TESTING FOR SERIAL INDEPENDENCE OF GENERALIZED ERRORS

Transcription

1 TESTING FOR SERIAL INDEPENDENCE OF GENERALIED ERRORS aichao Du Idiaa Uiversiy April 4, 2008 Absrac I his paper, we develop a Neyma-ype smooh es for he serial depedece of uobservable geeralized errors. Our es is uisace parameer-free i he sese ha model parameer esimaio uceraiy has o impac o he limi disribuio of he es saisic. Keywords ad Phrases: Serial depedece; Goodess-of-Fi; Time series; Empirical processes; Parameer esimaio uceraiy; Smooh es. I m grealy idebed o my advisor Prof. Escaciao for his umerous guidace ad help. I also hak Prof. Jacho-Chavez, Prof. Trivedi ad Prof. Tyuri for heir commes o a early draf of his paper. All he errors are my ow.

2 . INTRODUCTION Tesig for serial depedece or lack hereof) coiues o be a maor ieres of ime series aalysis. May saisical ifereces, icludig may asympoic resuls, rely o he idepede ad ideically disribued iid) assumpio o raw daa or errors. Therefore i is impora o develop some speci caio ess for iid. So far here have bee may researches o esig iid, bu mos of hem have bee focused o raw daa bu o o uobservable geeralized errors, which hampers he applicaio of hese exisig procedures o more geeral cases; see he examples below. I his paper, we propose ess for serial idepedece of geeralized errors, hereby cosiderably broadeig he scope of possible applicaios of he exisig mehods. Before we prese our moivaioal examples, le us iroduce some geeral oaios. Suppose ha we observe daa fy ; g =; where Y is some real-valued depede variable ad may coai lagged values of Y ad curre ad lagged values of oher exogeous variables, say : Furhermore, 0 is some ukow parameer i a compac se R p. Example. Our leadig moivaioal example is he codiioal mea ad variace model, Y = ; 0 ) + ; 0 )u ; where ; 0 ) = E[Y ] ad 2 ; 0 ) = V ar[y ] almos surely a.s.). This ype of models, wih he popular ARMA-GARCH model as a special case, are widely used i acial applicaios, for example, i modelig sock reurs, erm srucure of ieres raes, ec. I hese models, oe usually makes he iid assumpio for he sadardized errors u = u 0 ) = Y ; 0 ) : ; 0 ) The iid assumpio o u is a crucial assumpio for he validiy of ifereces abou he parameer 0 ad for speci caio ess of he parameric mea ad variace. I paricular, he iid assumpio is a ecessary codiio for he Value a Risk V ar) model a level o be expressed as V ar = ; 0 ) + ; 0 )Fu ); where Fu ) is he quaile fucio of u : V ar models play a impora role i he assessme of marke risk a commercial baks ad oher acial isiuios. Example 2. A Codiioal) Goodess-of-Fi GOF) es problem arises whe oe assumes ha Y follows a cerai codiioal parameric disribuio G; ; 0 ): Oe way o cosruc a es Aoher impora assumpio is he idepedece bewee u ad ; which implies he iid of u ; as fu ; u 2; :::g are fucios of : I some cases, such as he saioary ad iverible ARMA models, hey are equivale. 2

3 is o use he well-kow fac ha for a coiuous ad correcly speci ed G; he geeralized error u = u 0 ) = GY ; ; 0 ) follows a iid U[0; ] disribuio. Bai 2003) uses his idea o es he codiioal disribuios of dyamic models, alhough his es is ailored oly for he U[0; ] disribued par, o for he iid par. Hog ad Li 2005) also use his idea o cosruc a speci caio es for he erm srucure of ieres raes. Alog hese lies, our proposed es for iid of geeralized errors ca be used for codiioal Goodess-of-Fi purpose. Moivaed by he above examples, i his paper we wa o es H 0 : fu 0 )g 2 is a sequece of iid r.v, for some 0 2 R p ; ) where u 0 ) HY ; ; 0 ) is a geeralized error, give by a kow measurable rasformaio H ad a ukow parameer 0 i R p : The aleraive hypohesis is he egaio of he ull ). Noice ha he ull hypohesis ) is composie, sice we do o kow 0. Give a p -cosise esimaor of 0 ; say b ; we ca cosruc geeralized residuals bu = u b ); ad base our es o bu, bu he, he ull limi disribuio of he es will geerally lose he uisace parameerfree propery, which is a ee i empirical process heory whe parameers are esimaed, see Durbi 973). To circumve his problem, hree mai soluios have bee proposed. Oe is usig marigale rasformaio due o Khamaladze 98)) o ge asympoically disribuio-free es, see, e.g., Bai 2003) i a codiioal GOF seup. Aoher approach is usig boosrap o overcome he problem of he daa-depede asympoic disribuios, see Corradi ad Swaso 2003) or Hidalgo ad a aroi 2007) i codiioal GOF seups. The oher is he smoohig mehod, which ivolves badwidh selecios, see Hog ad Lee 2003) i a seup similar o Example. I his paper, we propose a ew mehod o solve he aforemeioed problem i a more geeral seup ha he exisig oes. We cosider Hoe dig-blum-kiefer-rosebla-ype empirical process applied o residuals bu a di ere lags, ad esablish a uiform expasio of he process, hereby derivig a explici expressio for he esimaio e ec i a geeral seup. The proposed ess will he be smooh ess based o orhoormal polyomials i he spiri of Neyma 937), coupled wih daa-drive mehod as i Kalleberg ad Ledwia 999). We choose he compoes i he smooh es i such a way ha he esimaio e ec is aihilaed ad asympoically disribuio-free ess are obaied. Our proposed ess have a wide variey of applicaios. Whe applied o esig serial idepedece of errors i locaio-scale models, our es is relaed o Camero ad Trivedi 993), ye a compleme o heir ess as ours ca deal wih he esimaio e ec. As a aleraive o Hog ad Lee 2003), our es does o require selecios of weighig fucios ad badwidh, ad is simple o impleme, ye sill powerful. Whe applied o codiioal GOF, our es is closely relaed o 3

4 Boemps ad Meddahi 2005, 2007), who use a GMM approach coupled wih orhoormal polyomials o rea he parameer esimaio uceraiy problem, bu we are usig a oally di ere approach o heirs. As a resul, our ess ca be applied o more geeral codiioal disribuios ha he Pearso family, for example skewed- Hase 994)), o-ceral Havey ad Siddique 999)), ec. Moreover, our Neyma-ype smooh es does o require a esimae of he variace of he momes as GMM does. Also worh of meioig are some applicaios of our uiform expasio i Secio 2. Whe applied o oliear regressio models wih heerogeeous errors, i exeds he digs i Delgado ad Mora 2000). Whe applied o he Box-Lug-Pierce Tes, i gives a expressio for he parameer esimaio e ec. The res of he paper is orgaized as follows: i Secio 2 we esablish a uiform expasio of he Hoe dig-blum-kiefer-rosebla-ype empirical process applied o residuals ad show some applicaios. I Secio 3 we cosruc Neyma-ype smooh ess for he idepedece of geeralized errors robus o he parameer esimaio uceraiy i a geeral seup. I Secio 4 we apply he geeral heory o our moivaioal examples. I Secio 5 we do some Moe Carlo simulaios o sudy he performace of our proposed ess, ad he apply he ess o some real daa. I Secio 6 we coclude ad sugges some direcios for fuure research. All he proofs are i he appedix. 2. THE GENERAL THEORY A Asympoic Uiform Expasio I he sequel, we simplify he oaios as follows: u = u 0 ) ad bu = u b ); where b is a p -cosise esimaor for 0 : Furhermore, le F ad F x; y) := Pru x; u y) deoe he margial ad oi disribuio fucios of u ; u u ; ad le IA) be he idicaor fucio for he se A. ); respecively; le f be he desiy fucio of Sice we wa o es he depedece of he geeralized errors u, we sar wih he followig depedece measure x; y) = Cov[Iu x); Iu y)] = F x; y) F x)f y) which is proposed by Hoe dig 948). Ulike correlaio ha oly measures liear depedece, x; y) capures all ypes of pairwise depedecies. The sample couerpar of x; y) based o a sample fu g = is x; y) = X Iu x)iu y) 8 < ) 2 : 9 8 X = < Iu x) ; : 9 X = Iu y) ; 4

5 Noice ha uder he ull hypohesis ), x; y) = 0 8 ; 8x; y) 2 R 2 ; hece we ca es ) based o he disace bewee ad 0. Hoe dig 948) used his idea o es he idepedece bewee wo iid radom variables. Alog he same lie, Hog 2000) proposed a geeralized specral es for serial idepedece of saioary ime series. However, i our prese coex fu g = is uobservable as 0 is ukow. The we subsiue bu for u i ad ge b x; y) = X Ibu x)ibu y) 8 < ) 2 : 9 8 X = < Ibu x) ; : 9 X = Ibu y) ; ; bu he asympoic disribuio of b x; y) is geerally di ere o ha of x; y). heorem gives a explici expressio for he di erece uder he followig codiios: The ex Assumpio A: sup z;z 2; Fz;z2) z i < ad supz;z 2; 2 F z ;z 2) z iz k < ; i; k = ; 2. Assumpio A2: P = = o P ): Assumpio A3: p b 0 ) = O p ). Theorem Uder Assumpios A-A3 ad he ull hypohesis ), we have sup x;y)2r 2 p [b x; y) x; y)] p b 0 ) 0 E x; y) = o p ) where F 0 ; x) E x; y) := E [Iu y) F y)] ; F ; x) := Pr[u ) x ] 2) Remark. Due o he presece of his addiioal erm p b 0 ) 0 E x; y); wha we call esimaio e ec, b x; y) does o have he same asympoic disribuio as x; y) i geeral; herefore, ulike ess based o x; y); ess based o b x; y) are o loger asympoic disribuio-free. The above heorem gives us a explici expressio for he esimaio e ec i a very geeral seig, which has may ice applicaios as show i he followig examples. Example. Coiued) Locaio-Scale Models: Y = ; 0 ) + ; 0 )u : 5

6 For hese models, we wa o es u = u 0 ) = Y ; 0 ) are iid for some 0 2 R p : ; 0 ) We ca ge may ieresig resuls i his seup by applyig Theorem. Cases Wihou Esimaio E ec. Suppose Y = X ; 0 )+X ; 0 )u ; where he exogeous covariaes X are idepede over, he i is easy o see ha i his case E x; y) = 0. By Theorem, sup x;y)2r 2 Therefore, ess for idepedece of u p [b x; y) x; y)] = o p ): based o b x; y); such as he Cramer-vo Mises ad Kolmogorov-Smirov ype of ess, Hog s 2000) geeralized specral ess, have he same asympoic disribuio as hose based o x; y): To he bes of our kowledge his resul is ew i he lieraure ad exeds he digs i Delgado ad Mora 2000), who proved a similar resul for he liear regressio models Y = X u wih xed regressors. Box-Lug-Pierce Tes. Oe may es he iid assumpio o u usig he es proposed by Box ad Pierce 970) ad Lug ad Box 978) mx BLP m) = + 2) ) b 2 ); where b) is he sample auocorrelaio fucio of fbu g =; bu = Y ;b ) ; b : ) Bu oe eeds o oice ha due o he parameer esimae errors i b ; BLP m) does o have a 2 limi disribuio for a xed m. As a maer of fac, here b) = R xydb x; y); where b x; y) is de ed i he previous subsecio, usig he uiform expasio i Theorem, oe has p b) = C + p b 0 ) 0 xyde x; y) + o p ); where C = p R xyd x; y) = p = u u ) 3=2 u! u! has a sadard ormal ull limi disribuio. Calculae E x; y) give i 2) ad plug i io he above display, oe ges p b) = C + p b 0 ) 0 E[ u u ] + o p ); where u = u) =0 : Therefore uder H 0 ; BLP m) = mx fc + p b 0 ) 0 E[ u u ]g 2 + o p ): = 6

7 Hece he Box-Lug-Pierce es saisic BLP m) does o have a sadard 2 limi disribuio as he parameer esimae errors brig i he addiioal erm p b 0 ) 0 E[ u u ], which is derived from he resul i Theorem. Example 2. Coiued) Codiioal Goodess-of-Fi: Oe way o es ha Y follows a coiuous codiioal disribuio G; ; 0 ) is o es he followig equivale hypohesis H 0 : u = u 0 ) = GY ; ; 0 ) is iid U[0; ] disribued. To es u beig boh iid ad U[0; ] disribued, we accordigly adus he sample depedece measure o x; y) = X Iu x)iu y) xy The correspodig versio of Theorem gives us F 0 ; y) E x; y) = E x + F 0 ; x) Iu y) ; where F0;x) = gg x; ; 0 ); ; 0 ) G x; ; 0) ; g; ; 0 ) is he desiy fucio correspodig o G; ; 0 ). To summarize his secio, ess for idepedece of geeralized errors u 0 ) based o b x; y) are geerally o asympoically disribuio-free due o he esimaio error i b. Theorem gives us a explici expressio for he esimaio e ec. Alhough we ca d some special cases where he esimaio e ec vaishes, he esimaio e ec does exis i geeral. For hese geeral cases how o elimiae he esimaio e ec ad ge asympoic disribuio-free ess is he problem we ackle i he ex secio. 3. DATA-DRIVEN SMOOTH TESTS I his secio, we iroduce ess for idepedece of geeralized errors immue o he esimaio e ec. The proposed ess are smooh ess i he spiri of Neyma 937), coupled wih daa-drive mehod as i Kalleberg ad Ledwia 999). 7

8 We sar wih he observaio ha ) implies he followig codiio covgu ); hu )) = 0; 8g; h 2 L 2 F ); 8 ; where we focus o he space L 2 F ): he Hilber space of all fucios g such ha R gx) 2 df x) < ; where F is he cdf of u : If we ca d a orhoormal basis fb k g; k = 0; ; 2; ::: for L 2 F ); he he display above is equivale o covb l u ); b m u )) = 0; 8l; m 0; 8. Furhermore, we ca always choose b 0 = ; he he orhoormaliy implies ha for ay l; m ; R b l x)df x) = 0; R b l x) 2 df x) = ad R b l x)b m x)df x) = 0 if l 6= m: Remark 2. If he desiy fucio of u ; f is kow ad belogs o he Pearso family, we ca easily d such basis fb k g as show i Boemps ad Meddahi 2007). Oherwise, le fp k g be ay orhoormal basis for L 2 [0; ]); for example, he Legedre polyomials, he b k = p k F ) will be a orhoormal basis for L 2 F ). Nex, le us sudy he quaiy covb l u ); b m u )) i deail. Noice ha covb l u ); b m u )) = b l x)b m y)d[f x; y) F x)f y)] = b l x)b m y)d x; y); where F, F ad x; y) are de ed i Secio 2. The sample aalogue of he above display based o a sample fu g = is b l x)b m y)d x; y) = X b l u )b m u ) 0 ) 2 0 X b l u ) A X b m u ) A : Whe 0 is ukow, we subsiue bu for u ad ge b l x)b m y)db x; y) = X b l bu )b m bu ) 0 ) 2 0 X b l bu ) A X b m bu ) A : To simplify oaios, we de e C lm := p b l x)b m y)d x; y); b C lm := p b l x)b m y)db x; y): 3) I urs ou C lm has a sadard limi disribuio as give i he ex lemma. Lemma Le fb k g; k = 0; ; 2; ::: be a orhoormal basis for he space L 2 F ) wih b 0 = ; he uder he ull hypohesis ), for ay ; l; m we have ) C lm!d N0; ) 2) covc l m ; C 2 l 2m 2 )! i = 2 ; l = l 2 ad m = m 2 ;! 0 oherwise. Sice fu g is o available i geeral, we cosider C b lm isead. From Theorem we obai bc lm = C lm + p b 0 ) 0 b l x)b m y)de x; y) + o p ): 8

9 Therefore, if he bases fb k g saisfy he followig codiio: Codiio E: R b l x)b m y)de x; y) = 0; 8l; m; ; he C b lm will have he same asympoic disribuio as C lm ; whose limi disribuio is give i Lemma. The quesio he arises how o d orhoormal bases fb k g saisfyig Codiio E. Oe way o do his is usig Gram-Schmid mehod, which ca be doe hrough simple recursive liear regressios. We will show he deailed procedure i he ex secio, bu here are a couple of problems we wa o poi ou here. Firs, i some cases he esimaio e ec p b 0 ) 0 E x; y) exiss oly a ie umber of lag s, bu i may oher cases he esimaio e ec exiss a i iely may lags, he we will focus o ie of hem here, say he rs J lags. Secod, like ay oher smooh es, we ca o use i ie umber of b k s eiher, isead we cocerae o he rs L ad M elemes of he basis. We ll discuss how o choose J ad K afer we iroduce our es saisic. Based o our discussio so far, we ca es he ull hypohesis ) by cosiderig he saisic T JLM := JX LX MX = l= m= bc lm 2 ; 4) where b C lm is de ed i 3). The ex heorem gives us he ull limi disribuio of T JLM : Theorem 2 Uder Assumpios A-A3, Codiio E ad he ull hypohesis ), we have T JLM! d 2 JLM); Proof: Codiio E implies C b lm has he same asympoic disribuio as C lm ; ad he laer has a idepede sadard ormal limi disribuio over l; m; s by Lemma. Now le us ge back o he problem of how may bases ad lags should be used. This is a model selecio problem, ad i should be deermied by he daa. As Kalleberg ad Ledwia 999) did, we adop a modi ed Schwarz s 978) rule ad choose J; L ad M as follows J; L; M) 2 Here d) is a sequece of umbers goig o i iy as! : arg max ft JLM JLM log g: 5) J;L;Md) To summarize hie secio, we provide a ew mehod o deal wih he parameer esimaio uceraiy i a geeral seup. Our proposed ess are Neyma-ype smooh ess based o orhoormal polyomials. This work is relaed o Camero ad Trivedi 993), who also provide ess of idepedece for a wide variey of models. Their ess are score-ype ess based o orhogoal polyomial 9

10 expasios for he oi pdf. Their ess are robus o parameer esimaio uceraiy i may cross-secio models, bu o for ime series models. I his sese, our es is a good compleme o heirs. 4. APPLICATIONS I his secio, we apply he geeral heory developed above o Examples ad 2 give i he Iroducio, ad cosruc ess for idepedece of geeralized errors immue o he esimaio e ec. Example.Coiued) Locaio-Scale Models: Y = ; 0 ) + ; 0 )u : For hese models, we wa o es u = u 0 ) = Y ; 0 ) are iid for some 0 2 R p : ; 0 ) We cosider he depedece measure x; y) wih sample aalogue x; y): As fu g is geerally o observable, we subsiue bu for u ; ad cosider b x; y) isead, bu b x; y) has a limi disribuio di ere o x; y):the di erece is p b e ec, give by Theorem. Some algebra shows ha i his seup ) + x E x; y) = fx) E [Iu y) F y)] : where = ; 0 ); = ;0) ad = ;0) : 0 )E x; y); wha we call esimaio To cacel he esimaio e ec ad ge asympoically disribuio-free es, we rasform b x; y) usig orhoormal basis fb k g ad ge bc lm = C lm + p b 0 ) b l x)b m y)de x; y) + o p ) where C b lm ad C lm are de ed i 3), ad he laer has a sadard ormal limi disribuio as give i Lemma. If Codiio E: R b l x)b m y)de x; y) = 0; 8l; m; ; holds, he C b lm will have he same limi disribuio as C lm. Some algebra shows ha i his seup b l x)b m y)de x; y) = where e y) = E[ u b l x)f 0 x)dx b m y)e y)df y)+ b l x)f 0 x)xdx = y]; s y) = E[ u = y], ad f 0 is he derivaive of f. b m y)s y)df y); Therefore, o kill he esimaio e ec we eed o d orhoormal basis fb k g perpedicular o e ad s ; which ca be doe hrough he followig procedure: 6) 0

11 We sar wih a orhogoal basis fp k g for L 2 F ) wih p 0 = ; ad ru he followig liear regressios p bu ) = 0 + p 2 bu ) = 0 + p 3 bu ) = 0 + ::: ::: JX [ e bu ) + s bu )] + ; he le b bu ) = = q b P b2 JX [ e bu ) + s bu )] + b bu ) + 2 ; he le b 2 bu ) = = ; where b is he regressio residual, q b 2 P b2 2 JX [ e bu ) + s bu )] + b bu ) + 2 b 2 bu ) + 3 ; he le b 3 bu ) = = where J is some umber ha we are goig o choose laer. By cosrucio, for ay l; m ; P b mbu ) = 0; ; q b 3 P b2 3 P b mbu ) 2 = ad P b lbu )b m bu ) = 0 if l 6= m. Furhermore, 0 = X b m bu )e bu )! E[b m u )e u )] = b m y)e y)df y); 0 = X b m bu )s bu )! E[b m u )s u )] = b m y)s y)df y); which, ogeher wih 6), implies Codiio E: R b l x)b m y)de x; y) = 0; 8 l; m K; 8 J. Wih he basis fb k g cosruced above; we ca calculae our es saisic T JLM give i 4), which follows a 2 JLM) disribuio by Theorem 2. Fially, he umber of lags J ad he umber of elemes of he basis L ad M are chose i a daa-drive maer described by 5). ; Example 2.Coiued) Codiioal Goodess-of-Fi Tess We wa o es ha Y follows a coiuous codiioal disribuio G; ; 0 ) by esig ha GY ; ; 0 ) is iid U[0; ] disribued. Bai 2003) uses his idea o es he codiioal disribuios of dyamic models, bu his es is ailored oly for he U[0; ] disribued par, o for he iid par. Our proposed es here is boh for he iid ad U[0; ] disribuio of u : I wha follows, we will focus o GOF for he locaio-scale models: Y = ; 0 ) + ; 0 ) ; where ; 0 ) = E[Y ] ad 2 ; 0 ) = V ar[y ] a.s. This geeral class of models, wih he popular ARMA-GARCH model as a special case, are widely used i acial applicaios, for example, i modelig sock reurs, erm srucure of ieres raes, ec. For heses models, we es he ull hypohesis ha follows a coiuous disribuio G by esig he followig equivale hypohesis H 0 : u = u 0 ) = G 0 )) is iid U[0; ] disribued, where ) = Y ;) ;).

12 As we es boh iid ad U[0; ] disribuio of u ; we adus he sample depedece measure i secio 2 accordigly o x; y) = X Iu x)iu y) xy: As fu g is geerally o observable, we subsiue bu = u b ) for u ; wih b beig a p -cosise esimaor of 0 ; ad base our es o b x; y). Applyig Theorem o his seup, we ge ) E x; y) = E gg y)) + G y) x + gg x)) + G x) Iu y) ; where = ; 0 ); = ;0), = ;0) ad g is he desiy fucio correspodig o G: As show i he previous secio, if we ca d a orhoormal basis fb k g for L 2 [0; ]) wih b 0 = such ha Codiio E: R b l x)b m y)de x; y) = 0; 8l; m; ; is sais ed, he we will ge a asympoically disribuio-free es T JLM de ed i 4). Some algebra shows ha i his seup b l x)b m y)de x; y) = b l x)q x)dx b m y)e y)dy + b l x)q 2 x)dx b m y)s y)dy; 7) x)) where q x) = g0 G gg x)) ; q 2x) = g0 G x))g x) gg x)) ; e y) = E[ u = y]; s y) = E[ u = y], ad g 0 is he derivaive of g. Therefore, for Codiio E o hold, oe oly eeds o d orhoormal bases fb k g perpedicular o q ad q 2 ; which ca be doe hrough he followig procedure: We sar wih ay orhogoal basis fp k g for L 2 [0; ]) wih p 0 polyomials; ad ru he followig liear regressios p bu ) = 0 + q bu ) + 2 q 2 bu ) + ; he le b bu ) = q = ; for example he Legedre b P b2 p 2 bu ) = 0 + q bu ) + 2 q 2 bu ) + b bu ) + 2 ; he le b 2 bu ) = ; where b is he regressio residual, q b 2 P b2 2 p 3 bu ) = 0 + q bu ) + 2 q 2 bu ) + b bu ) + 2 b 2 bu ) + 3 ; he le b 3 bu ) = ::: ::: ; q b 3 P b2 3 By cosrucio, for ay l; m ; P b P lbu ) = 0; b lbu ) 2 = ad P b lbu )b m bu ) = 0 if l 6= m. Furhermore, X b l bu )q bu )! b l x)q x)dx = 0; which, ogeher wih 7), implies Codiio E. X b l bu )q 2 bu )! b l x)q 2 x)dx = 0; ; 2

13 Wih he bases fb k g cosruced above; we ca calculae bc lm = p b l x)b m y)db x; y) = X p b l bu )b m bu ) ad our es saisic T JLM give i 4), which follows a 2 JLM) disribuio by Theorem 2. Fially, he umber of lags J ad he umber of bases L ad M are chose i a daa-drive maer described by 5). Remark 3. Our proposed codiioal GOF-es here ca be used o es ay coiuous disribuio G, such as ormal, sude-, skewed- Hase 994)), o-ceral Havey ad Siddique 999)), ec., i ay locaio-scale model seup ARMA-GARCH, oliear regressio, hreshold auoregressive TARs) models, ec.). 5. MONTE CARLO STUDY AND REAL DATA ANALYSIS To assess he ie sample performace of our proposed es i he previous secio, we do some Moe Carlo sudies. The we apply our es o NYSE Compariso wih Bai s 2003) Tes Followig he simulaio seup i Bai 2003), we cosider he model Y = ad es follows a disribuio G based o bu = Gb ) = G Y b b ); where b ad b are sample mea ad sample sadard deviaio, respecively. I his seup we have E x; y) = xgg y)) G + y) ygg x)) G ; x) where g is he desiy fucio correspodig o G: Therefore Codiio E holds here for ay orhoormal basis fb k g for L 2 [0; ]) wih b 0 calculae ou es saisic de ed by 4) ad 5). 5.. Size = : We will ake he Legedre polyomials ad We rs geerae Y from N 0 ; 2 0) ad es follows a N0; ) disribuio based o bu = Y b b ) where is he cdf of sadard ormal. We he geerae Y from ad es [= 2)] =2 follows a disribuio based o bu = T Y b b [= 2)] =2 ) where T is he cdf of. The empirical size of 000 simulaios are repored i Table a. Bai s 2003) resuls are give i Table b Power We geerae Y from ad 2 ) ad he es follows a N0; ) disribuio based o bu = Y b b ): Table 2a repors he power of our es from 000 simulaios ad Table 2b repors Bai s 2003) resuls. 3

14 5.2 Tesig he codiioal disribuio of u i a ARMA-GARCH Model 5.3 Real Daa: ARMA-GARCH o S&P 500 reurs, ad es he goodess-of-. 6. CONCLUSIONS 7. PROOFS Proof of Theorem : Noaio: u ) = GY ; ; ); u = u 0 );ad bu = u b ) where b is a p -cosise esimaor for 0 ;i.e. p b 0 ) = O p ): x; y) = X Iu x)iu y) 8 < ) 2 : 9 8 X = < Iu x) ; : 9 X = Iu y) ; b x; y) = X Ibu x)ibu y) 8 < ) 2 : 9 8 X = < Ibu x) ; : 9 X = Ibu y) ; p fb x; y) x; y)g = p fibu x)ibu y) Iu x)iu y)g #) p Ibu x) Ibu y) Iu x) Iu y) To hadle A, de e he process =: A A 2 K c; x; y) = X p fiu 0 + p c ) x) E[Iu 0 + p c ) x) ]giu 0 + p c ) y); idexed by c; x; y) 2 C K R 2 ;where C K = fc 2 R p : c Kg;ad K > 0 is a arbirary bu xed cosa. Lemma A: K c; x; y) is asympoically igh wih respec o c; x; y) 2 C K R 2 :?Uder Assumpio A, followig he seps i Escaciao ad Olmo 2007).? 4

15 We ca also show ha, for ay xed c; x; y) 2 C K R 2 ; E[K c; x; y) for ay bc = O p ); we have sup K bc; x; y) K 0; x; y) = o p ); x;y)2r 2 K 0; x; y) 2 ] = o):hece Apply his argume o bc = p b 0 ) = O p ); ad we ge fibu x) E[Ibu x) ]gibu y) sup p x;y)2r 2 fiu x) E[Iu x) ]giu y) Hece A = p fibu x)ibu y) Iu x)iu y)g = p fe[ibu x) ] E[Iu x) ]gibu y)+ = o p ) p =: B + B 2 + o p ) E[Iu x) ]fibu y) Iu y)g + o p ) Iroduce he oaio F ; x) = Pr[u ) x) ];he B = p ff b ; x) F 0 ; x)gibu y) = p = p b 0 ) p p F e ;x) b 0 )Ibu y) F e ;x) Ibu y) ULLN = p b 0 )E[ F0;x) Iu y)] + o p ) B 2 = p = p F 0 ; x)fibu y) Iu y)g F 0 ; x)ff b ; y) F 0 ; y)g + o p ) same rick as we use for A ) = p b 0 )F 0 ; x) p p F e ;y) + o p ) eed u idepede of ; which ca be added o H 0 ) ULLN = p b 0 )F 0 ; x)e[ F0;y) ] + o p ) Now, ur o A 2 A 2 = p = p Ibu x) Ibu x) Ibu y) # Ibu y) # Iu x) Iu x) #) Iu y) #) Ibu y) + 5

16 p =: C + C 2 C = p = Iu x) = [F 0 ; y) + o p )] Ibu x) Ibu y) ) p Ibu y) p = p b 0 )F 0 ; y) p p # Ibu y) # [Ibu x) Iu x) Iu x)] ) [F b ; x) F 0 ; x)] + o p ) ) same rick as we use for B 2 ) F e ;x) + o p ) ULLN = p b 0 )F 0 ; y)e[ F0;x) ] + o p ) Iu x) Iu y) #) Ibu y) #) Similarly, C 2 = = p Iu x) Iu x) ) p = p b 0 )F 0 ; x)e[ F0;y) ] + o p ) Ibu y) # [Ibu y) Iu y)] Iu x) ) Iu y) #) Therefore, p fb x; y) x; y)g = A A 2 = B + B 2 C C 2 + o p ) = p b o 0 ) E[ F0;x) Iu y)] F 0 ; y)e[ F0;x) ] + o p ) = p b o 0 )E F 0;x) [Iu y) F 0 ; y)] + o p ) =: p b 0 )E x; y) + o p ) o where E x; y) := E F 0;x) [Iu y) F y)] ; F ; x) := Pr[u ) x ]: 6

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