Consistency of the KPSS unit root test against fractionally integrated alternative

Transcription

1 Econoics Leers 55 (997) 5 60 Consisency of he KPSS uni roo es agains fracionally inegrae alernaive a b, * Hyung S. Lee, Chrisine Asler a Regional Econoy Division, Econoic Policy Bureau, Minisry of Finance an Econoy,,Joongang-ong, Kwacheon, Kyunggi-o, , Korea b Econoics Dearen, Marshall Hall, Michigan Sae Universiy, Eas Lansing, MI 4884 USA Receive Ocober 996; accee 4 January 997 Absrac We erive he asyoic isribuion of he Kwiakowski e al. (99) saisic uner nonsaionary long eory (/,,). Is orer in robabiliy is he sae uner nonsaionary long eory as uner a uni roo. I canno, herefore, isinguish consisenly beween he wo cases. 997 Elsevier Science S.A. Keywors: Nonsaionary long eory; Uni roo; Saionariy es saisic JEL classificaion: C. Inroucion he enorous lieraure on esing for a uni roo seeks o eerine wheher an econoic ie series is saionary or has a uni roo. Dickey, Fuller (979) ess are os coonly use o es he hyohesis of a uni roo agains he alernaive of saionariy. In his aer we consier he es of Kwiakowski e al. (99) hereafer KPSS which is esigne o es he hyohesis of saionariy agains he alernaive of a uni roo. he KPSS saisic wih ifferen criical values can also be use as he basis for a uni roo es, as suggese by Shin, Schi (99). More recenly, i has been recognize ha he null hyohesis for ess of saionariy (an he alernaive for uni roo ess) is acually shor eory (or weak eenence) since he series or is ifference is require o saisfy a funcional cenral lii heore for convergence o Brownian oion, an his resul requires a lii on he eory or eenence of he series. Long eory series are also eirically relevan, an for a ile groun beween shor eory an uni roo series; see, e.g., he survey by Baillie (996). A oular an useful oel is he fracionally inegrae, or I(), oel of Granger (980), Granger, Joyeux (980) an Hosking (98). In his fraework shor eory corresons o 5 0 an a uni roo o 5, bu fracional values are *Corresoning auhor: el.: ; fax: ; e-ail asler@ilo.su.eu / 97/ $ Elsevier Science S.A. All righs reserve. PII S (97)

2 5 H.S. Lee, C. Asler / Econoics Leers 55 (997) 5 60 eaningful. In aricular, he series is saionary bu has long eory for 0,, /, while i is nonsaionary an long eory (bu sill ean-revering) for /,,. he asyoic isribuion of he KPSS saisic uner shor eory an uner a uni roo were given by KPSS an by Shin, Schi (99). Lee, Schi (996) erive he asyoic isribuion of he KPSS saisic for he saionary long eory case (0,, /). In his aer we erive he asyoic isribuion for he nonsaionary long eory case (],, ). We show ha he orer in robabiliy of he es saisic is he sae uner nonsaionary long eory as uner a uni roo, so ha he KPSS saisic canno isinguish consisenly beween hese wo cases. In oher wors, of he four cases iscusse above, he KPSS saisic can isinguish consisenly beween shor eory, saionary long eory, an eiher nonsaionary long eory or uni roo. We also resen soe siulaions ha show he relevance of hese asyoic resuls in finie sales.. Asyoic resuls A. Noaion We consier he aa generaing rocess: y5 f j e, 5,,...,, () where hyj is he observe series an hej is he eviaion fro ren. Le e, be he resiuals fro a regression of y on inerce an ren (), an le S be heir arial su: S5oj5e j. Le s (,) be he Newey Wes esiaor of he long-run variance of he e :, s 5 s5 5s s (,)5 O e O [ s/(, )] O ee. () For he case,50, he secon er on he righ han sie of () is sily se o zero. hen he KPSS saisic ˆ h (,) is efine as 5 ĥ (,)5 O S /s (,). (3) he KPSS saisic h(,) ˆ is efine siilarly exce ha we se j 50 in (), which ilies use of he resiuals e5yy in efining S an s (,). o es he null of shor eory, KPSS require, ` bu,/ 0 as `. For he KPSS uni roo es, Shin, Schi (99) sugges,50. We will consier boh of hese cases. B. KPSS uner shor eory Le Z5oj5 ej reresen he arial su of he e. Following Lee, Schi (996), we can say ha e is a shor eory rocess if i saisfies: ` / [r] (A) s5li E(Z ) exiss an is non-zero (A) ; r[[0, ], Z sw(r) Here [r] enoes he ineger ar of r, enoes weak convergence, an W(r) is he sanar

3 H.S. Lee, C. Asler / Econoics Leers 55 (997) Wiener rocess (Brownian oion). KPSS assue he ixing an oen coniions of (Phillis, Perron, 988,.336), which ily (A) an (A). Uner he hyohesis ha e is a shor eory rocess, KPSS show ha ĥ (, ) E V (r) r, (4) 0 rovie, ` an,/ 0 as `. Here V (r) is a secon-level Brownian brige, as efine by KPSS, Eq. 6. For,50, ˆ h (0) converges o (s /s ) ies he lii in (4). hus, in eiher case ˆ h,is e O () uner shor eory. Siilar saeens hol for ˆ h, wih V (r) relace by he sanar Brownian brige, V (r)5w(r)rw(). C. KPSS uner uni roo Suose ha e has a uni roo, an ore secifically ha De saisfies he shor-eory coniions (A) an (A). hen, for he case ha, ` an,/ 0 as `, KPSS show ha a (,/ )h ˆ (l) E EW*(s)s a/ew*(s) s, (5) where W*(s) is a eeane an erene Weiner rocess, as efine in (Park, Phillis, 988,. 474). A siilar saeen hols for ˆ h (,), wih W*(s) relace by he eeane Brownian oion, W(s)5W(s)e W(r)r. hus ˆ h (,) an ˆ ] 0 h (,) are O (/,) for he case ha, `,,/ 0, as `. Shin, Schi (99) show ha ˆ h (0) has he sae isribuion as given in (5), wih a siilarly oifie resul for ˆ h (0). hus ˆ h (0) an ˆ h (0) are O ( ). D. KPSS uner saionary long eory Lee, Schi consier he case ha e follows an I() rocess: (L) e5u, where u is noral whie noise: his rocess is saionary for /,,/. Here we are concerne only wih in he range 0,,/, bu he resuls cie hol for /,,/. If Z is he arial su of he e,i saisfies he invariance rincile ( ]) Z k()w (r), (6) [r] where k() is a scalar whose value is irrelevan for our uroses, an W (r) is a fracional Brownian oion as efine by, e.g., (Beran, 994,. 56). Lee, Schi (996) esablish he following resuls: ˆ e 0 h (0) (k()/s )EV (r) r (, 50), (7a) ˆ 0 (,/) h (,) EV (r) r (, `bu,/ 0as `), (7b) ˆ where V (r) is efine by Lee, Schi, Lea,. 9. Siilar saeens hol for h, wih V (r)

4 54 H.S. Lee, C. Asler / Econoics Leers 55 (997) 5 60 ˆ ˆ ˆ ˆ relace by B (r)5w (r)rw (). hus h (0) an h (0) are O ( ); an h (,) an h (,) are O ((, / ) ) when, `,,/ 0 as `. Each of he KPSS saisics has a ifferen orer in robabiliy uner shor eory, saionary long eory, an uni roo, resecively. hus he KPSS es can isinguish consisenly beween hese hree ossibiliies. E. KPSS Uner Nonsaionary Long Meory We now urn o he ain heoreical conribuion of his aer, which is he erivaion of he asyoic isribuion of he KPSS saisics when e is a nonsaionary long eory rocess. hus we wish o consier he case ha e is I() wih /,,. Define *5, so ha De is I(*) wih u*u,/; ha is, De is a saionary long eory rocess. hen, assuing ha u 5 ( L) he invariance rincile * ( /) (/) [r] [r] * De 5( L) e is a noral whie noise rocess, we have * e 5 e k(*)w (r). (8) his is essenially he sae resul as (6) above, reaing e as he cuulaion of he De, which are I(*). Using his resul, we can rove he following heore. heore. Suose ha Eq. ()hols, wih j 50, an ha (L) e5u, /,,, where u is noral whie noise. hen r ˆ * * h (0) E E W (a)a r/ E W (a) a (, 50), (9a) ] ] r ˆ * *, h (,) EE W (a)a r/ E W (a) a (, `an,/ 0as `), (9b) ] ] W (b) b. Furherore, he sae resuls hol for ˆ h (0) an ˆ h (,), wih * * 0 * where W ] (a)5w (a)e W (a) relace by ] * W * * (a)5w * (a)(6a 4)E W * (b)b(a 6)E bw * (b)b. (0) 0 0 he resuls for ˆ h o no require he assuion ha j 50. Proof. See Aenix A. he heore ilies ha ˆ h (0) an ˆ h (0) are O (), while ˆ h (,) an ˆ h (,) are O (/,) for he case ha, ` an,/ 0. herefore he KPSS uni roo es is no consisen agains nonsaionary long eory alernaives, I() for /,,, because he KPSS saisics have he sae orers in robabiliy uner boh he null an alernaive hyoheses. In fac, for /,#, he orers in robabiliy of he KPSS saisics are ineenen of he value of, even hough he for of heir asyoic isribuions is affece by he value of. his is in conras o he case of a saionary

5 H.S. Lee, C. Asler / Econoics Leers 55 (997) long eory rocess, where boh he orer in robabiliy an he for of he asyoic isribuion een on. he KPSS shor eory es is known o be consisen agains saionary shor eory alernaives an uni roo alernaives. Our resul shows ha i is also consisen agains nonsaionary long eory alernaives. his is no surrising, bu he inabiliy o isinguish consisenly beween uni roo an nonsaionary long eory is erhas surrising. 3. Siulaion resuls In his secion we rovie siulaion evience on he ower of he KPSS saionariy (shor eory) an uni roo ess. he lag runcaion araeers are chosen as,050,,45ineger[4(/ /4 /4 00) ], an,5ineger[(/ 00) ] as in Schwer (989), KPSS (99) an Lee, Schi 996. We consier sale sizes 50, 50, 50, 500 an 000, an he nuber of ieraions is All of our ess are base on he 5% significance level. he eho of aa generaion is as escribe in Lee, Schi (996). able gives he owers of he 5% uer ail KPSS shor eory ess agains he alernaives 50.0,.,.,...,.9,.0, an also an hese are an elaboraion of he values consiere by Lee, Schi (996). For fixe, ower increases wih, reflecing he consisency of he ess. For fixe, ower increases wih, a resul ha is no surrising, even hough i is no ransaren fro he relevan asyoics for /,#. able gives he ower of he lower ail KPSS uni roo es agains I() alernaives wih /##. he os ioran resul is ha, wih fixe, ower oes no aroach one as increases. For exale, for 50.7, ower grows fro 0.69 wih 550 o only 0.56 wih his is a reflecion of he inconsisency of he KPSS uni roo es agains nonsaionary long eory rocesses; ower woul no be exece o aroach one even for arbirarily large values of. We can also see ha, for fixe, ower increases as ecreases. his is no surrising, bu also no ransaren fro he relevan asyoics for /,#l. 4. Conclusions In his aer we have aske wheher he KPSS saisic can be use o isinguish beween he following ossibiliies: (i) shor eory; (ii) saionary long eory; (iii) nonsaionary long eory; an (iv) uni roo. We fin ha i can consisenly isinguish (i) fro (ii) fro he cobinaion of (iii) an (iv), bu i canno isinguish consisenly beween (iii) an (iv). An ineresing quesion is wheher anoher single saisic can isinguish consisenly beween hese four ossibiliies. he cenral feaure of our resul is ha he orer in robabiliy of he KPSS saisic oes no een on for in soe range, an his ossibiliy is no eculiar o he KPSS saisic. For exale, Sowell (990) shows ha he orers in robabiliy of he Dickey Fuller (979) saisics een on for, bu no for #,3/. A broaer invesigaion of his henoenon woul cerainly be worhwhile.

6 56 H.S. Lee, C. Asler / Econoics Leers 55 (997) 5 60 able Power of KPSS Shor Meory es Agains I(), [[0.0,.0] ˆ h es ˆ h es , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

7 able. Coninue H.S. Lee, C. Asler / Econoics Leers 55 (997) ˆ h es ˆ h es , , , , , , , , , , , , , , , Acknowlegens We hank Peer Schi for helful coens an suggesions Aenix A In his Aenix we give a skech of he roof of heore for he ˆ h (0) an ˆ h (,) saisics. More eail, an he siilar roof for ˆ h (0) an ˆ h (,), can be foun in Lee (995). able Power of KPSS Lower ail Uni Roo es Agains I(), [[0.5,.0] ˆ h (0) es ˆ h (0) es

8 58 H.S. Lee, C. Asler / Econoics Leers 55 (997) 5 60 As in he ain ex, Z is he arial su of he e an S is he arial su of e ; for he ˆ h e 5e e. Lea. ess, ( * 3/) r (* /) [r] 0 * [r] * (* 3/) ] r ] * * 0 * [r] 0] * (* ) (* ) 5 0] * (* 4) r *. Z k(*) e W (a) a;. e k(*)w (a), where W (a)5w (a)e W (b) b; 3. S k(*) e W (a) a; 4. o e 5 S (0) k(*) he W (a) aj 5. o S k(*) e [e ] W (a) a]r * ( /) Proof. We begin wih Eq. (8) of he ex: e[r] k(*)w * (r); (i) follows irecly. o esablish (ii), noe (* /) (* /) (* /) [r] [r] j j5 e 5 e O e, (A) which ilies (ii). hen (iii), (iv) an (v) follow fro (ii). j he roof of he firs ar of heore is now sraighforwar. We have r (* 4) (* ) ˆ ] * ] * h (0) 5 O S / S (0) E E W (a)a r/ E W (a) a (, 50) using (iv) an (v) of Lea l. his esablishes (9A) of he ex. Lea. (A) ( *) 5 l s ( *) s j50 5s s j ( *) 5s s 0 * ( *) 0 *. o e De 0 for any nonnegaive ineger s.. o o e De 0 for any nonnegaive ineger s. 3. o ee k(*) e W (a) a. 4. When, ` an,/ 0 as `, hen [s (,)][, ] k(*) e ] W (a) a. Proof. Le gj be he jh auocovariance of De. We will rove (i) for he case s50; he case of s.0 is siilar bu ore colicae. Since o e De 5/(e e o (De )), ( *) (* /) (* /) 0 5 * ( ) 5 ( * /) * * Oe De 5() [ ][e]() [ ][e] ( )[ O (De )] 0, (A3) because [ ][e] v W (), [ o (De )] g an e is O (). o rove (ii),

9 H.S. Lee, C. Asler / Econoics Leers 55 (997) s ( *) ( *) s j s s s s j50 5s 5s OO e De 5 O [e Dee De???e De ] (A4) an each er 0 by (i). o rove (iii), wrie s ( *) ( *) ( *) s s s j 5s 5s j50 5s O ee 5 O e OO e De. (A5) he firs er has he esire isribuion by (iv) of Lea, while he secon er 0 by(ii). Finally, o rove (iv), efine v(s,,)5s/(,), recall e 5e e, an o soe algebra o obain H F GJ, ( *) (* /) (* /) 5 5 s5, ( *) H s s J 5s s5 [s (,)][ ] 5 O [( )e ] O ( )e O v(s,, ) 3 O [( )(ee eeee e )] 5 (A) O v(s,, ) 3(B s). (A6) Par (A) is he sae as in he consieraion of s (0) an has he isribuion given in (iv) of Lea. Furherore, Bs has his sae isribuion for all s, using (iii) an he sae arguen ha le o (iv), of Lea. Since o v(s,,)5,, for fixe, we have s5 ( *) * * 0 s (,) (, )k( ) E W (a) a. (A7) hen (iv) follows by iviing by,, an leing, ` as `. jhe roof of he secon ar of heore is now sraighforwar. We have (* 4) ( *) 5 (,/)h ˆ (,)5 O S /, s (,) (A8) an we obain (9B) of he ex using (v) of Lea, an (iv) of Lea. References Baillie, R.., 996. Long Meory Processes an Fricional Inegraion in Econoerics. Journal of Econoerics 73, Beran, J., 994. Saisics for Long-Meory Processes. Chaan an Hall, New York. Dickey, D.A., Fuller, W.A., 979. Disribuion of he Esiaors for Auoregressive ie Series wih a Uni Roo. Journal of he Aerican Saisical Associaion 74, Granger, C.W.J., 980. Long Meory Relaionshis an he Aggregaion of Dynaic Moels. Journal of Econoerics 4, Granger, C.W.J., Joyeux, R., 980. An Inroucion o Long-Meory ie Series Moels an Fracional Differencing. Journal of ie Series Analysis, 5 9. Hosking, J.R.M., 98. Fracional Differencing. Bioerika 68,

10 60 H.S. Lee, C. Asler / Econoics Leers 55 (997) 5 60 Kwiakowski, D., Phillis, P.C.B., Schi, P., Shin, Y., 99. esing he Null Hyohesis of Saionariy Agains he Alernaive of a Uni Roo: How Sure Are We ha Econoic ie Series Have a Uni Roo?. Journal of Econoerics 54, Lee, D., Schi, P., 996. On he Power of he KPSS es of Saionariy Agains Fracionally-Inegrae Alernaives. Journal of Econoerics 73, Lee, H.S., 995. Analysis of Econoic ie Series wih Long Meory. Ph.D. isseraion, Michigan Sae Universiy, Eas Lansing, MI. Park, J.Y., Phillis, P.C.B., 988. Saisical Inference in Regressions wih Inegrae Processes, Par I. Econoeric heory 4, Phillis, P.C.B., Perron, P., 988. esing for a Uni Roo in ie Series Regression. Bioerika 75, Schwer, G.W., 989. ess for Uni Roos: A Mone Carlo Invesigaion. Journal of Business an Econoic Suies 7, Shin, Y., Schi, P., 99. he KPSS Saionariy es as a Uni Roo es. Econoics Leers 38, Sowell, F., 990. he Fracional Uni Roo Disribuion. Econoerica 58,