Testing for a unit root in a process exhibiting a structural break in the presence of GARCH errors

Transcription

1 Tesing for a uni roo in a process exhibiing a srucural break in he presence of GARCH errors Aricle Acceped Version Brooks, C. and Rew, A. (00) Tesing for a uni roo in a process exhibiing a srucural break in he presence of GARCH errors. Compuaional Economics, 0 (3). pp ISSN doi: hps://doi.org/10.103/a: Available a hp://cenaur.reading.ac.uk/4164/ I is advisable o refer o he publisher s version if you inend o cie from he work. To link o his aricle DOI: hp://dx.doi.org/10.103/a: Publisher: Springer All oupus in CenAUR are proeced by Inellecual Propery Righs law, including copyrigh law. Copyrigh and IPR is reained by he creaors or oher copyrigh holders. Terms and condiions for use of his maerial are defined in he End User Agreemen.

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4 Tesing for a Uni Roo in a Process Exhibiing a Srucural Break in he Presence of GARCH Errors Chris Brooks and Alisair G Rew ISMA Cenre, Deparmen of Economics, Universiy of Reading April 001 This paper considers he effec of GARCH errors on he ess proposed by Perron (1997) for a uni roo in he presence of a srucural break. We assess he impac of degeneracy and inegraedness of he condiional variance individually and find ha, apar from in he limi, he esing procedure is insensiive o he degree of degeneracy bu does exhibi an increasing over-sizing as he process becomes more inegraed. When we consider he GARCH specificaions ha we are likely o encouner in empirical research, we find ha he Perron ess are reasonably robus o he presence of GARCH and do no suffer from severe over- or under-rejecion of a correc null hypohesis. Keywords: Uni roos, srucural breaks, Perron Tes, GARCH JEL References: C1, C15, C Address for correspondence: Chris Brooks, ISMA Cenre, Deparmen of Economics, PO Box 4, Whieknighs Park, Universiy of Reading RG6 6BA. Phone: Fax: ; C.Brooks@ismacenre.reading.ac.uk.

5 (1). Inroducion A considerable number of recen empirical sudies ha have analysed financial daa have found hem o be characerised by non-saionariy, and heir reurns by condiional heeroskedasiciy. I is now a widely acceped sylised fac ha he naural logarihm of almos all asse price series conains a uni roo. The pioneering developmen of echniques designed o es for and model uni roo processes was conduced by Dickey and Fuller (Dickey (1976), Dickey and Fuller (1979) and Fuller (1976)). Anoher imporan sylised feaure of he naural logarihm of asse prices, in heir firs differenced form, is ha hey exhibi volailiy clusering or auoregressive condiional heeroskedasiciy. The ARCH process, iniially suggesed by Engle (198), permis a class of ime series models for which he condiional variance is allowed o vary hrough ime as a funcion of curren and pas informaion. Bollerslev (1986) exends his formulaion o allow for a more general formulaion ha allows lags of he condiional variance o influence is curren value, ermed he generalised auo-regressive condiional heeroskedasiciy (GARCH) model. Numerous, oher exensions have been proposed such as exponenial GARCH (EGARCH), see Nelson (1990), which allows for posiive and negaive shocks o have an asymmeric effec on he condiional variance, see Bollerslev, Chou and Kroner (199) for a deailed survey. Since he uni roo in levels, condiionally heeroskedasic differences model has been found o be empirically relevan, here has been some invesigaion ino he impac ha he presence of heeroskedasiciy has on uni-roo esing mehodologies. In general, cerain ypes of heeroskedasiciy are known o have no influence on uni roo ess asympoically alhough he impac in finie samples is less clear. Recen sudies have considered he impac ha heeroskedasiciy has on uni roo esing mehodologies in finie samples. Kim and Schmid (1993) for example, analysing he Dickey-Fuller ess 1, and Haldrup (1994), analysing he Dickey-Fuller es, Phillips (1987) semi-parameric es and he Dickey-Fuller es using Whie s (1980) heeroskedasic consisen sandard errors, found ha hese uni roo ess were suscepible o GARCH error 1

6 processes, paricularly in cerain limiing cases. For example, Haldrup (1994) finds ha he Dickey- Fuller -es and Phillips (1987) semi-paramerically correced Z-es have similar properies, as he degree of degeneracy and inegraedness increases he empirical disribuions are shifed lefwards, Kim and Schmid (1993) confirm he resuls for he Dickey-Fuller es. In addiion o he problems experienced by he sandard uni roo mehodologies caused by he presence of GARCH, i is furher known ha he exisence of a srucural break in he daa could also have an impac on he es saisics in finie samples. For example, he mos common approach for esing for non-saionariy, he Dickey-Fuller es, has been shown o be biased owards non-rejecion of he null hypohesis of a uni roo if he series in quesion exhibis a srucural break. As a resul, a researcher may incorrecly classify a series ha is saionary and exhibis a srucural break as a nonsaionary process. Incorrecly classifying a series could easily lead o inappropriae modelling and policy decisions and herefore i is of imporance o ensure ha researchers do no make such errors due o he inappropriaeness of he esing mehodologies. Allowing for srucural breaks in financial ime series is likely o become increasingly imporan since here have been some major srucural changes in he world economy in recen years. For example, he wihdrawal of he UK from he European Exchange Rae Mechanism (ERM) in 199 or he collapse of he Tiger Economies in 1997 and associaed currency and sock marke urmoil are likely o have had a major impac on he behaviour of financial ime-series relaing o hese counries. Furher, he inroducion of a single European currency and he convergence o a single Europe are likely o have a significan impac on he ime series properies of relaed financial ime series. The exisence of a srucural break will preven he radiional Dickey-Fuller approach from being validly used o analyse he exisence or oherwise of uni roos in hese ime series and as such, researchers will have o explicily allow for he breaks in he esing mehodologies. 1 The procedure is developed in Dickey (1976), Fuller (1976) and Dickey and Fuller (1979).

7 To his end, Banerjee, Lumsdaine and Sock (199) and Zivo and Andrews (199) develop esing mehodologies ha allow for a uni roo and he possibiliy of a srucural break - a uni roo (saionary-srucural break) process under he null (alernaive) hypoheses. However, Perron (1989) argues ha we may wan o allow for a break under he null hypohesis and proposes a mehodology ha allows for a null (alernaive) hypohesis of a uni-roo wih a srucural break (saionary wih a break). This approach was criicised by Chrisiano (199) on he basis ha he daes for he srucural breaks are chosen such ha hey are in fac imposed on he daa, i.e. he ess assume ha he breaks are known a priori. If he break daes are chosen in such a manner, he underlying asympoic disribuion is no valid. In response o his, Perron (1997) develops a framework, henceforh he Perron procedure 3, which allows for he choice of he break dae o be endogenised and herefore deermined by he daa. However, he Perron procedure, whils allowing for srucural breaks, sill suffers from he problem ha i assumes ha he errors in he formulaed regressions are whie noise. To his end, he purpose of he presen paper is o assess he finie sample properies of he Perron procedure in he presence of GARCH processes, so ha we can deermine how robus he mehodology is likely o be in empirical applicaions where boh srucural breaks and condiional heeroskedasiciy are presen. In doing so, we will be able o deermine he consequences of using he mehodology wih high frequency financial daa. The res of he paper is srucured as follows. In Secion we review he Perron procedure and ouline Bollerslev s (1986) GARCH process, while Secion 3 considers he simulaion framework ha we employ in he analysis. In Secion 4 we discuss daa generaing processes and heir accompanying resuls while in Secion 5 we draw our conclusions. In a separae paper, we have analysed he impac of he UK resigning from he ERM and found ha his even precipiaed a significan srucural break in shor erm Euro Serling ineres raes. 3 Perron develops he necessary esing framework o allow for a break in mean, break in slope or a combinaion of he wo. 3

8 () Theoreical review (i) The Perron Model and Tes Saisics The model and esing framework ha we consider is based on ha proposed by Perron (1997), ermed he innovaional oulier model. In paricular he daa generaing process (DGP) allows for a uni roo process ha exhibis a gradual shif in he mean of he series in a way ha depends on he correlaion of he error process. Perron (1997) and Perron (1989) discuss several alernaive specificaions ha can be considered. As well as he innovaional oulier model described in deail below and which allows for a shir in he inercep only, a second model allows for a combinaion of a break in he mean and he slope of he DGP. Finally, a hird model allows for a change in he slope, bu wih he segmens of he rend funcion being joined a he ime of he break. Perron erms his laer specificaion he addiive oulier model, where he break is assumed o ake place quickly. This paper only analyses he mean break or innovaional oulier model in order o mainain he paper a a manageable lengh. The es equaion ha is considered in he innovaional oulier Perron procedure is given by: y DU ( ) D(T ) b y k 1 c j y j u (1) j 1 Where, DU ( ) = 1 if > T, zero oherwise, D(T b ) = 1 for = T b+1, zero oherwise, and u ~ i.i.d. N(0, 1). The regression equaion is esimaed sequenially by ordinary leas squares and we are ineresed in he -saisic for he es of = 1. The procedure requires a mehodology o endogenise he break and Perron proposes wo mehods 4. The firs approach chooses he minimum value for he -es on = 1 as we change he dae of he hypohesized break. The es saisics are defined as 4 For a more deailed explanaion we direc he reader o Perron (1997) 4

9 * * (1) ˆ (Tb,k), where T * b is such ha Min (k 1,T) ˆ (Tb,k) 5. The second mehod considered involves minimizing he -saisic on he coefficien for he change in he inercep parameer,, T b denoed * *. This es saisic is denoed (1) ˆ (1,T b,k), where T * b is such ha ˆ, * (T ) Min T (k 1,T) ˆ (Tb,k). This mehod assumes ha he break is negaive, i.e. i imposes a b b mild a priori ha here is a crash where he mean of he series falls. An alernaive specificaion where his resricion is no imposed is where we choose he maximum absolue values of, using ˆ he same approach o choose he break dae as above, for which he es saisics are given by *, (1). The analysis ha we underake is based solely on he * (1) es saisics as Perron remarks ha he es saisics * (1) and (1) *, are likely o have he same properies and consequenly, we expec ha similar conclusions would be made if we considered he oher es saisics. Finally, Perron considers wo mehods o selec he runcaion lag parameer k; a -es, k(-sig), and an F-es, k(f-sig), for which he former was shown o have slighly more power and herefore we execue our analysis using his mehod. (ii) Allowing he errors o follow a GARCH Process The exension ha we wish o make is o assess he Perron procedure when he DGP of he series in quesion exhibis GARCH errors. Bollerslev (1986) defines he GARCH model as: q j p h u h, where = 1,..T () 0 1,j j 1 i 1,i i where 0, 1,j and,i are non-negaive for all i and j. If,i = 0 for all i hen h collapses o Engle's (198) ARCH formulaion. In paricular, he form of heeroskedasiciy considered in his paper is ha ermed GARCH(1, 1) such ha he condiional variance is given by: h u h, (3) I can be seen ha he above es is analogous o he sequenial es saisic proposed by Banerjee e al (199), ~ min* ~ min (k / T), and o he es of Zivo and Andrews (199), bu is execued on a differen model ha allows for a DF k 0 k T k 0 DF srucural break under he null. 5

10 he implicaion for u in () and (3) is ha i is now disribued N(0, h ) If 0 > 0 and 1 + < 1 hen we will have a well defined uncondiional variance, given by 0 /( 1 1 ). Ineresing implicaions arise when hese condiions are no saisfied, and here are wo specific cases ha are ypically considered when GARCH modelling is being underaken he degenerae and inegraed cases. The degenerae case occurs when 0 = 0 causing he uncondiional variance, UV(u ), o be zero. In he inegraed case, ofen referred o as IGARCH, and where 1 + = 1, and consequenly shocks o h will persis indefiniely causing he uncondiional variance o be infinie. Similarly, we may expec here o be a high degree of persisence, hough no infinie, when 1 + is close o uniy, i.e. near inegraed. Given ha UV(u ) as and UV(u ) 0 as 0 0, hen o preven he limis being approached, herefore he uncondiional variance being unbounded, here mus be some ineracion beween 0 and ( 1 + ). Consider he case where 0 is large bu less han hen if 1 + is close o 1, UV(u ). Similarly, if 1 + = 0 and 0 is close o zero hen UV(u ) 0. Thus if one of he limiing condiions is being approached and he oher no hen he UV(u ) will end owards a violaion of he exisence of an uncondiional variance. Consequenly, if 0 or 1 + is close o heir limis hen he non-exisence of an uncondiional variance is approached. However, if 0 and 1 + are close o heir limis simulaneously, hen i can be seen ha he UV(u ) is no oo small or oo large and i will no herefore be approaching he limis of he exisence of an uncondiional variance. For example if 0 = i implies ha he UV(u ) is near degenerae, if 1 + = hen UV(u ) is near inegraed, bu in combinaion UV(u ) = 1 and hence UV(u ) is well defined, i.e. here is a mixing of near degeneracy and near inegraedness ha prevens UV(u ) from being unbounded. In addiion, Nelson (1990, 199) defines a heoreical framework for showing ha degeneracy and inegraedness are likely o occur ogeher by considering GARCH as an approximaion o a 6

11 coninuous ime diffusion process. For a deailed explanaion we direc he reader o he original aricles. In brief, Nelson (199) shows ha GARCH can be seen o be a consisen approximaion o a coninuous ime diffusion process in he limi as 0 if: 0 =, 1 = (1/), = 1-1 (4) where is he ime inerval of he daa, i.e. he frequency, and and are fixed. If we consider high frequency daa hen is small and hence we would expec ha 0 and 1 would be small and would be approximaely equal o 1, considering he limi = 0 hen 0 = 0, 1 = 0 and = 1. Consequenly, if we accep Nelson s heoreical framework, hen i can be seen ha we would necessarily expec degeneracy and inegraedness o exis in union. The empirical evidence supporing Nelson s framework is somewha mixed. Baillie and Bollerslev (1989), in a comprehensive analysis of exchange rae volailiy, provide a large amoun of evidence ha can be used o assess he applicabiliy of he framework. They consider six exchange raes sampled daily, weekly, fornighly and monhly from which log-reurns are consruced. In he monhly daa no ARCH effecs are deeced, whils he fornighly resuls indicae ha ARCH is presen. Once he frequency reaches he weekly and daily levels, GARCH(1,1) is found o be presen. Concenraing, herefore on he weekly and daily daa, he resuls indicae ha for four of he currencies whils he Yen showed srong persisence wih a value for 1 + = According o Nelson s framework, we would expec he Yen o exhibi a small value for he inercep, 0 and for 1 and his was indeed he case, wih he smalles 0 and second smalles 1 belonging o he Yen. If we consider he daily daa hen we can sae ha we would expec persisence o be greaer relaive o he weekly daa if Nelson s framework is o be suppored. Baillie and Bollerslev (1989) indeed found ha he persisence was generally sronger for he daily daa wih for each of he six currencies. Furher if Nelson s framework is accurae hen 0 and 1 should be smaller han in he weekly daa analysis and indeed his was he case. In summary, he daily resuls 7

12 found ha he larges persisence was accompanied by he smalles inercep and volailiy and vice versa. Consequenly, here is a heoreical grounding and empirical suppor for he exisence of (near-) degeneracy and (near-) inegraedness simulaneously. The implicaion of his is ha we will need o invesigae he effec of inegraedness and degeneracy as well as each one separaely on he Perron Tess. To summarise, we have discussed he implicaion of degeneracy and inegraedness and showed ha he uncondiional variance does no exis in hese limiing cases. We hen reviewed heoreical and empirical evidence suggesing ha degeneracy and inegraedness could generally be expeced o occur ogeher and if his was found o be he case hen i will prove beneficial o separae he effecs ou o see he impac of each on he Perron procedure as well as invesigaing he simulaneous presence of boh. We will now proceed o discuss he framework of he simulaions. (3) The Simulaion Framework The purpose of his paper is o consider he impac ha an error srucure of GARCH(1,1) has on he Perron Tess. Consequenly, we need o redefine he DGP ha Perron considered such ha we explicily allow he error srucure o follow a GARCH(1,1). Perron defined he DGP for his ess as: y DU D(T b ) y (i) y (1 L) e 1 i (5) From his specificaion of he DGP, i is easy o consider differen error srucures by varying he parameers (i) and 6. To allow for he GARCH(1,1) process we replace he error srucure componens ( i) y (1 L) e wih u, where u follows a condiionally normal GARCH(1,1) i giving u ~ N(0,h ). Consequenly, he DGP ha we will consider is given by: y DU D(T ) y u (6) b 1 8

13 where u ~ N(0,h ). For ransparency of resuls, he specificaion of he GARCH process follows ha of Kim and Schmid (1993), i.e. we incorporae he same coefficien values for 1,, 3 and h 0. In paricularly, we consider varying 1 and o assess he impac ha he degree of inegraedness, degeneracy and volailiy of he variance process has on he ess. Consequenly, he values of 1 are held consan and is varied and vice versa. In all cases, o simplify he analysis, h 0 = 1, however his does no adversely affec he resuls as Kim and Schmid (1993), pp. 9 noe ha he resuls are invarian o changes in h 0, so long as 0 is changed proporionally, so ha 0 /h 0 is held consan. In his firs se of ess Kim and Schmid se he iniial variance equal o he long-run variance (i.e. 0 = h 0 (1-1 - )). Whils, his prevens us from disinguishing wheher or no degeneracy or inegraedness has he greaes impac on he ess, i is a leas consisen wih he heoreical and empirical work of Nelson (1989) and Baillie and Bollerslev (1989). Furher, once we have discovered how degeneracy and inegraedness impac he Perron Tess, we can hen consider he individual componens. To do his we se 1 + = 1, hus imposing he degree of inegraedness and hen vary 0. Therefore we are no longer imposing he earlier relaionship beween 0 and h 0, raher we are direcly invesigaing he impac of 0 on he Perron Tess and hence he impac of degeneracy on he ess. Similarly, we will consider fixing 0, h 0 and 1 whils varying. Addiional experimens are included o assess he individual impac of 1 and. We will consider sample sizes ha are consisen wih Perron s size and power ess and also wih Kim and Schmid (1993) and consequenly we consider samples of 100, 500 and 1000 wih 10,000 simulaions. In doing so, we can compare he resuls wih hose of he oher error srucures considered by Perron and also have a direc comparison wih he analysis of Kim and Schmid. In he 6 Perron explicily considers (i) iid errors; (ii) posiive auocorrelaion; (3) negaive auocorrelaion; (4) wo specificaions of higher-order correlaion; (5) wo specificaions of MA(1) errors 9

14 exra experimens, due o he compuaional inensiy of he simulaions we only consider sample sizes of Descripion of he Daa Generaing Processes and Resuls Due o space consrains we only repor he full analysis for he experimens based on he saisic *. Furher, since Perron (1997) shows ha he procedure exhibis good size and power properies when he magniude of he srucural break is less han 5 sandard deviaions, which is ypically he case in he daa series analysed, he following analysis considers = = 0 unless oherwise saed. We spli he following secion in o separae sub-secions ha deal wih separae characerisics wih respec o he GARCH process being analysed however in each case he numbers in he ables refer o he proporion of rejecions under he null hypohesis for 1%,.5%, 5% and 10% ess. (i) The Impac on he Perron Procedure as when 0 = 1-1 -, = 1 In he firs se of experimens we vary he value of 1 + from a non-inegraed case, 1 + = 0.9, o an inegraed case, 1 + = 1, where 0 = Firsly we fix he value of 1 and vary he value of such ha he process becomes increasingly inegraed. We hen repea he experimen for a fixed and vary he value of 1. The iniial variance is se o 1, herefore h 0 = 1, hence he iniial variance is equal o he long run variance, i.e. 0 = h 0 (1-1 - ) and consequenly as we increase he degree of inegraedness he process also becomes degenerae. Table 1 summarises he resuls obained from he experimens for he hree samples sizes when 1 is fixed and is varied and Table for when is fixed and 1 is varied. The GARCH coefficiens repored are (0.1, 0.3, 0.6), (0.05, 0.3, 0.65) and (0, 0.3, 0.7) in he firs insance and (0.1, 0.6, 0.3), (0.05, 0.65, 0.3) and (0, 0.7, 0.3) for Table. I can be seen ha for a given sample size, as he GARCH process approaches inegraedness and degeneracy he Perron procedure becomes increasingly over sized. In he limi, when he GARCH 10

15 process is boh inegraed and degenerae he procedure suffers from a very serious over-sizing problem. If we consider he effec as we change he sample size we can see ha in he inegraed and degenerae case he proporion of rejecions repored acually increases as he sample size increases. Consequenly, we can see ha he sandard asympoic heory is inapplicable in his case, however we have no isolaed wheher i is he degree of inegraedness of he degree of degeneracy ha is driving he over-rejecion problem. Furher, even when he process is no degenerae/inegraed, no only is here sill a serious over sizing of he es bu i is no clear ha he es saisics are converging o heir asympoic disribuions as T, herefore i appears ha wih cerain non-degenerae/noninegraed GARCH specificaions he sandard asympoic disribuions are also no applicable. I can also be seen ha for given values of 1 + and 0, he over rejecion is higher in he case where 1 is higher and is lower, i.e. he over-rejecion is greaer for he simulaions repored in Table. For example, in he case of 100 observaions, nominal 5% significance level, he rejecion rae is approximaely 3% in he case of 1 = 0.6 and 1 = 0.3, where as he rae is only 15% in he case when 1 = 0.3 and 1 = 0.6. Once again he difference could be caused by wo facors, eiher 1 or, and his will be assessed shorly. (ii) Assessing he Impac of he Volailiy of he Variance Process, 1, and Following Kim and Schmid (1993), we considered how he degree of volailiy of he variance process and is persisence affec he degree of over rejecion. Kim and Schmid sae ha in he GARCH(1,1) model, i is roughly accurae o say ha 1 deermines he degree of volailiy of he variance process, while 1 + deermines is persisence. To analyse he effecs, we consider he GARCH coefficiens given by 1 = 0.1 for = 0.5, 0.8, 0.85, 0.9 (Table 3) and hen 1 = 0.3, 0.6, 0.65, 0.7 where is held consan a 0.3 (Table 4). The resuls confirm hose in Tables 1 and in ha as he degree of inegraedness and degeneracy increases owards heir limis he over rejecion problem becomes increasingly significan. 11

16 Addiionally, in he degenerae and inegraed case, he size disorion once again increases in sample size, confirming ha he asympoic heory is inappropriae in hese cases. Similarly, for some cases of non-degenerae and non-inegraed processes i is no clear ha he asympoic heory is appropriae. From Table 3, we can see ha if we allow he degree of inegraedness/degeneracy o increase by varying hen, alhough he rejecion rae increases, i does no do so significanly, excep in he limi. Consequenly, i appears ha hough does have an impac on he rejecion rae, i is no oo significan. If we now consider he cases where 0 and 1 + are he same across Tables 1,, 3 and 4, for example 0 = 0.1and 1 + = 0.9, hen i can be seen ha as 1 decreases in value, he degree of overrejecion falls. In he case where 0 = 0.1and 1 + = 0.9, for a sample of 100 observaions a a nominal 5% significance level, he rejecion rae was approximaely 3%, 15% and 7% for 1 = 0.6, 0.3 and 0.1 respecively. In he limi when he GARCH process is boh degenerae and inegraed, he rejecion raes are considerably differen - being approximaely 98%, 59% and 11% for 1 = 0.6, 0.3 and 0.1 respecively. Furher, he increase in over-rejecion as 1 + increases falls as 1 falls. For example moving from 0 = 0.1 and 1 + = 0.9, hrough o 0 = 0 and 1 + = 1 causes he rejecion rae o increase approximaely 1.6 and 3.9 fold for 1 = 0.1 and 0.3, respecively. Similarly, if we consider 0 = 0.1 and 1 + = 0.9, hen we can see ha he rejecion rae for 1 = 0.3 is approximaely.7 imes ha when 1 = 0.1. As we approach he inegraed and degenerae case, hese muliples increase such ha when he limis are reached he rejecion rae for 1 = 0.3 is approximaely 5.4 imes ha when 1 = 0.1. We can herefore conclude ha, for a consan degree of degeneracy and inegraedness, he magniude of he volailiy of variance coefficien deermines he exen of over-rejecion - he greaer he value he greaer he over-rejecion. In addiion, he relaive rejecion raes for differen DGPs 1

17 increases in: (1) he difference beween he 1 s; () he degree of degeneracy and (3) he degree of inegraedness. I is ineresing o noe ha even in he degenerae and inegraed case, he Perron procedure appears o have reasonable size as he volailiy parameer ends o zero. Wih respec o he coefficien, we have seen ha alhough he rejecion rae for he es saisic increases as he coefficien increases, i does no appear o have as much of a significan impac as 1. Consequenly, i appears ha he volailiy of variance coefficien has a considerable influence on he size of he es and no jus he degree of inegraedness and degeneracy. We now consider he individual impac of degeneracy and inegraedness. (iii) The Individual Impac of he Degree of Degeneracy on he Size of he Perron Procedure To assess he impac ha he degree of degeneracy has on he size properies of he Perron procedure we consider a series of simulaions ha hold he degree of inegraedness consan and vary he degree of degeneracy. The firs case ha we consider is he non-inegraed one where 1 = 0.3 and 1 = We allow he degree of degeneracy o vary from 0 = 0 o 0 = 100, he resuls are summarised in Table 5. I can be seen ha he rejecion rae appears o be very insensiive o he degree of degeneracy excep in he limi and consequenly we can sae ha he Perron procedure is robus o degeneracy excep in he limi. Some addiional experimens were conduced where 0 = and and i was found ha, alhough hese were slighly differen o hose repored in he able, he difference was no significan and as a whole hey confirmed he insensiiviy of he es o he degree of degeneracy. Similar experimens were carried ou on ( 1, ) = (0.3, 0.7), (0.1, 0.9), (0.1, 0.85), (0.1, 0.5), (0.5, 0.1), (0.5, 0.45), (0.5, 0.5) for which Table 6 summarises he resuls a he 5% nominal significance level for 100 observaions. I can be seen ha he rejecion rae is insensiive o he degree of degeneracy. I is worh noing ha when he series is near degenerae ( 0 = ), hen, as long as he volailiy of variance coefficien is small, he ess have good size, and 0.07 for ( 1, 1 ) = (0.1, 0.5) and (0.1, 0.85) respecively. As can be seen, he increase in he rejecion rae is no oo 13

18 large as we approach inegraedness, and even in he limi when ( 1, 1 ) = 1, he rejecion rae is sill only a he nominal 5% significance level, if 1 = 0.1 and = 0.9, compared o 0.58 when ( 1, ) = (0.5, 0.5). In he limi, when he GARCH process is degenerae, he rejecion rae is oo high bu is sill highly dependen on he volailiy of he variance. Having considered varying he degree of near-degeneracy, i is now necessary o consider he impac of holding he degree of near-degeneracy consan whils allowing he degree of near-inegraedness o vary. (iv) The Individual Impac of he Degree of Inegraedness on he Size of he Perron Procedure In assessing he impac ha he degree of inegraedness has on he size of he Perron es, we hold he degree of degeneracy and he volailiy of he variance consan and hen allow o vary such ha we go from an a non-inegraed case o he limi of an inegraed GARCH process. We hen consider a differen value of 1 and hen repea he exercise o assess he impac of he volailiy of he variance and he degree of inegraedness. Table 7 repors he resuls for he simulaions where 0 = 0.01 and 1 = 0.3, where is allowed o ake he values 0.6, 0.65 and 0.7. I can be seen ha as we increase he degree of inegraedness, he rejecion rae increases and his is more marked as he sample size increases. For example, he rejecion rae for 100 observaions a he 5% nominal significance rae increases from o as increases from 0.6 o 0.7, an increase of approximaely 30%, whils for a sample of 1000 observaions, he rejecion rae rises from o 0.436, an increase of over 150%. As would be expeced he variance of he volailiy also plays a significan role in deermining he rejecion rae, and his can be seen by considering he resuls in Table 8 where 1 = 1 and he degree of inegraedness is increased. The variance of he volailiy parameer is lower han ha used in Table 7, and as would be expeced, here has been a corresponding decrease in he rejecion rae. For 14

19 example if we consider he nominal 5% significance level when 1 + =0.9, he rejecion raes are and for a sample of 100 observaions for 1 = 0.3 and 0.1 respecively. The above analysis has considered various experimens o pinpoin he main facors in deermining he over-rejecion problem. However, from an empirical poin of view i is of ineres o consider GARCH specificaions ha are likely o be encounered in applied work. Specifically, hough we have shown ha he Perron es exhibis size problems when he error specificaion is allowed o be a GARCH(1,1), are we likely o experience problems in pracice? (v) The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure Alhough we have analysed wha he impac of he volailiy of variance, degeneracy and inegraedness has on he size of he es, we need o also ensure ha we consider cases ha are likely o be empirically relevan. To analyse his, we consider various parameerisaions as defined by Nelson (1990, 199) and Nelson and Foser (1991). Employing he coefficien values given in Kim and Schmid, we have = 0.01, = 0.3 for = 1, 0.5, 0.09, and The resuling values of he GARCH specificaion are ( 1,, 3 ) = (0.01, 0.3, 0.7), (0.005, 0.15, 0.85), (0.0009, 0.09, 0.91) and (0.0001, 0.03, 0.97) respecively. Table 9 summarises hese resuls. As decreases, he process becomes increasingly degenerae and inegraed and 1 falls. I can be seen ha he over-rejecion problem decreases and he values appear o approach heir asympoic values as oulined in Perron (1997). In paricular, given hese resuls and hose previously repored, he asympoic values appear o be approached as he volailiy coefficien falls, indicaing ha he majoriy of he over-rejecion oulined in he previous experimens is indeed caused by his parameer. In realiy, as he process becomes increasingly degenerae and inegraed, he volailiy parameer becomes smaller and he over-rejecion problem is reduced o he exen ha Perron s asympoes are approached. The effecs of he inegraedness and degeneracy ogeher appear o counerac each oher o some exen, so ha he degree of over-rejecion is reduced in such cases. 15

20 We also conduc experimens for = 0.1 and 0 for which he relaed GARCH specificaions are ( 1,, 3 ) = (0.1, 0.3, 0.7), (0.05, 0.15, 0.85), (0.009, 0.09, 0.91), (0.001, 0.03, 0.97) and ( 1,, 3 ) = (0.0, 0.3, 0.7), (0.0, 0.15, 0.85), (0.0, 0.09, 0.91) and (0, 0.03, 0.97) respecively. These resuls are repored in Tables 10 and 11. The resuls in hese ables make i apparen ha he main deerminan of he over-rejecion problem is in fac he volailiy of he variance parameer. This can be seen since even in a degenerae case, he over-rejecion decreases as he volailiy parameer falls. (vi) The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure as he Magniude of he Break Increases To complee he Mone Carlo analysis, we considered allowing he magniude of he break o increase such ha in he DGP akes a value of 0, 1,, 5 or 10 when he GARCH process is given by 1 = 0.03 and = 0.97 and 0 is allowed o vary, aking he values 0 = 0.001, and 0. Once again, we se = 1 in he DGP. Perron (1997) repors ha in he presence of normal errors, wih consan condiional variance, he es has reasonable size characerisics when is less han 5 sandard deviaions and his conclusion is shown o be applicable when we allow he errors of he DGP o be a GARCH process as shown in Table 1 7. For example, when = hen he rejecion rae in he presence of normally disribued errors is compared o 0.045, and when 1 = 0.03, = 0.97 and 0 = 0.001, and 0 respecively. If we consider he case when = 5, we can see ha, alhough here is an impac even in he case of normally disribued errors, he over-sizing increases as he process becomes more degenerae, from a rejecion rae of o when 1 = 0.03, = 0.97 and 0 = and 0 respecively. Consequenly, we can conclude ha he presence of an increasingly degenerae GARCH process does no have a significan impac on he size of he es excep when he break in he mean of he series is in excess of 5 sandard deviaions. (5) Conclusions 7 We only repor he rejecion raes for a sample size of 100 a he 5% nominal level due o space consrains. 16

21 The aim of his paper was o invesigae he impac of GARCH on he Perron procedure for a uni roo in he presence of a srucural break. Kim and Schmid (1993) and Haldrup (1994) boh repor severe over-sizing wih respec o he Dickey-Fuller and Perron Z-es es, rendering he es poenially problemaic in he presence of GARCH processes. A similar conclusion can be made wih respec o he impac of GARCH on he Perron procedure, which allows for a srucural break. Our analysis shows ha he main facor ha influences he rejecion rae for he Perron procedure is he volailiy of he variance of he GARCH process. If his value is large, here is a serious over-sizing of he es; however i is unlikely ha he value of his will be oo large in empirical work and consequenly he es should have appropriae acual size. This conclusion remains valid even in he degenerae and inegraed case. We show ha he degree of degeneracy does no have an impac on he size of he es excep in he limi, whereas he over-sizing increases as he degree of inegraedness increases, boh are however dependen on he volailiy of he variance. The resuls for he coninuous ime approximaion GARCH confirmed ha, when he GARCH process is similar o ha which we would ypically find in empirical work, he es has reasonable size characerisics. When we allow for differen magniudes of he break coefficien in he presence of GARCH, he resuling size characerisics are consisen wih hose repored by Perron (1997) only if he magniude of he break is less han 5 sandard deviaions. If he break is of a greaer magniude han 5 sandard deviaions, hen over sizing becomes progressively worse as he degree of degeneracy increases. Consequenly, we can conclude ha he Perron procedure, hough severely affeced in cerain cases, appears o be robus o he exisence of GARCH when i approximaes ha which we ypically find in pracice as long as he magniude of he break is small. An ineresing direcion for fuure research would be an analysis of he impac of differen error srucures on he abiliy of he Perron es o accuraely dae he break. The reason ha his is of relevance is ha, alhough we have shown ha he es for a uni roo and srucural break agains an alernaive of a saionary process wih a srucural break appears o be reasonably robus in he presence of GARCH of he form ha we are likely o encouner in empirical work, his sudy has no 17

22 assessed he effec of GARCH and /or srucural breaks on he es s abiliy o dae he break. I would also be of ineres o assess which of he various uni roo esing procedures available is mos robus o various empirically relevan sylised regulariies, including he impac of srucural breaks, GARCH and uncondiional fa ails. In doing so, one could poenially highligh which ess are bes in cerain circumsances and which ones are mos problemaic. This would prove o be beneficial as researchers would have a beer undersanding of he ools ha hey employ, and which of a se of conradicory resuls o favour in a given seing. 18

23 Table 1 - The impac on he Perron Procedure as for 0 = and 1 = 0.3 (0.1, 0.3, 0.6) (0.05, 0.3, 0.65) (0, 0.3, 0.7) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table - The impac on he Perron Procedure as for 0 = and = 0.3 (0.1, 0.6, 0.3) (0.05, 0.65, 0.3) (0, 0.7, 0.3) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 3 - Assessing he Impac of, for 1 = 0.1 as for 0 = (0.4, 0.1, 0.5) (0.1, 0.1, 0.8) (0.05, 0.1, 0.85) (0, 0.1, 0.9) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 4 - Assessing he Impac of 1, for = 0.3 as for 0 = (0.4, 0.3, 0.3) (0.1, 0.6, 0.3) (0.05, 0.65, 0.3) (0, 0.7, 0.3) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 5 - The Impac of he Degree of Degeneracy on he Perron Procedure, 1 = 0.3 and = 19

24 0.65 for 0 = 0 and 0.01 (0, 0.3, 0.65) (0.01, 0.3, 0.65) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Simulaions were also conduced for 0= 1.0 and 100, bu he resuls were idenical o he corresponding resuls for 0 = 0.01 and hence hese are no presened. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 6 - The Impac of he Degree of Degeneracy on he Perron Procedure, where ( 1, ) = (0.3, 0.7), (0.1, 0.9), (0.1, 0.85), (0.1, 0.5), (0.5, 0.1), (0.5, 0.45), (0.5, 0.5) and 0 = , , 0.01, 1.0 and 100 Degree of Degeneracy ( 1, ) (0.1, 0.9) (0.3, 0.7) (0.5, 0.5) (0.1, 0.85) (0.3, 0.65) (0.5, 0.45) (0.1, 0.5) (0.5, 0.1) The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Simulaions were also conduced for 0= 0.01, 1.0 and 100, bu he resuls were idenical o he corresponding resuls for 0 = and hence hese are no presened. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 7 The Impac of he Degree of Inegraedness on he Perron Procedure 0 = 0.01, 1 = 0.3 and = 0.6, 0.65, 0.7 (0.01, 0.3, 0.6) (0.01, 0.3, 0.65) (0.01, 0.3, 0.7) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 0

25 Table 8 The Impac of he Degree of Inegraedness on he Perron Procedure 0 = 0.01, 1 = 0.1 and = 0.5, 0.8, 0.85 and 0.9 (0.01, 0.1, 0.5) (0.01, 0.1, 0.8) (0.01, 0.1, 0.85) (0.01, 0.1, 0.9) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1 Table 9 - The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure, = 0.01 = 1 = 0.5 = 0.09 = 0.01 (0.01, 0.3, 0.7) (0.005, 0.15, 0.85) (0.0009, 0.09, 0.91) (0.0001, 0.03, 0.97) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1. The values for he GARCH coefficiens are derived from he Nelson (199) formulaion given by 0 =, 1 = (1/), = 1-1, where = 0.01, = 1, 0.5, 0.09, and Table 10 - The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure, = (0.1, 0.3, 0.7) (0.05, 0.15, 0.85) (0.009, 0.09, 0.91) (0.001, 0.03, 0.97) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1. The values for he GARCH coefficiens are derived from he Nelson (199) formulaion given by 0 =, 1 = (1/), = 1-1, where = 0.1, = 1, 0.5, 0.09, and Table 11 - The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure, = 0.0 1

26 (0.0, 0.3, 0.7) (0.0, 0.15, 0.85) (0.0, 0.09, 0.91) (0, 0.03, 0.97) % % % % The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1. The values for he GARCH coefficiens are derived from he Nelson (199) formulaion given by 0 =, 1 = (1/), = 1-1, where = 0.0, = 1, 0.5, 0.09, and Table 1 - The Impac of Coninuous Time Approximaion - GARCH on he Perron Procedure, where he coefficien on he srucural break variables is allowed o ake he values 0, 1,, 5 and 10 ( 0, 0.03, 0.97) Magniude of Break Normal 0 = = = The above able repors he rejecion raes when he nominal 5% criical values, as oulined in Perron (1997), are employed o es for he presence of a uni roo and srucural break. Sample sizes of 100, 500 and 1000 are considered and he GARCH coefficiens are expressed as ( 0, 1, ) where 0, 1 and refer o he value of he coefficiens in h 0 1u 1 h 1. The values for he GARCH coefficiens are derived from he Nelson (199) formulaion given by 0 =, 1 = (1/), = 1-1, where = 0.1, 0.01 and 0.0 and = 0.01.

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28 Phillips, P. C. B. (1987). Time series regression wih a uni oo and infinie variance errors. Economerica, Vol. 55, pp Whie, H. (1980). A heeroskedasiciy-consisen covariance marix esimaor and a direc es for heeroskedasiciy. Economerica, Vol. 48, pp Zivo, E., and Andrews, D. W. K. (199). Furher evidence on he Grea Crash, he oil-price shock and he uni-roo hypohesis. Journal of Business and Economic Saisics, Vol. 10, pp