ) TOPIXD
23 IIPD CPID : ( ) : ( ) 23
24 : ( ) VAR FF : ( ) 24
25 : ( ) 5 Nelson-Siegel TOPIX 10 FF FF
26 VAR
27 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
28 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
29 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, [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, , [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 , [4] Diebold, F. X., and C. Li, Forecasting the term structuring of government bond yields, Journal of Econometrics, Vol. 130, , [5] Diebold, F. X., M. Piazzesi, and G. D. Rudebusch, Modeling Bond Yields in Finance and Macroeconomics, The American Economic Review, Vol. 95, , [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, ,
30 [7] Duffie, D., and R. Kan, A Yield-Factor Model of Interest Rates, Mathematical Finance, Vol. 6, No. 4, , [8] Frye, J., Principals of Risk: VAR through Factor-Based Interest Rate Scenarios, VAR : understanding and applying value-at-risk, , [9] Litterman, R., and J. Scheinkman, Common factors affecting bond returns, Journal of Fixed Income, Vol. 1, 54 61, [10] McCulloch, J. H., Measuring the term structure of interest rates, The Journal of Business, Vol. 44, 19 31, [11] McCulloch, J. H., The tax-adjusted yield curve, The Journal of Finance, Vol. 30, , [12] Nelson, C. R.and A. F. Siegel, Parsimonious Modeling of Yield Curves, The Journal of Business, Vol. 60, No , [13] Piazzesi, M., Bond Yields and the Federal Reserve, Journal of Political Economy, Vol. 113, no.2, , [14] Steeley, J. M., Estimating the gilt-edged term structure: basis splines and confidence intervals, Journal of Business Finance and Accounting, Vol. 18, , [15]
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