004 6-3 CARR(Condiional Auo-Regression Range) CARR GARCH CARR GARCH Chou(00) S P500 CARR CARR GARCH
004 6 volailiy Hull Whie(987) vega( kappa) innovaion erm Poerba Summers(986) French, Schwer Sambaugh(987) Bollerslev, Engle Wooldridge (988) Bailie Degennaro(990) Andersen Bollerslev(997) Alizadeh, Brand Diebold(00) Morgan(976) heeroscedasiciy Mandelbro(963) Fama(965) lepokuric fa ail Mandelbro(963) clusering Cassuo(995) Engle(98) ARCH(Auo-Regression Condiional Heeroskedasiciy) Bollerslev(986) GARCH(Generalized Auo-Regression Condiional Heeroskedasiciy) vega kappa
3 (parsimony) ARCH/GARCH Bollerslev, Chou Kroner(99) Bollerslev, Engle Nelson(994) range proxy Mandelbro(963) Parkinson(980) Brand Jones(00) EGARCH Chou(00) CARR(Condiional Auo-Regressive Range) GARCH CARR dominance S&P500 Brand Jones(00) Chou(00) Chou(00) Brand Jones(00) CARR CARR Chou(00) Chou(00) CARR
4 004 6 ARCH/GARCH Hull Whie(987) (Sochasic Volailiy) Parkinson (980) Chou(00) Gallan, Hsu Tauchen(999) Alizadeh, Brand Diebold(00) Chou(00) GARCH CARR GARCH Brand Jones(00) EGARCH EGARCH CARR Engle Russell (998) ACD (Auoregressive Condiional Duraion) CARR CARR 3 GARCH CARR CARR(p,q) () R α R ε ~ iid f (.) = λ ε λ = ω + p + q i i i= = β λ () range duraion
5 High Low R P R = ln P ln P λ R λ E( R I ) λ 0 ε ω w > 0 α i α i 0 i =,..., p β β 0 =,..., q p α + i= i q = β CARR α + β p i= i q = < ω = ω /( ( p i= q α i + β ) α i + β = p i= q = ( ω ) Chou(00) ε CARR (log likelihood funcion) 4 i T,..., RT ) = [ln( λ ) + ] = λ R L( ω, α, β ; R () CARR (3) p q α i R i + β λ + i= = L λ = ω + φ X (3) l= l, l 3 4 Engle Russell(998)
6 004 6 CARRX (3) ( ) (leverage effec) 5 CARR (Quasi-Maximum- Likelihood Esimaion mehod QMLE) Engle Russell(998) QMLE Bollerslev Wooldridge(99) Robus 6 (realized volailiy) (Sum of Square Daily Reurn SSDR) (Weekly-Reurn-SQuared WRSQ) (Weekly RaNGe WRNG) (Absolue Weekly -RETurn AWRET) CARR GARCH CARR GARCH MV = γ + FV ( CARR) + u (4) 0 γ MV = γ + FV ( GARCH ) + u (5) 0 γ 0 + γ FV ( CARR) + γ MV = γ FV ( GARCH ) + u (6) 5 Black (976) Chrisie(98) leverage effec D E raio 6
7 MV (measured volailiy) (proxy) FV (CARR) FV (GARCH ) CARR GARCH (4) γ γ 0 ˆ γ CARR γ 0 CARR ˆ (4) (5) Ad.R-squared CARR GARCH (6) γ γ CARR GARCH 0 ˆ0 γ CARR GARCH CARR(,) k 7 λ f, + = E( λ+ I ) = ω + αr + ˆ ˆ ˆ βλ λ = ( λ ) ˆ ˆ ( ) ˆ I = ω + αe R I βe( λ f, + E + + + + I = ˆ ω + ˆ αe ( λ ) ˆ + I + βe( λ+ I = ˆ ω + ( ˆ α + ˆ) β E ( λ + I ) = ˆ ω + ( ˆ α + ˆ) β ˆ ω + ( ˆ α + ˆ) β λ ) ) 7 λ
8 004 6 k ˆ ω( ( ˆ α + ˆ) β ) λ ) ( ˆ α + ˆ) β k f k k E λ k I ˆ α ˆ, + = ( + ) = + ( + β λ (7) λ f, + k k GARCH(,) (8) ˆ ω( ( ˆ α + ˆ) β ) h ) ( ˆ α + ˆ) β k f k k E h k I ˆ ˆ, + = ( + ) = + ( α + β h (8) h f, + k k Roo Mean Square Error RMSE Mean Absolue Error MAE CARR GARCH = T +, T +,..., T +τ V FV (9) (0) = T + T + τ RMSE = ( FV V ) (9) τ MAE = τ T + τ = T + FV V (0) V MV CARR GARCH FV forecased volailiy MV (scale) FV adused forecased volailiy AFV () MV =ψfv + u =,..., T () AFV = ψfv
9 988 6 3 003 0 3 795 8 CARR GARCH CARR range GARCH reurn ( ) high low = 00 [(ln( P ) ln( P )] close close = 00 [ln( P ) ln( P )] WRNG WRET 795 794 6.304 0.0 5.336 0.376 3.865.06 0.5-5.34 3.89 4.86.60-0.333 0.00 6.477 Jarque-Bera 50.89(0.000) 44.50(0.000). Jarque-Bera p-value. WRNG WRET 3. 988 6 3 003 0 3 795 4. 8 988 6 3 799 4 994 998 00 00
0 004 6 Jarque-Bera(JB) 9 JB 50.89 JB 44.50 JB 3 lepokuric CARR GARCH 40 30 0 0 0-0 -0-30 00 00 300 400 500 600 700 WRNG WRET 9 Jarque-Bera= N ( S + ( K 3) ) χ ( ) 6 4 N S K α = 5% χ α () = 5. 99 95% Jarque-Bera 5.99
CARR CARR GARCH CARR GARCH 988 6 3 003 0 3 795 CARR GARCH LR Likelihood Raio es 0 CARR GARCH CARR CARR(,) CARR(,) CARR(,) LR es CARR(,) LLF(Log Likelihood Funcion) -835.809 CARR(,) -835.308 CARR(,) LLF -835.30 CARR(,) β 0.07 0.0 CARR(,) α -0.0-0.95 0 CARR(,) 0 LR = ( L L ) ~ null alernaive χ ( k) k k= α = 5% LR χ () 3. 84 > α =
004 6 988/06/03~003/0/03 CARR R ε ~ iid = λ ε λ = ω + p α R i i i= = f (.) + q β λ log likelihood func. CARR(,) CARR(,) CARR(,) -835.30-835.809-835.308 ω 0.387(3.409) 0.387(3.37) 0.37(.84) α 0.63(8.394) 0.65(6.47) 0.67(6.54) α -0.0(0.95) β 0.674(6.744) 0.655(3.498) 0.684(.63) β 0.07(0.0) ρ 0.04 0.0 0.00 ρ 0.050 0.049 0.049 Q() 7.30(0.836) 7.365(0.833) 7.399(0.830). R. -value wih robus sandard error p-value 3. ρ ρ 4. Q() Q 5. GARCH 3 GARCH(,) GARCH(,) GARCH(,) 3 LR es GARCH(,) LLF -94.343 GARCH(,) -94.38 GARCH(,) LLF -94.350 GARCH(,) β 0.030 0.08 GARCH(,) α -0.04-0.36 0 GARCH(,) GARCH
3 3 CARR(,) GARCH(,) CARR GARCH CARR(,) α 0.63 GARCH(,) α 0.3 CARR GARCH 3 988/06/03~003/0/03 GARCH y h ε = ε = ω + Ω p q α iε i + i= = ~ N(0, h ) β h GARCH(,) GARCH(,) GARCH(,) log likelihood func. -94.350-94.343-94.38 ω.63(.98).83(.76).098(.877) α 0.3(3.798) 0.34(.36) 0.39(.43) α -0.04(0.36) β 0.85(6.577) 0.78(.808) 0.85(6.044) β 0.030(0.08) ρ 0.036 0.036 0.036 ρ -0.7-0.07-0.08 Q().388(0.045).448(0.044).555(0.043). y. -value wih robus sandard error p-value 3. ρ ρ 4. Q() Q 5.
4 004 6 CARR(,) GARCH(,) 4 CARR GARCH FV (CARR) FV (GARCH) FV (CARR) FV (GARCH) MV SSDR WRSQ WRNG AWRET 0 FV (GARCH) γ -value -0.8 -.05-0.40 -.34 FV (CARR) γ 0 CARR(,) MV GARCH(,) FV (CARR) FV (GARCH) Ad.R-squared MV SSDR WRSQ WRNG AWRET FV (CARR) 0.56 0.97 0.453 0.77 FV (GARCH) 0.468 0.36 0.395 0.44 FV (CARR) FV (GARCH) MV CARR GARCH Chou(00) S&P500 S&P500 CARR GARCH
5 4 CARR(,) GARCH(,) MV = γ + FV ( CARR) + u 0 γ MV = γ + FV ( GARCH ) + u 0 γ MV = γ FV ( GARCH ) + u 0 + γ FV ( CARR) + γ MV γ 0 γ γ Ad.R-squared Durbin-Wason SSDR 0.57 (0.484) 0.49 (3.885) 0.56.73 SSDR 0.088 (0.06) 0.859 (.994) 0.468.39 SSDR 0.668 (0.484) 0.44 (6.33) -0.09 (-0.8) 0.56.79 WRSQ -.867 (-0.369) 0.56 (4.65) 0.97.885 WRSQ 0.588 (0.30) 0.99 (4.05) 0.36.807 WRSQ.0 (0.557).54 (3.08) -.37 (-.05) 0..838 WRNG -0.355 (-0.87).060 (4.96) 0.453.807 WRNG -0.594 (-.98).56 (3.883) 0.395.504 WRNG -0.87 (-0.68).0 (6.533) -0.096 (-0.40) 0.453.85 AWRET -0.055 (-0.5) 0.575 (6.85) 0.77.996 AWRET -0.063 (-0.) 0.795 (6.47) 0.44.944 AWRET 0.43 (0.30) 0.750 (3.935) -0.86 (-.34) 0.78.985. Whie Heeroskedasiciy-Consisen Sandard Errors & Covariance -value. 4 MV(Measured Volailiy) (calendarweek) (Sum of Square Daily-Reurn SSDR) (Weekly-Reurn-SQuared WRSQ) (Weekly RaNGe WRNG) (Absolue Weekly-RETurn AWRET) 3. FV (CARR) FV (GARCH ) CARR(,) GARCH(,) MV SSDR WRSQ MV WRNG AWRET 4. 988 6 3 003 0 3 795 5. 4 Durbin-Wason u FV (CARR) FV (CARR) FV (GARCH) Durbin-Wason
6 004 6 CARR(,) GARCH(,) CARR(,) GARCH(,) GARCH 644 CARR(,) GARCH(,) 645 646 696 5 645 5 3 647 4 648 00 743 3 00 5 FV(CARR) FV(GARCH) MV RMSE MAE 5 5 CARR RMSE MAE GARCH CARR GARCH 5 00
7 5 RMSE = T + 00 00 = T + ( AFV MV ) MAE = 00 + 00 T = T + AFV MV RMSE SSDR WRSQ WRNG AWRET CARR GARCH CARR GARCH CARR GARCH CARR GARCH 4.63 6.00 47.03 48.358.396.566 3.0 3.80 6.008 7.97 50.43 5.865.637.848 3.53 3.354 3 5.787 7.98 50.059 5.74.653.898 3.44 3.35 4 5.755 7.464 45.480 45.779.644.840 3.070 3. 3 3.46 4.694 43.804 44.96.498.674.983 3.046 6.873 3.99 43.508 43.677.46.495.943.958 5.00.776 40.939 4.078.35.467.809.839 MAE SSDR WRSQ WRNG AWRET CARR GARCH CARR GARCH CARR GARCH CARR GARCH 0.389.96 4.46 4.783.799.948.60.37.0.678 5.43 5.76.950.48.83.366 3.9.797 5.475 5.9.0.9.99.373 4.373.40 3.440 3.554.06.7.05.60 3 9.758.067.03.878.885.999.70.7 6 9.343 0.84.60.948.775.90.7.66 5 9.085 0.333 0.55.367.848.04.09.68. 988 6 3 003 0 3 795. CARR(,) GARCH(,) 3. AFV Robusness Over The Couner OTC 996 4 003 0 3 40 3 6 6 3 405 3 998 00 00 996
8 004 6 RMSE CARR GARCH 6 OTC RMSE = T + 50 50 = T + ( AFV MV ) MAE = 50 + 50 T = T + AFV MV RMSE SSDR WRSQ WRNG AWRET CARR GARCH CARR GARCH CARR GARCH CARR GARCH 4.395 6.078 54.49 55.40.74.95 3.430 3.503 5.445 6.454 57.34 57.564.903 3.040 3.68 3.673 3 5.70 6.656 57.95 57.7.888 3.03 3.63 3.660 4 6.068 6.978 57.908 58.38.985 3.58 3.683 3.75 3 3.93 4. 30.670 3.4.456.655.905 3.04 6.85 3.434 8.539 8.58.596.704.76.740 5.68 3.66 9.636.975.873 3.30.43.55 MAE SSDR WRSQ WRNG AWRET CARR GARCH CARR GARCH CARR GARCH CARR GARCH 0.97.367 8.066 30.379.979.83.564.709.59.874 30.45 3.3.73.74.740.87 3.97.977 30.54 3.554.48.95.733.868 4.987 3.9 3.76 33.750.3.49.777.940 3.04.94 3.59 5.875.05.6.496.66 6 0.840.705.865 3.58.97.39.366.399 5 0.7.090 8.89 0.57.586.834.0.39. 996 4 003 0 3 40 OTC. CARR(,) GARCH(,) 3. AFV 50
9 CARR Chou(00) CARR 4 GARCH Black (976) Chrisie(98) leverage effec D E raio CARR CARRX () λ ω + αr + βλ φ( re ) () = + r φ TGARCH TCARR (3) LR es λ (3) = ω + αr + βλ + γkr K K=0 R () (3) 7 CARRX(,) φ -0.05-4.79 LR es LR 9.706 φ -0.05 CARR TCARR(,) 4
0 004 6 7 988/06/03~003/0/03 CARR CARR(,) CARRX(,) TCARR(,) log likelihood funcion -7400.566-7395.73-7398.035 ω 0.059(6.53) 0.063(6.636) 0.058(6.6) α 0.06(7.343) 0.0(7.43) 0.84(4.897) β 0.765(55.49) 0.767(56.446) 0.77(57.633) φ γ -0.05(-4.79) 0.03(3.8) ρ 0.04 0.045 0.038 ρ 0.09 0.03 0.00 ρ -0.00 0.00 0.00 Q().03(0.037).504(0.043) 7.53(0.3). -value wih robus sandard error p-value. ρ ρ 3. Engle(98) ARCH 0 ARCH Chou(00) GARCH Parkinson(980) CARR CARR CARR GARCH (candlesick plos) K
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