Eliza Buszkowska Universiy of Poznań, Poland Linear Combinaions of Volailiy Forecass for he WIG0 and Polish Exchange Raes Absrak. As is known forecas combinaions may be beer forecass hen forecass obained wih single models. The purpose of he research is o check if linear combinaion of forecass from models for of he WIG0 Index and differen currency exchange raes is a good soluion when searching for he bes forecass. We check if he forecasing models are highly correlaed wih response variable and poorly correlaed wih each oher so if hey fulfill he Hellwig assumpions. Key words: volailiy, forecass, linear regression, MCS JEL classificaion: C5, C53. INTRODUCTION According o Sock and Wason (004) he combinaion of he models generaes beer forecas hen he single model. A combinaion of forecass is a good choice when i is no possible o disinguish one dominan model (Timmermann 006). Anoher argumen for a combinaion is ha he combinaions of forecass are more sable hen individual forecass (Sock, Wason 004) The aim is o verify if linear combinaion of forecass of volailiy for WIG0 nad differen exchange raes are a good soluion when searching for bes volailiy forecasing models. We check if he forecass are highly correlaed wih response variable and poorly correlaed wih each oher so if hey fulfill he
Hellwig s assumpions. We compare he volailiy forecass wih daily realized volailiy. We invesigae he resuls for differen measures of realized volailiy and differen bes forecasing models for differen funcions of error.. FORECASTS COMBINATIONS The simples combinaion is linear wih he idenical coefficiens and he sum of he weighs equels one. N gyˆ h ; h, yˆ h,, j () N j where y ˆ h, is he forecas, and h, is he weigh The forecas error is defined by: c e h, yh g yˆ h, ; h, () The paramers of he opimal combinaions of he forecass in his case are he soluion of he following problem c * argmin E L e (3) W where L denoes mean squared error (MSE) loss. Under MSE he combinaion weighs only depend on he firs wo momens of he join disribuion of y h and y ˆ h, y yˆ h h ~ ˆ y h, y h, y ˆ h, yy h, ' yyˆ h, yy ˆˆ h, (4) For MSE Timmermann (006) obained he following opimal weighs:
3, (5) 0 y h, yˆ h, yy ˆˆ h yyˆ h, Consider he combinaion of wo forecass ˆy, yˆ. Le e ie denoe he forecas errors. Assume ~ 0,, Var e. and is heir correlaion e ~ 0 e, where Var,, e is he covariance beween e and e and The opimal weighs for his combinaion by Timmermann (005) have he form * *,. (6) The idenical weighs are opimal if he forecas variances are he same independenly of he correlaion beween forecass on condiion ha he forecass are unbiased (Timmermann 006). The naural example is he following scheme of wo forecass: / y. ˆ yˆ (7) When he forecas are unbiased Timmermann (006) propose he combinaion ha gives he inverse weighs o he forecass wih he assumpion ha he correlaion is zero:, - inv. inv (8) For N forecass one can assume 0, ni i,..., N o make he values of he combinaion forecass be in he inerval of values of he individual forecass. Le yˆ c yˆ yˆ, y yˆ e ~ 0,, y yˆ e ~,, so ˆy is he biased forecas and assume cove, e. Using he formulas e c e e
4 c. Timmermann obained So if MSE y MSE y ˆ, hen MSEy c MSEy. ˆ ˆ The condiion allways holds for. In his case he forecas of he combinaion of models doesn ouperform he unbiased forces of he simple model. Wha is more he bigger is he bias of he forecas he smaller is he advanage of he combinaion. If he forecass are biased hen idenical weighs are opimal when he forecas errors have he same variance and idenical correlaion beween forecass (Timmermann 006). The opimal weighs problem may be formulaed as he opimalizaion ask of minimalizaion of expeced forecas error variance E ee' e, where e y lyˆ wih he condiion ha he sum of weighs is one and he individual forecass are unbiased: min'. (0) e ' l. () where l is he vecor of ones. For he inverible covariance marix e Timmerman, (005) obains he following opimal weighs: l ' l l. c ˆ e e () The problem of he opimal combinaion can be solved as he following es H H 0 A A A B ˆ, h EL ˆ, f h, h, A A B ˆ, h E L ˆ, f h, h, : E L : E L. (3) (9)
5 The es saisic of Diebold-Mariano and Wes (DMW) can be used in he es. Le define he difference d A A B ˆ, h L ˆ, f h, h, L (4) Then he DMW es saisic is he following: DMW T where T d T T d avar ˆ. T d T Td, (5) (6) Under he null hypohesis he es saisic has normal disribuion. If y y y y ˆ (7) ˆ y y, y y y y y, cov ˆ ˆ ˆ yˆ (8) he opima model is he combinaion of forecass, Timmermann (006). Anoher scheme can be creaed on he base of he ranking of models by Aiolfi and Timmermann (006). Le The weighs od he combinaion are he following: R i be he posiion of he i-model in ranking. N ˆ / R i R i. (9) i 3. HELLWIG S IDEA In goood linear regression model:. explanaory variables are highly correlaed wih response variable.. explanaory variables are poorly correlaed wih each oher.
6 Wha is more big correlaions beween exsplanaory variables cause big paramers average errors. 4. DATA In he empirical invesigaion we used daily observaions of he WIG0 Index, from May 8, 00 ill May 8, 009 for model esimaion. On he nex 56 daa from 9 April 008 ill 8 May, 009 we calculaed day volailiy forecass. To evoluae he qualiy of our forecass we compared hem wih daily realized volailiy calculaed for 5, 0 and 30 minue inraday reurns. We considered he following ypes of GARCH (, ) wih differen disribuions of error: RiskMerics, GARCH, EGARCH, GJR, APARCH, IGARCH, FIGARCH-BBM, FIGARCH-CHUNG, FIEGARCH, FIAPARCH- BBM, FIAPARCH-CHUNG, HYGARCH. The models esimaed wih differen disribuions of error: GAUSSIAN, STUDENT-, and GED, SKEWED STUDENT 5. THE TEALIZED VOLATILITY The realized volailiy can be calculaed by summing he squares of inraday reurns. Wih he use of he equaion which allow for he nigh reurn i is defined as follow: N, r, i, i0 (0) where he inraday reurn in he day n and in he momen d is : r ln P P, ln P P 00 n, d n, d ln n, d r n, 0 00 n, ln n, N, () N is he numer of periods in a day. The alernaive approach was proposed by Andersen and Bollerslev in 997. They suggesed reprezening he daily volailiy as he sum of inraday reurns N, r, i. i ()
7 They sugges muliplying by c,, where c is he posiive consan (Marens 00). They choose Var r ) and Var ( ), co (, 0 realized volailiy can be expressed: oc N oc co 3, r, i oc i as he consan c, where co N r, n oc Koopman i e al, oc 005. Then he (3) In he aricle MSE means he mean squared error and MAD means mean absolue deviaion, where N is he number of forecass. N MSE = N MAD= N N ˆ, (4) l, k, ˆ, (5) l, k, where l,, 3, k,, m is he numer of models from he considered se. In he following formula is he forecas of volailiy from he model k ˆ k, on he momen, l, is he value of he realized volailiy of he ype l in he momen. 6. EMPIRICAL RESULTS The bes models obained wih Model Confidence Se mehod (MCS) for MAD loss funcion, realized volailiy of reurns are: GARCH (,) wih Gaussian disribuion of error AR()-GARCH wih Gaussian disribuion of error,, and 5 minue frequency,, 3,
8 3 MA()-GARCH wih Gaussian disribuion of error 4 HYGARCH wih Gaussian disribuion of error 5 AR()-HYGARCH wih Gaussian disribuion of error 6 MA()-HYGARCH wih Gaussian disribuion of error The marics of correlaions: Table. The values of correlaions beween forecass 3 4 5 6 0.999678 0.99969 0.98593 0.9833 0.98393 0.999999 0.984996 0.9884 0.98866 3 0.98498 0.9878 0.98836 4 0.99953 0.99967 5 0.999999 6 The bess model obained wih MCS mehod for MSE loss funcion, realized volailiy,, and 5 minue frequency of reurns is:,, RiskMerics wih skewed Suden disribuion of error. 3, The MCS for MAD, realized volailiy frequency of reurns is:, GARCH (,) wih Gaussian disribuion of error AR()-GARCH wih Gaussian disribuion of error 3 MA()-GARCH wih Gaussian disribuion of error 4 HYGARCH wih Gaussian disribuion of error, and 0 minue 3, The marics of correlaions
9 Table. The values of correlaions beween forecass 3 4 0.999678 0.99969 0.98593 0.999999 0.984996 3 0.98498 4 The MCS for MSE, realized volailiy,, and 0 minue,, 3, frequency of reurns is:. FIGARCH wih GED. AR()-RiskMerics wih Gaussian disribuion of error 3. RiskMerics wih skewed Suden disribuion of error 4. GARCH wih skewed Suden disribuion of error The marics of correlaions : Table 3. The values of correlaions beween forecass 3 4 0.985936 0.987563 0.987633 0.9997 0.995763 3 0.996994 4 The bes models obained wih MCS mehod for MAD loss funcion, realized volailiy,, and 30 minue frequency of reurns are: 3, GARCH (,) wih Gaussian disribuion of error AR()-GARCH wih Gaussian disribuion of error 3 MA()-GARCH wih Gaussian disribuion of error 4 HYGARCH wih Gaussian disribuion of error
0 5 AR()-HYGARCH wih Gaussian disribuion of error 6 MA()-HYGARCH wih Gaussian disribuion of error The marics of correlaions : Table 4. The values of correlaions beween forecass 3 4 5 6 0.999678 0.99969 0.98593 0.9833 0.98393 0.999999 0.984996 0.9884 0.98866 3 0.98498 0.9878 0.98836 4 0.99953 0.99967 5 0.999999 6 The bes models obained wih MCS mehod for MAD loss funcion, realized volailiy and 30 minue frequency of reurns are:,. GARCH wih GED. FIGARCH wih GED 3. ARNA(,) GARCH wih GED 4. GARCH wih skewed Suden The marics of correlaions: Table 5. The values of correlaions beween forecass 3 4 0.987 0.999998 0.999795 0.9876 0.987633 3 0.99985 4
The MCS for MSE, realized volailiy,,,, and 30 minue 3, frequency of reurns is:. AR() RiskMerics wih Gaussian disribuion of error. RiskMerics wih skewed Suden disribuion of error The marics of correlaions: Table 6. The values of correlaions beween forecass 0.9997 7. THE ESTIMATES OF THE PARAMETERS OF THE BEST MODELS Table 7. The esimaes of he parameers of he bes models Model GARCH AR()- GARCH MA()- GARCH HYGARCH AR()- HYGARCH MA()- HYGARCH Disribuion Gauss Gauss Gauss Gauss Gauss Gauss Parameers 0.077 (0.03309) 0.0754 (0.03499) 0.076 (0.03486) 0.0584 0.05588 a (0.04) (0.0349) b (0.0398) (0.035) -0.0539-0.05554 0.07077 (0.056) 0.0695 (0.054) 0.06933 (0.054) 0.77 (0.37) 0.3065 (0.9) 0.3096 (0.94) 0.06009 0.06007 0.06006-0.05078-0.50707-0.50696 (0.03) (0.08) (0.08) (0.095) (0.08936) (0.08956) 0.9083 (0.03599) 0.90895 (0.03475) 0.9089 (0.03477) 0.686 (0.03) 0.6759 (0.0977) 0.67568 (0.09744 0.86847 0.8785 0.8783
k (0,07) (0,074) (0,073) 0.59709 0.598 0.59698 d (0.0658) (0.0697) (0.066) Table 8. The esimaes of he parameers of he bes models Model GARCH(,) FIGARCH(,d,) ARMA(,) - GARCH(,) GARCH(,) Disribuion GED GED GED skewed Suden - Parameers 0.06983 (0.03343) 0.7860 a (0.383) b (0.3456) 0.7860 0.050 (0.0048) 0.08545 (0.095) 0.0466 (0.059) 0.03765 (0.09) -0.47563 0.0557 0.05744 (0.3607) (0.00976) (0.0099) 0.7345 0.95 0.986 (0.76) (0.0786) (0.0394) 0.50567 d (0.454).35756 (0.0838).3834 (0.07).4053 (0.0783) 7.565 (.40).0476 (0.0305) Table 9. The esimaes of he parameers of he bes models Model RiskMerics AR()- RiskMerics GARCH(,) Disribuion skewed - Suden Gauss skewed - Suden
3 Parameers a 0.06906 (0.0387) 0.06 (0.005) 0.05468 (0.030) 0.06983 (0.003343) 0.03765 (0.09) 0.05744 (0.0099) or 0.94 0.986 6.77 (.878).04346 (0.0305) (0.0394) 7.565 (.4009).0476 (0.0305) 8. CONCLUSION W conclude ha linear combinaion of volailiy forecass doesn ouperform he forecas from he single model, because of he big correlaions beween forecass for WIG0 Index. The deducion is he same for main Polish exchange raes volailiy forecass, no prezened in he aricle. 9. THE REFERENCES Aiolfi M., Timmermann A., (006), Presisence in forecasing performance and condiional combinaion sraegies, Journal of Economerics 35, 3-53. Hansen P. R., Lunde A., Nason J. M., (003), Choosing he Bes Volailiy Models: The Model Confidence Se Approach, Oxford Bullein of Economics and Saisics 65, 839-86, 003.
4 Sock J. H., Wason M., (004), Combinaion forecass oupu growh in seven-counry daa se, Journal of Forecasing 3, 405-430. Timmermann A, (006), Forecas Combinaions, [in:]: Handbook of Economic Forecasing, Norh-Holland, Amserdam.