THE IMPACT OF MISDIAGNOSING A STRUCTURAL BREAK ON STANDARD UNIT ROOT TESTS: MONTE CARLO RESULTS FOR SMALL SAMPLE SIZE AND POWER

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1 THE IMPACT OF MISDIAGNOSING A STRUCTURAL BREAK ON STANDARD UNIT ROOT TESTS: MONTE CARLO RESULTS FOR SMALL SAMPLE SIZE AND POWER E Moolman and S K McCoskey * A Absrac s discussed by Perron (989), a common problem when esing for uni roos is he presence of a srucural break ha has no been accouned for in he esing procedure. In such cases, uni roo ess are biased o non-rejecion of he null hypohesis of nonsaionariy. These resuls have been discussed using asympoic heory and large samples in papers by Leybourne and Newbold (2000), Monanes and Reyes (998) and Lee, Huang, and Shin (997). In his paper we invesigae he impac of ignoring srucural breaks on sample sizes of more ineres o empirical economiss and show he resuls on power and size for boh ess of he null of nonsaionariy (ADF and Phillips-Perron) and he null of saionariy (KPSS). We are also able o give some guidelines on break placemen which can cause he rapid flipping of rejecion probabiliies as discussed in Leybourne and Newbold (2000). Finally, we provide examples from ime series daa in Souh Africa o show he danger of misdiagnosis and he resuling misspecificaions ha can occur.. Inroducion The mos imporan implicaion of he uni roo revoluion is ha under his hypohesis random shocks have a permanen effec on he sysem. However, many such series may also conain srucural breaks, and herefore i is imporan o assess carefully he reliabiliy of uni roo ess in he presence of a srucural break. This paper compares he performance of uni roo and saionariy ess in he presence of a srucural break, allowing for boh he null of saionariy and he null of non-saionariy. Such a comparison can be of use as he debae coninues as o he appropriae null hypohesis for differen applicaions. * Respecively Senior Lecurer a he Deparmen of Economics, Universiy of Preoria, Preoria 000, Republic of Souh Africa and Associae Professor a he Unied Saes Naval Academy, Maryland, USA and Honorary Professor a he Universiy of Preoria. emoolman@hakuna.up.ac.za J.STUD.ECON.ECONOMETRICS, 200, 27() 57

2 The mos influenial conribuion wih respec o he effec of srucural breaks on uni roo ess is found in Perron (989). He considers he null hypohesis ha a ime series has a uni roo agains he alernaive ha he process is rend-saionary. Under boh he null and alernaive hypoheses a one-ime change in he level and/or in he slope of he rend funcion is allowed, assuming ha he ime of he break is known. He shows ha asympoically sandard ess of he uni roo hypohesis agains rend saionariy canno rejec he uni roo hypohesis if he rue daa generaing process (DGP) is ha of saionary flucuaions around a rend funcion which conains a one-ime break. Therefore hese ess are biased owards acceping he null wheher i is rue or no, which means ha i decreases he poenial power of he ess. Monanes and Reyes (998) exend he resuls of Perron (989) for he Dickey Fuller (DF) es and show wih Mone Carlo sudies ha he uni roo ess are biased in favour of non-rejecion of he uni roo hypohesis. In conras wih Perron (989) where i is supposed ha he effecs of differen ypes of srucural breaks are idenical, hey showed ha he DF ess reac differenly o differen ypes of srucural breaks. They also show ha he ess reac differenly o breaks a he beginning of he sample and breaks a he end of he sample, and ha he ess also reac differenly o differen sizes of he breaks. The paper by Leybourne and Newbold (2000) also considers he ADF -es and, like Monanes and Reyes, considers he imporance of break placemen. In heir paper, he auhors find ha here is a poenial break placemen which causes a rapid flip from very many rejecion o very few wih a saionary DGP conaining a srucural brea. They furher predic ha in smaller samples, which hey do no invesigae, his flipping could occur in breaks which are in he firs half of he daa se. Idenifying he exac placemen of his break flipping could be an imporan ool for applied economericians in inerpreing esing resuls. When he uni roo is he null hypohesis o be esed, hen he way in which classical hypohesis esing is carried ou ensures ha he null hypohesis is acceped unless here is srong evidence agains i. Therefore an alernaive explanaion for he common failure o rejec a uni roo is simply ha mos economic ime series are no very informaive abou wheher or no here is a uni roo, or equivalenly, ha sandard uni roo ess are no very powerful agains relevan alernaives. For example, De Jong e al. (989) provide evidence ha he Dickey-Fuller ess have low power agains sable auoregressive alernaives wih roos near uniy, and Diebold and Rudebusch (990) show ha hey also have low power agains fracionally inegraed alernaives. By esing boh he uni roo hypohesis and he saionariy hypohesis, i is possible o disinguish series ha appear o be saionary, series ha appear o have a uni roo and series for which he daa (or he ess) are no sufficienly informaive o be sure wheher hey are saionary or inegraed. Kwaikowski e al. (992) (hereafer KPSS), considered Lagrange Muliplier (LM) ess wih a saionary or rend saionary null hypohesis raher han a uni roo hypohesis. However, he KPSS es of he null of saionariy faces a parallel problem as he null of non-saionariy in he presence of a srucural break. 58 J.STUD.ECON.ECONOMETRICS, 200, 27()

3 Lee, Huang and Shin (997) clarify he naure of he effecs of a srucural break on he KPSS saionariy ess. They show ha saionariy ess ignoring he exising break diverge and are biased oward rejecing he null hypohesis of saionariy in favor of he false alernaive uni roo hypohesis. This resul can be compared wih he findings of Perron (989) and Monanes and Reyes (998) ha uni roo ess are biased oward acceping he false uni roo null hypohesis. The size disorion problem of saionariy ess parallels he power loss problem of uni roo ess. However, like Perron (989) hey only looked a he asympoic consequences and no he small sample case. In addiion, Lee e al. (997) only consider breaks in he firs halve of he sample, bu since he KPSS saisic does no rea he errors symmerically he effec of a break in he second half of he sample may be differen. Therefore, in his sudy he effec of breaks in boh halves of he sample will be esed. However, all he resuls described above have been esablished asympoically and migh differ subsanially for small samples ha are of pracical ineres o he applied economis. In his paper we expand on he resuls of Perron (989) for uni roo ess, and he resuls of Lee e al. (997) for saionariy ess o show he small sample resuls on power and size in he presence of a srucural break. Our esing will be more general han hese ess, allowing for differen ypes of srucural breaks, differen break magniudes, differen placemens of he break and also more general error srucures. The paper is oulined as follows: Secion 2 summarizes ess and explains he Mone Carlo design for he comparison of he ess. Secion summarizes he resuls of he Mone Carlo experimens. Secion 4 presens an applicaion of he ess for Souh Africa, and Secion 5 provides some concluding houghs. 2. The Mone Carlo design The goal of his Mone Carlo sudy is o compare he size and power of uni roo and saionariy ess in he presence of a srucural break, wih special reference o he small sample properies. Three ess are considered: he Dickey Fuller (DF), Phillips-Perron (PP) and Kwaikowsky e al. (KPSS). The firs wo ess are consruced under he null of a uni roo and he las one under he null of saionariy. The power and size of hese ess are compared across boh he null of saionariy and he null of non-saionariy. 2. The es saisics The DF and PP ess are ess of he null hypohesis of non-saionariy agains he alernaive of saionariy. The DF es involves esimaing he auoregressive (AR) coefficien (ρ) of he dependen variable and hen deermining wheher o accep or rejec he null hypohesis of a uni roo (ρ=) by comparing he following saisic wih he appropriae DF criical value: J.STUD.ECON.ECONOMETRICS, 200, 27() 59

4 DF -saisic: ( ρ ˆ ) σˆ ρ ˆ T () By definiion, he DF es assumes ha he daa generaing process (DGP) is an AR() process. To allow for a higher AR order, he augmened Dickey-Fuller (ADF) es is used. Boh DF ess assume ha he errors are i.i.d. Said and Dickey (984) showed ha he ADF es can be used when he error process is a moving average (MA). In his sudy he DF es will be used excep when he errors conain an AR componen, in which case he ADF es will be used. The PP es is a generalizaion of he DF-procedure ha allows for fairly mild assumpions concerning he error disribuion, by allowing he errors o be weakly dependen and heerogeneously disribued. The es is performed by comparing he following es saisic o he relevan criical value: / 2 2 ( ω γ ) γ0 0 0 Ts0 = ω 2ωσˆ (2) where: ω 2 = γ 0 q j + 2 γ j q = + j () γ j = T T εˆ εˆ = j+ j (4) Alhough he ADF and PP ess have very low power especially for a near uniy AR erm or rend saionariy, Mone Carlo sudies have shown ha he PP es has greaer power han he ADF es (Enders 995:242). However, Mone Carlo sudies have also shown ha in he presence of negaive MA erms, he PP es ends o rejec he null of a uni roo wheher or no he acual DGP conains a negaive uni roo (Enders 995, p. 242). I is herefore preferable o use he ADF es when he rue model conains negaive MA erms and he PP es when he rue model conains posiive MA erms. However, in pracice he rue DGP is never known, and herefore a safe choice is o use boh ess. Kwaikowsky e al. (992) (hereafer KPSS) developed a es of he null hypohesis ha an observable series is saionary around a deerminisic rend, agains he alernaive ha he series is difference-saionary. They derived a one-sided LM-es under he assumpion ha he errors are whie noise, however, since his is assumpion is no credible in many applicaions hey derived a modified version of he LM saisic ha is valid under more general condiions. In he modified version hey use a similar auocorrelaion correcion o he PP correcions. Tes saisic: ˆ µ = ηµ / s ( ) = T S / s η ( ) (5) 60 J.STUD.ECON.ECONOMETRICS, 200, 27()

5 T 2 2 where s ( ) = T ε + 2T w(s, ) T ε ε (6) = s= = s Barle window: w(s, ) = s /( + ) (7) Wih i.i.d. errors, he KPSS es has approximaely he correc size, excep when T is small and he lag runcaion parameer ( ) is large. The power of he es increases as T increases. There is a rade-off beween correc size and power: choosing large enough o avoid size disorions in he presence of realisic amouns of auocorrelaion will make he ess have very lile power. 2.2 Consrucion of he hypoheses As in Perron (989), hree differen models are considered under boh he null hypoheses: one ha permis an exogenous change in he level of he series, one ha permis an exogenous change in he ime rend, and a mixed model ha allows boh changes. These hypoheses are parameerized as follows: Under he null of a uni roo: s Model A: Model B: Model C: y y y = µ + dd(tb) + y + ε (8) = µ + y + δ DU + ε (9) = µ + y + dd(tb) + δ DU + ε (0) Under he null of rend saionariy: Model A: y = µ + β + δ DU + ε () Model B: y 2 * = µ + β + δ DT + ε (2) Model C: y = µ + β + δ DU + δ DT + ε () 2 where: T B is he ime of he break D(TB) = if = T B +, 0 oherwise (4) DU = if >T B, 0 oherwise (5) DT * = T B if > T B, 0 oherwise (6) J.STUD.ECON.ECONOMETRICS, 200, 27() 6

6 DT = if > T B, 0 oherwise (7) In model A, he null hypohesis of a uni roo is characerized by a dummy variable ha akes he value one a he ime of he break. Under he alernaive hypohesis of rend saionariy, he model allows for a one-ime change in he inercep of he rend funcion. Model B allows a change in he drif parameer a he ime of he break under he null, as opposed o a change in he slope of he rend under he alernaive. Model C allows for boh effecs o ake place simulaneously, in oher words a break consising of a change in he level as well as a change in he ime rend. All he esing will be performed under all hree models, o es wheher differen ypes of breaks have significanly differen effecs on he uni roo and saionariy esing. 2. Experimen dimensions Differen dimensions of he srucural break will be esed in four experimens. The dimensions ha will be considered are he sample size, he placemen of he break, he size of he break, differen error srucures and he disance beween he null and he alernaive. To es he disance beween he null and he alernaive, a more general form of he saionary models will be used by adding an auoregressive dependen variable. The coefficien of his variable (ρ in (8) o (20)), will hen be varied o change he disance beween he null and he alernaive. The focus of his paper is on sample sizes ha are of pracical ineres o he applied economis. In Experimen, he power and size of he ess are compared for differen sample sizes, as he sample size (T) will ake on he following values: T {5, 25, 50, 00}. Monanes and Reyes (998) and Leybourne and Newbold (2000) showed ha he power and size of he DF ess are influenced by he placemen of he break. To es wheher small samples have he same resuls, and o es wheher his is also he case wih he PP and he KPSS ess, he effec of he placemen of he break will also be esed. This is done by assigning he following values o he break fracion (λ), i.e. he raio of pre-break sample size o oal sample size: λ {0,, 0,2, 0,, 0,4, 0,5, 0,6, 0,7, 0,8, 0,9}. Again, he resuls of Monanes and Reyes (998) and Leybourne and Newbold (2000) indicae ha he magniude of he break has a significan impac on he rejecion rae of he DF ess. In Experimen 2 he magniude of he break will be varied o es wheher i is also he case in small samples and for he oher ess, by using he following values for he break magniudes δ and δ 2 : δ {, 2,, 4}, δ 2 {0,2, 0,4, 2, 4}. The AR() parameer ρ is a convenien nuisance parameer o consider, since i naurally measures he disance of he null from he alernaive. Since Mone Carlo sudies have shown ha uni roo ess can no disinguish beween non-saionary and saionary series when he AR coefficien (ρ) is close o uniy (Enders 995: 25), i will be esed wheher ρ has a significan impac on he ess. The 62 J.STUD.ECON.ECONOMETRICS, 200, 27()

7 coefficien (ρ) will ake on he following values: ρ {0, 0,25, 0,5, 0,6, 0,7, 0,8, 0,9}. In Experimen, a lagged dependen variable will be added o model A, B and C under saionariy, which give he following null hypoheses: Model A: Model B: Model C: y y y = µ + β + δ DU + ρy + ε (8) 2 * = µ + β + δ DT + ρy + ε (9) 2 = µ + β + δ DU + δ DT + ρy + ε (20) Wihou a srucural break he ess perform differenly wih differen error srucures, for example, PP ends o always rejec he null when he errors are negaive MA (Enders 995, p. 242), and Lee e al. (997) have shown ha he power of KPSS increases as σ increases wih i.i.d. N(0, σ) errors. Therefore he ess performances in he presence of a srucural break will be esed wih differen error srucures. In Experimen 4 differen error srucures will be used: an AR(p) srucure, a MA(q) srucure and an i.i.d. N(0, σ) srucure where: p {0,75, 0,85, 0,95}, q {-0,8, 0, 0,8}, σ {0,25,, 4}. Unless oherwise specified, he sudy considers λ=0.5 and T=25, and he errors ha are i.i.d. N(0,). The random number generaor used for his sudy is URN22 from Karian and Dudewicz (99). All he experimens were done wih a 5% level of significance, and each experimen was repeaed imes.. Mone Carlo resuls All he resuls of he Mone Carlo sudy are presened as rejecion raes, where a rejecion rae is he percenage of imes ha a es rejecs he null hypohesis. Because he ess are no all derived under he same null hypohesis, i is difficul o compare heir performances direcly. When he DGP is saionary, he rejecion rae of he KPSS measures is size, while he rejecion rae of he DF and PP measure heir power. On he oher hand, when he DGP is non-saionary, he rejecion rae of he KPSS measures is power while he rejecion raes of DF and PP measure heir size. Table provides an overall summary of he resuls.. Experimen In Experimen he size and power of he ess were compared for differen sample sizes (T) o see wheher he resuls of small samples corresponds o he asympoic resuls. A he same ime he influence of he break fracion (λ) was also esed. The rejecion raes of he ADF, PP and KPSS ess for he models, A, B and C explained in Secion, are given for a saionary DGP in Table 2 and for a non-saionary DGP in Table. Wih model A, he ADF and PP ess performs very well under boh hypoheses, as boh has consan power of 0,9999 while he size was also wihin reasonable ranges. J.STUD.ECON.ECONOMETRICS, 200, 27() 6

8 For example, he size of ADF (0,048, 0,076) and he size of PP (0,05, 0,2). For boh ess, he size disorions ge worse as he T decreases. In oher words, a break in he inercep doesn enirely mislead he ess alhough i is a lile bi more misleading in smaller samples. However, i seems as if he break makes he daa look non-saionary o KPSS, since i is biased owards rejecing saionariy wheher he DGP is saionary or no. For example, under saionariy he rejecion rae of KPSS (i.e. size) (0,22, 0,99), while he rejecion rae under nonsaionariy (i.e. power) (0,45, 0,99). In model A, he break fracion (λ) did no have a significan impac on any of he ess, apar from sligh size disorions wih KPSS. Table : Summary of resuls of experimens Experimen Parameer(s) esed Resuls T, λ Model A: Tess perform consisen wih asympoic resuls. Model B & C: ADF and PP rejec when break a beginning, ADF and PP don rejec when break a end. KPSS almos always rejecs. 2 δ 2 δ Insignifican Only significan under saionariy: δ bias of ess oward finding series non-saionary. ρ ρ bias oward finding series non-saionary 4 AR(p) MA(q) N(0,σ) PP beer han DF. DF higher power han PP. DF beer for negaive q, PP beer for posiive q. DF beer han PP; size and power of KPSS increase as σ increases For daa consruced according o model B, he ADF and PP ess seem o always rejec he null, wheher i is rue or no, for breaks a he beginning of he sample. This resul is consisen wih he findings of Monanes and Reyes (998) ha asympoically he es saisics converge o values ha only permi he rejecion of he uni roo null hypohesis if he break is very close o he beginning of he sample. For breaks a he beginning of he sample ADF and PP end o rejec less as T decreases, because i converges slower o he heoreical values in smaller samples. For example, when λ=0,2 he rejecion rae of DF decreases from 0,99 o 0,24 under saionariy while i decreases from 0,82 o 0,0 under non-saionariy, when T decreases from 00 o 5. However, when he break occurs in he firs or second year in a very small sample (T=5 or T=25), he break doesn mislead he ess and hey have he normal characerisics, which means approximaely he correc size and he usual low power. For example, when T=5 and λ=0, (i.e. a break in he firs year) he size of DF is 0,06 while he power is 0,. For breaks 64 J.STUD.ECON.ECONOMETRICS, 200, 27()

9 ha appear laer on, he ADF and PP ess converge o nearly zero rejecion in larger samples (as prediced by Monanes and Reyes (998)), bu again hey converge slower o he heoreical values in smaller samples. This discussion allows us o idenify crucial break placemen values for which he rejecion probabiliies quickly flip from high o low raes of rejecion. The presence of such flip values is an imporan resul of Leybourne and Newbold. I is also ineresing o see ha hese flip values, according o our resuls, depend on T and even he es used. For example, consider he DF es, if T=00, he rejecion rae dramaically decreases from 0,99 o 0,0 as λ decreases from 0,2 o 0,4. For T=50, he rejecion rae does no reach 0,0 unil λ=0,5. Overall, he PP es seems o flip a higher λ han he DF es. For example, for T=00, he rejecion rae is sill as high as 0,57 when λ=0,5 and decreases o 0,07 for λ=0,6. In addiion, in our simulaions for small T (eiher 25 or 5) here is a reurn o high rejecion raes for λ=0,9. This does no happen in he samples where T=50 or 00. Undersanding he imporance of he break placemens may give applied researchers more incenive o use he PP es (a leas in conjuncion wih he DF ess) as he PP ess have beer power and are more useful under a wider variey of break placemens. Noe ha in Model C he flip values occur for earlier break values, especially in he case of he DF es. In model B, KPSS has very high power, which seems o decrease as he T decreases. For example, is power decreases from 0,99 o 0,54 when T decreases from 00 o 5, when λ=0,. However, i shows some serious size disorions, consisen wih he resuls of Lee e al. (997). The size disorions end o be smaller in very small samples, for example i decreases from 0,99 o 0,9 when T decreases from 00 o 5, when λ=0,4. In model C, he change in he inercep (model A) is added o he change in he ime rend (model B), and he resuls are consisen wih ha of only a change in he ime rend (model B). This is consisen wih he iniial resul for model A, namely ha he change in he level doesn mislead he ess. For all hree ess, he power and size only change slighly bu sill have he same properies han wih model B. J.STUD.ECON.ECONOMETRICS, 200, 27() 65

10 Table 2: Rejecion raes wih saionary DGP, for differen T and λ (Experimen ) Model A Model B Model C T= Tes λ=0, DF 0,99 0,99 0,99 0,99 0,99 0,98 0,7 0, 0,99 0,80 0,08 0,06 PP 0,99 0,99 0,99 0,99 0,99 0,99 0,54 0,45 0,99 0,99 0,5 0,25 KPSS 0,96 0,74 0,62 0,54 0,45 0, 0,86 0,54 0,99 0,9 0,99 0,98 λ=0,2 DF 0,99 0,99 0,99 0,99 0,99 0,9 0,78 0,24 0,4 0,20 0,08 0,09 PP 0,99 0,99 0,99 0,99 0,99 0,99 0,95 0,75 0,99 0,96 0,64 0,4 KPSS 0,99 0,92 0,70 0,52 0,99 0,4 0,4 0,2 0,99 0,99 0,89 0,66 λ=0, DF 0,99 0,99 0,99 0,99 0,54 0,65 0,08 0,0 0,0 0,07 0,00 0,00 PP 0,99 0,99 0,99 0,99 0,99 0,99 0,42 0,8 0,9 0,86 0,0 0,04 KPSS 0,99 0,89 0,86 0,80 0,99 0,89 0,95 0,65 0,99 0,99 0,99 0,99 λ=0,4 DF 0,99 0,99 0,99 0,99 0,0 0,22 0,27 0,2 0,00 0,02 0,04 0,06 PP 0,99 0,99 0,99 0,99 0,76 0,95 0,86 0,70 0,26 0,6 0,49 0,9 KPSS 0,97 0,77 0,5 0,9 0,99 0,99 0,47 0,9 0,99 0,99 0,96 0,68 λ=0,5 DF 0,99 0,99 0,99 0,99 0,00 0,0 0,0 0,08 0,00 0,00 0,00 0,00 PP 0,99 0,99 0,99 0,99 0,4 0,57 0,28 0,5 0,00 0,7 0,02 0,05 KPSS 0,9 0,66 0,69 0,66 0,99 0,99 0,99 0,69 0,99 0,99 0,99 0,99 λ=0,6 DF 0,99 0,99 0,99 0,99 0,00 0,00 0,0 0,0 0,00 0,00 0,00 0,02 PP 0,99 0,99 0,99 0,99 0,00 0,07 0,42 0,49 0,00 0,00 0, 0,2 KPSS 0,97 0,77 0,5 0,9 0,99 0,99 0,99 0,54 0,99 0,99 0,99 0,95 λ=0,7 DF 0,99 0,99 0,99 0,99 0,00 0,00 0,0 0,05 0,00 0,00 0,00 0,00 PP 0,99 0,99 0,99 0,99 0,00 0,00 0,9 0,5 0,00 0,00 0,0 0,07 KPSS 0,99 0,90 0,62 0,50 0,99 0,99 0,99 0,72 0,99 0,99 0,99 0,99 λ=0,8 DF 0,99 0,99 0,99 0,99 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,00 PP 0,99 0,99 0,99 0,99 0,00 0,00 0,08 0,27 0,00 0,00 0,00 0,04 KPSS 0,99 0,92 0,69 0,52 0,99 0,99 0,99 0,80 0,00 0,00 0,67 0,5 λ=0,9 DF 0,99 0,99 0,99 0,99 0,00 0,00 0,5 0,2 0,00 0,00 0,67 0,5 PP 0,99 0,99 0,99 0,99 0,00 0,00 0,86 0,84 0,00 0,00 0,76 0,90 KPSS 0,96 0,74 0, 0,22 0,00 0,99 0,97 0,8 0,99 0,99 0,8 0,99 66 J.STUD.ECON.ECONOMETRICS, 200, 27()

11 Table : Rejecion raes wih non-saionary DGP, differen T and λ (Experimen ) Model A Model B Model C T Tes λ=0, DF 0,05 0,05 0,06 0,08 0,75 0,4 0,05 0,06 0,75 0,40 0,04 0,06 PP 0,06 0,07 0,08 0, 0,8 0,5 0,9 0,0 0,82 0,52 0,8 0,0 KPSS 0,99 0,99 0,79 0,45 0,99 0,99 0,85 0,54 0,99 0,99 0,84 0,54 λ=0,2 DF 0,05 0,05 0,05 0,08 0,82 0,46 0, 0,0 0,82 0,46 0,2 0,0 PP 0,06 0,07 0,08 0,2 0,90 0,56 0,7 0,4 0,90 0,56 0,7 0,42 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,77 0,99 0,99 0,99 0,75 λ=0, DF 0,05 0,05 0,05 0,08 0,27 0, 0,2 0,20 0,27 0, 0,2 0,2 PP 0,06 0,07 0,08 0,2 0,42 0,7 0,7 0,26 0,42 0,8 0,8 0,27 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,92 0,99 0,99 0,99 0,92 λ=0,4 DF 0,05 0,05 0,05 0,08 0,0 0,0 0,0 0,05 0,0 0,0 0,0 0,05 PP 0,06 0,07 0,08 0,2 0,0 0,0 0,02 0,06 0,0 0,0 0,02 0,07 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,99 0,99 0,99 0,99 0,99 λ=0,5 DF 0,05 0,05 0,05 0,08 0,00 0,00 0,00 0,02 0,00 0,00 0,00 0,02 PP 0,06 0,07 0,08 0,2 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,0 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,99 0,99 0,99 0,99 0,99 λ=0,6 DF 0,05 0,05 0,05 0,08 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 PP 0,06 0,07 0,08 0,2 0,00 0,00 0,00 0,00 0,00 0,00 0,00 0,00 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,99 0,99 0,99 0,99 0,99 λ=0,7 DF 0,05 0,05 0,06 0,08 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,0 PP 0,06 0,07 0,08 0,2 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,0 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,98 0,99 0,99 0,99 0,99 λ=0,8 DF 0,05 0,05 0,05 0,07 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,0 PP 0,06 0,07 0,08 0,2 0,00 0,00 0,00 0,0 0,00 0,00 0,00 0,0 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,99 0,9 0,99 0,99 0,99 0,94 λ=0,9 DF 0,05 0,05 0,05 0,08 0,00 0,00 0,04 0,06 0,00 0,00 0,04 0,06 PP 0,06 0,07 0,08 0,2 0,00 0,00 0,0 0, 0,00 0,00 0,02 0,0 KPSS 0,99 0,98 0,79 0,45 0,99 0,99 0,92 0,59 0,99 0,99 0,9 0,62 J.STUD.ECON.ECONOMETRICS, 200, 27() 67

12 .2 Experimen 2 In Experimen 2 he effec of he magniude of he break was esed. In Table 4 and 5, he resuls of he ess are summarized under he null of saionariy and he null of non-saionariy respecively. As δ, he size of he once-off change in he inercep, was increased, he power of ADF and PP decreased while he size disorions of KPSS go worse. When he daa is already non-saionary, an increase in he magniude of he break doesn make he daa appear more non-saionary. The resuls for he ADF and PP ess for model B and C are consisen wih he resuls of Lee e al. (997), namely ha he power of he ess are no affeced when differen values of he magniude of srucural breaks (δ and δ 2 ) are used.. Experimen In Experimen he effec of he disance beween he null and he alernaive was esed by varying he AR() coefficien (ρ) of he dependen variable when i was added o each of he hree saionary models. The resuls are summarized in Table 6. The resuls of he power of he ADF and PP ess in model A are counerinuiive as i decreases for ρ<0,5, and increases for ρ>0,5. Wih he changing ime rend model, he power of ADF and PP decreases as ρ increases, since he daa gives less evidence of no being saionary and herefore he ess rejec less ofen. Overall he PP es seems o perform beer han he ADF es, for example wih model A power of PP (0,95, 0,682) while he power of DF (0,02, 0,59). Table 4: Rejecion raes wih rend saionary DGP, for differen δ (Experimen 2) Tes DF PP KPSS DF PP KPSS DF PP KPSS DF PP KPSS δ = δ =2 δ = δ =4 Model A 0,7 0,9 0,2 0, 0,67 0,69 0,02 0,2 0,97 0,00 0,0 0,99 B 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 C 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 Table 5:Rejecion raes wih non-saionary DGP, for differen δ (Experimen 2) Tes DF PP KPSS DF PP KPSS DF PP KPSS DF PP KPSS δ = δ =2 δ = δ =4 Model A 0,05 0,08 0,79 0,05 0,08 0,79 0,05 0,08 0,79 0,05 0,08 0,79 B 0,02 0,0 0,98 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 C 0,02 0,0 0,98 0,00 0,00 0,99 0,00 0,00 0,99 0,00 0,00 0,99 68 J.STUD.ECON.ECONOMETRICS, 200, 27()

13 Table 6: Rejecion raes wih rend saionary DGP, for differen ρ (Experimen ) Model A Model B Model C ADF PP KPSS ADF PP KPSS ADF PP KPSS ρ 0,00 0, 0,67 0,694 0,027 0,284 0,989 0,00 0,022 0,999 0,25 0,070 0,287 0,85 0,0 0,07 0,994 0,000 0,00 0,999 0,50 0,04 0,080 0,98 0,006 0,02 0,997 0,000 0,000 0,999 0,60 0,026 0,048 0,960 0,004 0,007 0,998 0,000 0,000 0,999 0,70 0,02 0,048 0,994 0,00 0,004 0,998 0,000 0,000 0,999 0,75 0,024 0,08 0,999 0,002 0,00 0,998 0,000 0,000 0,999 0,80 0,044 0,95 0,999 0,00 0,002 0,999 0,000 0,000 0,999 0,90 0,59 0,682 0,999 0,000 0,000 0,999 0,000 0,000 0,999 As expeced, he size disorions of KPSS ge worse as ρ comes closer o, which means ha he growing AR coefficien made he daa appear even more nonsaionary o he es. Wih model A, for example, he size of KPSS increased from 0,694 o 0,999 when ρ increased from 0 o 0,9. Wih model C all hree ess perform really badly, since he complexiy of break srucure, he inclusion of boh ypes of breaks and he AR componen, is compleely misleading. In his case he power of DF are always less han 0,00 and he power of PP are always less han 0,022, while he size of KPSS was consan a 0, Experimen 4 Mos of he sudies on uni roo esing in he presence of a srucural break assume ha he errors are i.i.d. N(0,). However, when differen error srucures were creaed in Experimen 4, he resuls (see Tables 7 and 8) indicae ha he performances of he various ess are significanly influenced by differen error srucures. When he errors are i.i.d. N(0,σ), DF has larger power and smaller size han PP. Consisen wih he resuls of Lee e al. (997), he power of KPSS increases as σ increases. The size of KPSS also increases as σ increases. For example, wih model A he power of KPSS increased from 0,7 o 0,787 when σ increased from 0,25 o, and hen o 0,997 when σ subsequenly increased o 4. Wih model A, he size of KPSS increased from 0,9 o 0,694 when σ increased from 0,25 o, and hen o 0,995 when σ increased o 4. Wih AR error erms, he size of DF and PP increases and he power decreases as he AR coefficien (p) increases. However, p does no influence he size or power of he KPSS es. When he errors conain an AR componen, PP has larger power and smaller size han DF. J.STUD.ECON.ECONOMETRICS, 200, 27() 69

14 Table 7: Rejecion raes wih rend saionary DGP, differen error srucures (Experimen 4) Model A Model B Model C ADF PP KPSS ADF PP KPSS ADF PP KPSS AR(0.75) 0,09 0,088 0,999 0,09 0,040 0,999 0,00 0,007 0,999 AR(0.85) 0,06 0,067 0,999 0,06 0,0 0,999 0,00 0,008 0,999 AR(0.95) 0,00 0,052 0,999 0,06 0,024 0,999 0,004 0,008 0,999 MA(-0.8) 0,004 0,99 0,999 0,000 0,90 0,999 0,000 0,487 0,999 MA(0.00) 0,02 0,70 0,999 0,00 0,28 0,999 0,000 0,05 0,999 MA(0.80) 0,07 0,52 0,999 0,025 0,058 0,999 0,002 0,008 0,999 N(0,0.25) 0,00 0,099 0,9 0,000 0,00 0,997 0,000 0,000 0,999 N(0,.00) 0, 0,67 0,694 0,027 0,284 0,989 0,000 0,022 0,999 N(0,4.00) 0,72 0,90 0,995 0,242 0,786 0,999 0,067 0,47 0,999 Table 8: Rejecion raes wih non-saionary DGP, differen error srucures (Experimen 4) Model A Model B Model C ADF 2 PP 2 KPSS ADF PP KPSS ADF PP KPSS AR(0.75) 0,067 0,05 0,999 0,056 0,02 0,999 0,055 0,02 0,999 AR(0.85) 0,070 0,022 0,999 0,06 0,025 0,999 0,060 0,025 0,999 AR(0.95) 0,070 0,04 0,999 0,062 0,09 0,999 0,06 0,09 0,999 MA(-0.8) 0,02 0,927 0,999 0,00 0,000 0,999 0,02 0,927 0,999 MA(0.00) 0,049 0,078 0,999 0,06 0,007 0,999 0,049 0,078 0,999 MA(0.80) 0,09 0,06 0,999 0,09 0,08 0,999 0,09 0,06 0,999 N(0,0.25) 0,055 0,08 0,7 0,000 0,000 0,999 0,000 0,000 0,999 N(0,.00) 0,054 0,08 0,787 0,002 0,004 0,999 0,002 0,004 0,999 N(0,4.00) 0,054 0,08 0,997 0,06 0,028 0,999 0,06 0,028 0,999 Normally he PP es rejecs H 0 wheher or no i is rue in he case of a negaive MA error srucure. Our resuls in he previous experimens ha he break of model A is no very misleading are confirmed, since PP has he normal propery of overrejecing wih negaive MA errors. Bu in model B and C, PP rejecs less if he rue DGP has a uni roo, which means ha he effec of he break is dominaing he effec of he negaive MA errors. When he errors conain a negaive MA componen, DF has smaller size han PP, while PP has smaller size hen DF for a posiive MA componen. Wih model A, for example, he size of DF and PP are 0,02 and 0,927 for MA(-0,8) errors, while heir sizes are 0,09 and 0,06 2 For he ADF and PP ess lags were used. 70 J.STUD.ECON.ECONOMETRICS, 200, 27()

15 respecively for MA(0,8). The power of DF is always higher han ha of PP when he errors conain a MA componen. Wih model A for insance, he power of DF (0,004, 0,07) while he power of PP (0,52, 0,99). The resuls of his experimen indicae ha he relaive performance of he ess is significanly influenced by he error srucure. This is addiional evidence ha more han one es should be used in applied work when he rue srucure of he errors is usually unknown. 4. Does he Souh African economy really walk randomly? The graph of he Souh African long-erm ineres rae (see Figure ) confirms ha a possible srucural break occurred in 985 when moneary policy in Souh Africa changed from direc conrol o marke orienaed moneary insrumens (Boha 997). The resuls of he Perron es for esing uni roos in he presence of a srucural break, which are summarized in Table 0, confirms ha he long-erm ineres rae is rend saionary wih a srucural break in 985. The naïve researcher ha misspecifies he break will do he DF, PP and KPSS ess, of which he resuls are given in Table 9. The DF and PP ess does no rejec he null of a uni roo when he variable is esed in levels, bu rejecs when he variable is firsdifferenced. In oher words he researcher will regard he variable o be inegraed of order one. The KPSS es, on he oher hand, rejecs he null of saionariy for he ineres rae in levels. This will confirm he researcher s conclusion from he DF and PP ess ha he ineres rae is inegraed of order one. However, if he break is correcly specified, and he Perron es for a uni roo in he presence of a srucural break is done accordingly, he researcher will rejec he null of a uni roo and herefore regard he variable rend saionary wih a srucural break Figure : The Souh African long-erm ineres rae J.STUD.ECON.ECONOMETRICS, 200, 27() 7

16 The resuls of he DF, PP and KPSS ess for consumer price inflaion are given in Table 9. Boh he ADF and PP ess do no rejec he null of a uni roo in levels, bu i rejecs he null when inflaion is firs-differenced. KPSS rejecs he null of saionariy in levels, herefore inflaion is rendered non-saionary, which does no conform o economic heory. However, if here is a srucural break in inflaion, i invalidaes he resuls of he ADF and PP ess. The resuls of he Perron es for uni roos in he presence of a srucural break renders inflaion rend saionary wih a srucural break in 985. The inflaion rae is herefore regarded as saionary, consisen wih expecaions ha i canno conain a uni roo. Table 9: Augmened Dickey-Fuller and Phillips-Perron uni roo ess and KPSS es, levels Series Model ADF Lags τ τ,τ µ,τ φ,φ PP 4 KPSS Long-erm ineres rae Trend Consan None -,28 -,6,59 2,40 2,8 2,22 -,24* -,05,2 4,66*** Inflaion rae Trend Consan None 0,6 -,69-0,6 2,70 2,0,50-0,5 -,6-0, 52*** long-erm ineres rae Trend Consan None -,9* -,25** -2,70*** 4,*** 7,4*** 2,0*** -8,75*** -8,84*** -8,7*** 0,8** inflaion rae Trend Consan None -,6** -2,5-2,55** 7,72*** 7,07*** 9,66*** -5,2*** -4,79*** -4,85***,28 */**/*** Significan a a 0%/5%/% level According o he Mone Carlo resuls in he previous secion, DF and PP will be biased oward non-rejecion and KPSS will be biased oward rejecion in he presence of a srucural break. The examples above are consisen wih his Mone Carlo resul, since DF and PP did no rejec while KPSS rejeced when he break was misspecified. The resuls of he Perron es for a uni roo in he presence of a srucural break were in boh examples consisen wih he a priori expecaions of saionariy in he presence of a srucural break. The number of lags used in he esimaed equaions was deermined according o he mehod suggesed by Said and Dickey (984), which means a maximum of T / lags. 4 The number of runcaion lags used in he Barle kernel was deermined as suggesed by Newey- Wes. For his sample size Newey-Wes suggesed. 72 J.STUD.ECON.ECONOMETRICS, 200, 27()

17 Table 0: Perron Tes 5 for non-saionariy in he presence of a srucural break 6, levels Series T B λ K µ ~ ) β ~ ( µ ~ ( β ~ ) ~ γ α ~ 7 ) ( α ~ ( γ ~ ~ T( ) α ) Longerm ineres rae 985 0, ,7 (-4,04) 0,44 (4,2) -0,42 (-4,75) 0,2 (-4,8***) -29,26* Inflaion rae 985 0,64-0,27 (-6,) 0,68 (6,2) -,9 (-,65) 0, (-4,68***) -26,6* */**/*** Significan a a 0%/5%/% level 5. Conclusion In his paper we have shown wih Mone Carlo simulaions and pracical examples from he Souh African economy ha misdiagnosed srucural breaks in small samples have serious consequences for uni roo and saionariy esing. In he presence of a srucural break he DF and PP ess are biased oward non-rejecion of non-saionariy, while he KPSS es is biased oward rejecion of saionariy. The DF and PP have low power in he presence of a srucural break, while he KPSS es has serious size disorions. However, he differen ypes of srucural breaks are no misleading he ess o he same exen. A srucural break consising only of a change in he inercep is generally less disrupive as a srucural break ha includes a change in he slope. The impac of cerain dimensions of possible srucural breaks has also been esed. When he break consiss only of a change in he inercep, he effec of he small sample is dominaing he effec of he break since DF and PP are sill low power ess while KPSS has serious size disorions. However, when he break includes a change in he ime rend, he effec of he break placemen is dominaing, since DF and PP are biased oward rejecion when he break is a he beginning of he 5 The version ha ess H0 : y = µ + y + δdu + ε agains H a : y = µ + β + δ2dt * + ε, where DU = if >T B and 0 oherwise and DT *= if >T B and 0 oherwise, was used. 6 The parameers given are from he model: y = µ + β + γdt * + y ; y K = αy + c i y i + ε i = 7 Phillips and Ouliaris (990) showed ha -raio procedures diverge under ha alernaive a a slower rae han direc coefficien ess, which means ha direc coefficien ess should have superior power properies over -raio ess. Therefore he T(α-) es migh have higher power han he α es so ha boh are repored. However, in his sudy hey gave he same resuls. J.STUD.ECON.ECONOMETRICS, 200, 27() 7

18 sample, while hey are biased oward non-rejecion when he break occurs a he end of he sample. This is rue regardless he sample size, alhough i is less biased in smaller samples. KPSS is no influenced by he placemen of he break, and has serious size disorions for any ype of break. The magniude of he break is only significan when he rue DGP is saionary, since he daa appears less saionary as he break increases. All he ess are increasingly biased owards finding he daa non-saionary as he magniude of he break increases. The performances of he ess are also significanly influenced by he error srucure. This is addiional moivaion o use a combinaion of ess raher han only one o es he saionariy of a series. References De Jong, D N, Nankervis, J C, Savin, N E and Whieman, C H (989): Inegraion versus Trend Saionariy in Macroeconomic Time Series, Working paper no , Deparmen of Economics, Universiy of Iowa, Iowa Ciy, IA. Diebold, F X and Rudebusch, G D (990): On he Power of Dickey-Fuller Tess Agains Fracional Alernaives, Economics Leers, 5, Enders, W (995): Applied Economeric Time Series, John Wiley & Sons, Inc., New York. Karian, Z A, and Dudewicz, E J (99): Modern Saisical, Sysems, and GPSS Simulaion: The Firs Course, W.H. Freeman and Company, New York. Kinderman, A J and Ramage, J G (976): Compuer Generaion of Normal Random Numbers, Journal of he American Saisical Associaion, 7, Lee, J, Huang, C J and Shin, Y (997): On Saionariy Tess in he Presence of Srucural Breaks, Economics Leers, 55: Leybourne, S J and Newbold, P (2000): Behaviour of Dickey-Fuller -ess When There is a Break Under he Alernaive Hypohesis, Economeric Theory, 6, Monanes, A and Reyes, M (998): Effec of a Shif in he Trend Funcion on Dickey-Fuller Uni Roo Tess, Economeric Theory, 4, Perron, P (989): The Grea Crash, he Oil Price Shock, and he Uni Roo Hypohesis, Economerica, 57, Said, S and Dickey, D (984): Tesing for Uni Roos in Auoregressive-Moving Average Models wih Unknown Order, Biomerics, 7, J.STUD.ECON.ECONOMETRICS, 200, 27()