Paper ID : Paper title: How the features of candlestick encourage the performance of volatility forecast? Evidence from the stock markets

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1 Paper ID : Paper ile: How he feaures of candlesick encourage he performance of volailiy forecas? Evidence from he sock markes Jung-Bin Su Deparmen of Finance, China Universiy of Science and Technology, 245, Yen-Chiu-Yuan (Academia) Road, Sec3, Nankang, Taipei 11581, Taiwan jungbinsu@cc.cus.edu.w ;jungbinsu@gmail.com Tel: ex 16; (Mobil phone); Fax: Absrac This sudy provides comprehensive analysis of he possible influences of he real body, and boh upper and lower shadows of candlesick of prior day on volailiy esimaion hrough he evaluaion of accuracy covering a range of sock indices. To his end, he asymmeric GJR-X and GARCH-X models were adoped o grasp he characerisics of he buying and selling pressure and examine he effec of hese abovemenioned exogenous variables on reurn and volailiy forecasing. Empirical resuls show ha, he upper (lower) shadow of prior day can diminish (enlarge) he reurn oday. Conversely, boh upper and lower shadows of prior day can augmen he volailiy oday. Noably, wih regard o hese wo shadows, here exiss an asymmeric response for boh volailiy and reurn esimaes. Moreover, he black (whie) real body of prior day can augmen (abae) he reurn and volailiy oday. Furhermore, from he viewpoin of accuracy, he influence of he real body is more crucial han upper and lower shadows on volailiy forecass in sock markes, irrespecive of wheher he GARCH-based or GJR-based model is used. Addiionally, he asymmeric volailiy specificaion, GJR-based model, has he beer ou-of-sample volailiy forecasing performance as compared wih he symmeric volailiy specificaion, GARCH-based model. These resuls seem o appear he same as he resuls of log-likelihood raio es. Jel classificaion: C52; C53; G15 Keywords: volailiy, accuracy, candlesick, asymmeric GJR-X model, exogenous variables 1

2 1. Inroducion Technical analysis is a mehod of evaluaing securiies by relying on he assumpion ha marke daa, such as chars of price, volume, and open ineres, can help predic fuure marke rends. Technical analyss hus believe ha hey can accuraely predic he fuure price of a sock by looking a is hisorical prices and oher rading variables. Moreover, candlesick 1, a echnical analysis charing ool oday, is a graphical represenaion of price movemen for a given period of ime. I is commonly formed by he opening, high, low, and closing prices of sock. These prices hen consruc he real body, and an upper and a lower shadow. The area beween he open and he close prices is called he real body, price excursions above (below) he real body is respecively called he upper (lower) shadow. These wo shadows illusrae he highes and lowes raded prices of a securiy during he ime inerval represened and he body demonsraes he opening and closing rades. If he securiy closed higher (lower) han i opened, he body is whie (black). The real body can be long, normal, or shor depending on is proporion o he line above or below i. Generally speaking, he longer he whie (black) body is, he more inense he buying (selling) pressure is. Turning o he upper and lower shadows of candlesicks, hey can provide valuable informaion abou he rading session. 1 According o Seve (2001), candlesick charing firs appeared someime afer Much of he credi for candlesick developmen and charing goes o a legendary rice rader named Homma from he own of Sakaa. I is likely ha his original ideas were modified and refined over many years of rading evenually resuling in he sysem of candlesick charing ha we use oday. 2

3 Candlesicks wih a long upper shadow and shor lower shadow indicae ha buyers dominaed during he session, and bid prices higher. However, sellers laer forced prices down from heir highs, and he weak close creaed a long upper shadow. The abovemenioned phenomenon implies ha he longer he upper shadow is, he more inense he selling pressure. Conversely, he longer he lower shadow is, he more inense he buying pressure. In finance, volailiy is a measure for variaion of price of a financial insrumen over ime. I can be used o derive he opion prices as well as quanify he risk of he financial insrumen over he specified ime period. Hence, accurae volailiy forecass are crucial o raders, invesors, and risk managers, as well as researchers who seeking o undersand marke dynamics. However, here are several sylized facs abou financial marke volailiy. These include fa-ails, lepokurosis and a moderae amoun of skewness for heir uncondiional disribuions 2 of risky asse reurns, volailiy clusering, asymmery and mean reversion, and comovemens of volailiies across asses and financial markes (Engle and Paon, 2001; Poon and Granger, 2003). Consequenly, various popular ime series volailiy models are developed o seize he above sylized facs of financial marke volailiy in he researches over he las wo decades. For example, he models which can capure volailiy persisence or 2 See Fama, 1965; Mandelbro, 1963; Theodossiou, 1998; Su and Hung, 2011 and so on for more deails. 3

4 clusering, are he exponenially weighed moving average (EWMA) mehod suggesed by he Risk Merics framework (JP Morgan, 1996); he auoregressive condiional heeroskedasiciy (ARCH) model of Engle(1982); a general version of he ARCH model, he generalized ARCH (GARCH) model of Bollerslev (1986). Addiionally, Bollerslev e al. (1992) and Arora e al. (2009) show ha GARCH is a more parsimonious model han ARCH and he GARCH(1,1) specificaion works well in mos applied siuaions. The oher kind of models which can ake ino accoun volailiy asymmery, are he exponenial GARCH (EGARCH) model of Nelson (1991); hreshold GARCH (TGARCH) model of Zakoian (1994) which is similar o he GJR-GARCH model of Glosen, Jagannahan, and Runkle (1993), quadraic GARCH (QGARCH) and various oher nonlinear GARCH models reviewed in Franses and Dijk (2000). Moreover, according o he foregoing saemen of candlesicks, we can predic he fuure marke rends based on he candlesick of prior day. Hence, I is an ineresing ha how he volailiy forecasing performance is influenced by he feaures of candlesick of prior day (i.e. he real body, and boh upper and lower shadows). Thus, we ry o involve hem as he exogenous variables of our surveyed volailiy models (i.e. GARCH-X based and GJR-X based models) o promoe he volailiy forecas performance. The similar researches are Chou (2005, 2006) and Chou and Liu (2010). In conras o Chou (2005), he proposed a condiional 4

5 auoregressive range (CARR) model which define he range as he difference of he naural log of high and low prices during a day (i.e. R = ln P High - ln P Low ). Bu in his paper, we use reurn-based GARCH model wih he real body, and boh upper and lower shadows of candlesick of prior day as he exogenous variables. In our model, we consider no only he high and low prices bu also he open and close prices during a day as compared wih Chou(2005, 2006). Unlike previous sudies, he objecive of his paper is he firs o analyze possible influences of he real body, and boh upper and lower shadows of candlesick on volailiy esimaion hrough he evaluaion of accuracy covering a range of sock indices. Moreover, wheher he black body of he prior day can far enlarge he volailiy oday han he whie body of he prior day does is also considered. To his end, we uilize he asymmeric erm in he exogenous variables of GJR volailiy specificaion, he asymmeric GJR-X and GARCH-X models wih asymmeric reurn innovaion, he skewed generalized error disribuion (SGED) of Theodossiou (2001), o grasp he characerisics of he buying and selling pressure caused by he feaures of candlesick and he sylized facs of financial marke volailiy menioned before, and hen examine he effec of hese abovemenioned exogenous variables on reurn and volailiy forecasing. The sock indices we included are he NYSE in he U.S.; he Brussels, CAC40, DAX and SWISS in Europe; and he NIKKEI in Asia. Furhermore, 5

6 he crieria of accuracy measure include boh mean absolue error (MAE) and roo-mean-square error (RMSE) which are frequenly used o measure he differences beween values prediced by a model or an esimaor and he values acually observed from he hing being modeled or esimaed. We hope ha he volailiy forecasing performance can be encouraged by he suiable exogenous variables be included. Our resuls show ha, he upper (lower) shadow of prior day can diminish (enlarge) he reurn oday. Conversely, boh upper and lower shadows of prior day can enlarge he volailiy oday. Noably, wih regard o hese wo shadows, here exiss an asymmeric response for boh volailiy and reurn. Furher, for GJR-based model, he asymmeric response may be seized by γ and causes ha he upper shadow of prior day does no influence boh he reurn and volailiy oday. Moreover, he black (whie) real body of prior day can augmen (abae) he reurn and volailiy oday. Furhermore, from he viewpoin of accuracy, he influence of he real body is more imporan han upper and lower shadows on volailiy forecass in sock markes, irrespecive of wheher he GARCH-based or GJR-based model is used. Addiionally, he asymmeric volailiy specificaion, GJR-based model, has he beer ou-of-sample volailiy forecasing performance as compared wih he symmeric volailiy specificaion, GARCH-based model. These resuls seem o appear he same as he resuls of log-likelihood raio es. 6

7 The res of his paper is organized as follows. Secion 2 presens he GJR-based models wih SGED disribuions. Secion 3 hen provides crieria used o evaluae volailiy forecasing. Secion 4 repors daa and descripive saisics. Subsequenly, he empirical resuls are lised. Conclusions are finally drawn in Secion Economeric mehodology Mos financial asse reurns series appear o exhibi properies of volailiy pooling 3, leverage effecs 4, fa-ails, lepokurosis and a moderae amoun of skewness for heir uncondiional disribuions(see Fama, 1965; Mandelbro, 1963; Theodossiou, 1998; Su and Hung, 2011 and so on). To capure hese sylized facs for financial asse reurns, his sudy considers he applicabiliy of he GJR (Glosen, Jagannahan and Runkle,1993) model wih skewed generalized error disribuion (SGED) of Theodossiou (2000), o esimae he corresponding volailiy in erms of differen sock indices. In addiion, according o he concep of echnical analysis, he candlesick paern of prior day can predic he fuure price of a sock. Hence, he real body and boh upper and lower shadows of candlesick char are involved as he exogenous variables of our surveyed volailiy models. Consequenly, he GJR-based 3 Volailiy pooling is he endency for volailiy in financial markes o appear in bunches. Thus large reurns are expeced o follow large reurns, and small reurns o follow small reurns. 4 Leverage effec is he endency for volailiy o rise more following a large price fall han following a price rise of he same magniude. 7

8 models (GJR, GJR-UD and Asy-GJR-Sd models) and he GARCH-based models (GARCH, GARCH-UD and Asy-GARCH-Sd models), oaling six models, are uilized o forecas volailiy in his paper and are described as follows. 2.1.GJR model wih no exogenous variables Le ( ln P ln P ) 100 r 1 =, where P denoes he close price of sock, r denoes he coninuously compounded daily reurns of he underlying asses on ime. The GJR(1,1) model wih SGED disribuion (hereafer, GJR) can be expressed as follows: r = μ e, e = z σ, ~ IID SGED(0,1; κ, λ) (1) + 2 z 2 2 ( α + I γ) e + βσ σ = ω+ (2) where e is he curren error, μ and σ 2 are he condiional mean and variance of reurn, respecively. Moreover, I 1 is an indicaor dummy ha akes he value 1 if e 1 < 0 and zero oherwise, and hus parameer γ can be used o capure he leverage effec of volailiy. Furhermore, he variance parameers ω, α and β are he parameers o be esimaed and obey he consrains ω, α, β > 0 and α + β < 1. IID denoes ha he sandardized errors z are independen and idenically disribued. As z is drawn from he sandardized SGED disribuion which allows reurns innovaion o follow a flexible reamen of boh skewness and excess kurosis in he condiional disribuion of reurns. The probabiliy densiy funcion for he 8

9 sandardized SGED disribuion 5 can be represened as follows: k z + δ f (z ) = Cexp k k [ 1+ sign(z + δ λ] θ (3) ) 1 κ where C = Γ, θ = Γ Γ S( λ), S( λ) 2θ κ κ κ λA A = Γ Γ Γ, δ =. κ κ κ S( λ) = λ 4A λ where he shape parameer κ governs he heigh and fa-ails of densiy funcion wih consrain κ > 0, while he skewness parameer λ conrols he rae of descen of he densiy around he mode of z wih 1 < λ < 1. Γ ( ) denoes he gamma funcion and he sign is he sign funcion. In he case of posiive ( negaive) skewness, he densiy funcion skews oward he righ (lef). Therefore, he log-likelihood funcion of he GJR model can be wrien as: ( ψ) = ln f ( r Ω ; ψ) L 1 κ ln C ln σ z δ 1+ sign( z + δ) λ = κ κ [ ] θ + where ψ = [ μ ω, α, β, γ, κ, λ], is he vecor of parameers o be esimaed, and Ω 1 (4) denoes he informaion se of all observed reurns up o ime 1. Paricularly, he GJR model generaes he GARCH model for γ = GJR-X model wih upper and lower shadows as exogenous variables 5 The sandardized SGED disribuion, which has zero mean and uni variance, was checked by Mahemaica sofware. 9

10 Since he longer he lower (upper) shadow is, he more inense he buying (selling) pressure is. Thus boh he lower and upper shadows are included as he exogenous variables of GJR model, and his model is named as GJR-UD. The GJR-UD model can be employed o seize boh he characerisics of he buying and selling pressure derived by hese wo shadows, and can be uilized o invesigae he influences of hese wo shadows on reurn and volailiy forecasing. The GJR-UD model can be represened as follows: r = μ + Bu U 1 + Bd D 1 + e, e z σ, z = ~ IID SGED(0,1; κ, λ) (5) σ = ω + (6) ( α + I 1 γ) e 1 + β σ 1 + A u U 1 + A d D 1 where U high P = ln 100 open close max( P, P ), D open close ( P, P ) min = ln 100 low P, I 1 if e 1 < 0 = 0 oherwise 1. where open P ( P ) denoes he opening (closing) price of sock, P high ( close low P ) represens he high (low) price of sock during a day ; U( D ) is he log of price beween upper (lower) shadow of Candlesicks and denoes he highes (lowes) raded prices of a securiy during he ime inerval represened; parameer γ can be used o capure he leverage effec of volailiy; B u (B d ) parameer can be uilized o seize he influence of he upper (lower) shadow of prior day on reurn esimae oday. 10

11 Conversely, A u (A d ) parameer can be applied o capure he influence of he upper (lower) shadow of prior day on volailiy assessmen oday. Therefore, he log-likelihood funcion of he GJR-UD model can be wrien as he same form as Eq. (4). Noably, ψ = [ μ B,B, ω, α, β, γ, A,A, κ, λ], d u d u is he vecor of parameers o be esimaed. Moreover, he GJR-UD model reduces o he GARCH-UD and GJR models given he resricions of γ = 0 and B u = B d = A u = A d = 0, respecively. Paricularly, he GARCH-UD model generaes he GARCH model for B u = B d = A u = A d = Asy-GJR-X model wih real body as exogenous variables On a candlesick char, if he securiy closed higher han i opened, he body is whie indicaing ha he buying pressure exiss. On he oher hand, if he securiy closed lower han i opened, he body is black indicaing ha he selling pressure subsiss. Generally speaking, he longer he whie (black) body is, he more inense he buying (selling) pressure is. Thus he real body is included as he exogenous variables of GJR model, and he siuaion of he real body (whie or black) is also considered. This model is assumed ha he whie body (or buying pressure) of he prior day can far increase he reurn oday han he black body (or selling pressure) of he prior day does. Moreover, he black body of he prior day can far increase he 11

12 volailiy oday han he whie body of he prior day does. Owing o he asymmeric erm in he exogenous variables of GJR volailiy specificaion, hence his model is given he name asymmeric GJR-Sd (hereafer, Asy-GJR-Sd). The Asy-GJR-Sd model can be used o seize he feaure of he buying or selling pressure derived by he real body, and can be uilized o invesigae he influences of he real body on reurn and volailiy forecasing. The Asy-GJR-Sd model can be represened as follows: + r = μ + (Bs + I Bs, 1 γ Bs ) Sd 1 + e, e z σ, z = ~ IID SGED(0,1; κ, λ) (7) σ = ω + (8) ( α + I 1 γ) e 1 + β σ 1 + (A s + I As, 1 γ As ) Sd 1 close P where Sd = ln 100 open P, I 1 if Sd 1 > 0 = 0 oherwise + Bs, 1, I As, 1 1 if Sd 1 < 0 = 0 oherwise where Sd is he log of price beween he real body of Candlesicks, he area beween he open and he close prices, and is drawn in black (whie) real body as he opening price is above (below) he closing price; + I Bs, 1 is an indicaor dummy ha akes he value 1 if Sd 1 > 0 (i.e. whie real body) and zero oherwise. Hence parameer γ Bs can es wheher he inensiy of buying pressure on prior day can increase he reurns of financial asse oday; I As, 1 is an indicaor dummy ha akes he value 1 if Sd 1 < 0 (i.e. black real body) and zero oherwise. Consequenly parameer γ As can examine wheher he volailiy of reurn oday can be enlarged by he inensiy of selling pressure on prior day, implying anoher kind of leverage effec; Bs (A s ) parameer can be uilized o seize he influence of he black (whie) real body 12

13 of prior day on reurn (volailiy) esimae oday. Therefore, he log-likelihood funcion of he Asy-GJR-Sd model can be wrien as he same form as Eq. (4). Bu [ μ B,A, ω, α, β, γ, γ, γ, κ λ] ψ =, s s Bs As, is he vecor of parameers o be esimaed. Furhermore, he Asy-GJR-Sd model reduces o Asy-GARCH-Sd and GJR models given he resricions of γ = 0 and B = A = γ = γ 0, respecively. s s Bs As = Paricularly, he Asy-GARCH-Sd model generaes he GARCH model for B = A = γ = γ 0. s s Bs As = 2.4. Tess for asymmeries in volailiy Engle and Ng (1993) have proposed a se of ess for asymmery in volailiy, known as sign and size bias ess. The Engle and Ng ess should hus be used o deermine wheher an asymmeric GJR or EGARCH model is required for a given series, or wheher he symmeric GARCH model can be deemed adequae. The es of Engle and Ng (1993) is a join es for sign and size bias based on he regression ẑ 2 + = φ0 + φ1 S 1 + φ2 S 1z 1 + φ3 S 1z 1 + υ (9) where S 1 if ẑ 1 < 0 = 0 oherwise 1, S. + 1 = 1 S 1 where S is an indicaor dummy ha akes he value 1 if ẑ 1 < 0 and zero 1 oherwise, + S 1 can pick ou he observaions wih posiive innovaions and υ is an iid error erm. If posiive and negaive shocks o ẑ 1 impac differenly upon he 13

14 condiional variance, hen φ 1 will be saisically significan. Significance of φ 1 indicaes he presence of sign bias, where posiive and negaive shocks have differing impacs upon fuure volailiy, compared wih he symmeric response required by he sandard GARCH formulaion. On he oher hand, he significance of φ 2 or φ 3 would sugges he presence of size bias, where no only he sign bu he magniude of he shocks is imporan. A join es saisic is formulaed in he sandard fashion by calculaing he sample size muliplied by coefficien of deerminaion (i.e. T.R 2 ) from regression (9), which will asympoically follow a 2 χ disribuion wih 3 degrees of freedom under he null hypohesis of no asymmeric effecs. 3 Evaluaion mehods of model-based volailiy To compare he forecasing abiliy of he aforemenioned models in erms of volailiy, his sudy firs uilizes he log-likelihood raio es (LR) o compare he fi abiliy of hese six models wih he empirical reurn series. Then wo loss funcions, mean absolue error (MAE) and roo-mean-square error (RMSE), are used o assess hese models in erms of volailiy forecasing The log-likelihood raio es The log-likelihood raio es (LR) is a saisical es employed o compare he fi of wo models, one of which, he null model, is a special case of he oher, he alernaive 14

15 model. This es is based on he likelihood raio, which expresses how many imes more likely he daa are under one model han he oher. The log-likelihood raio can hen be compared o a criical o decide wheher o rejec he null model in favour of he alernaive model. Consequenly, he log-likelihood raio es is applied o es he null hypohesis of he resriced model agains he alernaive hypohesis of he unresriced model, and is given as follows. LR i = 2(LR r - LR u ) ~ χ 2 (m), i= 1, 2 or g (10) where LR r and LR u are, respecively, he maximum value of he log-likelihood values under he null hypohesis of he resriced model and he alernaive hypohesis of he unresriced model, and m is he number of he resriced parameers in he resriced model. For example, LR 1 for GARCH-UD (GJR-UD) model could be used o es he null hypohesis ha volailiy specificaion is specified by GARCH (GJR) model agains he alernaive hypohesis ha i is specified by GARCH-UD (GJR-UD) model. Therefore he null hypohesis is H 0 : B u = B d = A u = A d = 0. Resae, LR 1 = 2(LR r - LR u ) ~ χ 2 (4) where LR r and LR u are respecively he maximum value of he log-likelihood values under he null hypohesis of resriced model, GARCH (GJR) model, and he alernaive hypohesis of unresriced model, GARCH-UD (GJR-UD) model, and m is he number of he resriced parameers in he resriced model and equal o 4 in his case. In oher word, LR 1 for GARCH-UD and GJR-UD models 15

16 follows he chi-squared disribuion wih four degree of freedom, χ 2 (4). A he same deduce, LR 2 for GARCH-Sd and Asy-GJR-Sd models also follows he χ 2 (4). Moreover, LR g for all GJR-based models follows he χ 2 (1). For he number of he resriced parameers in he resriced model, please see secion 2 for more deails Evaluaion of volailiy forecass To compare he forecasing abiliy of our surveyed volailiy models, his work considers he squared inraday reurns as he volailiy proxies, hen assesses he forecasing accuracy of compeing models in forecasing daily volailiy using he following wo saisical error loss funcions, he MAE and RMSE. The MAE is a quaniy used o measure how close forecass are o he evenual oucomes, and is given by T MAE = σ ˆ + h i + i 1 + i (11) T i= 1 where σ ˆ + i + i 1 is one-sep-ahead forecass of he sandard deviaion of he reurns condiional on all informaion upon he ime +i-1 and can be esimaed by GARCH-based or GJR-based models here; h + is he rue value of variance and is 2 i replaced by he squared inraday reurns as he proxies; T is he number of compuing 1-day-ahead volailiy and equals 500 in his sudy. Moreover, he RMSE is expressed as follows. 16

17 1 2 RMSE = (12) T T 2 2 ( σˆ h + ) + i + i 1 i i= 1 where 2 2 ˆ, h + i + i 1 + i σ and T are defined he same as Equaion (11). The RMSE is a frequenly used o measure he differences beween values prediced by a model and he values acually observed from he hing being modeled or esimaed. 4. Empirical resuls 4.1 Daa and descripive saisics\ This paper calculaes he daily volailiy for a range of inernaional sock indices, including he U.S. NYSE(6/4/1998-5/7/2010), he Belgium Brussels (7/28/1998-5/7/2010), he France CAC40(8/3/1998-5/7/2010), he Germany DAX(7/20/1998-5/7/2010), he Swizerland Swiss(6/19/1998-5/7/2010) and he Japan NIKKEI(2/13/1998-5/7/2010), oaling six sock indices. The numbers in parenheses are he sar and end daes for he sample. Daily closing spo prices for he sudy period, oaling observaions abou 12 years, were obained from he Yahoo finance websie hp://finance.yahoo.com. Sock reurns are defined as he firs difference in he logarihms of daily sock prices hen muliplied by 100. Table 1 summarizes he basic saisical characerisics of daily reurn series for overall period. Moreover, he daily upper (U) and lower (D) shadows, and real body (Sd) of candlesick are also lised and hen be assayed. Noably, excep for NYSE, he average daily reurns and real body are all negaive and very small compared wih he variable sandard deviaion, indicaing high volailiy. On he conrary, he average upper and lower shadows are all posiive and heir minimums are zero. This can be 17

18 seen from he definiion of wo shadows, see secion 2.2 for more deails. All reurns and real body series almos exhibi negaive skewness excep for DAX and SWISS. Conversely, all upper and lower shadows series exhibi posiive skewness. For all four kinds of series (i.e. R, U, D and Sd), he excess kurosis all significanly exceeds zero a he 1% level, indicaing a lepokuric characerisic. Furhermore, J-B normaliy es saisics are all significan a he 1% level and hus, rejec he hypohesis of normaliy and confirm ha hey are no normally disribued. Moreover, he Ljung-Box Q 2 (20) saisics for he squared reurns are all significan a he 1% level and hus indicae ha hey exhibi linear dependence and srong ARCH effecs. Therefore, he preliminary analysis of he daa suggess he use of a GARCH or GJR model wih SGED disribuion o capure he skewness, fa-ails and ime-varying volailiy found in hese reurns series of sock indices. 4.2 Esimaion resuls for GJR-based and GARCH-based models Table 2-3 respecively lis he esimaion resuls 6 of GARCH-based models (GARCH, GARCH-UD and Asy-GARCH-Sd ) and GJR-based models (GJR, GJR-UD and Asy-GJR-Sd), oaling six models, for he six sock indices a he 117 h in-sample period during he 500 imes rolling process. As observed in Table 2-3, he 6 The parameers in he GARCH-based models and GJR-based models are esimaed by QMLE (Quasi maximum likelihood esimaion) and he BFGS opimizaion algorihm using he economeric package of WinRATS

19 ω, α and β coefficiens of hese six models are almos posiive and significan almos a he 1% level. The sums of parameers α and β for hese models are less han one hus ensuring ha he condiions for saionary covariance hold and confirming he mean reversion 7 exiss. As o he shape parameers of SGED disribuion, he fa-ails parameer ( κ ) ranges from (GARCH of DAX) o (Asy-GARCH-Sd of CAC40) for GARCH-based models, and is beween (GJR of DAX) and (GJR-UD of CAC40) for GJR-based models. The skewness parameers ( λ ) of GARCH-based and GJR-based models respecively range from (GARCH-UD of NIKKEI) o (Asy-GARCH-Sd of SWISS), and from (GJR-UD of NIKKEI) o (Asy-GJR-Sd of SWISS), in which hese shape parameers are almos significan a he 1% level. These resuls indicae ha he reurns of hese sock indices exhibi skewed o lef and fa-ails and are in line wih he previous researches(such as Fama, 1965; Mandelbro, 1963; Theodossiou, 1998; Su and Hung, 2011 and so on) The influences of he upper and lower shadows on reurn and volailiy esimaes Now we will firs invesigae he influences of he upper and lower shadows on reurn and volailiy esimaes. However, boh he GARCH-UD and GJR-UD models can be 7 Mean reversion is a mahemaical concep someimes used for sock invesing. I assumes ha boh a sock's high and low prices are emporary and ha a sock's price will end o move o he average price over ime. 19

20 used o seize boh he characerisics of he buying and selling pressure derived by he wo shadows. In hese wo models, B u (B d ) parameer can be uilized o seize he influence of he upper (lower) shadow on reurn esimae. Conversely, A u (A d ) parameer can be applied o capure he influence of he upper (lower) shadow on volailiy assessmen. For GARCH-UD model, he parameer B u is significanly negaive a he 1% level and equals , while he parameer B d is almos posiive excep for Brussels and NIKKEI. Noably, he value of B d is far greaer han ha of B u. These resuls indicae ha he upper (lower) shadow of prior day can diminish (enlarge) he reurn oday, and here exiss an asymmeric response for hese wo shadows for reurn increase or decrease. Moreover, he parameer A u is significanly posiive a he 1% level and equals , while he parameer A d is significanly posiive almos a he 5% level. Noably, he value of A d is abou four imes as ha of A u excep NYSE. These resuls imply ha boh upper and lower shadows of prior day can augmen he volailiy oday, bu here sill exiss an asymmeric response for hese wo shadows for volailiy increase. This asymmeric phenomenon for volailiy appears o be he same as ha for reurn. Tha is, wih regard o he size affeced by heses wo shadows, he lower shadow is more influenial han upper shadow, irrespecively for reurn or volailiy esimaes. Turning o GJR-UD model, he parameer γ is significanly posiive a he 1% 20

21 level, indicaing ha he leverage effec exiss. Moreover, he parameer B u is significanly equal o zero a he 1% level, while he parameer B d is almos posiive excep for Brussels and NIKKEI. These resuls indicae ha he lower shadow of prior day can increase he reurn oday. Furher, he asymmeric response may be seized by γ and causes ha he upper shadow of prior day does no influence he reurn oday. As o he parameer A u, i is significanly equal o zero a he 1% level, while he parameer A d is significanly posiive almos a he 1% level excep for NYSE. These resuls imply ha he lower shadow of prior day can increase he volailiy oday. Furher, he asymmeric response may be seized by γ and moives ha he upper shadow of prior day does no influence he volailiy oday. This phenomenon for volailiy appears o be he same as ha for reurn. In oher words, wih regard o he size affeced by heses wo shadows, owing o he leverage effec is seized he upper shadow does no influence on boh reurn and volailiy esimaes. Conversely, he lower shadow has he same effec as GARCH-UD model, irrespecively for reurn or volailiy esimaes The influences of he whie or black real body on reurn and volailiy esimaes In his subsecion, we will invesigae he influences of he whie or black real body on reurn and volailiy esimaes. However, boh he Asy-GARCH-Sd and 21

22 Asy-GJR-Sd models can be used o seize he characerisics of he buying (selling) pressure derived by he whie (black) real body. In hese wo models, Bs (A s ) parameer can be uilized o seize he influence of he black (whie) real body of prior day on reurn (volailiy) esimae oday. Moreover, parameer γ Bs can es wheher he inensiy of buying pressure on prior day can increase he reurns of financial asse oday. Conversely, parameer γ As can examine wheher he volailiy of reurn oday can be enlarged by he inensiy of selling pressure on prior day. For Asy-GARCH-Sd model, he parameer B s is almos posiive excep Brussels, indicaing ha he black real body of prior day can increase he reurn oday. On he oher hand, he parameer γ Bs is almos negaive and significan a he 10% level excep Brussels. Noably, B s and γ Bs are differen a he sign and he value of γ Bs is greaer han ha of B s, implying ha he whie real body of he prior day ries o reverse he rend from increase o decrease on he reurn esimae oday. For example, for NYSE, he reurn oday is above he mean if he prior day is a black real body and is (= ) above he mean if he prior day is a whie real body. Moreover, he parameer A s is all significanly negaive, implying ha he whie real body of prior day can decrease he volailiy oday. On he conrary, he parameer γ As is all significanly posiive. Noably, A s and γ As are differen a he sign and he value of γ As is greaer han ha of A s, indicaing ha he black real body of he prior day ries 22

23 o reverse he rend from decrease o increase on he volailiy esimae oday. This phenomenon seems o be a kind of leverage effec. For insance, for NYSE, he coefficien of Sd is if he prior day is a whie real body and is ( = ) if he prior day is a black real body. Turning o Asy-GJR-Sd model, he parameer γ is significanly posiive a he 1% level, indicaing ha he leverage effec exiss. The parameer B s is almos posiive excep Brussels, indicaing ha he black real body of prior day can increase he reurn oday. On he oher hand, he parameer γ Bs is almos negaive excep Brussels. Noably, B s and γ Bs are differen a he sign and he value of γ Bs is greaer han ha of B s excep for NYSE and SWISS, implying ha he whie real body of he prior day ry o reverse he rend from increase o decrease on he reurn esimae oday. Moreover, he parameer A s is almos negaive excep DAX and NIKKEI, implying ha he whie real body of prior day can decrease he volailiy oday. On he conrary, he parameer γ As is almos significanly posiive. Noably, A s and γ As are differen a he sign excep for DAX and NIKKEI, and he value of γ As is greaer han ha of A s excep for NYSE, indicaing ha he black real body of he prior day ry o reverse he rend from decrease o increase on he volailiy esimae oday. This phenomenon seems o be a kind of leverage effec. The abovemenioned phenomena are he same as hose for Asy-GARCH-Sd model. 23

24 Neverheless, he consisency of Asy-GJR-Sd is no beer han ha for Asy-GARCH-Sd. I may aribue ha he γ parameer of Asy-GJR-Sd model can seize he leverage effec. I should be concluded, from wha has been said above, ha he black real body of prior day can increase he reurn oday, while he whie real body of he prior day ries o reverse he rend from increase o decrease on he reurn esimae oday. On he conrary, he whie real body of prior day can decrease he volailiy oday, bu black real body of he prior day ry o reverse he rend from decrease o increase on he volailiy esimae oday. This implies ha here exiss anoher kind of leverage effec causing by he selling pressure derived by he black real body of prior day. These phenomena appear o he same for boh Asy-GARCH-Sd and Asy-GJR-GARCH models The resuls of Engle and Ng ess Owing o he sylized fac of volailiy asymmery in financial marke, we will invesigae wheher he symmeric GARCH model can be deemed adequae, or an asymmeric model is required for a given series. This decision mus be made by Engle and Ng ess. As shown in Table 4, for GARCH-based model, he join es saisics are almos significan a 10% level, indicaing ha all reurn series rejec he null 24

25 hypohesis of no asymmeric effecs. This implies ha he asymmeric effec or leverage effec exiss in all reurn series of sock indices. On he oher hand, he join es saisics for GJR-based model are smaller han hose for GARCH-based model excep DAX, SWISS and NIKKEI of Asy-GJR-Sd. Moreover, some cases in GJR-based model, such as all GJR-based models of Brussels, boh GJR and GJR-UD models of NYSE, and boh GJR and GJR-UD models of DAX, he join es saisics are no significan, indicaing ha hese cases do no rejec he null hypohesis of no asymmeric effecs. Hence, wih regard o GJR-based model, he sylized fac of volailiy asymmery appears o be eliminaed. This implies ha he GJR-based model can seize he sylized fac of volailiy asymmery. 4.3 The resuls of volailiy performance assessmen This secion esimaes he GARCH-based and GJR-based models, oaling six models, for performing volailiy forecasing analysis. For each daa series, six models are esimaed wih a sample of 2500 daily reurns, and he esimaion period is hen rolled forward by adding one new day and dropping he mos disan day. In his procedure, he ou-of-sample volailiy is compued for he nex 500 days The log-likelihood raio es 25

26 Before o compare he forecasing abiliy of he aforemenioned models in erms of volailiy, his sudy firs uilizes he log-likelihood raio es (LR) o compare he fi abiliy of hese six models wih he empirical reurn series. As shown in Table 2-3, he LR g saisics for all GJR-based models are all highly significan excep Asy-GJR-Sd of DAX, indicaing ha rejec he null hypohesis ha volailiy specificaion is specified by GARCH-based models for eiher sock index. These resuls hus imply ha GJR-based models closely approximae he empirical reurn series as compared wih GARCH-based models. Moreover, he LR 1 saisics of GARCH-UD and he LR 2 saisics of Asy-GARCH-Sd are almos highly significan. Similarly, he LR 1 saisics of GJR-UD and he LR 2 saisics of Asy-GJR-Sd are almos highly significan, oo. Noably, he value of LR 2 is almos greaer han ha of LR 1, irrespecive of GARCH-based or GJR-based model. These resuls indicae ha he Asy-GARCH-Sd (Asy-GJR-Sd) model more closely approximaes he empirical reurn series as compared wih he GARCH-UD ( GJR-UD) model. In shor, in erms of he fi abiliy of hese six models, he GJR-based models are superior o GARCH-based models. Besides, he Asy-GARCH-Sd (Asy-GJR-Sd) model is mos excellen among he GARCH-based (GJR-based) models Assessmen of volailiy forecass 26

27 Table 5 and Table 6 respecively repor he predicive performances of six volailiy models based on he wo loss funcions MAE and RMSE. Now we invesigae he influences of he upper and lower shadows on volailiy forecasing by boh he GARCH-UD and GJR-UD models. Then, via boh he Asy-GARCH-Sd and Asy-GJR-Sd models, he influences of he whie or black real body on volailiy forecasing are explored. The deailed resuls are described as follows. As observed from Table 5 and Table 6, we can find ha, for mos of sock indices and for all day horizons (i.e. 1, 5 and 10 day horizons), he Asy-GARCH-Sd (Asy-GJR-Sd) model yields he lowes MAE and RMSE for all GARCH-based (GJR-based) models. On he conrary, he GARCH (GJR) model gives he highes MAE and RMSE for all GARCH-based (GJR-based) models. I seems reasonable o conclude ha, for all day horizons, he Asy-GARCH-Sd (Asy-GJR-Sd) model owns he bes ou-of-sample volailiy forecasing performance bu he GARCH (GJR) model has he wors ou-of-sample volailiy forecasing performance for all GARCH-based (GJR-based) models. This reveals ha he influences of he whie or black real body on volailiy are more crucial han hose of he upper and lower shadows on volailiy forecasing. Moreover, for mos of sock indices and for all day horizons, he GJR-UD (GJR-Sd) model yields lower MAE and RMSE han he GARCH-UD (GARCH-Sd) model. 27

28 Similariy, The GJR model yields lower MAE and RMSE han he GARCH model. Tha is o say, he GJR-based models are more accurae han he GARCH-based models for mos of sock indices and for all day horizons. This reveals ha he asymmeric volailiy specificaion has he beer ou-of-sample volailiy forecasing performance. To sum up, in erms of he evaluaion of accuracy, he Asy-GARCH-Sd model provides he mos accurae volailiy forecass followed by he GARCH-UD model while he GARCH model has he wors forecasing performance for GARCH-based models. Similariy, he Asy-GJR-Sd model provides he mos accurae volailiy forecass followed by he GJR-UD model while he GJR model has he wors forecasing performance for GJR-based models. Consequenly, i appears reasonable o conclude ha, from he viewpoin of accuracy, he influence of he real body is more imporan han upper and lower shadows on volailiy forecass in sock markes, irrespecive of wheher he GARCH-based or GJR-based model is used. Addiionally, he asymmeric volailiy specificaion, GJR-based models, has he beer ou-of-sample volailiy forecasing performance as compared wih he symmeric volailiy specificaion, GARCH-based models. These resuls seem o appear he same as he resuls of log-likelihood raio es. 28

29 5. Conclusion This sudy provides comprehensive analysis of he possible influences of he real body, and boh upper and lower shadows of candlesick of prior day on reurn and volailiy esimaion hrough he evaluaion of accuracy covering a range of sock indices. The measure of accuracy included boh MAE and RMSE. To his end, he real body, and boh upper and lower shadows of candlesick are included as he exogenous variables of GJR and GARCH models, he asymmeric GJR-X and GARCH-X models, wih SGED disribuion, o grasp he characerisics of he buying and selling pressure caused by he feaures of candlesick and he sylized facs of financial marke volailiy menioned before and hen examine he effec of hese abovemenioned exogenous variables on reurn and volailiy forecasing. The empirical findings can be summarized as follows. Firs, he upper (lower) shadow of prior day can reduce (increase) he reurn oday. Conversely, boh upper and lower shadows of prior day can increase he volailiy oday. Noably, wih regard o hese wo shadows, here exiss an asymmeric response for volailiy increase and for reurn increase or decrease. Furher, for GJR-based model, he asymmeric response may be seized by he leverage effec and causes ha he upper shadow of prior day does no influence boh he reurn and volailiy oday. Second, for Asy-GARCH-Sd model, he black (whie) real body of prior day can augmen (abae) 29

30 he reurn oday. On he conrary, he whie (black) real body of prior day can diminish (enlarge) he volailiy esimae oday, implying ha here exiss anoher kind of leverage effec causing by he selling pressure derived by he black real body of prior day. Noably, for Asy-GJR-Sd model, he above phenomena appear less significan as compared wih Asy-GARCH-Sd model. This may be owing o he leverage effec be seized by Asy-GJR-Sd model. Third, in erms of he fi abiliy of hese six models, he GJR-based models are superior o GARCH-based models. Besides, he Asy-GARCH-Sd (Asy-GJR-Sd) model is mos excellen among he GARCH-based (GJR-based) models. Fourh, from he viewpoin of accuracy, he influence of he real body is more imporan han upper and lower shadows on volailiy forecass in sock markes, irrespecive of wheher he GARCH-based or GJR-based model is used. Addiionally, he asymmeric volailiy specificaion (i.e. GJR-based model) has he beer ou-of-sample volailiy forecasing performance as compared wih he symmeric volailiy specificaion (i.e. GARCH-based model). These resuls seem o appear he same as he resuls of log-likelihood raio es. Reference Arora, R.K., Das, H. and Jain, P.K. (2009). Sock Reurns and Volailiy: Evidence from Selec Emerging Markes. Review of Pacific Basin Financial Markes and 30

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32 Fama, E. (1965). The behavior of sock marke prices. Journal of Business, 38, Franses, P. H. and Dijk, D.V. (2000). Non-Linear Time Series Models in Empirical Finance. Cambridge U. Press. Glosen, L.R., Jagannahan, R. and Runkle, D.E. (1993). On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance, 48, Jarque, C.M. and Bera, A.K. (1987). A es for normaliy of observaions and regression residuals. Inernaional Saisics Review, 55, JP Morgan (1996). RiskMerics - Technical Documen, 4h ediion, New York. Mandelbro, B. (1963). The variaion of cerain speculaive prices. Journal of Business, 36, Nelson, D.B. (1991). Condiional heeroskedasiciy in asse reruns: A new approach. Economerica, 59, Poon, S.H. and Granger, C.W.J. (2003). Forecasing volailiy in financial markes: a review. Journal of Economic Lieraure, 41, Seve, N. (2001). Japanese candlesick charing echniques. 2nd ediion, Publisher: Prenice Hall Press. Su, J.B. and Hung, J.C. (2011). Empirical analysis of jump dynamics, heavy-ails and 32

33 skewness on value-a-risk esimaion. Economic Modelling, 28, Theodossiou, P. (1998). Financial daa and he skewed generalized disribuion. Managemen Science, 44, Theodossiou, P. (2001). Skewed generalized error disribuion of financial asses and opion pricing. Working paper, School of Business and Rugers Universiy. Available a: hp://papers.ssrn.com/sol3/papers.cfm?absrac_id=219679, accessed on December 8, Zakoian, J.M. (1994). Threshold heeroskedasic models. Journal of Economic Dynamics and Conrol, 18, Table 1 Descripive saisics of daily reurn, upper and lower shadows, and real body Sd. Max. Min. Mean Skewness Kurosis J-B Q 2 (20) Dev. Panel A. R NYSE c c c c Brussels c c c CAC c c c DAX c c c SWISS c c c NIKKEI c c c c Panel B. U NYSE c c c c Brussels c c c c CAC c c c c DAX c c c c SWISS c c c c NIKKEI c c c c Panel C. D NYSE c c c c Brussels c c c c CAC c c c c DAX c c c c SWISS c c c c NIKKEI c c c c 33

34 Panel D. Sd NYSE c c c c Brussels c c c c CAC c c c DAX c c c c SWISS c c c NIKKEI c c c c Noes: 1. a, b and c denoe significanly a he 10%, 5% and 1% levels, respecively. 2. J-B saisics are based on Jarque and Bera (1987) and are asympoically chi-squared-disribued wih 2 degrees of freedom Q ( 20 ) saisics are asympoically chi-squared-disribued wih 20 degrees of freedom. 4. U, D and Sd respecively denoe he daily upper and lower shadows, and real body of candlesick and are defined in equaions (5)-(8). Table 2 Esimaion resuls for GARCH-based models NYSE Brussels CAC40 DAX SWISS NIKKEI Panel A. GARCH model μ b c b c ω (0.0156) (0.0151) (0.0199) (0.0188) (0.0153) (0.0228) c c c c c c α (0.0047) (0.0051) (0.0055) (0.0119) (0.0060) (0.0110) c c c c c c (0.0120) (0.0159) (0.0120) (0.0174) (0.0143) (0.0112) β c c c c c c κ (0.0125) (0.0156) (0.0120) (0.0175) (0.0144) (0.0115) c c c c c c (0.0556) (0.0579) (0.0650) (0.0457) (0.0575) (0.0624) λ c (0.0240) c (0.0244) c (0.0283) c (0.0157) c (0.0272) a (0.0220) Q 2 (20) a a LL Panel B. GARCH-UD model μ b (0.0192) (0.0214) (0.0285) (0.0277) (0.0233) (0.0326) B u c c c c c c B d c b b ω (0.0803) (0.0617) (0.0632) (0.0529) (0.0636) (0.0734) c c c α (0.0052) (0.0075) (0.0078) (0.0138) (0.0085) (0.0088) c c c c c c (0.0109) (0.0174) (0.0114) (0.0178) (0.0154) (0.0118) β c (0.0121) c (0.0211) c (0.0163) c (0.0274) c (0.0182) c (0.0177) A u c c c c c c A d c c b c c κ (0.0222) (0.0503) (0.0575) (0.0775) (0.0518) (0.0766) c c c c c c (0.0585) (0.0624) (0.0663) (0.0479) (0.0599) (0.0602) λ c (0.0260) c (0.0269) c (0.0296) c (0.0240) c (0.0252) (0.0251) Q 2 (20) LL LR c c b c 18.8 c Panel C. Asy-GARCH-Sd model μ b

35 (0.0195) (0.0232) (0.0202) (0.0205) (0.0226) (0.0336) B s b c b ω (0.0435) (0.0399) (0.0239) (0.0231) (0.0388) (0.0425) c b a c c c α (0.0073) (0.0084) (0.0026) (0.0034) (0.0082) (0.0117) c c c c c c (0.0118) (0.0161) (0.0030) (0.0040) (0.0121) (0.0134) β c (0.0129) c (0.0158) c (0.0028) c (0.0036) c (0.0139) c (0.0156) A s c b c c c a κ (0.0226) (0.0253) (0.0048) (0.0057) (0.0326) (0.0474) c c c c c c (0.0534) (0.0667) (0.0644) (0.0304) (0.0537) (0.0596) λ c (0.0242) c (0.0260) c (0.0253) c (0.0186) c (0.0250) b (0.0250) γ b c b b a Bs (0.0493) (0.0398) (0.0323) (0.0327) (0.0441) (0.0473) γ c c c c c c As (0.0425) (0.0331) (0.0114) (0.0140) (0.0298) (0.0670) Q 2 (20) c c LL LR c c c c c c Noes: 1. a, b and c denoe significanly a he 10%, 5% and 1% levels, respecively. 2. Numbers in parenheses are sandard errors. 3. LL indicaes he log-likelihood value. 4. The criical value of he LR 1 and LR 2 es saisics a he 10%, 5% and 1% significance level is 7.779, and , respecively. 5. The criical value of he LR g es saisics a he 10%, 5% and 1% significance level is 2.706, and 6.635, respecively Q ( 20 ) saisics are asympoically chi-squared-disribued wih 20 degrees of freedom. Table 3 Esimaion resuls for GJR-based models NYSE Brussels CAC DAX SWISS NIKKEI Panel A. GJR model μ b ω (0.0158) (0.0159) (0.0173) (0.0214) (0.0152) (0.0232) c c c c c c α (0.0051) (0.0041) (0.0027) (0.0113) (0.0055) (0.0106) c c c c c (0.0101) (0.0127) (0.0029) (0.0117) (0.0126) (0.0106) β c c c c c c κ (0.0122) (0.0135) (0.0027) (0.0157) (0.0127) (0.0117) c c c c c c (0.0571) (0.0649) (0.0596) (0.0468) (0.0563) (0.0565) λ c c c c c b γ (0.0252) (0.0264) (0.0248) (0.0238) (0.0256) (0.0233) c c c c c c (0.0236) (0.0210) (0.0068) (0.0278) (0.0251) (0.0196) Q 2 (20) c c LL LR g c c c c 60.1 c 29.5 c Panel B. GJR-UD model μ a b (0.0187) (0.0210) (0.0258) (0.0255) (0.0237) (0.0314) B u c c c c c c B d c c a ω (0.0739) (0.0620) (0.0605) (0.0471) (0.0637) (0.0716) c b c 35