Predictability and Model Selection in the Context of ARCH Models

Transcript

1 Predicabiliy and Model Selecion in he Conex of ARCH Models Savros Degiannakis and Evdokia Xekalaki Deparmen of Saisics Ahens Universiy of Economics and Business 76 Paission Sree 434 Ahens Greece echnical Repor no 69 July 999 Absrac Mos of he mehods used in he ARCH lieraure for selecing he appropriae model are based on evaluaing he abiliy of he models o describe he daa. An alernaive model selecion approach is examined based on he evaluaion of he predicabiliy of he models on he basis of sandardied predicion errors. Keywords and Phrases: ARCH models Model selecion Predicabiliy Correlaed Gamma Raio disribuion Predicion Error Crierion Corresponding auhor. el.: ; fax: address: exek@aueb.gr.

2 . Inroducion ARCH models have widely been used in financial ime series analysis and paricularly in analying he risk of holding an asse evaluaing he price of an opion forecasing ime varying confidence inervals and obaining more efficien esimaors under he exisence of heeroscedasiciy. In he recen lieraure numerous parameric specificaions of ARCH models have been considered for he descripion of he characerisics of financial markes. In he linear ARCH(q) model originally inroduced by Engle (98) he condiional variance is posulaed o be a linear funcion of he pas q squared innovaions. Bollerslev (986) proposed he generalied ARCH or GARCH(pq) model where he condiional variance is posulaed o be a linear funcion of boh he pas q squared innovaions and he pas p condiional variances. Nelson (99) proposed he exponenial GARCH or EGARCH model. he EGARCH model belongs o he family of asymmeric GARCH models which capure he phenomenon ha negaive reurns predic higher volailiy han posiive reurns of he same magniude. Oher popular asymmeric models are he GJR model of Glosen e al. (993) he hreshold GARCH or ARCH model inroduced by Zakoian (99) and he quadraic ARCH or QGARCH model inroduced by Senana (995). ARCH models go by such exoic names as AARCH NARCH PARCH PNP-ARCH and SARCH among ohers. he richness of he family of parameric ARCH models cerainly complicaes he search for he rue model and leaves quie a bi of arbirariness in he model selecion sage. he problem of selecing he model ha describes bes he movemen of he series under sudy is herefore of pracical imporance. he aim of his paper is o develop a model selecion mehod based on he evaluaion of he predicabiliy of he ARCH models. In secion of he paper he ARCH process is presened. Secion 3 provides a brief descripion of he mehods used in he lieraure for selecing he appropriae model based on evaluaing he abiliy of he models o describe he daa. In secion 4 Panareos e al. s (997) model selecion mehod based on a sandardied predicion error crierion is examined in he conex of ARCH models. In secion 5 he suggesed model selecion mehod is applied using reurn daa for he Ahens Sock Exchange (ASE) index over he period Augus 3 h 993 o November 4 h 996 while in secion 6 a selecion mehod based on he abiliy of he models describing

3 3 he daa is invesigaed. Finally in secion 7 a brief discussion of he resuls is provided.. he ARCH Process Le { ( θ )} y refer o he univariae discree ime real-valued sochasic process o be prediced (e.g. he rae of reurn of a paricular sock or marke porfolio from ime o ) where θ is a vecor of unknown parameers and E( y ( θ ) I ) E ( y ( θ )) µ ( θ ) denoes he condiional mean given he informaion se available a ime I. he innovaion process for he condiional mean { ( θ )} ε ( θ ) ( θ ) µ ( θ ) y ε is hen given by = wih corresponding uncondiional variance V ( ε ( θ )) E ε ( θ ) σ θ ero uncondiional mean and ( ε ( θ ) ε ( θ )) = = E s. he condiional variance of he process given I is defined by V ( y ( θ ) I ) V ( y ( θ )) E ε ( θ ) σ ( θ ). Since invesors would know he informaion se I when hey make heir invesmen decisions a ime he relevan expeced reurn o he invesors and volailiy are µ ( θ ) and σ ( θ ) An ARCH process { ( θ )} can be presened as: σ ε ε ( θ ) = σ ( θ ) i. i. d. ~ f [ E( ) = V ( ) = ] ( θ ) = g σ ( θ ) σ ( θ )...; ε ( θ ) ε ( θ ) (...; υ υ...) where E = V = (). f is he densiy funcion of ( θ ) s respecively. (.) σ is a ime-varying posiive and measurable funcion of he informaion se a ime υ is a vecor of predeermined variables included in I and g (.) is a linear or nonlinear funcional form. By definiion ε ( θ ) is serially uncorrelaed wih mean ero bu wih a ime varying condiional variance equal o σ ( θ ). he condiional variance is a linear or nonlinear funcion of lagged values of σ and ε and predeermined variables included in I ( υ υ...). he sandard ARCH models assume ha (.) f is he densiy funcion of he normal disribuion. Bollerslev (987) proposed using he suden disribuion wih an esimaed kurosis regulaed by he degrees of freedom parameer. Nelson (99)

4 4 proposed he use of he generalied error disribuion (Harvey (98) Box and iao (973)) which is also referred o as he exponenial power disribuion. Oher disribuions ha have been employed include he generalied disribuion (Bollerslev e al. (994)) he normal Poisson mixure disribuion (Jorion (988)) he normal lognormal mixure (Hsieh (989)) and a serially dependen mixure of normally disribued variables (Cai (994)) or suden disribued variables (Hamilon and Susmel (994)). In he sequel for noaional convenience no explici indicaion of he dependence on he vecor of parameers θ is given when obvious from he conex. Since very few financial ime series have a consan condiional mean of ero an ARCH model can be presened in a regression form by leing ε be he innovaion process in a linear regression: σ = g ε I ~ ( σ ) ( σ ( θ ) σ ( θ )...; ε ( θ ) ε ( θ )...; υ υ...) y = x β + ε f (.) where x is a k vecor of endogenous and exogenous explanaory variables included in he informaion se I and β is a k vecor of unknown parameers. Le us assume ha he condiional mean = E( y I ) h κ order auoregressive [ AR ( κ )] model: described by a y κ i= ( ci y i ) + ε µ can be adequaely = c +. (.3) Usually he condiional mean is eiher he overall mean or a firs order auoregressive process. heoreically he AR () process allows for he auocorrelaion induced by disconinuous (or non-synchronous) rading in he socks making up an index (Scholes and Williams (977) Lo and MacKinlay (988)). According o Campbell e al. (997) he non-synchronous rading arises when ime series usually asse prices are aken o be recorded a ime inervals of a fixed lengh when in fac hey are recorded a ime inervals of oher possible irregular lenghs. he Scholes and Williams model suggess he s order moving average process for index reurns while he Lo and MacKinlay model suggess an AR () form. Higher orders of he auoregressive process are considered in order o invesigae if hey are adequae o produce more accurae predicions.

5 5 Engle (98) inroduced he original form of g(.) pas q squared innovaions: q ( ε i ) a i i= σ as a linear funcion of he = σ = a +. (.4) For he condiional variance o be posiive he parameers mus saisfy α > a for i =...q. In empirical applicaions of ARCH(q) models a long lag lengh and a large number of parameers are ofen called for. o circumven his problem Bollerslev (986) proposed he generalied ARCH or GARCH(pq) model: q p ( aiε i ) + ( b jσ j ) σ = a + (.5) i= q p where α > a i for i =... q and b j for j =... p. If a + j < j= i= i b j= hen { ε } is covariance saionary and is uncondiional variance is equal o σ = a q p ( a ) j i= i j = b. Noe ha even hough he innovaion process for he condiional mean is serially uncorrelaed i is no independen hrough ime. he innovaions for he variance are denoed as: E ( ) E ( ε ) = ε σ v ε. (.6) he innovaion process { v } is a maringale difference sequence in he sense ha i canno be prediced from is pas. However is range may depend upon he pas making i neiher serially independen nor idenically disribued. he uncondiional disribuion of ε has faer ails han he ime invarian disribuion of. For example in he case of he ARCH process in (.) wih he densiy funcion f (). being he normal disribuion and he funcional form of σ denoed as in he ARCH() model he kurosis of 4 ε is ( ε ) E( ε ) = 3( α ) ( 3α ) E always greaer han 3 he kurosis value of he normal disribuion. he GARCH(pq) model successfully capures several characerisics of financial ime series such as hick ailed reurns and volailiy clusering firs noed by Mandelbro (963): large changes end o be followed by large changes of eiher sign and small changes end o be followed by small changes. On he oher hand he GARCH srucure imposes imporan limiaions. he variance only depends on he magniude and i

6 6 no on he sign of ε which is somewha a odds wih he empirical behavior of sock marke prices where a leverage effec may be presen. he erm leverage effec firs noed by Black (976) refers o he endency for changes in sock reurns o be negaively correlaed wih changes in reurns volailiy i.e. volailiy ends o rise in response o bad news ( ε < ) and o fall in response o good news ( > ) ε. In order o capure he asymmery exhibied by he daa a new class of models was inroduced ermed he asymmeric ARCH models. he mos popular model proposed o capure he asymmeric effecs is Nelson s (99) exponenial GARCH or EGARCH(pq) model: q p ε i ε i ε i ln σ = ( ( ) a ai E γ i b j ln σ j. (.7) i= σ i σ i σ i j= he parameer γ allows for he asymmeric effec. If γ = hen a posiive surprise ( ε > ) has he same effec on volailiy as a negaive surprise ( ε < ). Here he erm surprise a ime refers o he unexpeced reurn which is he rae of reurn from ime o minus he relevan expeced reurn o he invesors e.g. ε = y µ. If < γ < a posiive surprise increases volailiy less han a negaive surprise. If γ < a posiive surprise acually reduces volailiy while a negaive surprise increases volailiy. For γ < he leverage effec exiss. Because of he logarihmic ransformaion he forecass of he variance are guaraneed o be non-negaive. hus in conras o he GARCH model no resricions need o be imposed on he model esimaion. he number of possible condiional volailiy formulaions is vas. he hreshold GARCH or ARCH(pq) model is one of he widely used models: σ δ = a p δ δ δ ( aid( ε i > ) ε i + γ id( ε i ) ε i ) + ( b jσ j ) q + i= where d (). denoes he indicaor funcion (i.e. ( > ) = ( > ) = i i j= (.8) d ε if ε > and d ε oherwise). Zakoian s (99) model is a special case of he ARCH model wih δ = while Glosen e al. (993) consider a version of he ARCH model wih δ =. he ARCH model allows a response of volailiy o news wih differen coefficiens for good and bad news. i

7 7 A wide range of ARCH models proposed in he lieraure has been reviewed by Bollerslev e al. (99) Bollerslev e al. (994) Bera and Higgins (993) Hamilon (994) and Gourieroux (997). Henschel (995) considers a complee parameric family of ARCH models. his family ness he mos popular symmeric and asymmeric ARCH models hereby highlighing he relaion beween he models and heir reamen of asymmery. 3. Model Selecion Mehods Mos of he mehods used in he lieraure for selecing he appropriae model are based on evaluaing he abiliy of he models o describe he daa. Sandard model selecion crieria such as he Akaike Informaion Crierion (AIC) (Akaike (973)) and he Schwar Bayesian Crierion (SBC) (Schwar (978)) have widely been used in he ARCH lieraure despie he fac ha heir saisical properies in he ARCH conex are unknown. hese are defined in erms of l ( θ ) he maximied value of he log-likelihood funcion of a model where θ is he maximum likelihood esimaor of θ based on a sample of sie n and θ denoes he dimension of θ hus: AIC = l ( n θ ) θ (3.) SBC = l n θ θ ln n (3.). In addiion he evaluaion of loss funcions for alernaive models is mainly used in model selecion. When we focus on esimaion of means he loss funcion of choice is ypically he mean squared error (MSE): MSE = n n = ε. (3.3) When he same sraegy is applied o variance esimaion he choice of he mean squared error is much less clear. Because of high non-lineariy in volailiy models a number of researchers consruced heeroscedasiciy-adjused loss funcions. Bollerslev e al. (994) presen four ypes of loss funcions: L n = = ( σ ) ε (3.4) L n = = ε ln σ (3.5)

8 8 L n 3 = = ( ε σ ) σ 4 (3.6) L n 4 = = ε σ + ln( σ ). (3.7) Pagan and Schwer (99) used he firs wo of he loss funcions o compare alernaive esimaors wih in-sample and ou-of-sample daa ses. Andersen e al. (999) and Heynen and Ka (994) are some examples from he lieraure ha applied loss funcions o compare he forecas performance of various volailiy models. Moreover loss funcions have been consruced based upon he goals of he paricular applicaion. Wes e al. (993) developed such a crierion based on he porfolio decisions of a risk averse invesor. Engle e al. (993) assumed ha he objecive was o price opions and developed a loss funcion from he profiabiliy of a paricular rading sraegy. 4. Model Selecion Based on a Predicion Error Crierion (PEC) Le us assume ha a researcher is ineresed in evaluaing he abiliy of he ARCH models o forecas he condiional variance. Consider he simple case of a regression model: y β + = x ε where β is a vecor of k unknown parameers o be esimaed x is a vecor of variables included in he informaion se a ime i. i. d. and ε ~ N( σ ) ime he expeced value µ of y is esimaed on he basis of he informaion available a ime i.e. y = µ = x β where β ( ) ( = X X X Y ) is he Y is he ( ) X is he ( k) leas square esimaor of β a ime dependen variable y and. A l vecor of l observaions on he l marix of he k variables included in he informaion se. In a manner of speaking and y can be considered as in-sample y and ou-of-sample forecass respecively. In oher words is measured on he basis of y I he informaion se available a ime while he informaion se available a ime. y is measured on he basis of I

9 9 In he sequel he densiy funcion f (.) in equaion (.) is assumed o be ha of he normal disribuion. For an ARCH process being presened as σ ( σ ) ε I ~ N (...; υ υ...) ( θ ) = g σ ( θ ) σ ( θ )...; ε ( θ ) ε ( θ ) y = x β + ε and θ being he vecor of unknown parameers le ε denoe he σ sandardied one sep ahead predicion errors. he vecor θ denoes he se of parameers o be esimaed for boh he condiional mean and he condiional variance. he mos commonly used way o model he condiional variance is he GARCH(pq) process: q p = a + ( ai ε i ) + ( b j σ j ) i= j= he parameers ( a... a b b ) σ... are indexed by he subscrip o indicae a q p ha hey may vary wih ime. he GARCH(pq) process may be rewrien as: ( u η w )( v ζ ω ) σ = where u = ( ε... ε q ) η = w = ( σ... σ p ) v = ( a a... aq ) ζ = ω = ( b ). b p... he vecor θ = ( β ζ ) v ω denoes he se of parameers o be esimaed for boh he condiional mean and he condiional variance a ime. he residual ε y y reflecs he difference beween he forecas and he observed value of he sochasic process. Panareos e al. (997) suggesed measuring he predicive behaviour of linear regression models on he basis of he sandardied disance beween he prediced and he observed value of he dependen random variable. he esimae of he sandardied disance was defined by: y y r = V ( y ) Consider he case of he AR()GARCH() model as defined by equaions (.3) and (.5) for κ = and p = q = respecively. he esimaors of he one sep ahead predicion error and is variance condiional on he informaion se available a ime are given by ε = y c c y and σ a + a ε b σ respecively = +

10 ( x )( l ) = k where V ( y ) ( Y X β ) ( Y X β ) + x ( X X ). A scoring rule o rae he performance of he model a ime for a series of poins in ime (... ) = was defined by R = r = he average of he squared sandardied residuals. In he sequel his approach is adoped using he average of he squared sandardied one sep ahead predicion errors as a scoring rule in order o rae he performance of an ARCH model o forecas he condiional variance in paricular σ R = =. (4.) ε is he esimaed sandardied disance beween he prediced and he observed value of he dependen random variable when he condiional sandard deviaion of he dependen variable given I is defined by an ARCH model V σ y I. heorem : Le ( θ ) denoe he vecor of unknown parameers o be esimaed a ime. Under he assumpion of consancy of parameers over ime ( θ ) ( θ ) =... = ( θ ) = ( θ ) = he esimaed sandardied one sep ahead predicion errors +... are asympoically independenly sandard normally disribued i.e. i. i. d. ( y y ) ~ ( ) σ. (4.) N Proof: o prove he heorem we need he following lemmas. Lemma : (Slusky s heorem) (see e.g. Greene (997 p.8)): For a coninuous funcion g ( x n ) ha is no a funcion of n p g( xn ) g( p lim x n ) (Here lim =. p lim denoes he limi in probabiliy as n.) he following wo Lemmas are implicaions of Slusky s heorem.

11 Lemma : (see e.g. Hamilon 994 p. 8): Le { X n } denoe a sequence of ( ) random vecors wih p lim X = n c i.e. X c p. Le (.) n g be a vecor-valued funcion g p m : R R which is coninuous a c and does no depend on n. hen g( X ) g( c) n. Lemma 3: (see e.g. Hamilon (994 p. 8)): Le { X n } denoe a sequence of ( ) p random marices wih X n C where C is a non-singular marix. Le X n denoe a sequence of ( ) p p random vecors wih X n c where c is a consan. hen ( X n ) X n ( C ) c or p lim( X n ) X n ( C ) c =. We now prove he following lemma. X for i =... denoe a sequence of random vecors wih Lemma 4: Le { } in p lim X in = W i where W i i =... are independenly and idenically disribued wih some disribuion funcion F ().. hen p ( X X... X ) ( W W... W ) lim n n n = and n X n X n are asympoically independenly and idenically disribued wih X... disribuion funcion F ().. ~ : Proof of Lemma 4: Le ~g (). be a vecor-valued real funcion g. : R R ~ ( x x... x ) g( x x... x ) ( g ( x x... x ) g ( x x... x )... g ( x x... x )) Assume ha (). ~g is coninuous a i i =... and does no depend on n. According o Slusky s heorem (Lemma ) for a coninuous funcion g ( x n ) ha is no a funcion of n p lim g( x ) = g( p lim ). hus ~ n x n p lim g( X n X n... X n ) ( g( X X... X ) g ( X X... X )... g ( X X... X )) By seing g ~ ( x x... x ) = ( x x... x ) (i.e. gi ( x x... x ) xi i =... =. = ) and applying Slusky s heorem we obain p lim g~ X n X n X n p X n X n X n g~... lim... = W W... W W W... W.

12 Le X X X x x x F n n n denoe he join densiy disribuion of he random variables n n n X X X.... As convergence in probabiliy implies convergence in disribuion we have = = W W W X X X n x x x F x x x F n n n lim X n X n X n W W W x F x F x F x F x F x F n n n = = lim... lim lim... As he join densiy is asympoically he produc of he marginal densiies n n n X X X... are asympoically independenly disribued each wih disribuion funcion (). F. Le us now reurn o he proof of heorem : A ime he expeced value of y is esimaed on he basis of he informaion available a ime i.e. = x y β and he expeced value of he condiional variance is esimaed on he basis of he informaion available a ime i.e. = v w u ω ζ η σ. Noe ha he elemens of he vecor w u η belong o he I so are considered as known values. he can be wrien as: = + = = + = = = x x x y y σ β β σ ε σ β ε β σ = + = x σ β β σ σ / / / + = v w u x v w u v w u ω ζ η β β ω ζ η ω ζ η We assume ha a sample of n observaions has been used o esimae he vecor of unknown parameers. According o Bollerslev (986) he maximum likelihood esimae θ is srongly consisen for θ and asympoically normal wih mean θ. In oher words

13 3 v v p p ω ζ β ω ζ β θ θ = = lim lim where lim p denoes limi in probabiliy as he sie of he sample n goes o infiniy. According o Lemma : = lim p = + = / / / lim lim v w u x p v w u v w u p ω ζ η β β ω ζ η ω ζ η hen based on Lemma 3: = + = / / / lim lim lim v w u p p x v p w u v w u ω ζ η β β ω ζ η ω ζ η = + = / / / lim lim v p w u p x v w u v w u ω ζ η β β ω ζ η ω ζ η v w u x = = + = / ω ζ η As convergence in probabiliy implies convergence in disribuion he... + are asympoically sandard normally disribued: ~ N d p his resul combined wih Lemma 4 implies ha he... + are asympoically independenly sandard normally disribued i.e. ~... N d i i d. Hence he heorem has been esablished. he resul of he heorem is valid for all he condiional variance funcions wih consisen esimaors of he parameers. Remark: As concerns he EARCH and he ARCH models he maximum likelihood esimaor v ω ζ β θ = is consisen and asympoically normal.

14 4 Consider he EGARCH(pq) model in he following form ln which can be wrien as: q p ε i ε i σ = + + a ai γ i + ( b ( i ) i ln σ σ σ i= i i i= ( u η w )( v ζ ω ) lnσ = where u = ( ε σ... ε q σ q ) η = [ ε σ ]...[ ε q σ q ] w = ( lnσ... σ ) = ( a a a ) ζ = ( γ...γ ) ω ln p v... q q = ( b... ) b p he parameers ( a... a γ... γ b b ) a q q... p are indexed by he subscrip o indicae ha hey may vary wih ime. According o Nelson (99) under sufficien regulariy condiions he maximum likelihood esimaor θ ( β ζ ) = is consisen v ω and asympoically normal. Also for he ARCH(pq) process he condiional variance can ake he form: q p ( ai ε i ) + γ ε d + ( bi σ i ) σ = a + which can be wrien as: ( u η w )( v ζ ω ) = σ i= where u = ( ε... ε q ) η = ( d ε ) w = ( σ... σ p ) v = ( a a... aq ) ζ = ω = d = if ε < and d = oherwise. γ b... b p i= As poined ou by Glosen e al. (993) as long as he condiional mean and variance are correcly specified he maximum likelihood esimaes will be consisen and asympoically normal. According o Lemma if plim = ~ N() a coninuous funcion hen p lim ( ) = ( ) = and g( ) = ( ) = convergence in disribuion ( ) ( ) ~ χ = d = = which is. As convergence in probabiliy implies. Hence as are asympoically sandard normal variables he variable degrees of freedom i.e. R is asympoically χ disribued wih

15 5 R d χ. (4.3) Also for wo processes A and B wih and observaions respecively he raio of he scoring rules R = ( A) ( A) degrees of freedom i.e. and ( A) R = ( B) ( B) ~ F R ( B) is F disribued wih and R R (4.4) if ( A) R and ( B) R are independenly disribued. According o Kibble (94) if for =... ( A) and are sandard B normally disribued variables following joinly he bivariae sandard normal disribuion ( A) ( B) hen he join disribuion of R R has a bivariae gamma disribuion wih probabiliy densiy funcion (p.d.f) given by: f ( A) ( B) ( R R ) ( A) ( B) R + R exp ρ = Γ ( )( ρ ) i= Γ ( ρ ( ρ ) i ( i + ) Γ( i + ( ) ) ( A) ( B) ( R R ) where Γ (). is he gamma funcion and ρ is he correlaion coefficien beween i (4.5) ( A) and ( B) ( A) ( B) ρ Cor( ). Panareos e al. (997) showed ha when he join disribuion ( A) ( B) ( A B) ( A) ( B) of ( R R ) is Kibble's bivariae gamma he disribuion of he raio Z R R is defined by he following p.d.f.: f ( A B Z ) ( A B) ( ρ ) Z = ( ) B where B = Γ Γ( ) Z Z ( + Z ) ( A B) ( A B) ( A B). Z ( A B) ( B) ( A) ~ CGR( k ρ) = = ρ Z + ( A B) + (4.6) (4.7) where k =. Panareos e al. (997) referred o he disribuion in (4.6) as he Correlaed gamma raio (CGR) disribuion. (A sample of ables of is percenage poins and of graphs depicing is probabiliy densiy funcion is given in he Appendix).

16 6 As poined ou by Panareos e al. (997) ( A) R and ( B) R could represen he sum of he squared sandardied predicion errors from wo regression models (no necessarily nesed) bu wih a common dependen variable. hus wo regression models can be compared hrough esing a null hypohesis of equivalence of he models in heir predicabiliy agains he alernaive ha model ( A ) produces beer predicions. Here he noion of he equivalence of wo models wih respec o heir predicive abiliy is considered in Panareos e al. s (997) sense o be defined implicily hrough heir mean squared predicion errors. Following Panareos e al. s (997) raionale he closes descripion of he hypohesis o be esed is using Versus H : Models A and B have equal mean squared predicion errors H : Model A has lower mean squared predicion error han model B ( A B) Z as a es saisic i.e. using he raio of he sum of he squared sandardied one sep ahead predicion errors of he wo compeing models. he null hypohesis is rejeced if Z ( A B) > CGR( k ρ a) where CGR ( k a) ( a) percenile of he CGR disribuion. In he case of independence beween he form: ( A) R and ρ is he ( B) R he CGR densiy funcion reduces o ( A B) ( A B) ( A B) ( Z ) = Z + Z (4.8) ( ) f ( A B ) Z B which is he p.d.f. of he F disribuion wih and degrees of freedom. Since very few financial ime series have a consan condiional mean of ero in order o esimae he condiional variance he condiional mean should have been defined. hus boh he condiional mean and variance are esimaed simulaneously. According o he PEC model selecion algorihm he models ha are considered as having a beer abiliy o predic fuure values of he dependen variable are hose wih he lowes sum of squared sandardied one-sep-ahead predicion errors. I becomes eviden herefore ha hese models can poenially be regarded as he mos appropriae o use for volailiy forecass oo.

17 7 5. Empirical Resuls he suggesed model selecion procedure is illusraed on daa referring o he daily reurns of he Ahens Sock Exchange (ASE) index. Le ( P P ) y denoe he = ln coninuously compound rae of reurn from ime o where P is he ASE closing price a ime. he daa se covers he period from Augus 3 h 993 o November 4 h 996 a oal of 8 rading days. able presens he descripive saisics. For an esimaed kurosis equal o 7.5 and an esimaed skewness equal o.8 he disribuion of reurns is fla (playkuric) and has a long righ ail relaive o he normal disribuion. he Jarque Bera (JB) saisic (Jarque and Bera (98)) is used o es wheher he series is normally disribued. he es saisic measures he difference of he skewness and kurosis of he series from hose of he normal disribuion. he JB saisic is compued as: ( S 4 ) JB = n K (5.) where n is he number of observaions S is he skewness and K is he kurosis. Under he null hypohesis of a normal disribuion he JB saisic is χ disribued wih degrees of freedom. able (). Descripive Saisics of he daily reurns of he ASE index (3h Augus 993 o 4h November 996 (8 observaions)) Observaions 8 Mean 5.7E-5 Median -.8 Sandard Deviaion. Skewness.8 Kurosis 7.5 Jarque Bera (JB) 6.38 probabiliy <. Augmened Dickey Fuller (ADF) -.67 % criical value Phillips Perron (PP) % criical value he skewness of a symmeric disribuion as he normal disribuion is ero. Posiive skewness implies ha he disribuion has a long righ ail. Negaive skewness implies a long lef ail disribuion. he kurosis of he normal disribuion is 3. If he kurosis exceeds 3 he disribuion is peaked (lepokuric) relaive o he normal. If he kurosis is less han 3 he disribuion is fla (playkuric) relaive o he normal. Under he null hypohesis of a normal disribuion he JB saisic is χ disribued wih degrees of freedom. he repored probabiliy is he probabiliy ha he JB saisic exceeds in absolue value he observed value under he null hypohesis. ADF: he null hypohesis of non-saionariy is rejeced if he ADF value is less han he criical value. (4 lagged differences). PP: he null hypohesis of non-saionariy is rejeced if he PP value is less han he criical value. (4 runcaion lags).

18 8 From able he value of he JB saisic obained is 6.38 wih a very low p-value (pracically ero). So he null hypohesis of normaliy is rejeced. In order o deermine wheher { y } is a saionary process he Augmened Dickey Fuller es (ADF) (Dickey and Fuller (979)) and he nonparameric Phillips Perron (PP) es (Phillips (987) Phillips and Perron (988)) are conduced. he ADF es examines he null hypohesis H : γ versus he alernaive H : γ < in he following regression: = y = c + γy + κ ϕ y + ε i i (5.) i= where denoes he difference operaor. According o he ADF es he null hypohesis of non-saionariy is rejeced a he % level of significance for any lag order up o κ =. he es regression for he PP es is he AR() process: y = c + γ + ε. (5.3) y While he ADF es correcs for higher order serial correlaion by adding lagged differenced erms on he righ hand side he PP es makes a correcion o he saisic of he γ coefficien from he AR() regression o accoun for he serial correlaion in ε. he correcion is nonparameric since an esimae of he specrum of ε a frequency ero ha is robus o heeroscedasiciy and auocorrelaion of unknown form is used. According o he PP es he null hypohesis is also rejeced a he % level of significance. able (). Lagrange muliplier (LM) es. es he null hypohesis of no ARCH effecs in he residuals up o order q. ε = β + β iε q i= ε = y c Q LM saisic p-value he LM saisic is compued as he number of observaions imes he R from he auxiliary es regression. I converges in disribuion o a χ q. i + u

19 9 he mos commonly used es for examining he null hypohesis of homoscedasiciy agains he alernaive hypohesis of heeroscedasiciy is Engle s (98) Lagrange muliplier (LM) es. he ARCH LM es saisic is compued from an auxiliary es regression. o es he null hypohesis of no ARCH effecs up o order q in he residuals he regression model wih q ε = β + β iε i + u (5.4) i= ε = c is run. Engle s es saisic is compued as he produc of he number of y observaions imes he value of he coefficien of variaion R of he auxiliary es regression. From able he values of he LM es saisic for q =... 8 are highly significan a any reasonable level. As according o he resuls of he above ess he assumpions of saionariy and ARCH effecs seem o be plausible for he process { y } of daily reurns several ARCH models are considered in he sequel. I is assumed specifically ha he condiional mean is considered as a h κ order auoregressive process: y µ = c = µ + σ + i= i. i ~. d. κ ( c y ) i N( ) i (5.5) and he condiional variance σ is assumed o be relaed o lagged values of ε and according o a GARCH(pq) model an EGARCH(pq) model or a ARCH(pq) model. In paricular σ is assumed o be deermined by one of he following models: he GARCH(pq) model q p ( aiε i ) + ( b jσ j ) σ = a + (5.6) i= he EGARCH(pq) model j= q p ε i ε i ln σ = a + ai + γ i + ( b j ln( σ j ) (5.7) i= σ i σ i j= σ

20 he ARCH(pq) model q p ( aiε i ) + γε d + ( b jσ j ) σ = a + (5.8) i= j= where d = if ε < and d = oherwise. hus he AR(κ )GARCH( p q ) AR(κ )EGARCH( p q ) and AR(κ )ARCH( p q ) models are applied for κ =... 4 p = and q = yielding a oal of 9 cases. Since in esimaing non-linear models no closed form expressions are obainable for he parameer esimaors an ieraive mehod has o be employed. he value of he l θ he log likelihood conribuion for each parameer vecor θ ha maximies observaion is o be found. Ieraive opimiaion algorihms work by saring wih an ( ) iniial se of values for he parameer vecor θ say θ and obaining a se of parameer () values l θ. his process is repeaed unil θ which corresponds o a higher value of he objecive funcion ( θ ) l no longer improves beween ieraions. In he sequel he Marquard algorihm (Marquard (963)) is used. his algorihm modifies he Bernd Hall Hall and Hausman or BHHH algorihm (Bernd e al. (974)) by adding a correcion marix o he Hessian approximaion (i.e. o he sum of he ouer produc of he gradien vecors for each observaion s conribuion o he objecive funcion). he Marquard updaing algorihm is compued as: n () i () i n () i ( i+ ) ( i) l l l θ = θ + (5.9) = θ θ where I is he ideniy marix and a is a posiive number chosen by he algorihm. he effec of his modificaion is o push he parameer esimaes in he direcion of he gradien vecor. he idea is ha when we are far from he maximum he local quadraic approximaion o he funcion may be a poor guide o is overall shape so i may be beer off o simply follow he gradien. he correcion may provide a beer performance a locaions far from he opimum and allows for compuaion of he direcion vecor in cases where he Hessian is near singular. he quasi-maximum likelihood esimaor (QMLE) is used as according o Bollerslev and Wooldridge (99) i is generally consisen has a limiing normal disribuion and provides asympoic sandard errors ha are valid under non-normaliy. ai = θ

21 In order o compue he sum of squared sandardied one sep ahead predicion errors a rolling sample of consan sie equal o 5 is used or s = 5 so 3 one sep ahead daily forecass are esimaed. he ou-of-sample daa se is spli ino 5 subperiods and he PEC model selecion algorihm is applied in each subperiod separaely. hus he model selecion is revised every 6 rading days and he informaion se includes daily coninuously compound reurns of he wo mos recenly years or 5 rading days. he choice of a 6 day lengh for each subperiod is arbirary. he sum of he squared one sep + = s+ s ahead predicion errors ( ) is esimaed for each model and presened in able 3 in he Appendix. he models seleced for each subperiod and heir sums of he squared sandardied one sep ahead predicion errors are: ( ) + = s+ s Subperiod Model Seleced min ( ). 5 Augus November 995 AR() EGARCH() November February 996 AR() EGARCH() February May 996 AR() EGARCH() May Augus 996 AR(3) EGARCH() Augus November 996 AR() EGARCH() 43.9 According o he PEC selecion mehod he exponenial GARCH() model describes bes he condiional variance for he oal examined period of 3 rading days. I is seleced by he PEC selecion mehod in each subperiod. Figure shows he daily value of he ASE index and he one sep ahead condiional sandard deviaion of is reurns. Daily Condiional Sandard Deviaion 4.% 3.5% 3.%.5%.%.5%.% Figure. he ASE index and he one sep ahead condiional sandard deviaion of is reurns esimaed by he EGARCH() models Value of he ASE index.5% 7 Aug-95 Oc-95 Dec-95 Feb-96 Apr-96 Jun-96 Aug-96 Oc-96 Dae EGARCH() daily one sep ahead condiional sandard deviaion of reurns Ahens Sock Exchange (ASE) Index Despie he fac ha an asymmeric model is seleced by he PEC algorihm here are no asymmeries in he ASE index volailiy. According o Figure he major episodes of high

22 volailiy are no associaed wih marke changes of he same sign. Figure presens he values of he parameers a and γ of he 3 esimaed EGARCH() models while Figure 3 depics he relevan sandard errors for he parameers a and γ. Obviously he γ parameer which allows for he asymmeric effec is posiive bu saisically insignifican. herefore he asymmeric relaion beween reurns and changes in volailiy does no characerie he examined period. An ineresing poin is ha he higher order of he condiional mean auoregressive process is chosen as adequae o produce more accurae predicions for he firs and he fourh subperiods. As concerns he firs subperiod he AR()EGARCH() model y = c + c y + c y + ε ε ε ln = + + a a γ σ σ 56 is he one wih he lowes value of ( ) σ = 5 equal o.96. he hypohesis: H : he model AR()EGARCH() has equivalen predicive abiliy o model X is esed versus (5.) H : he model AR()EGARCH() produces beer predicions han model X wih X denoing any one of he remainder models..6 Figure. he parameers of he esimaed EGARCH() models Value of he Parameer α Value of he Parameer γ. -.4 Aug-95 Oc-95 Dec-95 Feb-96 Apr-96 Jun-96 Aug-96 Oc-96 Dae Value of he parameer α Value of he parameer γ

23 3.6 Figure 3. he sandard error for he parameers of he esimaed EGARCH() models Sandard Error of he Parameers Aug-95 Oc-95 Dec-95 Feb-96 Apr-96 Jun-96 Aug-96 Oc-96 Dae Sandard Error of he parameer α Sandard Error of he parameer γ Noe ha he correlaion beween he sandardied one sep ahead predicion errors is 6.96 = 5 AR() EGARCH () X 56 ( X ) greaer han.9 in each case. If ( >.9 = 3 a) > CGR ρ he null hypohesis of equivalen predicive abiliy of he models is rejeced a a % level of significance and he AR()EGARCH() model is regarded as beer han model X. able 4 in he Appendix summaries he resuls of he hypohesis ess for each subperiod. Figure 4 in he Appendix depics he one sep ahead 95 per cen predicion s inervals for he models wih he lowes Z + = s+ in each subperiod. he predicion inervals are consruced as he expeced rae of reurn plus\minus.96 imes he condiional sandard deviaion boh measurable o informaion se: µ ±.96σ. So each ime nex day s predicion inerval is ploed only informaion available a curren day is used. Remark ha around November 995 a volaile period he predicion inerval in Figure 4 racked he movemen of he reurns quie closely (seven ouliers or.33% were observed). 6. An Alernaive Approach In his secion an in-sample analysis is performed in order o selec he appropriae models describing he daa. hen he seleced models are used o esimae he one sep ahead forecass. Having assumed ha he condiional mean of he reurns follows a h κ

24 4 order auoregressive process as in (.3) Richardson and Smih (994) developed a es for auocorrelaion. I is a robus version of he sandard Box Pierce (Box and Pierce (97)) procedure. For p i denoing he esimaed auocorrelaion beween he reurns a ime and i he es is formulaed as: RS () r = n p r i i= + ci (6.) where n is he sample sie and c i is he adjusmen facor for heeroscedasiciy which is calculaed as: c i Cov ( y y ) i = (6.) Var ( y ) where y = y n = y n. Under he null hypohesis of no auocorrelaion he saisic is asympoically disribued as χ wih r degrees of freedom. If he null hypohesis of no auocorrelaion canno be rejeced hen he reurns process is equal o a consan plus he residuals ε. In oher words { y } follows he AR() process. If he null of no auocorrelaion is rejeced hen { y } follows he AR() process. In order o es for he exisence of a higher order auocorrelaion he es is applied on he esimaed residuals from he AR() model. In his case he saisic under he null hypohesis is asympoically disribued as χ wih r degrees of freedom. he es is calculaed on 7 auocorrelaions ( r = 7) for 8 observaions yielding a value equal o RS ( 7) = 486 > χ 7.5. As he null hypohesis of no auocorrelaion is rejeced he es is RS 6 = 33 < χ. run on he esimaed residuals from he AR() model ha gives 6.5 hus a firs order auocorrelaion is deeced for he reurns process. Noe ha he AR() form allows for he auocorrelaion imposed by disconinuous rading. Having defined he condiional mean equaion he nex sep is he esimaion of he condiional variance funcion. he AIC and he SBC crieria are used o selec he appropriae condiional variance equaion. Noe ha he AIC mainly chooses as bes he less parsimonious model. Also under cerain regulariy condiions he SBC is consisen in he sense ha for large samples i leads o he correc model choice assuming he rue model does belong o he se of models examined. hus he SBC may be preferable o use. As concerns he specific daase boh he AIC and SBC selec he

25 5 GARCH() model as he mos appropriae funcion o describe he condiional variance. So performing an in-sample analysis he AR()GARCH() model is regarded as he mos suiable which is he model applied in mos researches. Figure 5 in he Appendix presens he in-sample 95 per cen confidence inerval for he AR()GARCH() model. here are foureen observaions or 4.66% ouside he confidence inerval. In order o compare he model selecion mehods he choice of he models should be conduced a he same ime poins. hus he Richardson Smih es for auocorrelaion deecion and he informaion crieria for model selecion are used in each subperiod separaely. he models seleced for in each subperiod are: Subperiod Richardson Smih SBC AIC Model selecion Model Selecion Model Selecion. AR(3) GARCH() EGARCH(). AR() GARCH() GARCH() 3. AR() GARCH() GARCH() 4. AR() GARCH() GARCH() 5. AR() GARCH() ARCH() Based on able 4 he hypohesis ha he model seleced by he in-sample analysis is + = s+ s equivalen o he model wih minimum value of is rejeced in he majoriy of he cases. Proceeding as in he previous secion he one sep ahead predicion inervals for he models seleced in each subperiod are creaed. As in secion 5 nex day s predicion is based only on informaion available a curren day. Figures 6 and 7 in he Appendix presen he one sep ahead 95 per cen predicion inervals for he models seleced by he SBC and AIC respecively. here are hireen observaions or 4.33% ouside he predicion inerval for he models seleced by he SBC whereas here are foureen ouliers or 4.66% for he models seleced by he AIC. herefore he imporance of selecing a condiional variance model based on is abiliy o forecas and no on fiing he daa gains a lead over. Of course he consrucion of he predicion inervals is a naïve way o examine he accuracy of our mehod s predicabiliy. 7. Conclusion An alernaive model selecion approach based on he CGR disribuion was inroduced. Insead of being based on evaluaing he abiliy of he models o describe he daa (Akaike informaion and Schwar Bayesian crieria) he proposed approach is based on evaluaing he abiliy of he models o predic he condiional variance. he mehod was

26 6 applied o 8 daily reurns of he ASE index a daase covers he period from Augus 3 h 993 o November 4 h 996. he firs s observaions were used o esimae he one sep ahead predicion of he condiional mean and variance a s +. For s = 5 a oal of 3 one sep ahead predicions of he condiional mean and variance were obained. he ou-of-sample daa se were spli o 5 subperiods and he PEC model selecion algorihm were applied in each subperiod separaely. hus he model selecion was revised every 6 rading days. he idea of jumping from one model o anoher as sock marke behavior alers is inroduced. he ransiion from one model o anoher is done according o he PEC model selecion algorihm. Each ime he model selecion mehod is applied he model is used o predic he condiional variance is revised. Of course he idea of swiching from one regime o anoher has been already applied o he class of swich regime ARCH models inroduced by Cai (994) and Hamilon and Susmel (994) and exended by several auhors such as Dueker (997) and Hansen (994). However hese models allow he parameers of a specific ARCH model o come from one of several differen regimes wih ransiions beween regimes governed by an unobserved Markov chain. Using an alernaive approach based on evaluaing he abiliy of fiing he daa he condiional mean is firs modeled and subsequenly an appropriae form for he condiional variance is chosen. Applying he PEC model selecion algorihm he null hypohesis ha he model seleced by he in-sample analysis is equivalen o he model + = s+ s wih minimum value of ( ) is rejeced in he pluraliy of he cases a less han 5% level of significance. he in-sample model selecion mehods and he predicabiliybased mehod do no coincide in he sifing of he appropriae condiional variance model. Moreover.33% and 4.33% of he daa were ouside he µ.96σ predicion ± inerval consruced based on he PEC and he SBC model selecion mehods respecively. he predicive abiliy of he PEC model selecion algorihm has o be furher invesigaed. Among he financial applicaions where his mehod could have a poenial use are in he fields of porfolio analysis risk managemen and rading opion derivaives.

27 7 References Akaike H. (973). Informaion heory and he Exension of he Maximum Likelihood Principle. Proceeding of he Second Inernaional Symposium on Informaion heory. Budapes 67-8 Andersen.. Bollerslev and S. Lange (999). Forecasing Financial Marke Volailiy: Sample Frequency vis-à-vis Forecas Horion. Journal of Empirical Finance Bera A.K. and M.L. Higgins (993). ARCH Models: Properies Esimaion and esing Journal of Economic Surveys Bernd E. R. B. H. Hall R. E. Hall and J. A. Hausman (974). Esimaion and Inference in Nonlinear Srucural Models. Annals of Economic and Social Measuremen Black F. (976). Sudies of Sock Marke Volailiy Changes. Proceedings of he American Saisical Associaion Business and Economic Saisics Secion 77-8 Bollerslev. (986). Generalied Auoregressive Condiional Heeroscedasiciy. Journal of Economerics Bollerslev. (987). A Condiional Heeroskedasic ime Series Model for Speculaive Prices and Raes of Reurn. Review of Economics and Saisics Bollerslev. R. C. Chou and K. Kroner (99). ARCH Modelling in Finance: A Review of he heory and Empirical Evidence. Journal of Economerics Bollerslev. R. F. Engle and D. Nelson (994). ARCH Models. Ch.49 in R. F. Engle and D. L. McFadden eds. Handbook of Economerics IV Elsevier Bollerslev. and J. M. Wooldridge (99). Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih ime Varying Covariances. Economeric Reviews 43 7 Box G.E.P and D. A. Pierce (97). Disribuion of residual auocorrelaion in ARIMA ime series models. Journal of he American Saisical Associaion Box G.E.P. and G.C. iao (973). Bayesian Inference in Saisical Analysis. Reading Mass Cai J. (994). A Markov Model of Swiching-Regime ARCH. Journal of Business and Economic Saisics Campbell J. A. Lo and A. C. MacKinlay (997). he Economerics of Financial Markes New Jersey: Princeon Universiy Press

28 8 Dickey D.A. and W.A. Fuller (979). Disribuion of he Esimaors for Auoregressive ime Series wih a Uni Roo. Journal of he American Saisical Associaion Dueker M.J. (997). Markov Swiching in GARCH Processes and Mean-Revering Sock Marke Volailiy. Journal of Business and Economic Saisics Engle R. F. (98). Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of he Unied Kingdom Inflaion. Economerica Engle R. F. C. H. Hong A. Kane and J. Noh (993). Arbirage Valuaion of Variance Forecass wih Simulaed Opions Advances in Fuures and Opions Research Glosen L. R. R. Jagannahan and D. E. Runkle (993). On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. Journal of Finance Gourieroux C. (997). ARCH models and Financial Applicaions Springer Series Greene W.H. (997). Economeric Analysis 3rd ediion Prenice Hall Hamilon J. (994). ime Series Analysis New Jersey: Princeon Universiy Press Hamilon J. and R. Susmel (994). Auoregressive Condiional Heeroskedasiciy and Changes in Regime. Journal of Economerics Hansen B.E. (994). Auoregressive Condiional Densiy Esimaion. Inernaional Economic Review Harvey A.C. (98). he Economeric Analysis of ime Series Oxford Henschel L. (995). All in he family: Nesing symmeric and asymmeric GARCH models. Journal of Financial Economics Heynen R. and H. Ka (994). Volailiy Predicion: A Comparison of he Sochasic Volailiy GARCH() and EgARCH() Models. Journal of Derivaives Winer Hsieh D.A. (989). Modeling Heeroscedasiciy in Daily Foreign-Exchange Raes. Journal of Business and Economic Saisics Jarque C. M. and A. K. Bera (98). Efficien ess for Normaliy Heeroscedasiciy and Serial Independence of Regression Residuals. Economic Leers Jorion P. (988). On Jump Processes in he Foreign Exchange and Sock Markes. Review of Financial Sudies Kibble W. F. (94). A wo Variae Gamma ype Disribuion. Sankhya

29 9 Lo A. and C. MacKinlay (988). Sock marke prices do no follow random walks: Evidence from a simple specificaion es. Review of Financial Sudies 4-66 Mandelbro B. (963). he variaion of cerain speculaive prices. Journal of Business Marquard D.W. (963). An algorihm for Leas Squares Esimaion of Nonlinear Parameers. Journal of he Sociey for Indusrial and Applied Mahemaics Nelson D.B. (99). Condiional Heeroscedasiciy in Asse Reurns: A New Approach. Economerica Pagan A.R. and G.W. Schwer (99). Alernaive Models for Condiional Sock Volailiy. Journal of Economerics Panareos J. S. Psarakis and E. Xekalaki (997). he Correlaed Gamma-Raio Disribuion in Model Evaluaion and Selecion. Ahens Universiy of Economics and Business Deparmen of Saisics echnical Repor 33 Phillips P.C.B. (987). ime Series Regression wih A Uni Roo. Economerica Phillips P.C.B. and P. Perron (988). esing for a Uni Roo in ime Series Regression. Biomerika Richardson M. and. Smih (994). A unified approach o esing for serial correlaion in sock reurns. Journal of Business Scholes M. and J. Williams (977). Esimaing beas from non-synchronous daa. Journal of Financial Economics Schwar G. (978). Esimaing he Dimension of a Model. Annals of Saisics Senana E. (995). Quadraic ARCH models. Review of Economic Sudies Wes K. D. H. J. Edison and D. Cho (993). A Uiliy Based Comparison of Some Models for Exchange Rae Volailiy. Journal of Inernaional Economics Zakoian J. M. (99). hreshold Heeroskedasiciy Models manuscrip CRES INSEE

30 3 Appendix able 3. Sum of squared sandardied one sep ahead predicion errors for each subperiod able 4. esing he null hypohesis ha he model wih he lowes sum of he squared sandardied one sep ahead predicion errors has equivalen predicive abiliy o model X wih X denoing any of he remainder models. Figure 4. One Sep Ahead 95% Forecased Inerval for he Models wih he Lowes Sum of he Squared Sandardied One Sep Ahead Predicion Errors Figure 5. In-Sample 95% Confidence Inerval for he AR() GARCH() Model Figure 6. One Sep Ahead 95% Forecased Inervals for he Models Seleced by he SBC Figure 7. One Sep Ahead 95% Forecased Inervals for he Models Seleced by he AIC Figures 8-4. he probabiliy densiy funcion of he Correlaed Gamma Raio Disribuion Pages Percenage Poins of he Correlaed Gamma Raio Disribuion

31 3 able 3. Sum of squared sandardied one sep ahead predicion errors for each subperiod. he AR(κ)GARCH(pq) AR(κ)EGARCH(pq) and AR(κ)ARCH(pq) models are applied for κ= 4 p= and q=. able 3.a able 3.b 5 Augus November 995 (s=[556]) 7 November February 996 (s=[566]) able 3.c 4 February May 996 (s=[668]) κ=* κ= κ= κ=3 κ=4 κ=* κ= κ= κ=3 κ=4 κ=* κ= κ= κ=3 κ=4 GARCH(pq) GARCH(pq) GARCH(pq) p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= ARCH(pq) ARCH(pq) ARCH(pq) p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= EGARCH(pq) EGARCH(pq) EGARCH(pq) p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= p= q= ** p= q= ** 4899 ** *Regress he depeden variable on a consan. ** Model fails o converge a leas once. AR(κ) GARCH(pq) EGARCH(pq) ARCH(pq) y σ ln σ = a + q p ( a iε i ) + ( b jσ j ) i= j = q p ε i ε i σ = + + a a i γ i + ( b j ln( σ j ) = a + κ = c + i= σ σ i= i i j = q p ( a iε i ) + γε d + ( b jσ j ) i= ( ci y i ) + ε j =