004 May, 0 pp99-8 leehero00@yahoo.com.w 00 4 00 4 003 8 3 45 ( ) ADF ( ) Granger ( ) (VAR) (SSM) : Sudy o an invesigaion o he relaionship among Taiwan s Spo, Fuures and Opions prices -The Applicaion o Sae Space Model Absrac This paper Sudy o an invesigaion o he relaionship among Taiwan s uures, spo and opions prices Daily daa o hese series are colleced rom December 4,00 o Augus 3, 003 The research procedure includes ideniicaion o saionaliy o a ime series or each variable using Augmened Dickey-uller es (ADF) causaliy es using Granger s mehodology esimaion o vecor auoregressive model (VAR) and Sae Space Model (SSM) The resuls show ha many o he variables under invesigaion are inerrelaed o each oher In order o obain beer research resuls and predicion accuracy i is imporan o markes Keywords Granger es Impulse response analysis Forecas error variance Vecor auoregressive es Sae space model - 99 -
004 998 7 Taiwan Fuures Exchange, TAIFEX TAIFEX Taiwan Sock Index Fuures 00 4 expiraion eec S&P 500 ( ). Kawaller, Koch and Koch 987 S&P500 984 3 985 0~45-00 -
. Soll and Whaley 987 S&P500 985 ~985 3. Soll and Whaley 99 S&P500 985 ~989 6 Boosrapping 4. Edwards 998 S&P500 97 ~987 F-saisic 5. Lee Chun I 999 MMI NSE S&P500 sock index uures 5 986 ~99 7 U 0 ( ) 00 4. 86 SIMEX 5 997 9 ~997 0 ECM. 87 NIKKEI 5 5 993 ~997 6 997 ~997 6 GARCH GMM 5 5 3. 87 SIMEX TAIFEX Ljung-Box GARCH. TAIFEX SIMEX SIMEX 4. 89 998 7 ~999 9 0 GARCH - 0 -
004 5. 90 995 5 ~000 5 0 GJR 6. 90 SIMEX TAIFEX 500 GARCH. 50 500 7. 90 SIMEX TAIFEX TSE 5 998 7 ~999 3 0 GARCH / / - 0 -
00 4 003 8 3 0 45 ( ) R S, ( p = s, p P S, s, ) R S, P S, P S, - - ( ) R F, ( p = F, p P F, F, ) R F, : P F, P F, - - ( ) R C, ( p = C, p P C, C, ) R C, : P C, P C, - - - 03 -
004 3-3- Engle & Granger(987) (inegraion) X d (inverible) ARMA(Auoregressive Moving Average) d I(d) I(0) I() I(0) X=0 I() X=0 (uni roo es) - 04 -
ADF(Augmened Dickey-Fuller) DF (whie noise) Dickey-Fuller(98) Said & Dickey(984) X- ARMA DF X m + ρ X + ε i= = β X () X m + ρ X + ε i= = α + βx () X m + ρ X + ε i= = α + γt + βx (3) DF Dickey-Fuller(98) Granger Granger Granger Granger(980) ( ) X X X = { X =,,3..., } j = { =,,3..., } j = { X = 0,,,..., } j = { = 0,,,..., } j ( ) δ ( x X, ) < δ ( x X ) y (cause)x x δ (mean square error o orecasing) ( ) δ ( x X, ) < δ ( x X ) y (conemporaneous) x - 05 -
004 ( ) δ ( x X, ) < δ ( x X ) δ ( y X, ) < δ ( y ) y (cause)x x (cause) y (eedback eecs) ( ) δ ( x X, ) = δ ( x X, ) = δ ( x X ) δ ( x X, ) = δ ( y X, ) = δ ( y ) x y (independen) ( ) δ ( x X, ) < δ ( x X ) δ ( y X, ) > δ ( y ) y (cause)x x y (unidirecional cause and eec relaionship) y x (VAR) Sims(980) (VAR) (prior) (srucural model) (VAR) VAR Sim(980) VAR n = α + A µ (3-6-) i= i + i ' E ( µ ) = 0 E ( µ µ ) = 0 E ( µ iµ s ) = 0 (3-6-) (n ) (joinly covariance saionary ) (linearly sochasic process) µ (n ) (orecas error) (shock, innovaion or impulse) Ai (n ) m E ( µ iµ s ) = 0 ' E ( µ µ ) = 0-06 -
( ) (3-6-) (variance decomposiion) Sims (3-6-) Wold (Wold decomposiion heorem) (moving average) : = m m α + Aj j+ ε A j j = α + ε j= j= ( I A L ) m A L... Am L = α + ε = α ( I A L m m A L... Am L ) + ε ( I A L A L... Am L ) L lag operaor i= 0 = α ' + ciεµ ( 3-6-) α ' (n x ) c i (n x n) c 0 = I ( ) (3-6-) ( µ ) (conemporaneously uncorrelaed) µ µ ( ) VAR D i (innovaion) (3-6-) k (k-sep-ahead error) : Eˆ µ µ µ D W (3-6-3) k = c0 + c +... + ck k + = D0W + DW +... + (VAR) (MA) ARMA Vecor ARMA (Sae Space Model) k k + - 07 -
004 (SSM) (VAR) (MA) Granger & Newbold (990) (g x ) I X (s x ) E( I) X X = β + γ Z V ( g ) ( g s) ( s ) ( g u) ( u ) ( g ) X = τ X δ W µ ( s ) ( s s) + ( s ) + ( g ) ( ) + Ψ ( s m) ( m (3-7-) ) (3-7-) Z W V U X X (3-7-) (Sae Vecor) X Z V (3-7-) X Akaike(974) (ARMA) ARMA s-r Akaike ARMA(p q) Φ ( B ) = Θ ( B ) ε Φ... Φ p p = ε + Φ ε +... + Φ p (3-7-3) (3-7-3) B ( B = - ) ε ; Φ( B) = Θ( B), Φ (0) = Θ(0) = I (3-7-3) = Φ ( B) Θ( B) ε = s= 0 ψ ε s s (3-7-4) ψ (3-7-4) s (Impulse response marix) : = s= i ψ ε s + i s = + i + ψ i ε + (3-7-5) (3-7-6) - 08 -
(3-6-4) + p = Φ + p +... + Φ p + ε + p + Θε + p +... + Θ qε + p q (3-7-7) p>q ε ε 0( + I =0 I>0 ) i = p +p (3-7-6) +... + + + P + 0 0. =.. Φ p Φ 0... p Φ 0... p......... 0 0 +...... Φ + p I ψ. +.. ψ p ε + µ + Z + = FZ + Gε + (3-7-8) Z (sx) F (sxs) (Transiion Marix) G (sxr) (Inpu Marix) (Impulse Response Marix) ε + r Σ 0 Akaike(976) Z Granger Newbold(986). (VAR) AIC K AIC(Akaike s Inormaion Crierion) = nin( p ) pr (3-7-9) AIC p + p r n. (Sae Vecor) p = (,,..., k ) = (, +,..., + k ) (Canonical correlaion) Χ - 09 -
004 3. (.96) 4. (.96) 00 4 003 8 3 45 E-Views ( ) ADF(Augmened Dickey Fuller) ( ) Granger ( ) (VAR) (SSM) ADF H 0 3 SPR FPR CPR 45 4-- (4 ) Mackinnon Criical Value ADF Tes ( ) Saisic 0% 5% % SPR -.5706 -.8688-3.4484 45-8.86499 * FPR -.5706 -.8688-3.4484 45-9.677 * CPR -.5706 -.8688-3.4484 45-0.03568 * * 5% - 0 -
3 SPR FPR CPR 5% H 0 Granger VAR Vecor Auoregressive Granger VAR 6 AIC Akaike Inormaion Crierion VAR ADF VAR AIC, VAR 4 4-4- AIC Lag 3 4 5 6 AIC -9.06768-9.0703-9.09599-9.996* -9.088089-9.088375 * VAR (Impulse Response analysis) (Variance Decomposiion). SPR SPR. SPR CPR SPR CPR 3. FPR SPR FPR 4. FPR FPR 5. CPR CPR - -
004. (00%) (0%). (88.48%) (.5%) (0%) 3. (99.35%) (0.64%) (0.00073%) SSM SAS ( Sae Space Model, SSM) Granger & Newbold(986) (VAR) AIC(Akaike Inormaion Crierion) K (Canonical Correlaions Analysis) (Sae Vecor) > F AR G MA R S (T;T), R F (T;T), R C (T;T), R S (T+;T), R F (T+;T),R C (T+;T) 4-6- AIC χ R S (T;T), R F (T;T), R C (T;T), R S (T+;T) * -3.4557 *0.398 R S (T;T), R F (T;T), R C (T;T), R S (T+;T), R F (T+;T) -.4694 0.3907 R S (T;T), R F (T;T), R C (T;T), R S (T+;T), -3.45 6.49574 0 R F (T+;T),R C (T+;T) R S (T;T), R F (T;T), R C (T;T), R S (T+;T), -.588 6.348565 9 R F (T+;T),R C (T+;T), R S (T+;T) R S (T;T), R F (T;T), R C (T;T), R S (T+;T), -.045 5.90597 9 R F (T+;T),R C (T+;T), R F (T+;T) R S (T;T), R F (T;T), R C (T;T), R S (T+;T), R F (T+;T),R C (T+;T), R C (T+;T) -.938 6.0036 9 + + * - -
(Transiion Marix) (Inpu Marix) T T (.96 ) (.96 ) T 4-6- Sae Vecor RS (T ; T) RF (T;T) RC (T;T) RS (T+;T) RF (T+;T) RC (T+; T) Esimae o Transiion Marix 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0-0.5863 0.34646 0.003 0 0.73504 0.08066 0 0 0 37.6403-9.9743 0 Inpu Marix or Innovaion 0 0 0 0 0 0 0 0 0 0.577405-0.550 0 0 0-0.748 SSM Model - 3 -
004 4-6-3 SSM Model Parameer Esimae Sandard Error T-Value G(5,) 0.577405 0.09844 5.88 G(5,) -0.550 0.08703-6.03 G(6,3) -0.748 0.04430 -.66 F(5,) -0.5863 0.33 -.3 F(5,) 0.346460 0.4766.35 F(5,3) 0.003 0.0043.0 F(5,5) 0.73504 0.35995 3. F(5,6) 0.08066 0.00775.33 (Transiion Marix) ( Iupu Marix) (Innovaion) R R R R R R S, + F, + C, + S, + F, + C, + 0 0 0 0 0 R 0 0 0 0 0 R 0 0 0 0 0 R 0 0 0 0 0 0 RS 0.590.346 0.00 0 0.735 0.08 RF 0 0 0 0.096 9.974 0 RC S, F, C,, +, +, + 0 0 ε 0 0 0 0 ε F 0 0 0 0.577 0.55 0 0 0 07 ε R S, + = R S, + ε S, + (4-6-4) R F, + = R F, + + ε F, + (4-6-5) S,+, + C,+ - 4 -
R C, + =R C, + + ε c, + (4-6-6) R F, + = -0.59 x R S, +0.346 x R F, + 0.00 x R S, + +0.735 x R F, + +0.08 x R C, + + 0.577 x ε S, + - 0.55 x ε F, + (4-6-7) R C, + = 0.096 x R S, + + -9.974 x R F, + -0. x ε c, + (4-6-8) ( ) + (4-6-4) + + + + ( ) + (4-6-5) + + + + ( ) + (4-6-6) + + + + ( ) + (4-6-7) + + + + + + + + + - 5 -
004 ( ) + (4-6-8) + + + + + + 4-6-9 4-6-9 R S, R F, R C, R S, + R F, + R C, + ε S, + ε F, + ε c, + R S, +/ + R F, +/ + - + + + + + - R C, +/+ + - + - Aksu&Gunay(995) (Error cross-corelaion maraix) 4-6-0 4-6-0 R S, R S, R F, R C,.00000 0.94336 0.08633 (0.00000) (<.000 ) (0.0790) R F, 0.94336 <.000 R C, 0.08633 (0.0790).00000 (0.00000) 0.0844 (0.0866) 0.0844 0.0866.00000 (0.00000) - 6 -
Granger VAR Granger SSM SSM - 7 -
004.. GARCH Model 3. 4. -Nikkei 5 9 3 pp.9-6 5. 6. 47 7. GARCH SIMEX TAIFEX TSE / 4 pp.- 8. 9 3 pp.-8 9.. Edwards, F. R., Does uures rading increase sock marke volailiy? Financial Analyss Journal, 988, pp.63-69.. Kawaller, I. G., P. D. Koch, and T. W. Koch, The Temporal Price Relaionship Beween S&P 500 Fuures and he S&P 500 Index, Journal o Finance, 987, pp.309-39. 3. Lee, Chun I, The inluence o inormaion arrival on marke Microsrucure : Evidence rom hree relaed markes, The Financial Review, Feb. 999, Vol. 34, Iss., pp. -6. 4. Soll, H. R. and R. E. Whaley, Program rading and expiraion-day eecs, Financial Analyss Journal, 987(March-April), pp.6-8. 5. Soll, H. R. and R. E. Whaley, Expiraion-day eecs: wha has changed? Financial Analyss Journal, 99(January-February), pp.58-7. - 8 -