Robust Parametric Tests of Constant Conditional Correlation in a MGARCH Model

Transcription

1 Robus Parameric ess of Consan Condiional Correlaion in a MGARCH Model Wasel Shada yz and Chris Orme Economics, School of Social Sciences, Universiy of Mancheser, UK Seember 7, 205 Absrac his aer rovides a rigorous asymoic reamen of new and exising asymoically valid Condiional Momen esing rocedures of he Consan Condiional Correlaion assumion in a mulivariae GARCH model. Full and arial Quasi Maximum Likelihood Esimaion frameworks are considered, as is he robusness of hese ess o non-normaliy. In aricular, he asymoic validiy of he LM rocedure roosed by se (2000) is analyzed and new asymoically robus versions of his es are roosed for boh esimaion frameworks. A Mone Carlo sudy suggess ha a robus se es rocedure exhibis good size and ower roeries, unlike he original varian which exhibis size disorion under non-normaliy. JEL classi caion: C2; C32 Keywords: Mulivariae GARCH; Consan Condiional Correlaion; Condiional Momen ess; Robusness; Mone Carlo. Inroducion Wihin a Mulivariae GARCH (MGARCH) model, he condiional correlaion aroach has roved oular amongs alied workers when modelling volailiy. Iniially, he Consan Condiional Correlaion (CCC) model was emloyed (see for examle, Bollerslev (990), Kroner and Claessens (99), Kroner and Sulan (99, 993), Park and Swizer (995) and Lien and se (998)), whils recenly he Dynamic Condiional Correlaion (DCC) model (Engle, 2002) has become more revalen. Due o he simliciy and comuaional advanages of he CCC model, on he one hand, bu he increased generaliy of he DCC aroach, on he oher, esing he adequacy of he CCC assumion wihin a MGARCH model remains imoran from boh a racical and, herefore, heoreical oin of view. Indeed, he mos widely used es of he CCC assumion, among alied workers, is se s (2000) LM es (see, for examle, Lien, se and sui (2002), Andreou and Ghysels (2003), Lee (2006), Aslanidis, Osborn and Sensier (2008), among ohers) in reference o a number of oher roosals in he lieraure An earlier version of he aer eniled An Invesigaion of Parameric ess of he Consan Condiional Correlaion Assumion was resened a he Euroean and Asian Meeings of he Economeric Sociey in Oslo (Augus 25-29, 20) and Seoul (Augus -3, 20). Deailed and exhausive roofs are rovided in a freely available on-line comanion aer, Shada & Orme (205). We are graeful for he insighful commens of he aricians of hese meeings and also o Ralf Becker, Alasair Hall and Len Gill. wo anonymous referees also rovided consrucive and insighful commens which were of enormous hel in revising his aer. he sandard disclaimer alies. y Corresonding auhor: Dr Wasel Shada, Economics, School of Social Science, Universiy of Mancheser, Mancheser M3 9PL, UK. waselbin.shada@mancheser.ac.uk z he rs auhor s research is ar of his PhD hesis and was suored by a Commonwealh Scholarshi and Fellowshi Plan and a Mancheser School Award; boh of which are graefully acknowledged.

2 (e.g., Bollerslev (990), Longin and Solnik (995) and Bera and Kim (2002)). se s (2000) rocedure, unlike he informaion marix es aroach of Bera and Kim (2002), can be alied o high-dimensional daa bu i is redicaed on a Full Quasi Maximum Likelihood Esimaion (FQMLE) aroach, ogeher wih an exlici assumion of normaliy when consrucing he es saisic. his aer addresses four inferenial issues ha emerge from se (2000): (i) se s Ouer Produc of he Gradien (OPG) version of he LM es is only guaraneed o be asymoically valid under an exlici normaliy assumion. (ii) Even under normaliy he OPG varian of a LM es may demonsrae relaively oor nie samle erformance; see, for examle, Davidson and MacKinnon (983), Bera and McKenzie (986), Orme (990), Chesher and Sady (99). (iii) se s es rocedure may no be robus o misseci caion of he individual volailiy (GARCH) equaions. (iv) I is sill common racice, when esimaing a MGARCH model, o emloy a wo-sage or Parial Quasi Maximum Likelihood Esimaion (PQMLE) aroach, where he volailiy arameers in each equaion are rs esimaed using a univariae GARCH seci caion and, second, he correlaion arameers are hen esimaed using hese rs-sage volailiy arameer esimaes (see Engle and Sheard (2008), Hafner, Dijk and Franses (2005), Billio, Caorin and Gobbo (2006), among ohers); however, wihin his PQMLE framework, here aears o be no available es of he CCC assumion. hus, we roose, and rovide a rigorous analysis of, asymoically valid and non-normaliy robus ess of he CCC assumion, based on a Condiional Momen (CM) aroach. hese ess will be robus in he sense ha heir asymoic validiy does no deend on normaliy (unlike se (2002)); bu hey do require momen condiions which ensure sandard asymoic inferences can be alied. Such ess can be emloyed following eiher FQMLE or PQMLE and robus versions of se s es are given aricular aenion. he required derivaions require some sraighforward, ye edious, algebraic resuls bu lead o robus ess ha are easy o imlemen. In our Mone Carlo sudy, wih a moderae number of asses/ime-series (N = 5), hese ess demonsrae saisfacory size roeries in mos cases. Furhermore, whils no addressed analyically, he Mone Carlo sudy also sheds some ligh on he robusness of he various es saisics o GARCH misseci caions in he individual volailiy (GARCH) equaions. From he anoly of rocedures we consider, a robus version of se s LM es exhibis very good size and ower roeries under a variey of Daa Generaion Processes (DGPs). he res of his aer is organized as follows. he model, FQMLE, PQMLE and se s original LM es are reviewed in Secion 2. In Secion 3, a class of CM arameric ess is described, for boh esimaion frameworks, and robus varians roosed. his is exended in Secion 4 o rovide robus versions of se s LM es ha can be emloyed following eiher FQMLE or PQMLE. Secion 5 reors he ndings of a Mone Carlo sudy and Secion 6 concludes. he analysis follows sandard rs order asymoic heory, bu o avoid obfuscaing he main issues, echnical (bu fairly sandard) assumions Nakaani and eräsvira (2009) roosed anoher LM es for volailiy ineracion where he null model is CCC GARCH model agains he alernaive of Exended CCC (ECCC) GARCH model. 2

3 and roofs of he main resuls are relegaed o an Aendix; wih more deailed and exhausive roofs rovided in an accomanying freely available on-line aer, Shada and Orme (205), which also conains addiional informaion concerning he Mone Carlo exerimens underaken. he following noaion is emloyed: he vec (:) oeraor sacks he N columns of a (M N) marix as a (M N ) vecor; vech (:) sacks he lower riangular orion of a (N N) marix as a 2 N (N + ) vecor; and, vecl (:) sacks he sricly lower riangular orion of a (N N) marix as a 2 N (N ) vecor. Corresondingly, s J N = f(i; j) : i = j; :::; N; j = ; :::; N; and i changing more quickly han jg de nes he ordering of he elemens of a (N N) marix A = fa ij g ino vech (A) and s C N = f(i; j) : i = j + ; :::; N; j = ; :::; N ; and i changing more quickly han jg he corresonding ordering for he vecl (:) oeraor. 2 he CCC Model and se s LM es We consider he following sandard CCC-GARCH linear regression seci caion y i = w 0 i' i + " i ; i = ; :::; N = ; :::; ; () o model he (N ) ime-series vecor y = fy i g, wih large and N xed/small, where w i = (y 0 i; ; d0 i )0 is he (K ) vecor of regressor variables, conaining curren and lagged exogenous variables (d i ), and lagged deenden variables (y i; ) and ' i < K is an unknown vecor of regression arameers. he volailiy in he (N ) error vecor " = f" i g has a GARCH(; q) seci caion of h i = i0 + P q k= ik" 2 i; k + P j= ijh i; j wih i = ( i0 ; i ; :::; iq ; i ; :::; i ) 0 < K being an unknown vecor of volailiy arameers. he CCC model is described by " = H =2 0 ; and H = D D ; (2) where he 0 are indeendenly and idenically disribued (iid) random vecors, wih E [ 0 ] = 0 and E = IN, D = diag(h =2 i ) a (N N) diagonal marix and = ij ; a (N N) ime invarian symmeric osiive de nie marix wih ii = ; i = ; :::; N: hus, he condiional covariance marix, H ; has elemens h ij = h =2 i h =2 j ij; i; j = ; :::; N: o be more recise abou he arameerizaion emloyed, de ne i = (' 0 i ; 0 i )0 < K+K and! = 0 ; < N ; where N = N (K + K ) + 2 N(N ) wih = 0 ; :::; 0 N 0 < N(K+K ) and = vecl( ): 2 hen, " i = y i w 0 i ' i " i (' i ) wih h i h i ( i ); D D (), and H H (!): Leing! 0 = 0 0; denoe he rue arameer vecor, wih 0 = vecl( 0 ), we have " 0 = " ( 0 ); 2 For examle, for a AR() CCC seci caion wih N = 5 and individual GARCH (,) errors, we have K = 2; K = 3 and N = 35: 3

4 D 0 = D ( 0 ); and H 0 = H (! 0 ) = D 0 0 D 0 so ha E [" 0 jf ] = 0; and E [" 0 " 0 0jF ] = H 0, where F = (" 0; ; " 0; 2 ; :::): 3 Following Berkes, Horváh and Kokoszka (2003) and Ling and McAleer (2003), and given Assumion A3(i)(ii), he rocess for h i has he reresenaion h i = P l=0 ila i; l ; where, for all i; a i = i0 + P q k= ik" 2 i; k and il = P j= ij i;l j wih is = 0; s < 0; i0 = ; il > 0; l > 0; and 0 < P P l=0 il = j= ij < : he coe ciens, il ; decay exonenially fas, and here exis consans K > 0 and 0 < < ; indeenden of!; such ha il K l, for all i: hen, Assumion A in he Aendix, ensures he ideni abiliy, saionariy and ergodiciy of he rocess fy i; " 0i ; h 0i g ; where h 0i h i ( i0) ; see Ling and McAleer (2003). In he subsequen analyses, hree alernaive ransformed error vecors are emloyed: volailiy adjused errors ( ), fully sandardized errors ( ) and (se s) ransformed sandardized errors (" ). hese are, resecively, () = D " = f i ( i )g (3) (!) = H =2 " = f i (!)g (4) " " (!) = = D " = f" i(!)g (5) wih 0 = ( 0 ); 0 = (! 0 ), " 0 = " (! 0 ); and saisfying: (i) E [ 0 jf ] = 0; E 0 0 0jF = 0 (in he case of a CCC seci caion); (ii) E [ 0 ] = 0; E = IN ; from (2); and, (iii) E [" 0jF ] = 0; E [" 0" 0 0jF ] = 0. For some esimaor ^! = (^ 0 ; ^ 0 ) 0 of!; he esimaed counerars of (3)-(5) will be denoed ^ (^); ^ (^!); ^" " (^!) and similarly ^ i ; ^ i and ^" i: Finally, where here is no ambiguiy, his form of noaion will be adoed for general funcions of arameers m (!); so ha m m (!); m 0 m (! 0 ) and ^m m (^!); ec. 2. FQMLE and PQMLE Framework Given () and (2), he quasi-condiional log-likelihood er observaion, ; is given by l (!) = 2 ln j j 2 NX i= ln h i 2 0 : Assuming L (!) = P = l (!) is wice coninuously di ereniable, and g (!) = P = g (!) ; where g (!) (!)=@!; hen he FQML esimaor, ^! = arg max! L (!); sais es g (^!) = 0: However, he observed l (!) is consruced condiional on available re-samle values, because h i needs o be consruced recursively given iniial values, " + i0 = "2 i0 ; :::; "2 i; q ; h i0; :::; h i; 0. In order 3 Given he conex, here should be no confusion beween he random vecor " 0 ; which has elemens " 0i ; and he elemens of " ; denoed " i ; i = ; :::; N; = ; ::::; : 4

5 o simlify he algebra and asymoic heory, i is assumed (in addiion) ha he required re-samle observaions on w i are also available and ha h i = 0 for all i and 0: he simli caions derive from he fac ha h i can hen be exressed as h i = P l=0 ila i; l ; = ; :::; ; bu he rocesses h i and l (!) will no be saionary ergodic sequences. 4 Relacing h i by h i in l (!); hroughou, rovides an unobserved bu saionary and ergodic loglikelihood sequence l (!) = 2 ln j j 2 NX ln h i i= 2 0 ; where = " i = h i : hen, wih N nie and leing! ; and under Assumions A and B, B2 described in he Aendix, ^!!! 0 and d (^!! 0 )! N 0; J 0 ggj0 ; where J 0 = g E [@g (!)=@! 0 ]!=!0 and gg = E [g (! 0 )g (! 0 ) 0 ] are boh nie and osiive de nie and (!) (!)=@!: ha is o say, emloying he recursively consruced h i (raher han he rue bu unobserved h i ) makes no di erence asymoically. Adoing a PQMLE aroach, and following Engle (2002), we can wrie L (!) = L () + L C (!) where L () = P P N = i= l i( i ); wih l i ( i ) = 2 ln hi + h i "2 i and L C (!) = P = lc (!); wih l C (!) = 2 ln j j Here, P = l i( i ) is he average log-likelihood for he i h univariae GARCH regression model () and l C (!) models he CCC srucure. his a ords a wo-sage PQMLE rocedure where a sage one we obain ~ i = arg max i P = l i( i ), he consisen univariae GARCH QML esimaors. Equivalenly, ~ = arg max L (); which sais es g ( ~ ) = 0; where g () = P = g () and g () = P N i( i )=@; which is, of course, g () = (@l ( )@ 0 ; N ( N )=@ 0 N) 0. For he second sage, we emloy all he rs sage PQML esimaor ~ o obain ~ = arg max L C (~ ; ); which sais es P = ~" i~" j ~ ij = 0; j < i; where = ij : he resuling PQML esimaor, ~! = ( ~ 0 ; ~ 0 ) 0 ; is consisen, bu asymoically ine cien relaive o FQMLE. 5 If we ado Bollerslev s (990) alernaive arameerizaion of l i ( i ); i ransires ha he ensuing ~ has a closed form exression saisfying P ~i ~ = j ~ ij = 0; j < i: Even wihou his alernaive arameerizaion, i is sill he case ha he simle esimaor ~ ij = P = ~ i ~ j ; j < i will be consisen for he rue correlaion arameer value, and his will be he esimaor, ogeher wih ~ i ; ha we shall emloy in he PQMLE framework. Moreover, i urns ou ha he limi disribuions of he various es indicaors ha we shall consider, obained from PQMLE, are no in uenced by his choice of ~ and his leads o he consrucion of relaively simle asymoically valid es saisics. hus, we jus need he searae limi disribuions of ~i i0 and Assumions A and B, B2, in he Aendix, imly ( ~ 0 ) d! N(0; J 0 ggj0 ) where he (block diagonal) marix 4 Noe, ha his is no he same sar-u scheme emloyed by eiher Ling and McAleer (2003), who choose " + i0 = 0; Berkes e al (2003), or Francq and Zakoian (2004). In racice, and for all inferenial rocedures described in his aer, any consan value can be chosen for " + i0 ; in order o generae h i, = ; :::;. 5 Hafner and Herwarz (2008) rovided an analyical exression for he asymoic variance of he PQML esimaor, for boh he CCC and DCC models. 5

6 J 0 = ( 0 )=@ 0 = diag 2 l i ( i0)=@ 0 i and gg = E [g ( 0 )g ( 0 ) 0 ] are boh nie and osiive de nie, and g Halunga and Orme (2009, heorem ). () = P N i ( i)=@ wih l i ( i) = 2 ln h i + " 2 i =h i ; see, e.g., 2.2 se s LM es of he CCC Assumion For he uroses of consrucing se s es, a dynamic correlaion srucure of he form ij = ij + ij i; j; is assumed, where ii = 0; ij = ji ; even hough ij is no a well-de ned alernaive o he CCC since = ij is no necessarily a osiive de nie marix for all. here are N(N ) 2 addiional arameers in DCC alernaive model wih he null hyohesis of CCC being H 0 : ij = 0; for all disinc i; j; and i > j. se (2000) emloyed he Lagrange Mulilier (LM) rincile and roosed an OPG varian of he LM es saisic which, in his case wih es variables ^ ^0, is 6 where ^ is a N + dlm = 0 ^ ^0 ^ ^0 ; (6) N(N ) 2 marix, wih rows equal o (^g 0 ; vecl((^" ^" 0 ^ ) ^ ^0 ) 0 ); denoes he Hadamard roduc, and is he ( ) column vecor of ones. 7 Under he usual regulariy condiions LM d is asymoically disribued as 2 N(N ) : 2 he noaion LM d is used o emhasize ha (6) is consruced from ^! and canno be imlemened direcly using ~!. Furhermore, he OPG consrucion advocaed by se (2000) may be sensiive o non-normaliy, and some evidence for his is rovided by se (2000, Secion 5). In he nex secion we develo a Condiional Momen (CM) esing framework of he CCC assumion which accommodaes se s es. his framework rovides, in Secion 4, non-normaliy robus se es rocedures for boh he FQMLE and PQMLE cases. However, for his aricular choice of es variables, i; j; ; a sronger momen condiion of E j" 0i j 8 < for all i;, is hen required in order o jusify he asymoic validiy of hese robus ess. 3 A Class of Asymoically Valid CM es Procedures If he CCC seci caion is correc, hen E jF = 0 in which h0i = h i ( i0). he diagonal elemens of 0 corresond o he individual GARCH (or volailiy) seci caions, whereas he o -diagonal elemens corresond o he CCC assumion. Also due o he symmery here are 2 N (N + ) indeenden (disinc) resricions in his momen condiion; i.e., E J 0jF = 0; where J 0 = vech ; he suerscri J indicaing join esing of boh he CCC and of he individual 6 Here, cross-roducs of lagged sandardised residuals, are emloyed as es variables which is feasible using he LM rincile and, since his alernaive is an ari cial device simly emloyed o consruc a es saisic, Silvennoinen and eräsvira (2009) inerreed his as a general misseci caion es. 7 Deails of g (!); and hence g (!); are given in he Aendix, Proosiion 6. 6

7 volailiy seci caions. he yical elemen of his momen condiion can be wrien as E 0;i 0;j 0;ij jf = 0; i j; i = 2; :::; N: (7) When he underlying momen resricion is (7), he ensuing es will be referred as he Full CM (FCM) es and can be reaed as a join misseci caion es of he comlee MGARCH error seci caion. If we are only ineresed in esing he CCC assumion, he momen condiion is E C 0jF = 0; where C 0 = vecl and he suerscri C denoes esing only he CCC assumion; i.e., E 0;i 0;j 0;ij jf = 0; i > j; i = 2; :::; N: (8) he ensuing es based on (8) will be referred o as he CCC CM (CCM) es. he imlicaion of (7) is ha misseci caion ess of he CCC model can be consruced as ess of he following momen condiions E 0;i 0;j 0;ij rij; (! 0 ) = 0; (9) where he (q ij ) vecor r ij; (! 0 ) is a F measurable funcion, ossibly deending uon he rocesses h 0i and h 0j : 8 A CM es indicaor vecor can hen be consruced, u o a knowledge of!; as m (!) = P = m (!) wih he vecor m (!) consruced from he sacked sub-vecors m ij; (!) = ( i j ij )r ij; = ij; r ij; ; (q ij ) (0) where ij; = ( i j ij ); a scalar, and r ij; r ij; (!): Noe ha he m ij; (!) are arranged in m (!), ordered by (i; j); according o eiher s J N (as in he vech(:) oeraor) or sc N (as in he vecl(:) oeraor). In he former case, his will be denoed m J (!) = P = mj (!); q J, where q J = P ij q ij; whils in he laer case i will be m C (!) = P = mc (!), q C ; where q C = P i>j q ij = q J P i q ii: hen, m J (!) will be referred o as he FCM es indicaor, mc (!) he CCM es indicaor and boh can be consruced emloying eiher he FQMLE, ^!; or he PQMLE, ~!: he following secials cases emerge:. If r is a common vecor of es variables emloyed for all i; j; hen m J (!) = P = J r and m C (!) = P = C r ; where J = vech 0 and C = vecl 0 : 2. If r ij; is a scalar, wih = fr ij; g ; (N N) ; hen m J (!) = P = J r J and m C (!) = P = C r C, where r J = vech( ), r C = vecl( ). 8 Alhough F measurable, we wrie r ij; in de ning r ij; (! 0 ) raher han, say, r ij; : his is consisen wih he usual noaion h 0i ; which is also F measurable. 7

8 se s LM es can be inerreed as a es of he momen condiion E vecl " 0" jf = 0; where " is given in (5). Since " " 0 = ( 0 ) ; he (k; l) h elemen of his (N N) marix is k0 0 l = vec k l0 0 DN vech 0, where k is he k h column of and D N is he N 2 2 N(N + ) dulicaion marix. 9 Exloiing he roeries of D N, we can wrie " k " l kl = 0 kl J, where kl = vech k l0 + k l0 dg( k l0 ) ; ( 2N(N + ) ). For examle, for N = 2 we have 2 = 2 ; say, and = ( 2 ) 2 [ + 2 ]: Since 0 kl J is a scalar, he (k; l) h elemen of a se (LM es) indicaor emloying arbirary es variables, kl; ; (q kl ), wih indices (k; l) ordered according o s C N ; can be exressed as m LM kl; (!) = X 0 kl J kl; = ( 0 kl I ) mj qkl (kl) (!); (q kl ) () = where m J (kl) (!) = P = J kl;, ( q kl 2 N(N + ) ); is a join FCM es indicaor vecor, of he form m J (!); bu consruced wih a common (q kl ) vecor of es variables, kl;, for all elemens of J : herefore se s es indicaor is accommodaed in his CM framework since is (k; l) h elemen is simly he linear combinaion (hrough 0 kl I q kl ) of ALL he FCM es indicaors. Recall ha in se s original es, Secion 2.2, kl; = k; l;, and q kl = : o consruc asymoically valid CM ess of he CCC hyohesis we need o esablish he limi disribuions of he es indicaor vecors. his is done in he following wo secions for boh he FQMLE and PQMLE cases, for which we need o inroduce some more noaion. Le m (!) = P = m (!) denoe eiher m J (!) or m C (!); according o he es under consideraion, where m m (!); (q ) denoes eiher m J (!) or m C (!) as de ned above (wih q = q J ; or q = q C ; resecively) and he vecor 0 (! 0 ) will denoe eiher J 0; or C 0: In aricular, m 0 m (! 0 ) is consruced from he sacked (q ij ) sub-vecors, m ij; (! 0 ). We shall also emloy he following a various imes: z 0 i = h 0 i where 0 K = (c 0 i ; x0 i ); where c0 i = h 0 i and x 0 i i h 0 i is he (K ) vecor of zeroes. Finally, and similarly o l of signi es ha h i (and/or h j ) has been relaced by h i, and f 0 i = (w0 i = h i ; 0 0 K ); (!) and g (!); a suerscri (and/or h j ) where necessary. 3. ess based on FQMLE An asymoically valid 2 es saisic, and es rocedure, is jusi ed by he following resuls: Proosiion Suose Assumions A and B, as described in he Aendix, hold. hen, = E [u (i) (! 0 )u (! 0 ) 0 ] is nie, where u (!) 0 = (m (!) 0 ; g (!) 0 ) : Furhermore: P = u (! 0 )! d N(0; ), where = E [u (! 0 )u (! 0 ) 0 ] is nie; and, 9 For A = A 0 ; D N vech(a) = vec(a) whils for any A; D 0 N vec(a) = vech(a + A0 dg(a)); where dg(a) forms he diagonal marix from he diagonal elemens of he square marix A; see Magnus and Neudecker (986). 8

9 (ii) P = u (^!)u (^!) 0 = o (); for any ^!! 0 = o (); where u (!) 0 = (m (!) 0 ; g (!) 0 ) : Remark he choice of es variables r ij; = "i; "j; requires a srenghening of B, in he hi; hj; Aendix, o E j" 0i j 8 < : In he case of m J (! 0); for examle, mm E [m (! 0 )m (! 0 ) 0 ] is block h i ariioned wih (q ij q kl ) blocks E ij; kl; r ij; r0 kl; ; wih boh (i; j) and (k; l) ordered according!=! 0 o s J N : he modi caion is obvious for mc (! 0) which simly removes all enries 2, i = ; :::; N; from mm : he following secials cases emerge: ii; r ii; r0 ii;. If r is he same vecor of es variables emloyed for all i; j; hen mm = E [ r 0 r 0 0 ] : 2. If r ij; is a scalar, wih r being eiher r J or r C ; as aroriae: hen, mm = E [r 0 r ] : Proosiion 2 Under he assumions of Proosiion, m (^!) and A 0 = I q : B0J 0 h ; wih 0 = E he (q q) ideniy marix: (! 0 i ; osiive de nie, B 0 = E d! N (0; V ) ;where V = A 0 A 0 h ; and I q is 0 In he case of m J (^!); for examle, B 0 is block ariioned as B 0 = B 0;ij ; wih blocks B 0;ij sacked (verically) ordered by (i; j); according o s J N ; and given by 0 ; B 0;ij = rij; i j ij 0!=! 0 ; noing ij; =@!0 = r ij; (!)@ ij; =@!0 + ij; (!)=@!0 ; and E ij; (! 0)jF = 0, so ha E ij; (!)=@!0 jf = 0: he modi caion is obvious for m!=! C o (^!) and simly removes all B ii blocks, and in he secial case ha r is he same vecor of es variables emloyed for all i; j; hen B0 = E [@ (! 0 )=@! 0 r0 ]. From Proosiion 2, and rovided V is osiive de nie, he general form of he es saisic is ^S = m (^!) 0 n ^V o m (^!); (2) which has a limi 2 q disribuion, under he null, where ^V is any consisen esimaor for V : o consruc asymoically valid es saisics we need consisen esimaors for V : In doing so, we consider he cases of Gaussian and non-gaussian disribuions, resecively, for he fully sandardized error rocess, 0 : he rs case rovides he well-known OPG covariance marix esimaor, denoed ^V (o) : For he more general case, we develo a non-normaliy robus rocedure, in he similar siri of Wooldridge (990) 0, buil on a robus variance-covariance marix esimaor denoed (r) ^V : his esimaor will be robus in he sense ha is consisency asymoic does no deend on normaliy, bu i does require momen condiions which ensure sandard asymoic inferences can be alied. 0 Similar aroach was emloyed by Halunga and Orme (2009). 9

10 3.. he OPG-FQMLE es De ne he ( q) marix M M(!) o have rows m (!) 0 and G G (!) is a ( N ) marix wih rows g (!) 0 0 ; wih he undersanding ha ^M M (^!) and ^G G(^!): By Proosiion 6(ii) and (iv), Lemma imlies ha a consisen esimaor for is ^U 0 ^U = P = u (^!)u (^!) 0 ; where ^U = ( ^M; ^G ) has rows u 0 (!) = (m 0 (!); g 0 (!)) 0. However, under he addiional assumion of normaliy, 0 N (0; I N ) ; he generalized IM inequaliy holds (Newey, 985) so ha consisen esimaors of J 0 and B 0 will be ^G0 ^G and ^G0 ^M; resecively: In his case, a consisen esimaor for V can be obained as ^V (o) = 0 0 ^M ^M ^M ^G ( ^G 0 ^G ) ^G0 ^M : (3) his rovides he well-known OPG form formulaion of (2) as ^S (o) = 0 ^U ^U 0 ^U ^U 0 ; (4) which is simly of he form R 2 u; where R 2 u is he (uncenred) R 2 coe cien following a regression of on ^U : 3..2 he Robus-FQMLE es Here we consruc a (non-normaliy) robus esimaor for V = A 0 A 0 0 ; noing from above ha 0 ^U ^U = o (), bu wihou necessarily assuming normaliy. A robus esimaor for A 0 requires h i h i robus esimaors of = E 2 l and = E : he sraegy for consrucion (! 0 0 of such esimaors follows, e.g., Nakaani and eräsvira (2009): de ne he marix J (!); which is P h i consruced as = 2 l (! 0 F bu, once condiional execaions have been aken,! relaces! 0 and h i relaces h i. Similarly, B (!) is consruced from P = E 0 F i in he same way. We inroduce he following addiional noaion: (i) Z i is a ( K + K ) marix having rows z 0 i = h 0 i = (c 0 i ; x0 i ); (ii) F i is he ( K + K ) marix wih rows f 0 i = (w0 i = h i ; 0 0 K ); (iii) R ij is he ( q ij ) marix having rows r 0 ij; ; = ; :::; ; (iv) Z = diag (Z i) and F = diag (F i ) are ( N N(K + K )) block-diagonal marices; (v) e i is he i h column of he I N ; he (N N) ideniy marix, and e ij = vecl(( ij ) e i e 0 j ); so ha e ii is a vecor of zeros, for all i = ; :::; N; E N is he N 2 N marix; wih columns e i e i and L N is he (N 2 2N(N )) marix wih columns (e i e j ) + (e j e i ), ordered by (i; j) according o s C N ; (vi) A = I N + = 0 A ; (N N) ; and, (vii) P = I N + I N = P 0 ; N 2 N 2 : hen we have he following resul: Proosiion 3 Under he Assumions of Proosiion 6, in he Aendix 0

11 (i) J (^!) J 0 = o (); where J (!) = Z0 ( A I ) Z Z 0 (EN 0 P L N ) (L 0 N P E N 0 )Z 2L0 N L N F 0 I F (5) and boh J0 and J (!) are osiive de nie. (ii) B (^!) B 0 = o (); where, in he case of m J (^!); B (!) can be exressed in (verically-sacked) block-row form, as follows B (!) = B (!) 0 ; B 2 (!) 0 ; B 22 (!) 0 ; B 3 (!) 0 ; :::; B N;N (!) 0 ; B NN (!) 0 0 (6) and he B ij (!) are ordered by (i; j) according o sj N wih B ij (!) = 2 ijrij(e 0 0 j Z j + e 0 i Z i ); Rij(e 0 0 ij ) : (7) We exress B (!) in his way since i hen become ransaren how i is modi ed if, for a aricular (i; j) ; m ij; (!) is removed from m J (!): For examle, he modi caion for mc (^!) simly removes all B ii (^!) blocks from B (!). Using hese exressions, and since ^U 0 ^U = o (); we obain he following consisen robus esimaor ^V (r) = ^A(r) ^U 0 ^U ^A(r)0 ; (8) where ^A (r) = I q ; B (^!)J (^!) : ess based on (8) will be referred as he robus FQMLE ess and denoed as (r) ^S. De ning ^W (r) = ^U ^A(r)0 ; and noing ha ^W (r)0 = m (^!); ^S (r) = 0 ^W (r) ^W (r)0 ^W (r) ^W (r)0 : (9) 3.2 ess based on PQMLE As discussed in secion 2., he PQML esimaors of he i are ~ i = arg max i P = l i( i ), i.e., he consisen univariae GARCH QML esimaors; and he consan correlaions are esimaed as ~ ij = P ~ 0 = ~ i j; a funcion of ~ ; wih ~ ii : herefore, he CCM es indicaor m C (!), evaluaed a ~! = ( ~ 0 ; ~ 0 ) 0 ; is consruced from (q ij ) sub-vecors, wih (i; j) ordered according o s C N ; X ( ~ ~ i j ~ ij )~r ij; = = = X ( ~ ~ i j ~ ij ) (~r ij; r ij (~!)) = X ( ~ ~ i j 0;ij ) (~r ij; r ij (~!)) ; =

12 where r ij (~!) P = ~r ij;: Noe ha we reain he noaion r ij (!) and m C (!); for examle, since i is consisen wih he FQMLE case; however boh r ij (~!) and m C (~!) are sricly seaking a funcion of ~. his formulaion considerably simli es he derivaion of he limi disribuion of m C (~!): In view of his, and also o mainain simliciy in he consrucion of he various es saisics, we also emloy de-meaned es variables, ~r ij; r ij (~!); for he FCM es indicaor, m J (~!): hus, he (PQMLE) FCM and CCM es indicaors, resecively, are consruced from he following (q ij ) sub-vecors, wih (i; j) ordered according o s J N and sc N ; resecively, X ( ~ ~ i j ~ ij ) (~r ij; r ij (~!)) ; i > j; (20) = X ( ~ ~ i j ~ ij ) (~r ij; r ij (~!)) ; i > j: (2) = However, noe ha P = (~ 2 i ) 6= 0; so ha P = (~ 2 i ) (~r ii; r ii (~!)) 6= P = (~ 2 i )~r ij; : hus, in general, we consider he es indicaor m (~!) = P = m (~!) where m (~!); (q ) ; is consruced eiher from he (q ij ) sub-vecors (20), for he FCM es indicaor m J (~!), or (2) for he CCM es indicaor m C (~!): In order o rea eiher es indicaor, le n (!) = P = n (!) consruced in he same way from he (q ij ) sub-vecors n ij; (!) = = X ( i j 0;ij ) rij; ij (! 0 ) = X n ij;(! 0 ); = where ij (! 0 ) = E rij; and ij (!!=! 0 ) < ; by Assumion B4 in he Aendix. 0 he following jusi es an asymoically valid 2 es saisic, and rocedure, based on m (~!) : Proosiion 4 Suose, as described in he Aendix, Assumions A and B, wih B and B2, aroriaely srenghened for he aricular choice of r ij; ; hold. hen, = E [u (! 0 )u (! 0 ) 0 ] is nie, where u (!) 0 = (m (!) 0 ; g () 0 ) : Furhermore: (i) u (!) d! N(0; ), where = E [u (! 0 )u (! 0 ) 0 ] is nie; and, (ii) P = u (~!)u (~!) 0 = o (); for any ~!! 0 = o (); where u (!) 0 = (m (!) 0 ; g () 0 ) : Remark 2 Again, for he choice of es variables r ij; = "i; "j; we will require E j" 0i j 8 < : hi; hj; In he case of m J (! 0); for examle, nn E [n (! 0 )n (! 0 ) 0 ] is block ariioned in a similar way o mm ; given in Remark, bu wih r ij; ij (! 0 ) relacing rij; ; hroughou. he modi caion is obvious for m C (! 0) which simly removes all i = j enries from nn : Some secials cases emerge, however, for eiher m J (! 0) or m C (! 0) : 2

13 . If r is he same vecor of es variables emloyed for all i; j; so ha ij ; hen nn = E [ r (! 0 )r (! 0 ) 0 ] ; where r (! 0 ) = r (! 0 ) (! 0 ): 2. If r ij; is a scalar, le = r ij; (! 0 ) ij (! 0 ) ; (N N) and de ne r (! 0 ) o be eiher vech( ); in he case of m J (! 0); or vecl( ); in he case of m C (! 0): hen nn = E [r (! 0 )r (! 0 ) ] : Proosiion 5 Under he assumions of Proosiion 4, and rovided is osiive de nie, m (~!) N (0; V ) ; where V = A 0 A 0 0; A 0 = I q ; B 0 J0 h i h ; wih J0 = E ; B 0 = E ; and I q is he (q q) ideniy 0 0 In he case of m J (~!); for examle, B 0 is block ariioned as B 0 = [B 0;ij ] ; wih blocks B 0;ij, sacked (verically) for i j; i changing faser han j; and given by d! B 0;ij E E ij; (! 0 0 F = E r ij; ij (! 0 i 0!=! 0 : he modi caion is obvious for m C (~!) and simly removes all B 0;ii blocks from B 0 : From Proosiion 5, and rovided V is osiive de nie, he general form of he es saisic is ~S = m (~!) 0 n ~ V o m (~!); (22) which has a 2 q limiing disribuion, under he null, where ~ V = V + o (): Noe ha o obain he righ exression for he asymoic variance esimaor of he es indicaor one has o use vech( ~ ~ 0 ~ ) or vecl( ~ ~ 0 ~ )), raher han vech( ~ ~ 0 ) or vecl( ~ ~ 0 ); in he consrucion of m J (~!) or mc (~!) as given in (20) and (2), resecively. As earlier, ~ V (o) and ~ V (r) and robus variance-covariance marix esimaors, resecively, ha we migh use for ~ V : will denoe OPG 3.2. he OPG-PQMLE es De ne he ( q) marix M M(!) o have rows m 0 (!) wih he undersanding ha ~ M M (~!) : Also, le G = (G ; ; G N ) and G = (G ; ; G N ) ; where he G i G( i ) and G i G i (!) are ( K + K ) marices, i = ; ; N; wih i( i 0 i are ( N (K + K )) marices wih rows g() 0 = ( 0 ; N( N 0 N Firsly, and in general, U ~ 0 U ~ = o () by Proosiion 4(ii), where U ~ = 0 ; resecively. hen, G and G i ) (!) ; resecively. i 0 ~M; G ~ : Secondly, wih he addiional normaliy assumion of 0 N (0; I N ) ; he seci caion of he log-likelihood for he FQML esimaion of arameers is correc. hus, since E [g ( 0 )] = 0; he generalized IM equaliy ha is, G i is he marix having rows univariae GARCH scores; and is a funcion of i ; while he rows of G i conains he FQMLE scores; corresonding o he condiional mean and volailiy arameers, i, only, bu is a funcion of!: 3

14 imlies ha J 0 = diag E 2 l i i@ 0 i i = diag i ( i0 (! 0 0 i ; whils he blocks of B 0 are B 0;ij = E r ij; ij (! 0 i 0 = E n ij;(! 0 0 (! 0 0 : I hen follows ha, from Lemma, 6 and 7, in he Aendix, consisen esimaors J 0 and B 0 can be obained as diag G ~ 0 ~ i G i and M ~ 0 G ~ ; resecively. herefore, under normaliy, a consisen esimaor for V = AA 0 can be obained as ~V (o) = ~ A (o) ~ U 0 ~ U ~ A (o)0 ; (23) where he marix ~ A (o) = h I q ; i M ~ 0 G ~ diag ~G 0 ~ i G i. Again, exloiing he fac ha G ~ 0 0 so ha ~ A (o) ~ U 0 ~ M 0 = m (~!); he es saisic can be exressed in a R 2 u form, bu his ime following a regression of on ~ W (o) = ~ U ~ A (o)0 : ~S (o) = 0 ~ W (o) ~ W (o)0 ~ W (o) ~W (o)0 : (24) he Robus-PQMLE es o consruc a robus (o non-normaliy) es of (22), rs noe ha B (~!) (he robus esimaor for B 0 ) can be obained using he resuls of Proosiion 3, bu relacing ^r ij; by ~r ij; r ij (~!); he demeaned es variables. hus, if e R ij is he ( q ij ) marix having rows (~r ij; r ij (~!)) 0 ; = ; :::; ; hen B (~!) can be exressed in block form, bu wih a yical block now being B ij (~!) = 2 ~0 ij e R ij (e 0 j ~ Z j + e 0 i ~ Z i ); (25) ordered by (i; j) according o s J N ; or sc N ; for mj (~!); or mc (~!); resecively. In addiion, and as a h secial case of Proosiion 3(i) wih N = ; ii =, E 2 l i (0) can be consisenly esimaed i@ 0 i J i ( ~ i ) = 2 ~ Z 0 i ~ Z i + ~ F 0 i ~ F i ; which is osiive de nie, so ha J ( ~ ) = 2 ~ Z 0 ~ Z + ~ F 0 ~ F ; see, for examle, Halunga and Orme (2009, Lemma ). Combining he above wo resuls, we obain he following exression for he robus consisen variance esimaor where ~ A (r) = ~V (r) = ~ A (r) ~ U 0 ~ U ~ A (r)0 ; (26) h i I q ; B (~!)J (~ ). ess based on his esimaor will be referred as he robus PQMLE 4

15 es and will be denoed as S ~ (r) and can be consruced as ~S (r) = 0 ~ W (r) ~ W (r)0 ~ W (r) ~W (r)0 ; (27) where ~ W (r) = ~ U ~ A (r)0 : 3.3 Summary For each of he FCM and CCM es saisics, and deending on he esimaion framework and consrucion of he variance-covariance marix, we have a oal of eigh es saisics, namely, able : FCM and CCM es Saisics n o OPG-FQMLE CCM es: ^SC(o) = m C ^V (^!)0 C(o) m C (^!); see eqn (4) n o Robus-FQMLE CCM es: ^SC(r) = m C ^V (^!)0 C(r) m C (^!); see eqn (9) n o OPG-PQMLE CCM es: S ~ C(o) = m C ~V (~!)0 C(o) m C (~!); see eqn (24) n o Robus-PQMLE CCM es: S ~ C(r) = m C ~V (~!)0 C(r) m C (~!); see eqn (27) n o OPG-FQMLE FCM es: ^SJ(o) = m J ^V (^!)0 J(o) m J (^!); see eqn (4) n o Robus-FQMLE FCM es: ^SJ(r) = m J ^V (^!)0 J(r) m J (^!); see eqn (9) n o OPG-PQMLE FCM es: S ~ J(o) = m J ~V (~!)0 J(o) m J (~!); see eqn (24) n o Robus-PQMLE FCM es: S ~ J(r) = m J ~V (~!)0 J(r) m J (~!); see eqn (27) where he ha and ilde reresen FQMLE and PQMLE, resecively, he suerscri J and C denoe FCM and CCM es, resecively, and he suerscri o and r signify OPG and robus variance esimaor. 4 OPG and Robus se LM ess From (), i was noed ha m LM kl; (!) = (0 kl I q kl ) m J (kl) (!); (q kl ), where m J (kl) (!) = P = J kl; ; ( q kl 2 N(N + ) ); is a join FCM es indicaor vecor. Whils es variables kl;, (q kl ) ; are used for he FQMLE case, for he PQMLE case he sraegy of using de-meaned es variables, kl; kl; ; is mainained in order o consruc a modi ed se es. As a consequence of Proosiion 2 i is clear ha an asymoically valid 2 se saisic designed o es only he (k; l) h equaions for consan correlaion, using a vecor or es variables ^ ij; ; can be consruced as \ LM kl; = ^ 0 2 kl I qij m J (kl) (^!) ^ 0 kl I qij ^V J (kl) (^ kl I qkl ) ; 5

16 where eiher ^V J(o) (kl) or ^V J(r) (kl) can be emloyed, wih he laer roviding robusness o non-normaliy. Similar maniulaions can be used o consruc join se es saisics, bu he following (equivalen) rocedures see more sraighforward. Obain he ( q C ) marix M M(!); o have rows m LM (!) 0 ; where here q C = P k>l q kl: ha is, m LM (!) is formed by sacking he (q kl ) sub-vecors " k " l kl kl; = ( 0 kl I q kl ) ( J kl; ); for he FQMLE case, or " k " l kl ( kl; kl; ) = ( 0 kl I q kl ) ( J ( kl; kl; )); for he PQMLE case, wih (k; l) ordered according o s C N. hen, wih his de niion of M and mlm = P = mlm (!); he desired es saisic can be consruced as follows (wih he rs being jus se s original saisic):. he OPG-FQMLE se es saisic, d LM (o). Consruc ^V LM(o) using (3) and ^S (o) as in (4) giving an equivalen exression o (6) as dlm (o) = m LM n o (^!) 0 LM(o) ^V m LM (^!): (28) 2. he Robus-FQMLE se es saisic, d LM (r). Obain ^A (r) LM = I q ; D (^!)J (^!), where he marix D (^!) is consruced by verically sacking he marices Dkl; (^!) = ^0 kl I qkl B (kl) (^!); and B(kl) (^!) is de ned in Proosiion 3, bu where ^R ij is relaced ^R (kl) ; ( q kl ) ; having rows ^ 0 kl;: hen consruc and ^S LM(r) as in (9) giving ^V LM(r) using (8) dlm (r) = m LM n o (^!) 0 LM(r) ^V m LM (^!): (29) 3. he OPG-PQMLE se es saisic, g LM (o). Consruc ~ V LM(o) using (23) and ~ S (o) as in (24) giving glm (o) = m LM n o (~!) 0 V ~ LM(o) m LM (~!): (30) 4. he Robus-FQMLE se es saisic, LM g (r). Obain A ~ h i (r) LM = I q ; D (~!)J (~ ), where he marix D (~!) is consruced by verically sacking he marices D kl; (~!) = ( 0 kl I q kl ) B (kl) (~!); and B (kl) (~!) is de ned by (25), bu where e R ij is relaced by e R (kl) ; ( q kl ) ; having rows ( ~ kl; and ^S LM(r) as in (27) giving e kl; ): hen consruc ~ V LM(r) using (26) glm (r) = m LM n o (~!) 0 V ~ LM(r) m LM (~!): (3) he above derivaions also make i ransaren how o consruc a join se es of a subse of he 6

17 consan condiional correlaions, raher han for all 2 N(N ): However, if all 2N(N ) consan condiional correlaions are o be esed, he derivaions in he roof of Proosiion 3 (see Shada and Orme, 205) imly ha D (!) and D (!) can exressed as D (!) = 0 (L 0 N P E N I )Z; 2 0 (L 0 N 4 D (!) = 0 (L 0 N P E N I )Z ; 4 L N ) ; where = diag( kl ); = diag( kl ); 2 N(N ) qc wih kl ; ( q kl ) having rows kl; ; = ; :::; ; whils 2 N(N ) qc wih kl ; ( q kl ) having rows kl; kl; ; = ; :::; : 5 Mone Carlo Evidence In his secion, we resen Mone Carlo evidence on he nie samle behaviour of he 2, for boh FQMLE and PQMLE rocedures, for N = 5 equaions: he 8 CM ess described in able and he 4 se LM ess described in (28)-(3). he arameer values for he null and alernaive Daa Generaing Processes (DGPs), where ossible, are aken from he exising lieraure (e.g., Engle and Ng (993), se (2000), Lundbergh and eräsvira (2002), Halunga and Orme (2009)). For each exerimen, wo series of 200 and 700 daa realizaions were generaed wih he rs 200 observaions being discarded o avoid iniializaion e ecs, yielding samle sizes of = 000 and 500; resecively. Each model is relicaed and esimaed, 0; 000 imes (o obain emirical signi cance levels) and 2000 imes (for robusness o non-normaliy and ower exerimens). In racice, however, here is likely o be considerable uncerainy abou he recise form of misseci caion in he MGARCH CCC srucure and so ha any seleced alernaive may ofen be misseci ed, leading o an incorrec se of es variables. hus, he rimary urose of he Mone Carlo sudy, here, is o comare he nie samle erformance (emirical signi cance levels, robusness and ower) of he various ess, each consruced wih a common se of es variables, in order o see if a ranking emerges. Following se (2000), he common scalar es variable emloyed for his urose for is r ij; (!) = i; j;, alhough in a demeaned form following PQMLE. 2 All simulaion exerimens are conduced in GAUSS rogramming language. 2 Alhough no reored here, simulaions were also carried ou using i; 2 j; 2 as es variables yielding qualiaively similar resuls. hese are available from he auhors uon reques. 7

18 5. Emirical Signi cance Levels We emloy AR()-CCC-GARCH (,) DGP for N = 5 as our null model; viz., y i = ' i0 + ' i y i; +" i ; i = ; ; 5 V ar (" jf ) = H ) E " 2 ijf = hi ; " = H =2 (!) ; N(0; I); h i; = i0 + i " 2 i; + i h i; ; H = D D ; D = diag hi and = ij ; i; j = ; ; 5 wih ii = : (32) hree exerimens are considered E, E2 and E3 (and he rue arameer vecors emloyed are given in able A of Shada and Orme (205)). hese rovide models wih relaively low (ranging beween 0.20 and 0.37), mixed (ranging beween 0.30 and 0.80) and high (ranging beween 0.62 and 0.80) correlaion srucure, resecively, for. For all hree DGPs he same rue arameer values for ' 0 i and 0 i = ( i0; i ; i ) are used: Noe ha he exerimens considered various degrees of volailiy ersisence; however, o save sace, we reor he resuls only for + = 0:85 since he resuls are qualiaively similar in oher cases. se (2000) also reors correlaions seem o lay a role in deermining he rae of convergence o he nominal size. Models wih low correlaions are less subjec o over-rejecion in small samles...he ersisence of he condiional variance does no have much e ec (. 5). able 2 reors he rejecion frequencies when he null of he CCC is rue under boh Gaussian and non-gaussian errors. Aar from invesigaing he robusness of hese ess under non-normaliy, where he elemens of 0 are iid as (6); his also o ers some evidence on he robusness of he rocedure o violaions of he underlying momen assumions, since for his choice of es variables 8 h order momens are required. he resuls are reored for a nominal signi cance level of 5%: and Firs, under Gaussian errors he original se es and oher OPG-FQMLE ye ess ( LM d (o) C(o) ; ^S J(o) ^S ) end o over-rejec, for all DGPs, even wih = 000 (aricularly versions ( LM d (r) ; ~S C(o) ^S C(r) and J(o) ^S ). he robus J(r) ^S ) are much suerior. Ineresingly, he OPG-PQMLE ess ( LM g (o) ; and S ~ J(o) ) erform beer han he corresonding OPG-FQMLE ess, alhough, S ~ C(o) and S ~ J(o) are sill oversized. However, he emirical signi cance levels of heir robus counerars, boh FQMLE and PQMLE and including d LM (r) and LM g (r) ; are reasonably close o he nominal size of 5%, even when = 500. Second, in he case of exerimens wih mixed and high correlaion srucure (E2 and E3), he size disorions of OPG-FQMLE ess are relaively higher comared o he low correlaion srucure whils he robus version of hese saisics aears o correc his size disorion. On he oher hand, he rejecion raes for OPG-PQMLE ess under low correlaion srucure (E), in aricular for ~ S C(o) 8

19 and S ~ J(o) ; are higher han for E2 and E3, alhough heir robus version again correcs his deformiy. he nding from hese exerimens ha size erformance deends on correlaion is in line wih ha of se (2000) where he Mone Carlo exerimens were erformed for N = 2. hird, wih 0 (6) all OPG-FQMLE ess ( LM d (o), ^SC(o) and J(o) ^S ) over-rejec under all correlaion srucures, bu his disorion is more severe in high and mixed correlaion models. In aricular, se s original LM es ( LM d (o) ) is very sensiive o dearures from normaliy. he robus-fqmle version of all ess reduces he over-rejecion rae subsanially. he emirical signi cance levels of he robus versions of se s es (aricularly LM g (r) C(r) ; as for he case wih normal errors) and he robus CCM ess ( ^S and S ~ C(r) ), J(r) in general, are close o nominal level of 5% while he robus FCM ess ( ^S under-rejec. and S ~ J(r) ) can over or In summary: he OPG-FCM ess over-rejec; all es saisics erform beer in low correlaion exerimens; in general, he robus versions of ess erform beer han he OPG; and, in aricular,se s modi ed robus PQMLE es and he robus CCM PQMLE ess (i.e., g LM (r) and S ~ C(r) ) rovide quie reliable signi cance levels. All robus ess rovide signi can size correcion under non-normal errors. 5.2 Robusness o Misseci ed Univariae Volailiy In oal,we consider 2 exerimens (Ma-Mc, M2a-M2c, M3a-M3c and M4a-M4c), each wihin he regression conex o invesigae, via Mone Carlo simulaion, he imac of violaions in he univariae GARCH seci caion, bu when he rue correlaion srucure for 0 is consan wih Gaussian error. he condiional mean arameers and he correlaion srucures remain he same as hose reviously emloyed, as deailed in able A of Shada and Orme (205). For M, M2 and M3 he univariae volailiy seci caions of all ve variables are governed by he GJR, higher order GARCH (i.e., GARCH(2,2)) and he EGARCH models, resecively whereas for M4 all 5 variables are subjec o volailiy sillover via an ECCC model. he su x a, b or c associaed wih hese exerimens indicae low, mixed and high correlaion srucure, resecively, for. Seci cally, we emloy he following DGPs, for i = ; :::; 5 :. M (GJR): h i = a i0 + b i [j" i j b i2 " i ] 2 + b i3 h i ; wih arameer vecors, for each i; (0:005; 0:23; 0:23; 0:70) ; (0:005; 0:30; 0:7; 0:70) ; (0:005; 0:25; 0:20; 0:70) ; (0:005; 0:28; 0:5; 0:70) ; and (0:005; 0:20; 0:23; 0:70) : 2. M2 (GARCH(2; 2)): h i = a i0 + a i " 2 i; + a i2" 2 i; 2 + b ih i; + b i2 h i; 2 ; wih arameer vecors (0:0; 0:5; 0:05; 0:60; 0:5) ; (0:02; 0:25; 0:05; 0:50; 0:5) ; (0:5; 0:0; 0:05; 0:70; 0:0) ; (0:05; 0:0; 0:0; 0:70; 0:4) ; and (0:05; 0:20; 0:05; 0:65; 0:05) : 3. M3 (EGARCH): log (h i; ) = a i0 + b i log (h i; ) + b i2 i b i3 i ; wih arameer vecors, for each i; ( 0:23; 0:90; 0:25; 0:30) ; ( 0:20; 0:70; 0:25; 0:20) ; (0:23; 0:60; 0:25; 0:20) ; (0:20; 0:80; 0:28; 0:5) ; and ( 0:30; 0:90; 0:40; 0:5) : 9

20 4. M4 (ECCC): h i = i0 + i " 2 i; + iih i; + P j6=i ijh j; ; wih sillover arameer vecors, for each i; (0:0; 0:02; 0:05; 0:03) ; (0:02; 0:06; 0:03; 0:04) ; (0:03; 0:0; 0:025; 0:05) ; (0:05; 0:03; 0:02; 0:0) ; and (0:002; 0:035; 0:04; 0:02) : he GARCH arameers for he ECCC model remain he same as AR()-CCC-GARCH (,) null model. Alhough EGARCH and ECCC models are no formally wihin GARCH family of alernaives, as he oher DGPs considered here, hey reresen alernaive misseci caions of volailiy no caured by GJR and GARCH(2,2). In order o conserve sace, we reor in able 3 only he resuls for exerimens M (GJR) and M3 (EGARCH), since he resuls for M2 (GARCH(2; 2)) and M4 (ECCC) are qualiaively similar o M (GJR), bu summarise he main ndings for all exerimens. Full resuls are rovided in Shada and Orme (205). Rejecion frequencies are based on boh he 5% emirical and nominal criical values (wih he laer in he arenhesis) and wih 2000 relicaions where he daa are generaed wih normal errors; i.e., in he former case, and for each es rocedure, sizeadjused rejecion frequencies are reored, calculaed using he emirical criical value ha delivers a 5% signi cance level for he simulaions reored in Secion 5.. All robus ess and PQMLE-OPG ess are relaively insensiive o GJR, GARCH(2,2) and ECCC volailiy sillover DGPs and for all correlaion srucures; exce join ess ~ S J(r) hand, he FQMLE-OPG versions, aricularly and ~ S J(o) under he GARCH(2; 2) DGP. On he oher J(o) ^S ; over-rejec he null of CCC and he over-rejecion is subsanially higher when we use he nominal signi cance level. Since he FCM es indicaor enails he volailiy momen condiion, hese ess dislay some ower when his momen condiion is violaed. In case of he EGARCH alernaive, all ess, alhough o a much lesser exen d LM (r) and LM g (r) ; are quie sensiive o he volailiy missec caions embodied in M3b and M3c (i.e., wih mixed and high correlaion). In hese cases, all ess over-rejec signi canly he null of CCC and he rejecion raes are similar for boh emirical and nominal signi cance level. However, for M3a (low correlaion), all he robus ess are less sensiive o univariae condiional variance misseci caion. 5.3 Power Resuls o examine ower, we consider hree yes of MGARCH models wih ime varying correlaions. he AR() condiional mean seci caion, and arameers, remain as in (32) bu now we examine hree alernaive seci caions for he condiional variance marix H = V ar (" jf ). he rs is Engle s (2002) DCC-GARCH(,) model where he dynamic correlaion marix, ; is given as = (I ) =2 (I ) =2 = diag( ) =2 diag( ) =2 ; = ( ~ ~ ) + e 0 + e ; (33) 20

21 where ~ and ~ are nonnegaive scalar arameers and + < and is consan (ime invarian) 5 5 symmeric osiive de nie marix, wih ones on he diagonal. Secondly, we consider he following Varying Correlaion (VC) model of se and sui (2002) = ( a b) + a + b ; (34) where a and b are nonnegaive scalar arameers, saisfying a + b, and correlaion marix of ; ; 5 and is (i; j) h elemen is given by: is he 5 5 samle ij; = P5 P 5 m= i; m j; m =2 P5 =2 : m= 2 i; m m= 2 j; m Finally we consider he BEKK model of Engle and Kroner (995), H = C B + A 0 B " " 0 AB + B 0 BH B B : (35) In he following exerimens he diagonal BEKK (DBEKK) model is emloyed where he arameer marices A B and B B are 5 5 diagonal marices. Seven exerimens are considered: P, P2 and P3 follow he DCC DGP (33), P4 and P5 follow VC DGP (34) and remaining wo, P6 and P7, follow he DBEKK DGP (35). In all cases, he individual volailiy seci caion for all variables is reained from earlier size exerimen, whils for he DCC and VC DGPs he consan marix is se o he reviously de ned mixed correlaion srucure (see Secion 5.). he remaining rue arameer vecors are given in Shada and Orme (205). Again, o conserve sace, we only reor deailed resuls for he DCC DGP, P-P3, bu summarise he main ndings for all exerimens. Full resuls are rovided in Shada and Orme (205). able 4 resens he size-adjused ower (and nominal) resuls wih 2000 relicaions, based on a 5% emirical (resecively nominal) criical values and he daa are generaed assuming normaliy. As a measure of he variabiliy of he condiional correlaion coe ciens, in exerimens P o P7, we also reor in Shada and Orme (205) he average, maximum and minimum values of he rue condiional correlaion coe ciens across he 2000 Mone Carlo relicaions of each = 000 samle. When he rue DGP is he DCC, P3 has he larges variabiliy in correlaions followed by P2 and P; i.e., variabiliy increases as e increases and e decreases. In general, he FCM ess are found o have higher ower in all hree DCC exerimens. However, as he variabiliy in correlaion decreases ower decreases. he se and CCM ess also exhibi good ower roeries: even wih = 500, and all ess have high ower esecially for he P2 and P3 DGPs. In case of he VC and BEKK DGPs he conclusions are quie similar. P5 and P7 have larger variabiliy in correlaions han P4 and P6, resecively, and he erformance of all he ess re ec ha. 2

22 Alhough he OPG-FQMLE ess exhibi higher nominal ower, in erms of he size-adjused ower he robus-fqmle and robus-pqmle versions of hese do no cos much in his resec, esecially in view of he lack of robusness o non-normaliy of he former. 6 Concluding Remarks In his aer, we have considered a se of asymoically valid Condiional Momen (CM) ess designed o assess a consan correlaion assumion and/or he individual GARCH seci caions in a MGARCH model. In aricular, we consider boh he FQMLE and PQMLE framework for he CCC model, noing ha here is very lile in he exising lieraure for he laer case. hese ess are very easy o imlemen and include OPG versions - a oular varian in he alied lieraure bu whose asymoic validiy is based on an assumion of normaliy - and non-normaliy robus versions. In so doing, we also rovide a simle exression for a consisen esimaor for he hessian in he FQMLE framework. Our aroach accommodaes se s (2000) LM es, originally roosed as a OPG-FQMLE ye es, so ha we are able o rovide he PQMLE and robus version of his oular es, as well. We examine he nie samle erformance of hese asymoically valid ess via a small Mone Carlo sudy, wih N = 5 ime series raher han he usual bivariae model, which indicaes ha, in general, all ess have emirical signi cance levels ha are reasonably close o he nominal value of 5% bu ha he robus versions are slighly referred, even under normaliy. I also aears ha, under he null, whils he degree of univariae volailiy ersisence has lile derimenal e ec, low correlaion is associaed wih beer emirical signi cance levels. As aniciaed, hough, under non-normaliy (bu oherwise correc model seci caion), he robus version of any aricular es exhibis far suerior nie samle behaviour relaive o is OPG varian wih all OPG-FQMLE ess over-rejecing. Ineresingly, he OPG-PQMLE based ess exhibi more robusness han he corresonding OPG-FQMLE ess. When he GARCH error assumion of he null model is violaed by inroducing a volailiy sillover e ec he Mone Carlo evidence suggess ha here is lile imac on emirical signi cance levels. When here is no volailiy sillover bu simly one GARCH equaion misseci ed, and a high correlaion srucure, all ess exerience increased emirical rejecion raes. his is esecially rue in he case of he EGARCH alernaive wih he FCM ess (which es joinly he individual volailiy seci caions and he CCC assumion) being mos sensiive, as one migh exec. However, an imoran resul ha emerges for alied workers is ha alhough se s original FQMLE es is also a eced, as i emloys all he indicaors of he FCM ess, he modi ed and robus PQMLE version develoed in his aer aears o be much less sensiive o univariae volailiy misseci caion. urning o ower, which deends on he variabiliy of he rue correlaion arameer, i is found ha se s es and FCM ess have good ower, wih he former being slighly more owerful, even in models 22