On he oversized problem of Dickey-Fuller-ype ess wih GARCH errors Auhor Su, Jen-Je Published 2011 Journal Tile Communicaions in Saisics: Simulaion and Compuaion DOI hps://doi.org/10.1080/03610918.2011.575502 Copyrigh Saemen 2011 Taylor & Francis. This is an elecronic version of an aricle published in Communicaions in Saisics: Simulaion and Compuaion, Volume 40, Issue 9, 2011, pages 1364-1372. Communicaions in Saisics: Simulaion and Compuaion is available online a: hp:// www.informaworld.com wih he open URL of your aricle. Downloaded from hp://hdl.handle.ne/10072/41102 Griffih Research Online hps://research-reposiory.griffih.edu.au
On he oversized problem of Dickey-Fuller-ype ess wih GARCH errors Jen-Je Su Griffih Universiy Ausralia Absrac This paper examines he finie-sample size of a class of Dickey-Fuller-ype ess in he presence of GARCH errors, wih and wihou he influence of iniial condiions of he underlying simulaed pah. Oversizing is observed for all ess when he GARCH process is nearly degenerae and he volailiy parameer is large, bu he degree of size disorion varies across ess and is coningen on he iniial condiion. The resul due o he iniial effec is linked o he size disorion caused by a sequence of small downward variance breaks arising in he early sage of he underlying process. Keywords: GARCH; Uni roo ess; Size disorion; Iniial effec; Mone Carlo experimen JEL Codes: C22 Running Head: Tesing for a uni roo under GARCH Correspondence o: Jen-Je Su Deparmen of Accouning, Finance and Economics Nahan Campus, Griffih Universiy 170 Kesssels Road, Nahan Brisbane, Queensland 4111, Ausralia Email: j.su@griffih.edu.au Phone: 61-7-37353837 Fax: 61-7-37357760
[1] Inroducion I is a well known fac ha he Dickey-Fuller (DF) es ends o over-rejec he null of a uni roo in he presence of GARCH errors. Kim and Schmid (1993) firs invesigaed he reliabiliy of he DF es in his regard. They concluded ha he DF es is generally oversized and ha he problem becomes serious when he underlying GARCH process is nearly degenerae, nearly inegraed and he volailiy parameer is relaively large ha is, when he GARCH effec is srong. Similar resuls were found in Haldrup (1994) and Lin e al. (2003). Valkanov (2005) also demonsraed ha wih srong GARCH effecs, while he small-sample DF disribuion appears o converge o he asympoic disribuion, he convergence is raher slow. Because macroeconomic and financial ime series are ypically characerized by srong GARCH effecs, i may rise suspicion abou he resul in he sandard pracice of he DF es in empirical work. The over-rejecion problem appears in a class of modified DF ess oo. According o Cook (2006), where he sandard DF es and several modified DF ess were examined, while on he whole he modified ess appear o be robus o GARCH, he advanage of using hem over he sandard es is moderae. When he sample size ges larger, since he sandard DF es saisic moves closer o is asympoic disribuion, he advanage of he modified ess becomes even less noiceable. Similar o Kim and Schmid (2003), Cook (2006) idenified ha he over-sizing of all DF-ype ess is mainly driven by he volailiy raher han he persisence of he variance process. 1
An imporan bu commonly negleced or inadequaely reaed issue concerning he oversizing problem for uni-roo ess is he effec of iniial values of a simulaed I(1)-GARCH series. Kim and Schmid (1993), Lin e al. (2003), Valkanov (2005) and Cook (2006) did no deal wih such an effec in heir Mone Carlo exercises. Haldrup (1994) is he only paper in he uni roo es lieraure ha deal wih his effec. In order o ge rid of he effec Haldrup (1994) suggesed generaing addiional en observaions for each simulaing pah and hen deleed he firs en from he pah. Alhough such a rule seems o work well in he eliminaion of he iniial effec when he GARCH effec is moderae, i does no work well when he effec becomes srong. Lee and Tse (1996), in heir sudy regarding coinegraion ess, argued ha when he GARCH process is nearly inegraed and nearly degenerae he simulaed process akes a long while o sele down. In such a case discarding he firs 500 observaions is required o ge rid of he iniial effec. Though he same simulaion rule should be adoped for uni roo ess, i has been largely ignored in he uni-roo lieraure. In his paper, exensive Mone Carlo simulaions are conduced o examine he influence of iniial values of a simulaed I(1)-GARCH process o several DFype uni roo ess. Basically, i is found ha he over-rejecion problem for he sandard DF es wih GARCH errors can become much more serious if he iniial effec is no rimmed. In conras, he iniial effec does raher limied impac o he modified ess. The resul due o he mixed effec of GARCH and he iniial condiions is linked o he lieraure of Kim e al. (2002) and Cook (2003, 2004) 2
where he oversizing problem for he DF-ype ess is caused by an early break in he innovaion variance. The paper proceeds as follows. In secion 2, several DF-ype ess are presened. Secion 3 saes design and repors resuls of he Mone Carlo experimen. Secion 4 concludes. [2] Dickey-Fuller-ype Tess Consider he model y ( y ) µ = ρ µ + ε, (1) 1 where y, = 1,2,, T, are observaions, ε is a serially-uncorrelaed zero-mean process, and µ, ρ are unknown parameers. Le y = y y and 1 T y T = 1 y =. The sandard DF es ess he uni roo hypohesis H 0 : ρ = 1 agains he alernaive H : 1 1 ρ < using a es saisic ( ˆ ) τ = ρ 1/ se( ˆ ρ) (2) where ˆ / is he OLS esimaor of ρ in (1) and se( ˆ ρ ) is is T T 2 ρ = = 2yy 1 = 2y 1 sandard error. Despie is populariy, he DF es is known o have low power when ρ is close o one. Several aemps have been made o improve he power of he sandard DF es in he lieraure. Panula e al. (1994) inroduced a weighed symmeric (WS) esimaor ˆ ρ = yy /( y + T y ) (3) T T 1 2 1 T 2 ws = 2 1 = 2 = 1 3
of ρ in (1). Based on he WS esimaor, hey suggesed a modified DF es (DF- WS): T 1 T 1 2 1 2 ( ˆ WS = ws ws 1) y + T y = 2 = 1 τ σ ρ 1/2 (4) where σ = ( T 2) [ w( y ˆ ρ y ) + (1 w )( y ˆ ρ y ) ] wih 2 1 T 1 2 T 2 ws = 1 ws 1 = 2 + 1 ws + 1 w = ( 1) / T. Basically, he power gain of his es comes from he uilizaion of he reverse auoregressive (AR) represenaion in addiion o he forward AR represenaion. Power improvemen may also be achieved using differen mean-adjusmen schemes oher han OLS as in he sandard DF es. Ellio e al (1996) proposed a modified es using he applicaion of local-o-uniy demeaning via generalized leas squares (GLS) esimaion. The locally demeaned series y is derived as y ˆ = y β where ˆβ is obained by regressing yα = ( y1, y2 αy1,..., yt αyt 1) on z α = (1,1 α,...,1 α ) wih α = 1 (7 / T ). The resuling uni roo es, denoed as DF-GLS, is he -raio of φ from he regression: y = φ y 1 + ε. On he oher hand, Shin and So (2001) suggesed he use of recursively adjused mean 1 y i= 1 yi =. The recursively mean-adjused version of DF es (DF-REC) has he same forma as (2), excep ha ˆρ and of ( y y 1) on ( y 1 y 1). se( ˆ ρ ) are obained from he regression While he power advanage of he above menioned modified DF ess has been confirmed in subsequen sudies and become recognized in applied research 4
(see, for example, Leybourne e al. (2005)), heir robusness in he presence of GARCH has been discussed only very recenly in Cook (2006). [3] Mone Carlo Experimen and Resuls The daa-generaing process o be considered is a drifless inegraed process wih GARCH innovaions. The process is y = y + ε, =1,,T+d, (5) 1 where ε is assumed o be a GARCH process: ε = hη, where η is i.i.d. N(0,1), h = φ + φε + φ h. (6) 2 0 1 1 2 1 GARCH parameers ypically, φ 1 is know as he volailiy parameer and φ 2 he persisence parameer are se o reflec he empirical findings. Specifically, φ 1 is se beween 0 and 0.349 and φ 2 beween 0.65 and 0.95, bu φ 1 + φ 2 never goes larger han 0.999, and φ 0 is calibraed o make he uncondiional variance equal 1 in all cases: φ0 = 1 φ1 φ2. The iniial value of he uni-roo process is se y 0 = 0 and he iniial variance h 0 = 1. The pseudo random innovaions { } T + ε d = 1 are drawn from N(0, h ) and used o consruc he uni-roo process for T=100, 500, 1000, 2000 and d=500. Two ses of series are used for uni-roo esing: series [1] conains observaions from 1 o T, and series [2] from d+1 o T+d. In he firs series, he iniial effec is lef unreaed while he second series, according o Lee 5
and Tse (1996), he iniial effec is off-loaded. 1 All simulaions are based on 10,000 replicaions and done by GAUSS. All rejecion frequencies are calculaed a he nominal 5% significance level. 2 The Mone Carlo resuls are repored in Table 1. We summarize and commen here he resuls in Table 1. Firs, he simulaion resul shows ha in some cases he difference of size disorion beween he sandard DF es and he modified ess can be subsanially large. In paricular, as he sum of GARCH parameers becomes exremely close o he boundary of inegraion and he volailiy parameer is relaively large (say, φ1+ φ2 0.99 and φ1 0.24 ), while he oversizing problem of he sandard DF es deerioraes considerably, he modified ess only ge slighly worse. For example, when ( φ1, φ 2) = (0.349, 0.65) and T=500, he rejecion frequency of he sandard DF es is 0.358, bu 0.087, 0.146 and 0.109 for DF-GLS, DF-WS and DF-REC, respecively. Also, he ess end o show differen paerns of size disorion corresponding o T. DF es is less disored when T is larger if φ1+ φ2 0.995 and becomes irregular if φ 1 + φ 2 = 0.999 ; he DF-GLS es seems unaffeced by T; he DF-WS and he DF-REC ess appear more and hen less disored as T ges larger. Second, he Mone Carlo resul shows ha he influence of iniial condiions for he daa generaion process may conribue a significan proporion of he size disorion for he sandard DF es. Indeed, in some cases, he over-rejecing 1 The appropriaeness of his eliminaion rule has also been confirmed by he auhor using simulaion wih a wide class of GARCH models. The resul is available upon reques 2 Criical values are obained by simulaion wih 50,000 draws for each es a differen sample sizes and available upon reques. 6
problem due o GARCH migh look much more serious han i acually is provided ha such an influence has been eliminaed. For example, he rejecion rae drops from 0.358 o 0.108 for ( φ1, φ 2) = (0.349,0.65) a T=500 when he effec is gone. Accordingly, failing o noice he iniial effec, he severe over-rejecion of he sandard DF es wih srong GARCH errors is likely o be oversaed. On he conrary, he iniial effec does no inflae he size of he DF-GLS and he DF-REC ess and only causes a bi more size disorion for he DF-REC es. Besides, wihou he iniial effec all ess seem very sable over a wide range of sample size, implying ha he ess converge o he asympoic disribuion raher fas. Third, even if he iniial effec is removed, he GARCH effec iself sill brings abou some size disorion for he DF-ype ess. Again, since he uncondiional kurosis increases as φ 1 increases when φ 1 + φ 2 is fixed, given he value of φ 1 + φ 2, he size disorion is usually larger wih a larger φ 1. 3 Size improvemen of he modified DF ess (wih he DF-REC es as an excepion) over he sandard DF es is obvious, bu no very sriking. The DF-GLS es appears o be he mos robus one. In he presence of GARCH, he DF-GLS es ends o over-rejec bu he rejecion frequency never goes larger han 0.088 and his happens only when he GARCH effec is exremely srong. Finally, i is worh noing ha he simulaion resul due o he iniial effec can acually be relaed o he oversized problem for he same class of uni-roo ess considered previously when here exiss a variance break. According o Kim e al. (2002), he sandard DF es severely oversizes when applied o a uni-roo 3 2 2 1 2 The uncondiional kurosis for an GARCH(1,1) is 3[1 ( φ + φ ) 2 φ ] [1 ( φ + φ ) ]. 1 2 1 1 2 7
process experiencing an abrup decrease in innovaion variance if he decrease is large and occurs in he early sage of he process. In conras, as shown in Cook (2003, 2004), he modified DF ess seem quie robus o such a break. As for he case of GARCH, Lee and Tse (1996) noed ha when a GARCH model is nearly degenerae and wih a relaive large volailiy parameer he iniial variance h 0 = 1 is oo far in he righ ail of is saionary disribuion, so he simulaed imevarying variance h ends o decline as ges larger and his declining will las for a while. In oher words, if a simulaed GARCH series is unrimmed i will behave like a pah arisen from a seing where innovaion variance undergoes a sequence of downward breaks soon afer he sar and he breaks become larger and las longer when he GARCH process is closer o degenerae and wih a larger volailiy parameer. This should help o explain he puzzling size properies regarding he sandard DF es and he modified ess in he presence of GARCH, wih and wihou he iniial effec. Of course, Kim e al. (2002) and Cook (2003, 2004) obain heir resuls only concerning a sudden large drop in variance, bu i is no unreasonable o expec ha similar resuls should occur for a series of small bu prolonging variance reducion arising in he early sage of he uni-roo pah. As a maer of fac, according o he simulaion resul in Table 2 (please see Appendix I for deails) early downward can cause significan size disorion for he sandard DF es regardless of wheher he breaks are one-sho or sequenial. In conras, variance breaks of all sors resul in much less size disorion for he modified DF ess. 8
[4] Conclusion In his paper, exensive Mone Carlo simulaions are carried ou o sudy he size performance of a class of Dickey-Fuller ess in he presence of GARCH errors, wih and wihou he influence of iniial values of he underlying process. In addiion o he sandard DF es, hree modified DF ess are considered. Basically, simulaion resuls show ha he compound effec from GARCH and iniial condiions can cause significan upward size disorion for he sandard DF es bu he problem is much less severe if he effec is solely from GARCH. In conras, he modified ess seem insensiive o he iniial effec. Even if he iniial effec has been suiably conrolled, all ess suffer size disorion caused by GARCH o some degree. Among he ess, he DF-GLS es seems o have he leas size disorion. References Cook, S., 2003, Size and power properies of powerful uni roo ess in he presence of variance breaks. Physica A 217, 432-448. Cook, S., 2004, Finie-sample properies of he GLS-based Dickey-Fuller es in he presence of breaks in innovaion variance. Ausrian Journal of Saisics 33, 305-314. Cook, S., 2006, The robusness of modified uni roo ess in he presence of GARCH, Quaniaive Finance 6, 359-363. Dickey, D., Fuller, W., 1979. Disribuion of he esimaors for auoregressive ime series wih a uni roo. Journal of he American Saisical Associaion 74, 427-431. Ellio G., Rohenberg, T.J., Sock J.H. 1996. Efficien ess for an auoregressive uni roo. Economerics 64, 813-836. 9
Haldrup, N., 1994, Heeroskedasiciy in non-saionary ime series: some Mone Carlo evidence. Saisical Papers 35, 297-307. Kim K., Schmid, P., 1993. Uni roo ess wih condiional heeroskedasiciy. Journal of Economerics 59, 287-300. Kim T., Leybourne S., Newbold, 2002, Uni roo ess wih a break in innovaion variance, Journal of Economerics 109, 365-387. Lee, T.H., Tse, Y., 1996, Coinegraion ess wih condiional heeroskedasiciy. Journal of Economerics 73, 401-440. Leybourne, S., Kim, T. and Newbold, P., 2005. Examinaion of some more powerful modificaions of he Dickey-Fuller es. Journal of Time Series Analysis 26, 355-369. Ling, S., Li, W.K., McAleer, M., 2003. Esimaion and esing for uni roo process wih GARCH(1,1) errors: heory and Mone Carlo evidence. Economeric Reviews 22, 179-202. Panula, S. Gonzalez-Frarias, G., Fuller, W., 1994. A comparison of uni roo es crieria. Journal of Business and Economic Saisics 8, 153-162. Shin, D., So, B., 2001. Recursive mean adjusmen for uni roo ess. Journal of Time Series Analysis 5, 595-612. Valkanov, R., 2005. Funcional cenral limi heorem approximaions and he disribuion of he Dickey-Fuller es wih srongly heeroskedasic daa. Economics Leers 86, 427-433. 10
Appendix I To sudy he empirical size of he DF-ype ess wih breaks in innovaion variance, a uni-roo process is generaed according o equaion (5) wih ε = ησ, where η is an i.i.d. N(0,1) and σ is he scale of variance a : for an abrup break σ σ = σ for T, 1 1 for T <, 2 1 and for a series of smooh breaks beween T 1 and T 2, σ 1 for T1, σ2 σ 1 σ = σ1+ ( T1) for T1 < T2, T2 T1 σ 2 for T2 <. Denoing he break raio ( σ 2 / σ 1 ) as δ, he values δ {0.25,0.4,0.6} are considered. Four cases of breaks are examined: CASE I assumes a single big break arising righ afer T 1 ; CASE II~CASE IV winess a sequence of small breaks of same size ( ( σ2 σ1) /( T2 T1) ) beween T 1 and T 2. Breaks sar a eiher T 1 =1 or T 1 =0.1*T+1, and for CASE II~CASE IV hey end a T 2 = T 1 +0.1*T-1 (CASE II), T 2 = T 1 +0.2*T-1 (CASE III), or T 2 = T 1 +0.3*T-1 (CASE IV), where T is eiher 100 or 500. Since he iniial effec is no a concern, d=0 in (5). The simulaion resul of rejecion frequency for he DF-ype ess a he 5% significance level is given in Table 2. From Table 2 i is clear ha early downward breaks cause significan size disorion for he sandard DF es regardless of wheher he breaks are one-sho or sequenial. Acually, sequenial small breaks can cause more size disorion han 11
an abrup large break if he breaks begin a T 1 =1 while he opposie is rue if he breaks occur laer. On he oher hand, variance breaks of all sors bring abou much less serious size problems for he modified DF ess. 12
Table 1: Rejecion frequency a 5% level in he presence of GARCH [a] T=100 Iniial effec unreaed Iniial effec off-loaded (φ 1,φ 2) φ 1+φ 2 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC (0.300,0.650) 0.950 0.092 0.067 0.065 0.069 0.080 0.070 0.067 0.075 (0.200,0.750) 0.950 0.077 0.063 0.061 0.066 0.071 0.066 0.064 0.067 (0.100,0.850) 0.950 0.059 0.058 0.052 0.055 0.062 0.055 0.053 0.055 (0.000,0.950) 0.950 0.053 0.049 0.048 0.051 0.048 0.050 0.047 0.050 (0.340,0.650) 0.990 0.173 0.079 0.088 0.081 0.095 0.080 0.078 0.086 (0.240,0.750) 0.990 0.127 0.067 0.070 0.065 0.087 0.074 0.071 0.076 (0.140,0.850) 0.990 0.087 0.057 0.058 0.057 0.072 0.064 0.062 0.067 (0.040,0.950) 0.990 0.055 0.053 0.049 0.054 0.055 0.050 0.049 0.053 (0.345,0.650) 0.995 0.223 0.077 0.091 0.071 0.099 0.080 0.083 0.090 (0.245,0.750) 0.995 0.168 0.068 0.075 0.060 0.093 0.072 0.068 0.075 (0.145,0.850) 0.995 0.101 0.059 0.059 0.056 0.076 0.063 0.063 0.067 (0.045,0.950) 0.995 0.056 0.049 0.050 0.052 0.053 0.051 0.045 0.049 (0.349,0.650) 0.999 0.361 0.097 0.128 0.084 0.100 0.079 0.079 0.084 (0.249,0.750) 0.999 0.250 0.082 0.097 0.070 0.094 0.074 0.074 0.079 (0.149,0.850) 0.999 0.135 0.064 0.065 0.053 0.080 0.058 0.055 0.060 (0.049,0.950) 0.999 0.063 0.058 0.054 0.056 0.056 0.055 0.052 0.053 [b] T=500 Iniial effec unreaed Iniial effec off-loaded (φ 1,φ 2) φ 1+φ 2 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC (0.300,0.650) 0.950 0.077 0.065 0.072 0.075 0.075 0.063 0.070 0.074 (0.200,0.750) 0.950 0.071 0.059 0.068 0.068 0.069 0.059 0.066 0.067 (0.100,0.850) 0.950 0.055 0.054 0.053 0.054 0.056 0.052 0.051 0.052 (0.000,0.950) 0.950 0.048 0.045 0.049 0.047 0.050 0.050 0.050 0.049 (0.340,0.650) 0.990 0.142 0.073 0.093 0.088 0.101 0.080 0.094 0.101 (0.240,0.750) 0.990 0.123 0.065 0.081 0.081 0.092 0.079 0.086 0.088 (0.140,0.850) 0.990 0.090 0.065 0.075 0.074 0.084 0.069 0.075 0.077 (0.040,0.950) 0.990 0.054 0.053 0.053 0.052 0.055 0.056 0.056 0.056 (0.345,0.650) 0.995 0.188 0.077 0.102 0.098 0.104 0.082 0.093 0.098 (0.245,0.750) 0.995 0.174 0.077 0.104 0.093 0.104 0.078 0.091 0.097 (0.145,0.850) 0.995 0.133 0.069 0.080 0.073 0.090 0.069 0.076 0.078 (0.045,0.950) 0.995 0.060 0.053 0.054 0.051 0.065 0.058 0.056 0.057 (0.349,0.650) 0.999 0.358 0.087 0.146 0.109 0.108 0.082 0.099 0.102 (0.249,0.750) 0.999 0.336 0.082 0.130 0.094 0.105 0.084 0.098 0.102 (0.149,0.850) 0.999 0.239 0.075 0.101 0.075 0.096 0.068 0.080 0.081 (0.049,0.950) 0.999 0.089 0.055 0.061 0.057 0.069 0.053 0.057 0.057 13
[c] T=1000 Iniial effec unreaed Iniial effec off-loaded (φ 1,φ 2) φ 1+φ 2 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC (0.300,0.650) 0.950 0.076 0.062 0.074 0.083 0.077 0.070 0.074 0.082 (0.200,0.750) 0.950 0.064 0.063 0.067 0.073 0.068 0.062 0.065 0.073 (0.100,0.850) 0.950 0.057 0.054 0.055 0.060 0.058 0.054 0.057 0.064 (0.000,0.950) 0.950 0.049 0.055 0.050 0.053 0.054 0.054 0.051 0.055 (0.340,0.650) 0.990 0.127 0.077 0.094 0.100 0.104 0.084 0.099 0.109 (0.240,0.750) 0.990 0.115 0.074 0.089 0.094 0.096 0.083 0.089 0.101 (0.140,0.850) 0.990 0.081 0.065 0.073 0.079 0.080 0.067 0.076 0.081 (0.040,0.950) 0.990 0.058 0.054 0.056 0.061 0.062 0.053 0.057 0.060 (0.345,0.650) 0.995 0.160 0.083 0.108 0.109 0.103 0.088 0.098 0.110 (0.245,0.750) 0.995 0.153 0.080 0.102 0.105 0.107 0.084 0.096 0.107 (0.145,0.850) 0.995 0.120 0.070 0.087 0.092 0.094 0.074 0.085 0.093 (0.045,0.950) 0.995 0.066 0.057 0.059 0.066 0.062 0.059 0.058 0.064 (0.349,0.650) 0.999 0.302 0.083 0.130 0.115 0.115 0.084 0.101 0.116 (0.249,0.750) 0.999 0.304 0.092 0.131 0.113 0.108 0.083 0.097 0.109 (0.149,0.850) 0.999 0.243 0.079 0.109 0.097 0.106 0.081 0.090 0.099 (0.049,0.950) 0.999 0.106 0.060 0.068 0.068 0.081 0.063 0.064 0.068 [d] T=2000 Iniial effec unreaed Iniial effec off-loaded (φ 1,φ 2) φ 1+φ 2 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC (0.300,0.650) 0.950 0.065 0.058 0.064 0.068 0.069 0.057 0.065 0.070 (0.200,0.750) 0.950 0.060 0.056 0.057 0.057 0.058 0.058 0.058 0.060 (0.100,0.850) 0.950 0.055 0.056 0.057 0.057 0.056 0.050 0.052 0.053 (0.000,0.950) 0.950 0.046 0.051 0.049 0.049 0.050 0.050 0.051 0.050 (0.340,0.650) 0.990 0.108 0.078 0.091 0.095 0.094 0.074 0.092 0.098 (0.240,0.750) 0.990 0.102 0.073 0.089 0.091 0.091 0.076 0.087 0.093 (0.140,0.850) 0.990 0.083 0.064 0.073 0.077 0.081 0.068 0.074 0.080 (0.040,0.950) 0.990 0.054 0.050 0.052 0.053 0.054 0.053 0.054 0.053 (0.345,0.650) 0.995 0.128 0.076 0.094 0.097 0.105 0.083 0.094 0.103 (0.245,0.750) 0.995 0.128 0.080 0.098 0.101 0.103 0.080 0.093 0.099 (0.145,0.850) 0.995 0.106 0.075 0.087 0.089 0.083 0.070 0.080 0.085 (0.045,0.950) 0.995 0.065 0.057 0.057 0.059 0.059 0.058 0.059 0.059 (0.349,0.650) 0.999 0.229 0.080 0.112 0.101 0.106 0.082 0.099 0.110 (0.249,0.750) 0.999 0.231 0.079 0.111 0.101 0.107 0.088 0.103 0.112 (0.149,0.850) 0.999 0.214 0.082 0.103 0.095 0.109 0.087 0.098 0.104 (0.049,0.950) 0.999 0.104 0.061 0.071 0.067 0.086 0.065 0.069 0.069 14
Table 2: Rejecion frequency a he 5% level wih variance breaks [a] Variance breaks saring a T 1 =1 T=100 T=500 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC δ=0.25 CASE I 0.149 0.041 0.046 0.039 0.073 0.043 0.048 0.046 CASE II 0.366 0.076 0.093 0.048 0.367 0.064 0.095 0.051 CASE III 0.379 0.086 0.107 0.064 0.373 0.080 0.113 0.064 CASE IV 0.323 0.089 0.108 0.066 0.315 0.083 0.113 0.064 δ=0.4 CASE I 0.088 0.046 0.048 0.048 0.057 0.047 0.049 0.046 CASE II 0.184 0.051 0.057 0.041 0.180 0.048 0.059 0.040 CASE III 0.215 0.055 0.063 0.042 0.209 0.055 0.065 0.045 CASE IV 0.208 0.061 0.066 0.047 0.203 0.059 0.068 0.046 δ=0.6 CASE I 0.069 0.050 0.048 0.051 0.049 0.044 0.045 0.045 CASE II 0.087 0.046 0.047 0.047 0.074 0.040 0.045 0.042 CASE III 0.107 0.045 0.049 0.047 0.094 0.041 0.046 0.041 CASE IV 0.117 0.046 0.050 0.046 0.102 0.040 0.048 0.040 [b] Variance breaks saring a T 1 =0.1*T+1 T=100 T=500 DF DFGLS DFWS DFREC DF DFGLS DFWS DFREC δ=0.25 CASE I 0.400 0.079 0.102 0.055 0.406 0.080 0.120 0.061 CASE II 0.398 0.095 0.116 0.066 0.399 0.084 0.124 0.073 CASE III 0.336 0.095 0.114 0.070 0.331 0.083 0.117 0.071 CASE IV 0.267 0.083 0.099 0.069 0.260 0.077 0.108 0.068 δ=0.4 CASE I 0.228 0.057 0.066 0.045 0.225 0.053 0.067 0.045 CASE II 0.239 0.066 0.072 0.048 0.230 0.057 0.072 0.046 CASE III 0.223 0.066 0.073 0.051 0.213 0.060 0.073 0.045 CASE IV 0.199 0.069 0.075 0.053 0.190 0.058 0.072 0.047 δ=0.6 CASE I 0.122 0.050 0.049 0.043 0.116 0.051 0.055 0.044 CASE II 0.126 0.050 0.050 0.043 0.120 0.050 0.058 0.044 CASE III 0.128 0.050 0.052 0.043 0.122 0.053 0.058 0.045 CASE IV 0.131 0.051 0.054 0.044 0.120 0.053 0.059 0.044 Noe. CASE I: a one-sho variance break a T 1 ; CASE II: sequenial breaks saring a T 1 and ending a T 2 =T 1 +0.1*T-1; CASE III: sequenial breaks saring a T 1 and ending a T 2 =T 1 +0.2*T-1; CASE IV: sequenial breaks saring a T 1 and ending a T 2 =T 1 +0.3*T-1. 15