ESING FOR SERIAL CORRELAION: GENERALIZED ANDREWS- PLOBERGER ESS John C. Nankevis Essex Finance Cene, Essex Business School Univesiy of Essex, Colchese, CO4 3SQ U.K. N. E. Savin Depamen of Economics, Univesiy of Iowa, Iowa Ciy, IA 54. Augus 16, 8 ABSRAC his pape consides esing he null hypohesis ha a imes seies is uncoelaed when he ime seies is uncoelaed bu saisically dependen. his case is of inees in economic and finance applicaions. he GARCH (1, 1) model is a leading example of a model ha geneaes seially uncoelaed bu saisically dependen daa. he ess of seial coelaion inoduced by Andews and Plobege (1996, heeafe AP) ae genealized fo he pupose of esing he null. he aionale fo genealizing he AP ess is ha hey have aacive popeies fo case fo which hey wee oiginally designed: hey ae consisen agains all non-whie noise alenaives and have good all-ound powe agains nonseasonal alenaives compaed o seveal widely used ess in he lieaue. hese popeies ae inheied by he genealized AP ess. JEL classificaion: C1; C Keywods: Auoegessive moving aveage model; Lagange muliplie es; Nonsandad esing; Saisically dependen ime seies; Uncoelaedness. Coesponding auho: N. E. Savin, Depamen of Economics, 18 Pappajohn Business Building, Iowa Ciy, IA 54-1994. el.: (319) 335-855; fax: (319) 335-1956. Email addess: gene-savin@uiowa.edu.
1. INRODUCION As noed by Hong and Lee (3), hee has been gowing inees in developing consisen ess fo seial coelaion of unknown fom; examples include AP, Hong (1996), Chen and Deo (4) in esimaed egession esiduals and Dulauf (1991) and Deo () in he obseved aw daa. he ess assume independenly and idenically disibued egession eos unde he null excep fo Deo (), which genealizes Dulauf (1991) o allow fo a esicive fom of condiional heeoskedasiciy. his pape consides esing he null ha a imes seies is uncoelaed when he ime seies is uncoelaed bu saisically dependen. Fo a moe exensive lieaue eview, see Fancq, Roy and Zakoian (5). he case of uncoelaed dependen ime seies is of inees in economic and financial applicaions because many poblems such as financial (non-) pedicabiliy ae elaed o a maingale diffeence sequence (MDS) hypohesis afe demeaning, which implies seial uncoelaedness bu no seial independence. he GARCH (1, 1) model is a leading example of a model ha geneaes seially uncoelaed bu saisically dependen daa. he aionale fo genealizing he AP ess is ha hey ae consisen agains all non-whie noise alenaives and have good all-ound powe agains nonseasonal alenaives when compaed o seveal ess in he lieaue, including he Box-Piece (197, heeafe BP) ess. he genealized AP ess inhei he popeies of he AP ess in powe compaisons. In ou simulaion expeimens, he genealized AP ess ypically have subsanially bee powe han he genealized BP (Lobao, Nankevis and Savin ()) ess agains non-seasonal alenaives and powe equal o o bee han he Deo () es. 1
AP inoduced ess of seial coelaion designed fo he case whee he ime seies is geneaed by ARMA (1, 1) models unde he alenaive. As hey noed, i is naual o conside ess of his so because ARMA (1, 1) models povide pasimonious epesenaions of a boad class of saionay ime seies. ARMA(1, 1) models fo financial euns seies follow fom he mean-evesion models of Poeba and Summes (1988) and he pice-end models of aylo (5). Howeve, esing fo seial coelaion geneaed by an ARMA (1, 1) model is a nonsandad esing poblem because he ARMA (1, 1) model educes o a whie noise model wheneve he AR and MA coefficiens ae equal. he esing poblem is one in which a nuisance paamee is pesen only unde he alenaive hypohesis. Fo he poblem addessed by AP, he sandad likelihood aio (LR) saisic does no possess is usual asympoic chi-squaed disibuion o is usual asympoic opimaliy popeies. I is also possible ha an ARMA(1,1) geneaes whie noise when he AR and MA coefficiens ae no equal, as is he case fo he all-pass file model; see Andews, Davis and Beid (6). he LR es has he aacive feaue of being consisen agains all foms of seial coelaion (Posche (199)). AP show ha his feaue also holds fo ess inoduced ino he lieaue by Andews and Plobege (1994, 1995), namely, he sup Lagange Muliplie (LM) and aveage exponenial LM and LR ess. AP esablish he asympoic null disibuion fo he LR, sup LM and aveage exponenial es saisics unde he assumpion ha he geneaing pocess is a condiionally homoskedasic maingale diffeence sequence (MDS). he asympoic ciical values fo hese ess wee calculaed by simulaion. In Mone Calo powe expeimens, AP compaed he finiesample powes of he LR, sup LM, aveage exponenial, BP, and ohe ess. he
alenaives include Gaussian ARMA (1, 1) models. Agains his class of alenaives, he LR es was found o have vey good all-aound powe popeies fo non-seasonal alenaive models, especially compaed o BP ess. Fo seially uncoelaed bu saisically dependen ime seies, he ue levels of he LR, sup LM and aveage exponenial ess can diffe subsanially fom he nominal levels when he ess use he asympoic ciical values calculaed by AP. his pape genealizes a subse of he ess consideed by AP so ha hey have he coec level asympoically when he ime seies is seially uncoelaed bu saisically dependen. he subse consiss of LM-based ess, namely he sup LM es and he aveage exponenial LM ess. he genealizaion is obained by using he ue asympoic covaiance maix of he sample auocoelaions, o a consisen esimao. he same appoach has been used by Lobao, Nankevis and Savin () o genealize he BP ess o seings whee he ime seies is seially uncoelaed bu saisically dependen. he asympoic ciical values epoed by AP emain valid fo he genealized LM-based ess. he finie-sample levels of he genealized LM-based ess wih asympoic ciical values ae assessed by simulaion. In Mone Calo powe expeimens, hese ess ae compaed o genealized BP ess and he Deo () es. he genealized AP ess ypically have bee level-coeced powe agains non-seasonal alenaives. Hence, genealized AP ess can be ecommended fo use in economic and finance applicaions. he pape epos he esuls of an empiical applicaion o sock eun indexes.. ARMA (1, 1) MODEL AND ES SAISICS his secion eviews he model, hypohesis and es saisics consideed by AP. 3
he model is he ARMA (1, 1) model, Y = φy 1+ ε πε 1 fo =,3,..., (1) whee { Y : = 1,..., } ae obseved andom vaiables and { ε : = 1,,...} ae unobseved innovaions. AP epaameeize (1) as Y = ( β + π) Y + ε πε fo =,3,..., 1 1 whee β = φ π, he paamee space foπ is Π and ha fo β is B. hey assume ha Π and B ae such ha he absolue value of he auoegessive coefficien π + β <1, Π is closed and B conains a neighbohood of zeo. he fome condiion ules ou uni oo and explosive behavio of { Y : = 1,..., }. he null hypohesis is ha { Y : = 1,..., } is whie noise, and he alenaive is ha { Y : = 1,..., } is seially coelaed. hese hypoheses ae given by H : β = and H : β. () 1 When β =, he model (1) educes o Y = ε, and he paameeπ is no longe pesen. he esing poblem is non-sandad becauseπ is no idenified when he null hypohesis is ue. Le LR (π ) denoe he sandad LR saisic fo esing H vesus H 1 when π is known unde he alenaive. hen he LR saisic fo he unknown π is whee π = σ ˆ σ π LR ( ) log( Y / Y( )), LR = sup LR ( π ), π Π and σ 1 = Y Y = 1, 4
1 i i ˆ σy( π) = σy Y π Y i 1 π Y i 1. = i= = i= (3) Noe ha LR = LR ( ˆ π ), whee ˆ π is he ARMA (1, 1) ML esimae of π. AP poved ha an asympoically equivalen es saisic o he LR saisic (unde he null and local alenaives) is he sup LM saisic whee suplm ( π ), (4) π Π 1 i 4 LM ( π ) = Y π Y i 1 (1 π ) / σy. = i= (5) he LR and sup LM ess ae shown o saisfy an asympoic admissibiliy popey, and as a consequence, bea any given es in ems of weighed aveage powe agains alenaives ha ae local o, bu sufficienly disan fom he null; fo deails, see p. 1333 of AP. Andews and Plobege (1994) inoduced aveage exponenial ess. hese ess ae asympoically opimal in he sense ha hey minimize weighed aveage powe fo a specific weigh funcion. he weigh funcions fo he paamee β ae mean zeo nomal densiies wih vaiances popoional o a scala c >. he weigh funcion J fo he paamee π is chosen by he invesigao. Fo he simulaion esuls in AP, he funcion is aken o be unifom on Π, and similaly in his aicle. Fo each c (, ), he aveage exponenial LM es saisic is given by 1/ 1 c Exp-LM c = (1 + c) exp LM ( π ) dj( π ) 1+ c, (6) 5
whee LM (π) is as defined in (5), and J( ) is pobabiliy measue on Π. he aveage exponenial LR es saisic, Exp-LR c, is defined analogously wih LM (π) being eplaced by LR (π). he limiing aveage exponenial LM es saisics as c and c ae given by Exp-LM = LM ( ) ( π ) π dj and 1 Exp-LM = ln exp LM ( π ) dj ( π ). (7) Andews and Plobege (1994) show ha he aveage exponenial ess have asympoic local powe opimaliy popeies. 3. ASYMPOIC AND FINIE-SAMPLE NULL DISRIBUIONS OF AP ES SAISICS AP esablished he asympoic null disibuion of he es saisics peviously inoduced. his secion eviews he asympoic heoy. he asympoic null disibuions of he es saisics ae esablished by showing ha he sequences of sochasic pocesses {LR ( ) : 1} and {LM ( ): 1} indexed by π Π convege weakly o a sochasic pocess G( ) and hen by applying he coninuous mapping heoem. Le d denoe convege in disibuion of a sequence of andom vaiables. Le { Zi : i } be a sequence of iid N(,1) andom vaiables. Define (8) ( ) (1 i ) Zi fo i= G π = π π π Π. he following heoem is poved by AP unde a vaiey of assumpions. heoem 1. 6
a.lm G(), b. sup LM ( π) sup G( π), π Π c.exp-lm G( π) dj( π), d d π Π d 1 d. Exp-LM ln exp( G( π)) dj( π), and e. pas (a)-(d) hold wih LM eplaced by LR. heoem 1 holds fo ime seies whee he asympoic covaiance maix of he fis -1 of he sample auocoelaions is equal o he ideniy maix. A ime seies geneaed by a condiional homoskedasic maingale diffeence sequence is an example whee he asympoic covaiance maix of he sample auocoelaions is he ideniy maix. On he ohe hand, heoem 1 does no hold fo many models used in economics and finance, fo example, a GARCH (1, 1) wih nomal innovaions. he implicaions of he ideniy maix condiion fo esing H ae exploed in he nex secion. Fom heoem 1, he LR, sup LM and aveage exponenial LR and LM ess ae asympoically pivoal, ha is, he asympoic disibuions does no depend on any unknown paamees. Hence, he asympoic ciical values fo he ess can be simulaed by uncaing he seies i π Z i= i a a lage value. AP epo simulaed ciical values of he ess in hei able 1.he ciical values ae based on he paamee space Π = {, ±.1,, ±.79, ±.8}, = 5 and 4, epeiions. hey also calculae finie-sample ciical values fo he ess. he LR, sup LM and aveage exponenial LR and LM ess ae shown by AP o be consisen agains all deviaions fom he null hypohesis of whie noise wihin a class of weakly saionay song mixing sequences of andom vaiables. his consisency popey illusaes he obus powe popeies of he ess. 7
he ess inoduced by AP can also be used o es whehe egession eos ae seially coelaed. he ess ae consuced using he esiduals { Yˆ : } ahe han he andom vaiables{ Y : } ; see AP fo deails. Povided ha he egessos ae exogenous, he esuling LR, sup LM and aveage exponenial LR and LM es saisics have he same asympoic disibuions as when he acual eos ae used o consuc he saisics. hus, he asympoic ciical values peviously calculaed by AP ae applicable. Howeve, he ess ae no valid when applied o he esiduals of a dynamic egession model. 4. GENERALIZAION OF LM BASED ESS In his secion, he LM-based ess consideed by AP ae genealized such ha he ess have he coec asympoic level unde he null when he asympoic covaiance maix of he sample auocoelaions is no he ideniy maix. he asympoic disibuions of he genealized AP es saisics ae based on a cenal limi heoem fo he sample auocoelaions and a consisen esimao of he asympoic covaiance maix. We begin wih a eview of he asympoic disibuion heoy of he sample auocoelaions when { Y : = 1,,...} is a covaiance saionay sequence of saisically dependen bu uncoelaed andom vaiables wih mean zeo (o allow fo mean μ, as we do below). Define he lag-j auocovaiance by γ j = E(Y Y + j ) and he lag-j auocoelaion by ρ j = γ j /γ. he vaiance and lag-j auocovaiance ae given by ˆ = ( Y ) / - j γ j = 1 + j γ = 1 and ˆ = ( YY )/. We assume ha Y is a weak dependen pocess fo which he veco of sample auocovaiances γ = ( γ1, γ,..., γ K ) saisfies he following cenal limi 8
heoem: d ( ˆ γ γ ) N(, C), whee C (assumed o be finie and posiive definie) is π imes he specal densiy maix a zeo fequency of he veco wih elemens YY j. A saighfowad applicaion of he dela mehod leads o a cenal limi heoem fo he sample auocoelaions: Unde geneal weak dependence condiions, = (,,..., ) N(, V), (9) 1 K d whee ˆ / ˆ j = γ j γ and he ijh elemen of V is given in Lobao, Nankevis and Savin (, p. 73) and Romano and hombs (1996). A vaiey of weak dependence condiions ae eviewed in Lobao, Nankevis and Savin (). Using he idea of nea epoch dependence (NED), De Jong and Davidson () show ha he peceding cenal limi heoem fo ˆ γ holds unde he following assumpion: Assumpion 1. Le Y be a covaiance saionay pocess ha saisfies E Y s < fo some s > 4 and all and is L -NED of size 1/ on a pocess U whee U is an α-mixing sequence of size s/(s 4). Davidson () has poved ha many nonlinea imes seies models saisfy he NED assumpion. Nex conside esing he null hypohesis H (K): ρ = (ρ 1,, ρ K ) = agains he alenaive ρ. Suppose V is known. hen a es can be based on he es saisic 1 BPK( V) = ' V, (1) whee he saisic is asympoically chi-squae disibued wih K degees of feedom when H (K) is ue. In pacice, V is unknown. In he sandad BP saisic, V is eplaced by I, he ideniy maix. If V is no equal o he ideniy, he sandad BP es can poduce misleading infeences. 9
Unde he null, V simplifies o V = [( γ ) C ] whee C has as is ijh elemen d = ij i + d + d j d = c = E( Y μ)( Y μ)( Y μ)( Y μ) fo i,j = 1,,K, (11) whee he above fomula coves he case EY ( ) = μ. Lobao, Nankevis and Savin () use his simplificaion o consuc a genealized BP es saisic. his es saisic is ˆ ˆ 1 BP KV ( ) = ( V ), (1) whee ˆ V is a consisen esimao of V unde H (K). A consisen esimao can be obained by using ˆ γ o esimae γ and a consisen nonpaameic esimao of C. Ou genealizaion of he LM-based ess explois he fac ha he LM es saisic is a funcion of sample auocoelaions. Rewiing he LM (π) saisic in (5) gives 1 i i π = π YY i 1 σy π π π i+ 1 = i= = i+ i= LM ( ) / (1 ) (1 ), (13) whee he ih sample auocoelaion in (13) is i = ( i / ) Y =+ i 1 YY σ. Suppose ha he seies i π i= i+ 1 is uncaed a a lage value. We can hen wie he saisic in (13) as π = π p LM ( ) (1 )( ), whee 1 = (1,,,..., ), 1 p ππ π = (,,..., ). he veco is asympoically N (, V) whee V is a maix. If V = I, he LM-based ess of H : β = can be caied ou using he asympoic ciical values calculaed by AP. If V I, he ue levels of he ess wih AP asympoic ciical values can diffe subsanially fom he nominal levels in finie-samples and asympoically. 1
he level disoion of he LM-based ess when V I can be coeced asympoically by using L in place of in (13) whee V = LL. Ou poposed 1 genealizaion is o eplace V by V and consisenly esimae V by ˆ V. he genealized ess we conside ae: supπ Π LM ( π, V ) ˆ ˆ Exp-LM ( V ) = LM ( π, V ) dj( π), ˆ Exp-LM ( V ) ln exp LM ( π, V ) dj( π ), ˆ 1 ˆ = (14) whee ˆ ˆ LM ( π, V ) = (1 π )( p L ) wih ( Vˆ ) = Lˆ Lˆ. ` 1 A bief skech of poof ha he genealized AP ess have he same limiing disibuion as he AP ess is he following. In he AP case whee Va( ) = I, we have ha d (1 π ) ' (1 π ) (,1) (,1), p N = N 4 when max(π) =.8 and = ( 1.8 =.9999 ), he paamee values used in ou numeical calculaions. he asympoic coelaion beween π p and (1 k ) k (1 m ) π p is m (1 π k)(1 πm) (1 πk πm ) /(1 πkπm), 3 1 whee p = (1 π π π... π ). hus, he AP ess ae funcions of coelaed j j j j j asympoically sandad nomal vaiables, (1 j ) π p. In he case whee Va( ) = j V we have ha π pl ˆ N d (1 ) (,1), 11
and ha he asympoic coelaion beween (1 π ) p Lˆ and k k (1 π ) p Lˆ is m m (1 π k)(1 πm) (1 πk πm ) /(1 πkπm). his shows ha he genealized AP ess ae he same funcions of asympoically sandad nomal vaiables wih idenical asympoic coelaions. he asympoic ciical values epoed by AP emain valid fo ou genealizaion of he LM-based ess. We use ciical values based on =, wheeas AP use = 5. We have se, as AP did, max π =.8. Using = ahe han 5 has a negligible effec on he 1%, 5% and 1% asympoic ciical values hough ems 1 (1 π ). his is seen in able 1 whee we epo he asympoic ciical values fo he hee es saisics fo Π = {, ±.1,, ±.79, ±.8}, = and = 5, using 15 million eplicaions. of V he genealized vesions of BP and LM-based ess equie a consisen esimao. We use he VARHAC esimaion pocedue poposed by den Haan and Levin (1997). he consisency of he VARHAC esimao is poved by den Haan and Levin (1998) unde vey geneal condiions. hey demonsae ha, in many cases, he VARHAC esimao achieves a fase convegence ae han kenel-based mehods. Fancq e al. (5, p. 539) also pove he consisency of he VARHAC esimao. hei poof uses he exisence of he eighh momen of Y and a mixing condiion. he VARHAC pocedue uses a veco auoegessive (VAR) esimao of he covaiance maix whee he ode of equaion in he VAR is auomaically seleced. o pesen he explici fomula fo he VARHAC esimao of C, le wˆ i ( Y Y)( Y i Y) =, wˆ = ( wˆ,..., wˆ )', and le S be he maximum lag ode chosen fo he VAR. Fo 1, 1
consisency, den Haan and Levin (1998) also equie ha he maximum lag gows a ae 1/3. he esimaed esiduals fom he VAR egessions ae eˆ wˆ Aw ˆ ˆ s s s= 1 S =, whee A ˆs ae he maices of esimaed coefficiens fom he VAR, and he esimaed innovaion covaiance maix is hen he VARHAC esimao of Σ= ee ˆˆ'. = S+ 1 C is S S 1 1 s s s= 1 s= 1. (15) ˆ ˆ C = ( I A ) Σ ( I Aˆ ') We epo esuls fo he VAR wih he AIC (Akaike (1973)) and he SBC (Schwaz (1978)) cieia. he esuling esimaos ae denoed by V ˆ (AIC) and V ˆ (SBC). he maximum lag lengh is 3, 4, 5 and 8 fo sample sizes =, 5, 1 and 5, and, fo any equaion in he VAR, he same lag lengh is used fo each elemen of he veco pocess. o miigae he effec of occasional exeme esimaes we used he pocedue of Andews and Monahan (199), and se he minimum singula values of he invese of he ecoloing maix, S I ˆ A, o be.5. s= 1 s he fom of he V maix is simplified when he ime seies is a maingale diffeence sequence. Fo a MDS pocess, he only possible nonzeo elemens of C ae ems of he fom EY ( μ) ( Y μ)( Y μ). In (11) hese occu a d =. Guo and i j Phillips (1998) have developed a vesion of he BP es fo he MDS case. his special 13
fom of V maix can also be used in consucing a genealizaion of he LM- based ess fo geneal MDS pocesses. Fo ceain MDS pocesses, such as Gaussian GARCH pocesses, V is diagonal, wih he jh diagonal elemen equal o ( γ ()) EY ( μ) ( Y μ). Genealizaions of j he LM-based ess can be consuced fo he diagonal case. he BP es fo he diagonal case has been epeaedly einvened in he lieaue; see, fo example, aylo (1984), Diebold (1986), Lo and MacKinlay (1989), Lobao, Nankevis and Savin (1) and also Deo (). We denoe he esimao of V in he geneal MDS case byv GP. A consisen esimao of he ijh elemen of GP V is GP GP GP / ˆ ij = ij γ and ij = ( ) ( i )( j ) / ;, = 1,..., = + 1 vˆ cˆ cˆ Y Y Y Y Y Y i j. (16) he diagonal V maix is denoed by V *. A consisen esimao of he jh diagonal elemen of * V is vˆ cˆ cˆ Y Y Y Y j. (17) * * * jj = jj / ˆ γ and jj = ( ) ( ) /, 1,..., 1 = + j = As noed ealie, AP show ha hei ess apply o esiduals fom egessions wih exogenous egessos (Assumpions 4 and 5 in AP). he same holds fo he genealized ess. he eason hese assumpions ule ou an exension o dynamic egession models is because hey equie ha he condiional mean of he unobseved eos is zeo given pas values of he eos and pas and fuue values of he egessos. 5. MONE CARLO COMPARISONS his secion consides he ue levels of he LM-based ess, he BP ess and he Deo () es when he ess use asympoic ciical values. he LM-based ess ae he 14
sup LM, Exp-LM and Exp-LM ess. he finie sample level-coeced powes of hese ess and he ohe ess ae compaed. he BP ess ae included because hey ae widely employed in he economics and finance lieaue and he Deo () es is included because i is consisen agains all non-whie noise alenaives when V is diagonal in he MDS case, a popey no shaed by he BP ess. he model we conside in he level expeimens is he locaion model wih seially uncoelaed bu dependen eos, Y = μ + ε fo = 1,,...,, (18) whee V I. he models used fo he eos ε in he expeimens include wo MDS pocesses and wo non-mds pocesses. he wo MDS models fo he eos ae vaians of he GARCH model of Bolleslev (1986), namely, Gaussian GARCH (1, 1) and he exponenial GARCH (1, 1) o EGARCH (1, 1). Boh models ae descibed in Campbell, Lo and MacKinlay (1997). GARCH (1, 1). ε = Z σ, whee {Z }is an iid N(,1) sequence and σ = ω+ α ε + β σ. he consans α and β ae such haα + β < 1. his condiion 1 1 is needed so ha Y is covaiance saionay. He and eäsvia (1999) show ha he uncondiional mh momen of Y fo GARCH (1, 1) models of Y exiss if and only if m E( αz + β) < 1. We se ω =.1, α =.8, and β =.89. Wih his paamee seing, he He and eäsvia condiion fo he exisence of he fouh and eighh momens of Y ae saisfied. Fo his pocess, γ = EY ( μ) =.33, EY ( μ) / γ =, 3 3/ EY ( μ) / γ = 3.83, and V is diagonal. We noe ha ou esuls 4 4 15
ae invaian o he value of ω. Esimaes fom sock eun daa sugges ha α + βis close o one wih β also close o one; fo example, see Bea and Higgins (1997). EGARCH (1, 1). ε = Z σ, whee {Z }is an iid N(,1) sequence and whee ln( σ ) = ω+ ψ Z + α Z + β ln( σ ). We se ω =.1, ψ =.5,α =., and 1 1 1 β =.95. He, eäsvia and Malmsen () show ha Y is saionay if β < 1 and ha wih Gaussian {Z } all momens of Y exis. We have ha (he skewness is an esimae) γ = EY ( μ) = 1.8, EY ( μ) / γ =, and 3 3/ EY ( μ) / γ = 3.4, and V is 4 4 no longe diagonal. Ou esuls ae invaian o he value of he inecep and he vaiance. he wo models fo he non-mds eos ae he nonlinea moving aveage model, and he bilinea model. ong (199) consides he nonlinea moving aveage model, and Gange and Andesen (1978) he bilinea model. he moivaion fo eneaining non- MDS pocesses is he gowing evidence ha he MDS assumpion is oo esicive fo financial daa; see, fo example, El Babsii and Zakoian (1). Fo boh models consideed below, V is non-diagonal. Nonlinea Moving Aveage Model. Le ε = Z -1 Z - ( Z - + Z + c) whee { Z } is a sequence of iid N(, 1) andom vaiables and c = 1.. Fo his pocess all momens exis wih EY ( μ) = 5, EY ( μ) / γ =, EY ( μ) / γ = 37.8. 3 3/ 4 4 Bilinea Model. Le ε = Z + b Z -1 ε -, whee { Z } is a sequence of iid N(, σ ) andom vaiables, b =.5 and σ = 1.. he Y pocess is covaiance saionay povided ha b σ < 1. he fouh momen of his pocess exiss if 3b 4 σ 4 < 1. Fo his pocess, ( μ) σ (1 bσ ) EY = = 1.333, EY ( μ) / γ =, and 3 3/ EY ( μ) / γ 4 4 16
4 4 4 4 = 3(1 b σ ) /(1 3 b σ ) = 3.46. Bea and Higgins (1997) have fied a bilinea model o sock eun daa. We simulaed he finie-sample ejecion pobabiliies (pecen) of he nominal.5 LM, Exp-LM and Exp-LM ess of H : β = fo he MDS and non-mds eo models. he ejecion pobabiliies fo he sup LM, Exp-LM and Exp-LM ess ae compued using Π= {.8,.79,...,.79,.8}, which is he same se as used by AP. he ejecion pobabiliies ae based on =. Inceasing does no poduce a noiceable diffeence in he ejecion pobabiliies when using Π as defined above. Fo compaison, he simulaions included he BP6, BP1, BP ess and he Deo () es; BP6 and BP1 wee consideed by AP. he MDS and non-mds models ae simulaed fo sample sizes =, 5, 1 and 5 using 5, eplicaions. Resuls fo he non-mds models ae no epoed because hey do no change he conclusions fom he MDS models and also o save space. he esuls fo he GARCH models ae pesened in able. he esuls show ha he diffeences beween he ue and nominal levels ae subsanial when he ideniy maix is used. he diffeences ae lages fo he BP ess wih he EGARCH (1, 1) model. he diffeences end o incease as he sample size inceases. he incease is lage fo he EGARCH (1, 1). Nex we conside he genealized ess in able. We fis compae he LM based ess. he diffeences beween he ue and nominal levels end o be essenially eliminaed fo GARCH (1, 1) when he ess use he consisen esimaos ˆ * V, ˆ GP V o Vˆ(SBC) and 5. Fo EGARCH (1, 1), he diffeence is essenially eliminaed when 17
he ess use he consisen esimaos ˆ GP V o V ˆ(SBC) and 5. In boh cases lage sample sizes ae needed o eliminae he diffeence when Vˆ(AIC) is used, especially fo he sup LM es. Oveall, he Exp-LM and Exp-LM ess end o have bee conol ove he level han he sup LM ess. he esimao of case. he esuls fo ˆ * V is inconsisen in he EGARCH ˆ * V in able show only a small endency fo ove ejecion a = 1 because he aveage off-diagonal elemens of V ae close o zeo a his sample size. Nex conside ohe ess. he genealized BP ess geneally end o show less saisfacoy conol ove he level han he LM-based ess, especially BP. he levels of he Deo () es ae simila o hose of he genealized LM ess in he V* case. We also calculaed he finie-sample ejecion pobabiliies using he skewed (5) GARCH (1, 1) using he sandadized vesion given in Lambe and Lauen (1) and he mixues of nomal GARCH (1,1) poposed by Haas, Minik and Paolella, (4). he pevious conclusions ae no aleed by he esuls fo hese lae models. he locaion model (18) is also used fo he powe compaisons, bu now wih seially coelaed eos. Following AP, he models used fo he eos ε include AR(1) : ε = φε 1 + u, MA(1): ε = u + θu 1, 6 7 j AR(6): ε = φ ε + u, j j= 1 6 1 13 j AR(1) : ε = φ ε + u, j j= 1 1 7 j 6 j+ 1 AR(6) ± : ε = φ ( 1) ε j + u, j= 1 6 18
13 j (19) 1 j+ 1 AR(1) ± : ε = φ ( 1) j= 1 1 ε j + u. he above models wee chosen by AP because hey include a wide vaiey of paens of seial coelaion wih boh posiive and negaive seial coelaions. he models used fo he innovaions u ae he GARCH (1, 1) and EGARCH (1, 1) models and he nonlinea moving aveage and bilinea models. We calculaed he.5 level-coeced powes by simulaion fo sample size = 1 using 5, eplicaions. he finie-sample ciical values ae simulaed using 5, eplicaions. he paamee values ae chosen so ha he maximum powes ae appoximaely.4 and.8 fo he wo paamee values consideed. All models ae simulaed wih an appoximaely saionay saup by aking he las andom vaiables fom a simulaed sequence of he + 5 andom vaiables whee saup values ae se equal o zeo. able 3 pesens he.5 level-coeced powe of each of he ess fo he AR(1), MA(1), AR(6), AR(1), AR(6)±, and AR(1)± models wih GARCH(1,1) innovaions. he powes in able 3 pesen a mixed picue. A compaison of he LM-based ess shows ha Exp-LM ends o have he highes powe fo he AR(1) and MA(1) models and he lowes powe fo he AR(1), AR(6)±, and AR(1)± models. Fo he lae models, he sup LM es has he highes powe. We conclude ha he sup LM es has highe all-aound powe han he Exp-LM by a small magin. he same paen ends o hold when diffeen values of φ and θ ae used in he eo models. Among he BP ess, he BP6 has he highes powe. he sup LM es has highe powe han he BP6 es and by a consideable magin in many cases. he powes of he 19
BP6 es end o be highe han he powes of he Exp-LM fo he AR(1)± model. he powes of he BP6 es ae lowe han he powes of he Exp-LM es. he powe of he Deo () es is slighly highe han he powe of he genealized AP ess in he AR(1) and MA(1) cases, bu in ohe cases i is smalle and someimes subsanially so. When he powes of he ess ae compaed fo models wih EGARCH (1, 1) innovaions, ou conclusions ae essenially he same as hose fo GARCH (1, 1), and similaly fo he models wih nonlinea moving aveage innovaions and wih bilinea innovaions. Following AP, we also invesigaed he.5 level-coeced powes whee ARMA (1, 1) models ae used fo he eos. As peviously, he innovaions u ae geneaed by he MDS o non-mds models consideed peviously. In simulaion expeimens whee ARMA(1,1) models ae used fo he eos, he sup LM ess no longe have he bes all aound powe compaed o he Exp-LM and Exp-LM ess. Once he esuls fom ARMA eo models ae aken ino accoun, he sup LM, Exp-LM and Exp-LM ess all have bee powe han he BP ess, bu none is dominan. Of couse, he powe esuls ae influenced by he models and paamee values used in he simulaion expeimens. Noe ha he daa geneaion pocesses used in he above expeimens have declining weighs as he lag lengh inceases. As a consequence, he fis auocoelaion is dominan. his ype of design is elevan fo applicaions in economics and finance. As we have seen, fo his ype he genealized AP ess end o have highe powe han he genealized BP ess. Howeve, he evese can be hold fo designs whee he fis auocoelaion is no impoan. A simple example of a daa
geneaion pocess wih his popey is an AR() whee he coefficien on Y 1 is zeo and on Y is negaive. his cavea should be kep in mind when inepeing he esuls. AP consideed seasonal MA models fo he eos ε. he models ae MA( j) : ε = u + θu j, j = 1,...,6. () We calculaed he.5 level coeced powe fo hese eo models wih θ =.15 whee he u ae geneaed by he EGARCH (1, 1) model. Fo he seasonal models, he BP6 and B1 ess ae bes of hose consideed, wih BP6 es having he highes all aound powe. his conclusion is he same as ha eached by AP. AP implemen he ess using = 5 and we do so using =. We invesigaed whehe his affecs he consisency of he ess. We found ha wih Π.8 and a pocess wih ρ 1 =... = ρ =, ρ 1 inceasing he uncaion lag does no make any vey noiceable diffeence o powes and hus he consisency of he ess as he sample size inceases o 15,. Fuhe, we simulaed he level-coeced powes fo = 1 using = 4. he powes of he LM-based ess fo = 4 ae essenially he same fo =, and hence he conclusions fom he powe compaisons ae unchanged. Compuing. he andom numbe geneao used in he expeimens was he vey long peiod geneao RANLUX wih luxuy level p = 3; See Hamilon and James (1997). he pogam used fo VARHAC was he vesion of he pogam by den Haan and Levin (hp://econ.ucsd.edu/~wdenhaan/vahac.hml) modified o un subsanially fase. 6. EMPIRICAL APPLICAION As Campbell, Lo and MacKinlay (1997) noe, he pedicabiliy of sock euns is an acive eseach opic. hey illusae he empiical elevance of pedicabiliy by 1
applying he BP ess o CRSP sock eun indexes. In his secion, hei empiical applicaion is exended in wo ways: Fis, esuls ae pesened fo he AP ess as well as he genealized AP and BP ess; second, esuls ae pesened fo an exension of hei sample peiod. Campbell, Lo and MacKinlay (1997) epo he means, sandad deviaions, he fis fou sample auocoelaions (in pecen) as well as he BP 5 and BP 1 saisics in hei able.4 fo monhly, weekly and daily value-weighed (VWRED) and equalweighed (EWRED) sock eun indexes (NYSE/AMEX). he sample peiod is July 3, 196 o Decembe 31, 1994. We eplicaed he esuls by Campbell, Lo and MacKinlay (1997) fo his sample peiod and he sub-peiods hey seleced. In his secion, he sample peiod is July 3, 196 o Decembe 3, 5. Resuls wee calculaed fo he sup LM, Exp-LM and Exp-LM ess, and he genealized AP and BP ess, boh fo he sample peiod and sub-peiods consideed by Campbell Lo and MacKinlay (1997) and fo he exended sample peiod and seleced sub-peiods. he skewness and kuosis saisics wee also calculaed o povide a check on he nomaliy of he euns. As expeced, he kuosis saisics povide song evidence agains nomaliy. Fo he sake of beviy, we only epo esuls fo he monhly equal-weighed sock eun indexes. able 4 illusaes ha infeences fom he genealized AP ess can conflic wih hose fom he genealized BP ess. he genealized AP ess end o ejec and he genealized BP ess end no o ejec. Fo he daa used by Campbell, Lo and MacKinlay (1997), he BP ess end o no ejec a he nominal.5 level and similaly fo he genealized BP ess. he geae numbe of ejecions by he genealized AP ess
may be explained by he highe powe of he genealized AP ess compaed o he genealized BP ess. able 4 also shows ha boh he AP and BP ess end o ejec a he nominal.5 and o.1 levels fo he exended sample peiod and seleced sub-peiods. he same is ue fo he genealized AP ess. Again hee ae subsanially fewe ejecions by he genealized BP ess han he genealized AP ess. In ligh of ou Mone Calo expeimens, he esuls epoed by Campbell, Lo and MacKinlay (1997) fo he BP ess ae difficul o inepe in isolaion. Alhough he BP saisics ae ofen enomous fo weekly and daily daa, his alone does no povide song evidence ha he null of zeo coelaion is false. his is because he BP ess end o subsanially ove-ejec when daa ae geneaed by uncoelaed dependen pocesses such as a GARCH (1, 1) o EGARCH (1, 1) model. In paicula, he ove-ejecion is mos ponounced fo lage sample sizes, sizes simila o hose in his empiical applicaion. he moivaion of he empiical applicaion is pedicabiliy of sock euns, which is chaaceized by a MDS condiion in sock euns. he MDS hypohesis implies ha sock euns ae whie noises, so i is valid o use ess fo seial coelaion in esing he pedicabiliy of sock euns. Hence, obusness unde condiional heeoskedasiciy in he case of GARCH and EGARCH is an appealing popey of he genealized AP es. Howeve, obusness unde he non-mds cases (nonlinea MA and bilinea) may be inepeed as a dawback fo his pupose because non-mds pocesses which can be used fo pedicion will be missed. 7. CONCLUDING COMMENS 3
In he simulaions in his pape, he diffeences beween he ue and nominal levels of he genealized AP ess ae essenially zeo fo suiable sample sizes, and he genealized AP ess have good powe popeies fo nonseasonal alenaives compaed o he genealized Box-Piece ess and he Deo () ess. he Exp-LM es is ecommended fo nonseasonal applicaions in economics and finance. he pape includes an empiical applicaion o sock eun indexes ha is moivaed by he seach fo pedicabiliy in euns. he esuls illusae ha infeences fom he genealized AP ess can conflic wih hose fom he genealized Box-Piece ess and can make a diffeence o he infeences dawn fom he daa. Andews, Liu and Plobege (1998) exended hei appoach o esing whie noise agains muliplicaive seasonal ARMA (1, 1) models. A opic fo fuhe eseach would be o use ou appoach o genealize he LM-based es fo his case. he genealized LM-based ess do no apply o esiduals fom ARMA models. We plan o invesigae his opic in fuue eseach. ACKNOWLEDGEMENS he auhos gaefully acknowledge he helpful commens of he edio and efeees. Don Andews, John Geweke and Ignacio Lobao povided useful advice. REFERENCES Akaike, H. (1973), Infomaion heoy and he Exension of Maximum Likelihood Pinciple, in Second Inenaional Symposium on Infomaion heoy, eds. B. Peov and F.Casaki, Budapes: Akailseoniai-Kiudo, pp. 67-81. Andews, B., Davis, R. A. and Beid, F. J. (6), Maximum Likelihood Esimaion fo All-Pass ime Seies Models, Jounal of Mulivaiae Analysis, 97, 1638-1659. 4
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able 1. Asympoic Ciical Values sup-lm Exp-LM Exp-LM 1% 5% 1% 1% 5% 1% 1% 5% 1% 4.68 5.945 9.81.48 3.36 5.586 1.418 1.973 3.348 5 4.68 5.945 9.81.49 3.36 5.586 1.418 1.973 3.347 NOE: Ciical values obained by seing Π = {, ±.1,, ±.79, ±.8}. he numbe of eplicaions is 15 million. 9
able. Rejecion Pobabiliies (Pecen) of Nominal.5 ess: MDS Models GARCH(1, 1) EGARCH(1, 1) ˆ V 5 1 5 5 1 5 I sup LM 7.6 9.3 1.6 1.3 5. 37.9 47. 61.8 Exp-LM 7. 8.8 9.4 1.7 3.1 3.5 39.8 51. Exp-LM 7.7 9.4 1. 11.7 5.1 35.9 44.1 57.3 BP6 9.8 13.1 14.9 17.6 4. 57. 67.9 8.5 BP1 1.6 15.3 18.3.4 41.7 63.4 75.9 9.6 BP 1.3 16.5.6 5.4 37.8 6.7 76.8 9.9 ˆ * V sup LM 5.4 5.1 5. 5.3 7.9 6.3 6. 6. Exp-LM 5.1 4.9 4.9 5. 6.9 5.4 5.1 5.3 Exp-LM 5.3 5. 5. 5. 7.5 5.9 5.5 5.6 BP6 5.8 5.1 4.8 4.9 8.7 6.1 5.3 4.9 BP1 6. 5. 5. 5.1 9.3 6.7 5.7 5. BP 5.8 5. 5. 5. 8.9 6.5 5.9 5.4 DEO 4.6 4.7 4.8 4.9 4.8 4.5 4.6 4.8 ˆ GP V sup LM 5.3 4.8 5. 5.1 7.5 5.1 4.8 4. Exp-LM 5.1 4.7 4.9 5. 6.9 5.1 4.7 4.8 Exp-LM 5. 4.8 4.9 5.1 7. 5. 4.6 4.5 BP6 5.3 4.4 4.6 4.7 8.1 4.8 4.1 3.9 BP1 4.5 3.8 4.1 4.9 8.7 4.3 3.4 3.5 BP 3.3.9 3.4 4.6 8.7 3.7 3. 3.6 V ˆ (AIC) sup LM 6.5 5.9 6. 5.1 13.7 8. 6.4 6.7 Exp-LM 5. 4.7 4.7 4.7 9. 5. 4. 4.7 Exp-LM 5.6 5.1 5.1 4.9 11. 6. 4.9 5.4 BP6 7. 5. 4.7 4.5 18.8 8. 5.1 5.3 BP1 8.9 5.1 4.6 4.6 7.6 1.4 6. 6.4 BP 11.1 4.9 4. 4.4 37.5 13.5 7.3 8.7 V ˆ (SBC) sup LM 5.4 4.9 5.1 4.9 9.9 6. 5.6 6.3 Exp-LM 5. 4.6 4.8 5. 7.4 4.8 4.1 4.4 Exp-LM 5. 4.8 4.8 5. 8.5 5. 4.5 5. BP6 5.4 4.5 4.7 4.7 1.8 6.7 4.8 4.8 BP1 4.7 4. 4. 4.9 16.7 8.1 5.4 5.9 BP 3.7 3.1 3.6 4.7.9 9.7 6.5 7.8 NOE: he ess, as funcions of ˆ V, ae defined in (1) and (14) above. he esimao ˆ * V is consisen in he diagonal MDS case; ˆ GP V is consisen fo geneal MDS pocesses; he esimaos V ˆ(AIC) and V ˆ(SBC) ae consisen esimaos in boh MDS and non-mds cases. DEO efes o he Deo () es, which is consisen in he diagonal MDS case. he numbe of eplicaions is 5,. 3
able 3. Level-Coeced Powes of.5 ess fo AR and MA Models Wih GARCH (1, 1) Eos, =1 ˆ V AR(1) MA(1) AR(6) AR(1) AR(6) +- AR(1) +- φ.5.1.5.75.5.5.5.75.75.1 θ.5.1 I sup LM 67 66 53 88 5 81 9 64 79 97 Exp-LM 5 74 4 73 43 8 17 61 6 54 51 75 Exp-LM 3 7 3 71 5 86 1 75 9 61 68 9 BP6 1 46 1 45 39 79 16 67 45 54 8 BP1 9 33 9 3 9 69 14 64 14 3 39 65 BP 8 5 8 4 59 1 56 11 8 48 ˆ * V sup LM 1 68 67 54 88 6 81 9 64 79 96 Exp-LM 5 75 5 74 44 8 17 61 7 54 5 75 Exp-LM 4 73 4 7 5 86 73 9 61 65 89 BP6 13 48 13 47 4 8 17 67 1 47 56 8 BP1 1 36 1 35 31 7 15 65 15 3 41 67 BP 9 8 9 7 4 61 1 57 1 4 3 51 DEO 6 76 6 75 38 75 14 5 4 5 44 67 ˆ GP V sup LM 1 68 1 68 47 86 76 38 73 87 98 Exp-LM 5 75 5 74 41 78 15 57 9 58 57 81 Exp-LM 4 74 4 73 46 83 17 68 35 68 76 94 BP6 13 48 13 48 35 75 14 61 6 56 68 9 BP1 1 37 1 36 6 63 1 55 19 43 63 88 BP 9 9 9 8 5 1 45 15 33 51 79 ˆ (AIC) V sup LM 19 63 19 65 3 68 1 51 45 78 91 98 Exp-LM 5 74 5 74 33 67 1 43 35 65 67 88 Exp-LM 3 7 4 7 34 71 11 48 41 75 84 97 BP6 13 45 13 46 49 9 33 33 66 8 96 BP1 9 3 9 33 14 3 7 4 3 5 77 95 BP 8 3 8 4 11 6 16 17 4 65 9 V ˆ(SBC) sup LM 67 1 67 45 84 19 74 39 74 88 98 Exp-LM 6 75 6 75 4 76 15 56 31 6 6 83 Exp-LM 5 73 5 73 44 8 17 67 36 7 78 95 BP6 13 48 13 48 33 7 13 59 7 58 7 9 BP1 1 36 1 36 4 59 11 5 44 65 89 BP 9 8 9 8 19 49 9 4 15 34 5 81 NOE: See able. he numbe of eplicaions is 5, 31
able 4. ess of Seial Coelaion in Monhly CRSP EWRED Sock Index Reuns Sample 7:31:6 7:31:6 1:31:78 7:3:6 1:3:95 1:3:78 1:9:88 Peiod 1:3:94 9:9:78 1:3:94 1:3:5 1:3:5 1:3:5 1:3:5 Sample Size 39 195 195 5 13 36 16 Mean ( 1) 1.77 1.5 1.14 1.89 1.1 1.18 1.6 SD ( 1) 5.749 6.148 5.336 5.357 4.1 4.68 3.957 Skewness -.45.3-1.59 -.54-1.11-1.4 -.7 Kuosis 7.367 5.99 1.67 7.81 6.734 1.65 5.753 1 ˆρ ( 1) 17.1 18.4 15 17.5 19.7 18.8 3. ˆρ ( 1) -3.4 -.5-1.6-4.1-8.4-4. -3. ˆ V = I sup LM 11.8** 6.8* 4.4 16.6** 5.8 11.8** 11.8** Exp-LM 8.6** 5.7**.7 11.9** 3.8* 8.** 8.1** Exp-LM 5.**.9* 1.6 7.**.3* 4.9** 5.** BP5 1.8* 7.4 8.7 18.6** 1.3 19.9** 19.9** BP1.9* 1.3 13.7 7.8** 1.1 6.5** 1.3* BP 4.7** 3.6.4 47.7** 17.6 3.7* 7.5 ˆ V = V ˆ (AIC) sup LM 9.5** 6.4*.7 13.8** 3. 1.9** 8.3* Exp-LM 7.4** 4.7*. 1.3** 1.4 6.9** 5.* Exp-LM 4.1**.6* 1. 6.1**.9 4.1** 3.3* BP5 9.8 6.5 3.8 14.* 5.4 13.1* 1.6 BP1 16.5 8 8.6.* 7.3 19.1* 11.1 BP. 13.4 5.5 6. 11.1 4.5 15.9 ˆ V = V ˆ (SBC) sup LM 9.4** 5. 3.5 13.3** 5.3 11.3** 8.4* Exp-LM 6.3** 3.6*.3 9.** 3.3 6.9** 5.5* Exp-LM 3.8**. 1.3 5.6**.* 4.3** 3.4** BP5 11. 5.1 7.4 16.1** 7.5 15.3** 11.8* BP1 17. 9.7 1.6.6* 1.8.* 1. BP 4.4 16.9.7 9.4 15.4 9.3 14.6 NOE: See able fo definiions of funcions of V ˆ. In he VARHAC pocedue o compue he genealized saisics he maximum lag is se a in(.5 ). One and wo sas denoe ejecion a he nominal.5 and.1 levels, especively. 3