TESTS FOR VOLATILITY SHIFTS IN GARCH AGAINST LONG-RANGE DEPENDENCE

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1 JURNAL F TIME SERIES ANALYSIS Published olie 4 ctober 4 i Wiley lie Library (wileyolielibrary.com) DI:./jtsa.98 RIGINAL ARTICLE TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LNG-RANGE DEPENDENCE a TAEWK LEE, a MSUP KIM b AND CHANGRYNG BAEK c Deartmet of Statistics, Hau Uiversity of Foreig Studies, Yogi, Kyuggi-do, Korea b Deartmet of Statistics, Seoul Natioal Uiversity, Seoul, Korea c Deartmet of Statistics, Sugyuwa Uiversity, Seoul, Korea May emirical fidigs show that volatility i fiacial time series exhibits high ersistece. Some researchers argue that such ersistecy is due to volatility shifts i the maret, while others believe that this is a atural fluctuatio exlaied by statioary log-rage deedece models. These two aroaches cofuse may ractitioers, ad forecasts for future volatility are dramatically differet deedig o which models to use. I this article, therefore, we cosider a statistical testig rocedure to distiguish volatility shifts i geeralized AR coditioal heteroscedasticity (GARCH) model agaist log-rage deedece. ur testig rocedure is based o the residual-based cumulative sum test, which is desiged to correct the size distortio observed for GARCH models. We examie the validity of our method by rovidig asymtotic distributios of test statistic. Also, Mote Carlo simulatios study shows that our roosed method achieves a good size while rovidig a reasoable ower agaist log-rage deedece. It is also observed that our test is robust to the missecified GARCH models. Received 3 August 3; Revised May 4; Acceted 3 August 4 Keywords: Volatility shifts; log-rage deedece; GARCH models; chage oit aalysis. JEL. 6M; 6G; 6G8.. INTRDUCTIN Sice the semial articles of Egle (98) ad Bollerslev (986), geeralized AR coditioal heteroscedasticity (GARCH, i short) models have bee a oular tool to aalyse the stylized features o the volatility of fiacial time series. I the alicatios of GARCH models, however, the arameter estimatio frequetly idicates high ersistece. That is, the sum of the estimated coefficiets of the squared lagged returs ad the lagged coditioal variace terms i GARCH models becomes closer to for larger samle sizes. To icororate such observatios ito a model, the itegrated GARCH (IGARCH, i short) model was itroduced by Egle ad Bollerslev (986). Ufortuately, the ractitioers ad researchers have bee doubtful that IGARCH models have a ractical use for modellig ad log-term forecastig of volatility i fiacial alicatios. This is maily because IGARCH modellig of fiacial time series has bee reorted to be surious whe volatility shifts or model missecificatios are reset i GARCH models. See, for examle, Miosch ad Stărică (4), Beres et al. (5), Hillebrad (5), Jese ad Lage () ad refereces therei. A alterative way to icororate high ersistece is to use log-rage deedet modellig of time series for suitably trasformed data, for examle, absolute retur or squared retur. Amog may other literatures, for istace, Dig et al. (993) ad Baillie et al. (996) documet that log-rage deedece (LRD, i short) successfully catures the stylized facts of fiacial time series volatility. Here, LRD for series D¹ º Z refers to a secod-order statioary time series with a slowly decayig autocovariace fuctio for large lags as Corresodece to: Chagryog Bae, Deartmet of Statistics, Sugyuwa Uiversity, 5-, Sugyuwa-ro, Jogo-gu, Seoul, Korea crbae@su.edu

2 8 T. LEE, M. KIM AND C. BAEK.h/ D Cov. ; h / Ch d ; as h!; () where C>is a costat ad d.; =/ is the LRD arameter. bserve that uder coditio (), autocovariaces of LRD series are ot absolutely summable, P hd j.h/j DC.theotherhad,whethesum of autocovariaces is absolutely summable, that is, P hd j.h/j <, it is referred to as short-rage deedece (SRD, i short). However, it has a log history ad debates over the use of LRD modellig because of the o-statioary-lie features of LRD. I short, LRD series has slowly decayig ositive autocovariace fuctios, so a statioary LRD series exhibits aeriodic local shifts for fiite samles. It meas that volatility shifts ca roduce surious LRD (e.g. Klemeš, 974; Teverovsy, 999; Diebold ad Ioue, ; Miosch ad Stărică, 4 ad refereces therei). Therefore, rior to modellig the volatility of fiacial time series, it is crucial to mae a decisio o the selectio of the true data geeratig rocess amog IGARCH, volatility shifts ad LRD. For volatility shifts, i articular, we oly cosider GARCH models o subsamles, also ow as chagig-arameter GARCH models for ractical modellig ersectives. To be more secific, our test cosiders two hyotheses: H.R/ W VS Rmodel agaist H W LRDmodels; where VS-R reresets that volatility shifts have occurred at uow R locatios; hece, each GARCH model is fitted for (RC) regimes. Also, ote that the VS-R model is o-statioary, while LRD is secod-order statioary; hece, its imlicatios o the log-term forecastig of volatility could be dramatically differet. Therefore, their discrimiatio is a iterestig ad imortat roblem attractig may researchers for the last few decades. Sice the locatios of volatility shifts are uow, they eed to be estimated. We focus o the oular cumulative sum (CUSUM) tye of test statistic sice a exlicit test statistic is available ad very easy to calculate. For examle, Koosza ad Leius () first cosidered the CUSUM test of squared asset returs for a sigle volatility shift by adatig the idea of the CUSUM statistic i detectig mea chages with i.i.d. observatios. However, Adreou ad Ghysels () reorted that the CUSUM test of Koosza ad Leius () suffers from size distortios for fiite samles due to strog deedece o observatios. e may correct such a size distortio by cosiderig the so-called Bartlett log-ru variace estimator i the CUSUM test roosed by Beres et al. (6) ad Zhag et al. (7). However, the Bartlett log-ru variace estimator is very sesitive to the choice of erel badwidth ad cotributes to low ower agaist LRD as oited out by Teverovsy et al. (999) ad Bae ad Piiras (). To overcome such shortcomigs, we adat the so-called residual-based CUSUM test studied earlier. The residual-based CUSUM test uses the stadardized residuals from a estimated GARCH model. Such residuals mimic iovatio series i GARCH model; hece, deedece amogst observatios ca be dimiished i the residual-based CUSUM statistic. For examle, Lee et al. (4) cosidered a residual-based CUSUM test of arameter chages i regressio models with AR./ errors. Their statistic, however, eeded tuig arameters for trucatio ad cosidered test for o arameter chage oly. Kulerger ad Yu (5) studied may iterestig theoretical roerties o the high-momet artial sum of GARCH residuals ad roosed residual-based CUSUM test free from tuig arameter selectio for trucatio. However, their wor was still cofied to the test for o volatility shifts ad always trucated the first observatio. de Pooter ad va Dij (4) coducted extesive simulatios for volatility shifts ad showed that the residual-based CUSUM test erforms reasoably well amogst other methods. I our article, we study theoretical roerties of the residual-based CUSUM test uder the multile-volatility-shifts cotext. Note also that our roosed method does ot trucate ay observatios. The erformace of our roosed method is umerically evaluated for various tyes of volatility models icludig missecified models. The orgaizatio of this article is as follows. I Sectio, we itroduce our testig rocedure for o volatility shifts agaist LRD. At this stage, the IGARCH model ca also be dismissed from cosideratio. The, the testig rocedure is further examied for a sigle volatility shift i Sectio 3, ad it leads to atural geeralizatio to ow R umber of volatility shifts. The erformace of our roosed method i the simulatio study is reorted wileyolielibrary.com/joural/jtsa DI:./jtsa.98

3 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 9 i Sectio 4. I Sectio 5, we aly our roosed method to several real fiacial time series. Coclusios ad further discussios ca be foud i Sectio 6, ad roofs of theorems are rovided i Sectio 7.. TEST FR N VLATILITY SHIFTS AGAINST LRD Here, we cosider the testig rocedure of o volatility shifts o the observed volatility series ¹r t º tz agaist LRD. I may fiacial time series, volatility is measured by log-retur defied as r t D log P t log P t ; () where P t reresets the asset rice or stoc idex at time t. The, the celebrated GARCH.; q/ model for ¹r t º tz satisfies the followig relatioshi r t D t t ; t D! C q id i r ti C j D j tj ; (3) where the iovatio ¹ t º tz is a sequece of stadard, that is, E t D ad Et D, i.i.d. radom variables. It is also assumed that! >; i for all i D ; : : : ; q ad j for all j D ; : : : ;. The true arameter vector is deoted by D.! ; ;:::; q ; ;:::; /. Hereafter, the GARCH model i (3) is also deoted by VS- model for shortess s sae. For the LRD model, it is widely reorted that the retur series ¹r t º tz itself exhibits little autocorrelatios, while ower trasformatio of retur data such as ¹rt º tz exhibits roouced autocorrelatios. Therefore, we assume that ¹rt º tz is a LRD series satisfyig () for alterative hyothesis. Poular models of such tyes iclude the log-memory ARCH (LM-ARCH) model of Dig ad Grager (996), the closely related fractioally itegrated GARCH (FIGARCH) model of Baillie et al. (996) (although theoretical justificatio o log-memory roerty ad the existece of statioary solutio is ot fully aswered, see Giraitis et al., a) ad the log-memory liear ARCH (LM-LARCH) model of Giraitis et al. (b). We ca further cosider the log-memory stochastic volatility model sice it is well ow that, uder mild assumtios, the ower-trasformed series ¹jr t j º tz for ay >is agai LRD i the sese of () with LRD arameter d. WewillrefertoSurgailisadViao(), Robiso () ad refereces therei for more iterestig discussios ad geeralizatios. Now, we are iterested i testig the followig hyotheses: H./ :Theobserveddata¹r tº tz follow the VS- model. H :Theobserveddata¹r t º tz follow the LRD model. Note that GARCH.; q/ models are SRD because of the geometric ergodicity of radom differece equatios (e.g. Basra et al., ). I fact, Giraitis et al. (a) showed that all the ARCH./ models, which ecasule the class of GARCH.; q/ models ad eve hyerbolically decayig coefficiets, are SRD uder mild assumtios. It is deduced that ¹jr t j º tz ; >is also SRD, to be more recise, geometrically ergodic, sice f.x/djxj is acotiuoustrasformatio.therefore,theaforemetioedtestessetiallydistiguishessrdadlrdmodels. bserve also that the VS- model ecasules the IGARCH model, that is, P q id i C P j D j D, siceour model (3) oly assumed that arameters ¹ i º ad ¹ j º are o-egative. It idicates that our test ca be used to rule out the IGARCH model whe testig for H./ : VS-. For the test, oe may cosider the oular CUSUM test statistic based o ¹r t º tz,defiedas T D s max td r t td r t ; DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

4 3 T. LEE, M. KIM AND C. BAEK where s is a suitable estimator of the log-ru variace D P hd Cov.r ;r h /.However,theCUSUMtest for GARCH models is reorted to suffer from size distortios ad low owers i fiite samles (e.g. Lee et al., 4; de Pooter ad va Dij, 4). This is because of the widely observed fact that log-ru variace estimators such as heteroscedasticity ad autocorrelatio-cosistet estimators are sesitive to the choice of erel badwidth whe correlatios betwee observatios are strog as i the case of GARCH.; q/. Istead, we byass such deficit by taig ucorrelated observatios ito accout. e such atural cadidate is to cosider the CUSUM test based o the iovatio series ¹ t º i (3). That is, T D max t td so the log-ru variace is simly give by Var. / sice ¹ tº are a i.i.d. sequece of radom variables. However, ote that ¹ t º is uobservable ad hece eeds to be estimated. First, cosider a cosistet estimator of such that D P./; (4) td t ; where D.! ; Ǫ ;:::; Ǫ q ; ;:::; /.The,theiovatioseriesareestimatedfrom(3)by Q t WD r t Q t ; t D ; : : : ; ; (5) where Q t./ is calculated recursively from q./ D! C ir ti C id j D j j./; (6) where D.!; ;:::; q;;:::;/ is the arameter vector ad ¹r t º reresets the data at had, which is a realizatio of the statioary solutio i (3). We will also refer to (5) as the residuals i the cotext of Kulerger ad Yu (5). Fially, the CUSUM statistic based o the estimated iovatio series ¹Q t º is defied as where QT D max td td ; (7) D Q 4 t td td! is a method-of-momet estimator of Var. /. It is worth otig that ay -cosistet estimator ca be used to defie the residuals (5). I articular, we emloy the Gaussia quasi-maximum lielihood estimate (QMLE) based o the stadard ormal desity, which is show to be asymtotically ormally distributed by Fracq ad Zaoïa (4). Also ote that, for give GARCH.; q/ orders, recursio (6) requires iitial values for r ;:::;r q ad Q ;:::; Q.FollowigFra ad Zaoïa (4,. 6), iitial values could be ay costat values tae to be fixed, either radom or wileyolielibrary.com/joural/jtsa DI:./jtsa.98

5 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 3 fuctios of the arameters. I a comutatioal ersective, it is already available i the fgarch acage i R with all iitial values tae to be the samle average of r ;:::;r. Remar. Beres et al. (3) itroduced a differet way of costructig residuals from a ifiite-order movig average reresetatio of GARCH.; q/. I their method, however, it is ot ossible to obtai residuals u to first max.; q/ terms. Because (5) is comutatioally more efficiet ad simulatio results are oly little differet betwee two methods, we will use residuals defied as i (5) ad (6). So as to obtai the asymtotic roerties, we eed the followig regularity coditios. Assume that the arameter vector D.!; ;:::; q;;:::;/ belogs to the arameter sace Œc ;c CqC for some <c <c <. (A) is a iterior oit of,ad is comact. (A) The rocess ¹r t º tz is strictly statioary ad su j D j <: The, uder the ull hyothesis, we have the followig covergece result. Theorem. Assume that (4) ad coditios (A) ad (A) hold. The, uder H./,as!, where B.u/ is a stadard Browia bridge. QT d! su.u/ ; (8) u Remar. For GARCH.; / model, (A) holds if a to Lyauov exoet is egative, or equivaletly E log. t C / <. For a formal defiitio of a egative to Lyauov exoet i GARCH.; q/ model ad detailed discussio, readers are referred to (3) of Fracq ad Zaoïa (4), Giraitis et al. (7) ad refereces therei. Remar 3. To verify coditio (4), the followig three additioal coditios are assumed by Fracq ad Zaoïa (4) together with (A) ad (A). LetA. / D P q id i i ad B. / D P j D j j.covetioally,if q D ; A. / D, adif D ; B. / D. () t is a o-degeerate radom variable. () If >;A. / ad B. / have o commo root, A./, ad q C. (3) Et 4 <. We will assume that coditios () (3) hold wheever (4) is used i this article. It is also oteworthy that accordig to Fracq ad Zaoïa (7), (4) still holds whe the true arameter value is o the boudary of the arameter sace, ad thus, Theorem could be validated without the first art of Assumtio (A). 3. TEST FR A SINGLE VLATILITY SHIFT AGAINST LRD ce the hyothesis test for o volatility shifts agaist LRD is rejected, it is atural to cosider the testig rocedure o volatility shifts for ow R umber of times, although their locatios are uow, agaist LRD. Hece, the iterest of this sectio is i testig the followig hyotheses: DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

6 3 T. LEE, M. KIM AND C. BAEK H.R/ :Theobserveddata¹r t º tz follow the VS-R model. H :Theobserveddata¹r t º tz follow the LRD model. Sice the testig rocedure ca be easily exteded to multile volatility shifts, we will oly cosider R D, a sigle volatility shift, for a clear discussio of our method. Discussios o a uow umber of volatility shifts ca be foud i Sectio 6. We first itroduce the VS- model such that oe volatility shift occurs at a uow locatio by followig the framewor used by Koosza ad Leius (). Cosider two GARCH.; q/ models ¹r ;t º tz ad ¹r ;t º tz from the same iovatios, amely, 8 q r ˆ< ;t D ;t t ; ;t D! C id q ˆ: r ;t D ;t t ; ;t D! C id ;i r ;ti C j D ;i r ;ti C j D ;j ;tj ; ;j ;tj ; (9) where the iovatios ¹ t º tz are stadard i.i.d. radom variables. Assume that! m > ; m;i ; i D ; : : : ; q ad m;j ; j D ; : : : ; for m D ;. The true arameter vectors are rereseted by m D.! m ; m; ;:::; m;q ; ;:::;m; m; /; m D ;, for ¹r ;tº ad ¹r ;t º resectively. The, the VS- model refers to the model defied as r t D ² r;t ; if t ; r ;t ; if <t; where D Œ ;.; /, isauowchageoit. So as to costruct the test statistic, cosider the chage-oit estimator D argmax. td r t tdc r t A () () roosed by Koosza ad Leius (). The, for a sequece of residuals ¹Q t ;t D ; : : : ; º exlaied later, the test statistic is give by where QT ; D = max td td M D max QT ; ; QT ; ; () ; QT ; D = max < td C td C ; ad D td Q 4 td A ; D td C Q 4 td C A : wileyolielibrary.com/joural/jtsa DI:./jtsa.98

7 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 33 The residuals ¹Q t ;t D ; : : : ; º are obtaied basically by reeatig the rocedure i Sectio to two subsamles segmeted before ad after the estimated chage oit. To be more secific, let Qr ;t D r t ;t D ; : : : ;, ad Qr ;t D r t ;t D C ; : : : ;. Defie the recursio Q ;t. / D! C Q ;t. / D! C q ;i Qr ;ti C id q ;i Qr ;ti C id j D j D ;j Q ;tj. /; t D ; : : : ; ; ;j Q ;tj. /; t D C ; : : : ; for give fixed costat iitial vectors Qr ; ;:::;Qr ;q ; Qr ;:::;Qr ; Q ; ; qc ; ;:::; Q ;, ad Q ;:::; Q with arameter vectors deoted by m D.! m ; m; ;:::; m;q ;m; ;:::;m; / ;m D ; ; C ;. The,theGaussiaQMLEsfor ad are give by D argmi td Ql t. /; D argmi td C Ql t. /; (3) where is a arameter sace belogig to Œc ;c CqC for some <c <c < ad Ql t. m / D r t. m/ C log. m/;. m/ DQ ;t. m/i t CQ ;t. m/i t> ; m D ; : (4) Fially, the residuals ¹Q t ;t D ; : : : ; º are obtaied from the observatios ¹r t º by calculatig 8 ˆ< Q t D ˆ: r t ; Q ;t b t D ; : : : ; ; r t ; Q ;t b t D C ; : : : ; : (5) Remar 4. For the chage-oit estimator, we have used that of Koosza ad Leius (). I a theoretical ersective, ay chage-oit estimator of the VS- model satisfyig D P./ (6) will be sufficiet. Theorem 6 of Koosza ad Leius () verifies (6) for the chage-oit estimator i (). For the asymtotics of the test statistic () uder the VS- model, amely H./, we assume the followig regularity coditios. (B) ad are iterior oits of, which is comact. Moreover, there exist ositive costats c ad c such that Œc ;c CqC. (B) For m D ;, the rocesses ¹r m;t º tz are strictly statioary ad su m;j <: m j D DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

8 34 T. LEE, M. KIM AND C. BAEK (B3) Suose that D! P q id ;i P j D ;j! P q id ;i P j D ;j We first describe that the arameter estimators i (3) are, i fact, -cosistet estimators for the VS- model. Theorem. Suose that coditios (B) (B3) hold. The, for a chage-oit estimator satisfyig (6), the Gaussia QMLEs for ad i (3) are -cosistet estimators : m m D P./; m D ; : (7) Remar 5. We also assume that coditios () (3) i Remar 3 hold for ad. I fact, we will always assume those coditios to hold wheever (7) is used i the roofs of other lemmas ad theorems i this article. For the VS- model, we have the followig asymtotic result. Note that volatility shifts imly arameter chages, but arameter chages may ot ecessarily imly volatility shifts. The coditio (B3) rules out such a case; thus, our test is for volatility shifts due to arameter chages. Theorem 3. Suose that the coditios of Theorem hold. The, for the VS- model () uder H./,as!, µ d M! max su B.u/ ; su B.u/ ; (8) u u where B ad B are ideedet stadard Browia bridges. 4. SIMULATIN STUDY Here, we discuss the fiite-samle erformace of our roosed method through Mote Carlo simulatios. For volatility shifts, we cosider GARCH(, ) models with o-egative coefficiets satisfyig C < for the secod-order statioarity. For LRD models, we have cosidered LM-ARCH, FIGARCH, LM-LARCH ad AR fractioally itegrated movig average (FARIMA) models with the followig secificatios. LM-ARCH: The coditioal variace of LM-ARCH model is give by t N D w i it ; id N id w i D ; it D. i i/ C ir t C i it for some weights w i.wesetn D with equal weights, D ; i follows Beta(5, d/ distributio ad i D :99. i/ i simulatios. FIGARCH.; d; /: FIGARCH models coditioal variace as. L/ t D! C. L/ C. L/. L/ d r t ; wileyolielibrary.com/joural/jtsa DI:./jtsa.98

9 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 35 where L is a bacshift oerator. We set! D :6; D : ad D : i simulatios. LM-LARCH: Istead of coditioal variace, LM-LARCH models coditioal stadard deviatio by t D a C b j r tj ; j D ad we cosidered a D :3; b D. C d/=8;b j C D.j C d/=.j C /b j so that the coefficiets ¹b j º decay similar to FARIMA.; d; / coefficiets. FARIMA.; d; /: We also cosidered LRD modellig of the retur series ¹r t º tz by geeratig Gaussia FARIMA.; d ;/models. Uder Gaussiaity assumtios o ¹r t º tz, it follows from Taqqu (979) that the squared retur ¹r t º tz follows the LRD series with arameter d D d =. All results are based o relicatios of sizes, ad 5 with 5% sigificace level. Iovatio series ¹ t º tz are geerated from N.; / otherwise secified. First, emirical sizes ad owers of test for H./ : VS- agaist H : LRD are reseted i Tables I ad II. For emirical sizes reorted i Table I, we have cosidered GARCH(, ) models with two tyes of iovatios. Table I. Emirical size of test for o volatility shifts Iovatios N.; / t.5/.!; ; / D D D 5 D D D 5.:; :; :8/ :; :; :6/ :; :; :4/ :; :; :6/ :3; :; :8/ :3; :; :89/ GARCH(, ) models are cosidered. GARCH, geeralized AR coditioal heteroscedasticity. Table II. Emirical ower of testig H./ :VS-agaistH W r t is LRD d D D D 5 LM-ARCH LM-LARCH FIGARCH.; d; / FARIMA(; d; ) FARIMA, AR fractioally itegrated movig average; FIGARCH, fractioally itegrated geeralized AR coditioal heteroscedasticity; LM-ARCH, log-memory AR coditioal heteroscedasticity; LM-LARCH, log-memory liear AR coditioal heteroscedasticity. LRD, log-rage deedece. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

10 36 T. LEE, M. KIM AND C. BAEK They are i.i.d. N(,) ad t-distributio with a degree of freedom of 5, deoted by t.5/, which is stadardized to have zero mea ad uit variace. It ca be see that our method achieves excellet emirical sizes over the rage of samle sizes ad arameters. Eve for the case whe C D : C :89 D :99, o size distortio is observed. It is also observed that emirical sizes are gettig closer to the omial 5% sigificace level as samle size icreases. Eve with heavier tails realized by t.5/ distributios, there is o size distortio i our roosed method. The emirical ower of testig H./ W VS- agaist H :LRDis reorted i Table II. It is observed that our roosed method rovides a accetable ower i all cases cosidered. It is also observed that the ower aroaches as samle size icreases ad as deedece are gettig stroger. Now, we tur our attetio to test M i (8) for a sigle volatility shift. Table III shows the emirical size of test for H./ : VS-, ad the emirical ower for the LRD alterative is reseted i Table IV. Results are cosistet with the revious cases of test QT. The emirical size is slightly coservative tha the VS- testig rocedure, but aroaches the omial sigificace level as samle size icreases. The emirical ower agaist LRD is still owerful at this stage for moderate samle sizes. Therefore, it ca be deduced that our roosed method successfully distiguishes volatility shifts agaist LRD. Fially, we discuss the robustess of our tests to model missecificatios. We first cosider the erformace of our tests whe GARCH orders are missecified. Table V resets emirical size of test for H./ W VS- whe GARCH(, ) models are geerated, but GARCH(, ) models are fitted to obtai residuals. It ca be observed that all tests achieve a omial 5% sigificace level i geeral eve for model order missecificatio. Table III. Emirical size of test for a sigle volatility shift.! ; ;/ D D D 5.:; :; :8/ to.:; :; :6/ :; :; :8/ to.:; :; :4/ :; :; :8/ to.:; :; :6/ :; :; :8/ to.:3; :; :8/ Parameters chage at midoit. Table IV. Emirical ower of testig for H./ :VS-agaistH W r t is LRD d D D D 5 LM-ARCH LM-LARCH FIGARCH(; d; ) FARIMA(; d; ) FARIMA, AR fractioally itegrated movig average; FIGARCH, fractioally itegrated geeralized AR coditioal heteroscedasticity; LM-ARCH, log-memory AR coditioal heteroscedasticity; LM-LARCH, log-memory liear AR coditioal heteroscedasticity. LRD, log-rage deedece. wileyolielibrary.com/joural/jtsa DI:./jtsa.98

11 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 37 Table V. Emirical size of test for a sigle volatility shift whe geeralized AR coditioal heteroscedasticity orders are missecified.! ; ; ;/ to.! ; ; ;/ D D D 5.:; :; :; :7/ to.:; :; :; :5/ :; :; :; :7/ to.:; :; :; :3/ :; :; :; :7/ to.:; :; :; :4/ :; :; :; :7/ to.:3; :; :; :7/ Table VI. Emirical size of test for a sigle volatility shift whe asymmetric ower AR coditioal heteroscedasticity models are missecified.! ; ; ;; / to.! ; ; ;; / D D D 5.:35; :; :; :7; :6/ to.:35; :; :; :5; :6/ :35; :; :; :7; :6/ to.:35; :; :; :3; :6/ :35; :; :; :7; :6/ to.:35; :; :; :7; :6/ :35; :; :; :7; :6/ to.:35; :; :3; :7; :6/..9.3 Next, we cosider the asymmetric ower ARCH(, ) [A-PARCH(, )] model roosed by Dig et al. (993), which is differet from the coditioal variace i (3) of GARCH models, as follows: t D! C.j t j t / C t with >ad <<. This model is well ow to reflect the asymmetry of asset returs o volatility (e.g. Paolella, ; Giot ad Lauret, 4; Hartz et al., 6). Tables V VI reort the emirical sizes of testig for a sigle volatility shift whe the data are geerated from GARCH(, ) ad A-PARCH(, ) models with a arameter chage i the midoit, but GARCH(, ) model is fitted. Agai, the emirical sizes are close to the omial 5% sigificace level as samle size icreases. We have observed similar coclusios for over-secified model i GARCH models, but ot reorted here for brevity. I summary, our simulatio study strogly demostrates the validity of our tests i various tyes of model missecificatios. 5. REAL DATA APPLICATINS We illustrate here how to aly our roosed method to aalyse the volatility of a stoc idex. The volatility is measured by daily log-returs defied i (), where P t is the daily stoc idex of iterest. Four differet daily stoc idices, amely KSPI, KSDAQ, S&P5 ad Niei 5 from 4 Jauary 999 to 3 August, are cosidered. The data are lotted i Figure, ad samle autocorrelatio fuctios of squared log-returs are deicted i Figure. We emloy the GARCH(, ) model for the coditioal volatility model, ad arameter estimates are rovided i Table VII. First, we observe very high ersistece i volatility o all stoc idices from Table VII. Also, Figure reveals ossible volatility shifts, for examle, aroud the th observatio for KSDAQ idices. the other had, samle autocorrelatio fuctios of squared log-returs i Figure decay slowly i a hyerbolic fashio for all stoc idices cosidered excet Niei 5. I tur, it substatiates LRD modellig for squared log-returs. This exactly shows that we eed to distiguish betwee volatility shifts ad LRD models for a better exlaatio of observatios. We first aly the test for o volatility shifts, ad the results are show i Table VIII. Uder the 5% sigificace level, we do ot reject the ull hyothesis of H./ : VS- for KSPI, S&P5 ad NIKKEI5 idices. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

12 38 T. LEE, M. KIM AND C. BAEK KSPI log retur Idex log retur..5 KSDAQ = Idex log retur log retur S&P Idex NIKKEI Idex Figure. Time-series lot of log-returs of stoc idices from Jauary 999 to 3 August KSPI KSDAQ S&P5 NIKKEI Lag Lag Lag Lag Figure. Samle autocorrelatio fuctios for the squared S&P5 log-returs from Jauary to 3 December Table VII. Estimated arameters of GARCH(, ) models for stoc idices ad their ersistece! Ǫ Ǫ C KSPI : KSDAQ 8: S&P5 : NIKKEI5 4: GARCH, geeralized AR coditioal heteroscedasticity. wileyolielibrary.com/joural/jtsa DI:./jtsa.98

13 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 39 Table VIII. Test for o volatility shifts ad estimated chage oit T P-value Date of chage oit KSPI KSDAQ Aril 3 S&P NIKKEI Table I. Estimated arameters ad stadard deviatios from GARCH(, ) models together with samle stadard deviatios of two subsamles divided by chage-oit D 6 for KSDAQ! Ǫ Ǫ C Estimated SD Samle SD Subsamle 4: Subsamle : coditioal SD KSDAQ without chage idex coditioal SD KSDAQ with oe chage = Figure 3. Estimated volatilities, that is, the estimated value of t i the fitted GARCH(, ) models of KSDAQ idices with ad without structural chages idex Therefore, we coclude that there is o strog evidece agaist volatility shifts or LRD models for KSPI, S&P5 ad NIKKEI5 idices. However, we reject the ull hyothesis of o volatility shifts for the KSDAQ idex, so we coduct the testig of H./ :VS- cosequetly. The chage oit is estimated by D 6 (8 Aril 3) from Koosza ad Leius () with test statistics M D :648, adthecorresodig-value is.565. Hece, uder the 5% sigificace level, we coclude that the VS- GARCH(, ) model is referred to LRD models. The arameter estimates of the fial VS- GARCH(, ) model are reseted i Table I. bserve that the ersistece i two subsamles is weaeed comared with the estimated VS- GARCH(, ) model i Table VII. Furthermore, the estimated ad samle stadard deviatios idicate that the volatility shifts from high to low eriods for the KSDAQ idex. Figure 3 comares coditioal volatility estimated from VS- GARCH(, ) ad VS- GARCH(, ) with a chage oit at D 6. It is observed that the volatility from the estimated VS- GARCH(,) model teds to uderestimate high volatility eriod i VS- GARCH(, ) ad slightly overestimate after the chage-oit. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

14 4 T. LEE, M. KIM AND C. BAEK 6. CNCLUSINS AND DISCUSSINS I this article, we have itroduced a simle but effective testig rocedure o distiguishig volatility shifts i GARCH agaist LRD. ur testig rocedure is based o the CUSUM statistic from residual series obtaied after fittig GARCH models. Sice correlatios are sigificatly removed i our test, our test erforms better tha the usual CUSUM test based o origial observatios. Simulatios study cofirms that our testig rocedure imroves the usual CUSUM test by achievig good size ad rovidig reasoable ower agaist LRD alteratives. I additio, our method is robust to various tyes of model missecificatios such as GARCH orders, heavier tails ad asymmetry. We have assumed that the umber of volatility shifts is ow i riori. Here, we briefly discuss about estimatig the umber of chage-oits by alyig the so-called biary segmetatio (BS) method. BS method rovides acosistetestimateotheumberofchage-oitswhilecomutatioallyveryefficiet.see,forexamle, Vostriova (98) ad Bai (997). The algorithm is very simle due to its recursive structure. First, start with the whole samle ¹r ;:::;r º,aderformthetestigofH./ : VS-. The test statistic is give i (7) ad we reject the ull hyothesis based o (8) for a give level of sigificace. ce the hyotheses is rejected, we ext test H./ :VS-.Tothised,weestimatethefirstchage-oit as i () ad calculate test statistic i (). If it is ot rejected accordig± to asymtotics i (8), ± we set R D. therwise, we slit the whole samle ito two subsamles, r ;:::;r ad r ;:::; C. The, we ca reeat exactly the same testig rocedure to detect a sigle chage-oit ± for each subsamle. For examle, if the testig o a sigle chage-oit is rejected o r ;:::;r,wecaslititfurtheritotwosubsamlesadcotiuetestigtillofurtherchage-oits were foud. The fial estimated umber of chage-oits is give as the umber of subsamles mius. Sice the chage-oit estimator of Koosza ad Leius () i () satisfies (6), similar to Proositio of Bai (997) where chages i mea model has bee cosidered, uder suitable assumtios, we have the followig cosistecy result: P R D R! ; as!; (9) where the size of the test D./ is such that./! but lim if!./ >. e may cosider the volatility shifts i the sese of the followig model r t D ;t t ; if t ;t t ; if <t ; () where m;t D! m C P q id m;ir ti C P j D m;j m;tj for m D ;, isteadof(9)ad().thatis,the volatility beyod is determied by the observatios aterior to.themodel()seemstobemoreituitive modellig of volatility shifts. However, we cosider volatility shifts i the sese of (9) ad () by followig the framewor of Koosza ad Leius () due to its theoretical advatages. Also, our simulatio study, ot reorted here for brevity, shows that both models give almost the same results. It would be a iterestig future wor to examie theoretical roerties of chage-oit estimator of Koosza ad Leius () uder the model assumtio (). 7.. Proof of Theorem 7. PRFS Throughout this sectio, the symbol C>deotes a geeric costat which ca tae differet values from lie to lie. For a colum vector x R m ;m N, deotex D x x,adforamatrixa, leta WD su¹ax W x º. For a radom variable ; D E. /. wileyolielibrary.com/joural/jtsa DI:./jtsa.98

15 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 4 Let t./;,bethestrictlystatioary,ergodicado-aticiativesolutioof q t./ D! C ir ti C id j D j tj./; () where series ¹r t º are from (3). bserve also that, as it aears i Fracq ad Zaoïa (4), t./ i () ca be exlicitly rewritte as the first etry of the radom vector where B c t./; () D c t./ D C P id ir ti : C A ; B D B./ D : : :: : :: : : C A The, t. / D t a.s. imlies D from the roof of Theorem. i Fracq ad Zaoïa (4). We first recall the followig two lemmas whose roofs ca be foud i Lemma.3 of Beres et al. (3) ad Fracq ad Zaoïa (4) o ages Lemma. Assume that (A) holds. There exists s>such that E. /s < ad Ejr j s < : (3) Furthermore, let.b/ be the sectral radius of B, the su.b/ < : (4) Lemma. Uder (A)-(A), thereexistacostat.; / ad a fiite radom variable V such that for every t N, su Q t././ t t V: For matrix A, leta.i; j / deote.i; j /-th etry of A. For two matrices A ad A with the same dimesio, defie A A if A.i; j / A.i; j / for every i;j. For>,let N. / D R CqC W < be a eighborhood of. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

16 4 T. LEE, M. KIM AND C. BAEK Lemma 3. Suose that (A)-(A) hold. The, there exists >such that for every d N, E su N. / µ t. d / < ; E t./ su N. / µ t./ d t. < : / Proof This is a straightforward adatatio of the roof o age 6 of Fracq ad Zaoïa (4), hece omitted for brevity. Lemma 4. Suose that (A)-(A) hold. The, there exists >such that for every d N. E su N. t./ d < ; E su N. t./ d < ; Proof The roof is o age 63 of Fracq ad Zaoïa (4). Let r t t D ; for t D ; : : : ; : t The followig lemmas idicate that t is a good roxy of the residual Q t i (5). Lemma 5. Uder (4) ad (A)-(A), td td Q t t D op./; (5). t t / D o P./: (6) Proof We will oly rovide the roof of (5) sice (6) ca be roved similarly. Note that mi t Q t!.bservealsofromlemmathat td Q t t D td r t t D r t td t t t ± r t! r t td t! V t r t! : td Thus, (5) follows from the fact that!!! i robability ad P td t r t is fiite almost surely due to (3). wileyolielibrary.com/joural/jtsa DI:./jtsa.98

17 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 43 Lemma 6. Uder (A)-(A),itholdsthat Œu su t. t t / Œu t. / t t. / D o P./: u td td Proof Let 8 I.u/ D Œu < t. / t t. 9 / = t : t t. / C t t. t / ; ; td 8 I.u/ D Œu < t. / t t. 9 / = t : t t. / C t t. / ; ; td I 3.u/ D Œu td t. / t t. / t ; I 4.u/ D Œu td t. / t t. / : The, we ca show i a similar maer as i the roof of Lemma 5 that su ji.u/j_ji.u/j Do P./: u bserve that Œu t. t t / D Œu td td C t. / t t t. t D / C t td t. / t Œu td t. / t t t. t / C t t t. / C t D I.u/ C I.u/ C I 3.u/ C I 4.u/; hece it leads to Œu su t. t t / Œu t. / t t. / D u td td su u Œu t. / t t. t / C o P./: td Therefore, to comlete the roof, it suffices to rove that the leadig term is o P./. Alyig Taylor exasio yields DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

18 44 T. LEE, M. KIM AND C. BAEK Œu su t. / t td t. t / Œu ² D / u td t. t./ t su t. / u td t. t C t su./ u 3= td t. t su t. / u td t. t t./ 3= t. t ; u td where every D.;t/ is a itermediate oit betwee ad. From (4), it is show that su u Œu td t. t. t D o P./; sice the suremum is o P./ by the ergodic theorem. Similarly as i the roof of Lemma 5, it ca be show that so it comletes the roof. 3= td t. Proof of Theorem It ca be easily see that!. Togetherwith(5),observethat QT D D max su u td td Œu u td td D C o P./: su u Œu u td t D op./; td C su u u Œu From the ivariace ricile for artial sums for a i.i.d. sequece ad cotiuous maig theorem, it further leads to su u Œu t u td td t d! td su.u/ ; as!; (7) u wileyolielibrary.com/joural/jtsa DI:./jtsa.98

19 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 45 where B deotes a stadard Browia bridge. Moreover, ote that Œu su u Œu t C u t Œu Œu su td td td td u Œu t t. / t td td td t. / C Œu t su. / t t. / t u t. u / t. / : u td td Hece, it is sufficiet to show that Œu Œu su Œu t t. / t t. / D o P./; (8) u td td td ad su u Œu t. / t t. u / td t. / t t. / D o P./: (9) td bserve first from (5) that D D Œu Œu su Œu t t. / t td td td t. / Œu Œu su u Œu t C Œu t Œu t t. / t td td td td td t. / Œu Q t Œu t C su t td u Œu t t. / t td td td t. / Œu Œu Œu su. t t / t. / t C t. t t / u td td td t. / C o P./. t t / C Œu Œu t su. / t t. t t / u t. / C o P./: u td td td Therefore, (8) is established by (6) ad Lemma 6. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

20 46 T. LEE, M. KIM AND C. BAEK Next, there exists itermediate oits D.;t/ betwee b ad such that su u Œu t. / t t. u / td C su u su u Œu td t. / t t. / / t. u Œu. u/ t. / td t. / t. tdœuc t. where the first term coverges i robability to by ergodic theorem ad (4). Now, ote that the secod term is bouded by t. / : ; From Lemmas 3 ad 4, we ca tae a eighborhood N. / of such that E su N. / t./ t. / su N. t./ < ; which, together with (4) ad the ergodic theorem, imlies that with robability tedig to t. / E su td N. / su N. / t./ t. / t./ t. / su N. / su N. / Therefore, the secod term coverges i robability to. This comletes the t./ : 7.. Proof of Theorems ad 3 ±. The, (6) imlies that for ay We assume (), (6), ad (B)-(B3). LetA M D A M./ D M give >there exists M N such that P.A M / for sufficietly large. AsithecaseofTheorem, we defie m;t. m/; m ;m D ;, bethestrictlystatioary,ergodicado-aticiativesolutioofthe recursio m;t. m/ D! m C q m;i r m;ti C id j D m;j m;tj. m/; m D ; ; wileyolielibrary.com/joural/jtsa DI:./jtsa.98

21 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 47 where ¹r m;t º satisfy (9). Deote l m;t. m / D r m;t m;t. m/ C log m;t. m/; m D ; : We first reset some lemmas for the roof of Theorem i the below. Lemma 7. There exist.; / ad a strictly statioary sequece ¹V t º such that for N D CM with M N, max N<t t N su Q t. / ;t. / V N ; (3) ad max N<t su t Q ;t. _ V N ; (3) whe < N. Proof Sice. / DQ ;t. / ad Qr ;t D r ;t for t N whe < N, (3) ad (3) ca be roved i similar ways as i the argumets u to (4.6) ad (4.33) i Fracq ad Zaoïa (4) resectively. Lemma 8. For m D ;, there exists a strictly statioary sequece ¹U t º such that su Ql t. m / U t ; (3) m Q lt. m _ Ql m t. m / U t: (33) m Proof The roof of (3) is rovided for m D, sicetheothercasead(33)caberovediasimilarmaer.sice. /! D mi.! ;! /,wehavethat Ql t. / D rt Q t. / C log Q t. / C log!! r ;t _ r ;t! C. /! Cjlog!j: Moreover, there exist C>ad.; / such that su. / C C i r ;ti _ r ;ti! : id Hece, the roof is comleted. DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

22 48 T. LEE, M. KIM AND C. BAEK Proof of Theorem We will oly rove the case whe m D, D P./; (34) sice the other case, i D, comes similarly due to Lemmas 7-8. First, we will show that D o P./: (35) By followig the roof of Theorem. i Fracq ad Zaoïa (4), we already have that l ; is itegrable ad E ¹l ;. /º > E l ; for every 6D.Moreover,weclaimthat Note first that ^ ^ td Ql t! E l; : (36) ^ =! ad the argumet u to (4.7) i Fracq ad Zaoïa (4) imlies that From the ergodic theorem, we have that ^ td ^ td Agai, from ^ =!, (36) follows from Ql t l;t! : l ;t! E l; ; a.s. ^ ^ td l ;t! E l; : To comlete the roof, it suffices to show that for 6D,thereexistaeighborhoodN. / of such that if #N. / td Ql t.#/ > E l ; holds with robability tedig to, ad td Ql t! E l; : (37) wileyolielibrary.com/joural/jtsa DI:./jtsa.98

23 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 49 Let 6D ad Q D ^. AsitheroofofTheorem.iFracqadZaoïa(4),wecataea eighborhood N. / such that with robability tedig to if #N. / Q Q td Ql t.#/ > E l ; : Sice =! i robability, observe that if #N. / td Ql t.#/ D Q if #N. / Q Q if #N. / Q td Q td Ql t.#/ D Q Ql t.#/ C Q if #N. / if #N. / 8 < : 8 < : Q Q Q td td Ql t.#/ Q Ql t.#/ Q Q td Q td Ql t.#/ C Q 9 = Ql t.#/ ; : td 9 = Ql t.#/ ; We will show that su Q td Ql t. / Q Q td Ql t. / D o P./: (38) For >ad M>,wehavefromLemma8that su Q C td Ql t. / Q Ql Q t. / td > A su Ql t. / >ACP.A M / P C M! U t > C P.A M /: CM td C td Therefore, lim su su Q td Ql t. / Q Q td Ql t. / > A lim su P.A M /;! ad by lettig M!, we obtai (38), which imlies (37) by (36). Thus, (35) holds. Next, to show (34), we will verify asymtotic ormality of.alyigtaylorexasioofthescorevectorat gives that D lt Q td < lt Q @ 9 = Ql t./ ; ; DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

24 5 T. LEE, M. KIM AND C. BAEK where is a itermediate oit betwee ad.wehave lt Q lt D op./; @ Ql t./ @ Ql t./ D o P./; (4) which ca be checed i a similar fashio as i rovig (38). Moreover, we claim that ad Q lt Q d! N ; E @ ; (4) @ Ql @ l ; (4) where the limit is ositive defiite. First, we reset a verificatio of (4). It follows from the argumet u to (4.35) i Fracq ad Zaoïa (4) that Q l : (43) ± Let R CqC be a colum vector. l ;t W t D ; ; : : : are statioary ad square itegrable martigale differeces with resect to filtratio ¹. t ; t ;:::/W t D ; ; : : :º. Thus,fromthefuctioal cetral limit theorem for martigale differeces (cf. Theorem 8. i Billigsley (999)), we have that where./ D E (999) that Œ u l ;t d!./ i l ;. Moreover, sice = Q!, weobtaifromtheorem4.4ibilligsley Œ Q u l ;t d!./ i DŒ;; esecially, Q l d! N.;.// for every R CqC : (44) wileyolielibrary.com/joural/jtsa DI:./jtsa.98

25 TESTS FR VLATILITY SHIFTS IN GARCH AGAINST LRD 5 Furthermore, sice @ l ; D E 4 l ; ; (cf. (4.3) i Fracq ad Zaoïa (4)), we verify (4) by (43) ad (44). We ca also verify (4) i a similar fashio to (36), sice = Q! ad for ay shriig eighborhood V of, su @ l ;t. / l ;! a.s., (cf. stes (iv) ad (vi) i the roof of Theorem. of Fracq ad Zaoïa (4)). Hece, which comletes the roof. d! N ; ² E 4 ; ; Proof of Theorem 3 I light of Theorems ad, oce is relaced by the true value,covergecetotwoideedetbrowia bridges is a stadard argumet from a scalig roerty of a Browia motio, that is c = B.cu/ for u is agai a Browia motio, ad the ideedet icremets of a Browia motio. Hece, here we will comlete the roof by rovig the followig facts: max td td D max td t t td C o P./ (45) ad max < td C td C D max < td C t td C t C o P : (46) Note that (45) follows from that su u Œ u^ td ^ u td Œ u^ td ^ t C u td t D o P./; which ca be verified as i the roof of Theorem, ad su u Œ u tdœ u^ C u td ^ C D o P./ DI:./jtsa.98 wileyolielibrary.com/joural/jtsa

26 5 T. LEE, M. KIM AND C. BAEK by alyig (6) ad r ;t if¹! W º for t> : The latter relatioshi (46) ca be verified similarly by usig Lemma 7. Therefore, the cosistecy of! ad! fially rovides (8) by cotiuous maig theorem. ACKNWLEDGEMENTS We would lie to tha a Associate Editor ad aoymous Referees for their valuable commets ad suggestios which substatially imroved the resetatio. This research was suorted by Basic Sciece Research Program through the Natioal Research Foudatio of Korea (NRF), fuded by the Miistry of Educatio, Sciece ad Techology (NRF-3RAA6796), ad fuded by the Miistry of Sciece, ICT & Future Plaig (NRF- 4RAA65). REFERENCES Adreou E, Ghysels E.. Detectig multile breas i fiacial maret volatility dyamics. Joural of Alied Ecoometrics 7(5): Bae C, Piiras V.. Statistical tests for a sigle chage i mea agaist log-rage deedece. Joural of Time Series Aalysis 33(): 3 5. Bai J Estimatig multile breas oe at a time. Ecoometric Theory 3(3): Baillie RT, Bollerslev T, Mielse H Fractioally itegrated geeralized autoregressive coditioal heterosedasticity. Joural of Ecoometrics 74(): 3 3. Basra B, Davis RA, Miosch T.. Regular variatio of GARCH rocesses. Stochastic Processes ad their Alicatios 99(): Beres I, Horváth L, Koosza P. 3. GARCH rocesses: structure ad estimatio. Beroulli 9(): 7. Beres I, Horváth L, Koosza P. 5. Near-itegrated GARCH sequeces. The Aals of Alied Probability 5(B): Beres I, Horváth L, Koosza P, Shao Q. 6. discrimiatig betwee log-rage deedece ad chages i mea. Aals of Statistics 34(3): Billigsley P Covergece of Probability Measures, d editio, Wiley Series i Probability ad Statistics: Probability ad Statistics. New Yor: Joh Wiley & Sos Ic. Bollerslev T Geeralized autoregressive coditioal heterosedasticity. Joural of Ecoometrics 3(3): de Pooter M, va Dij D. 4. Testig for chages i volatility i heterosedastic time series - a further examiatio. Ecoometric Istitute Reort EI 4-38, Erasmus Uiversity Rotterdam, Ecoometric Istitute. Diebold F, Ioue A.. Log memory ad regime switchig. Joural of Ecoometrics 5(): Dig Z, Grager CW Modelig volatility ersistece of seculative returs: A ew aroach. Joural of Ecoometrics 73(): Dig Z, Grager CWJ, Egle RF A log memory roerty of stoc maret returs ad a ew model. Joural of Emirical Fiace (): Egle RF. 98. Autoregressive coditioal heteroscedasticity with estimates of the variace of Uited Kigdom iflatio. Ecoometrica 5(4): Egle RF, Bollerslev T Modellig the ersistece of coditioal variaces. Ecoometric Reviews 5(): 5. Fracq C, Zaoïa JM. 4. Maximum lielihood estimatio of ure GARCH ad ARMA-GARCH rocesses. Beroulli (4): wileyolielibrary.com/joural/jtsa DI:./jtsa.98

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