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1 Ctaton fo ublshed veson: Has, D, Havey, DI, Leyboune, S & Sakkas,, 'Local asymtotc owe of the Im-Pesaan-Shn anel unt oot test and the mact of ntal obsevatons' Econometc Theoy, vol. 6, no.,. -4. htts://do.og/.7/s DOI:.7/S Publcaton date: Document Veson Pee evewed veson Lnk to ublcaton Unvesty of ath Geneal ghts Coyght and moal ghts fo the ublcatons made accessble n the ublc otal ae etaned by the authos and/o othe coyght ownes and t s a condton of accessng ublcatons that uses ecognse and abde by the legal euements assocated wth these ghts. Take down olcy If you beleve that ths document beaches coyght lease contact us ovdng detals, and we wll emove access to the wok mmedately and nvestgate you clam. Download date:. Ma. 9

2 Local Asymtotc Powe of the Im-Pesaan-Shn Panel Unt Root Test and the Imact of Intal Obsevatons Davd Has, Davd I. Havey, Stehen J. Leyboune ;y and kolaos D. Sakkas Deatment of Economcs, Unvesty of Melboune School of Economcs, Unvesty of ottngham Mach 8 Abstact In ths note we deve the local asymtotc owe functon of the standadzed aveaged Dckey- Fulle anel unt oot statstc of Im, Pesaan and Shn (, Jounal of Econometcs, 5, 5-74), allowng fo heteogeneous detemnstc ntecet tems. We consde the stuaton whee the devaton of the ntal obsevaton fom the undelyng ntecet tem n each ndvdual tme sees may not be asymtotcally neglgble. We nd that owe deceases monotoncally as the absolute values of the ntal condtons ncease n magntude, n dect contast to the unvaate case. Fnte samle smulatons con m the elevance of ths esult fo actcal alcatons, demonstatng that the owe of the test can be vey low fo values of T and tycally encounteed n actce. Intoducton In ths note we consde the lage samle behavou of the standadzed aveaged Dckey-Fulle (DF) unt oot test of Im, Pesaan and Shn () (IPS) fo anels allowng heteogeneous detemnstc ntecet tems. We deve the local asymtotc owe functon fo ths statstc whee the tme sees dmenson T! followed by the coss-sectonal dmenson!. Allowance s made fo the fact that the devaton of the ntal obsevaton fom the undelyng ntecet tem (efeed to as the ntal condton) n each ndvdual tme sees may not be asymtotcally neglgble, theeby genealzng the unvaate model of Mülle and Ellott () to the anel envonment. We nd that the local asymtotc owe functon of the IPS statstc s a monotoncally deceasng functon of the magntude of the absolute values of the ntal condtons. Moeove, local asymytotc owe falls below the nomnal sze of the IPS test fo lausble values of the absolute ntal condtons. Ths behavou s n dect contast to the unvaate case, whee Mülle and Ellott () demonstate that the local asymtotc owe of the DF statstc s an nceasng functon of the absolute ntal condton. To show that ou lage samle esults ae of moe than just theoetcal nteest, we sulement ou asymtotc study wth a nte samle analyss. Ths clealy demonstates that the IPS test can also have vey low owe n stuatons whee T and assume the sot of values tycally encounteed n actce. Snce aled eseaches ae unable to stulate the natue of the ntal condtons they face, they should be fully awae of the otental fo oo owe efomance of the IPS test n such ccumstances. The authos would lke to thank Gusee Cavalee fo helful comments. y Addess fo Coesondence: Stehen J. Leyboune, School of Economcs, Unvesty of ottngham, Unvesty Pak, ottngham G7 RD, UK (emal: steve.leyboune@nottngham.ac.uk).

3 As a by-oduct of ou analyss, we also show that when the ntal condtons ae asymtotcally neglgble, the eesentaton of the lmt dstbuton of the IPS test as stated n etung and Pesaan (8) s actually ncoect, snce t omts motant contbutons fom a numbe of non-neglgble exectaton tems. The Panel Model and IPS Statstc Consde the followng data geneatng ocess fo coss-sectonal sees y t, = ; : : : ;, obseved ove t = ; : : : ; T tme eods y t = + w t w t = w ;t + v t ; t = ; : : : ; T w = () whee the nnovatons v t ae assumed to satsfy the followng assumton Assumton. The v t ae..d.(, v; ) acoss t = ; :::; T and ae ndeendently dstbuted acoss = ; :::;. We consde the case whee the ntal condtons ae govened by Assumton. Let the ntal condtons be = w;, whee w; denotes the shot un vaance of w t fo <,.e. = v;. Ths ntal value sec caton mles that each ntal condton s ootonal to the standad devaton of the coesondng ocess w t. Fo tactablty n the analyss, we assume that the constant of ootonalty s common acoss = ; :::;, and we teat as a xed aamete. The null and altenatve hyotheses fo the anel unt oot testng oblem ae H : = fo all H : < fo at least one. Denotng the standad DF statstc that allows fo an ntecet by t statstc s gven by Z = = ft E(t )g V (t ) fo sees, the IPS test whee t = P = t and whee E(t ) and V (t ) denote the mean and vaance, esectvely, of t unde the null hyothess. Unde Assumton these moments do not deend on and hence the subsct s omtted. We use ths conventon fo all exectaton tems thoughout the ae. Asymtotc Local Powe of the IPS Statstc We secfy the local altenatve hyothess by lettng be govened by the followng assumton Assumton. Let = + c T fo c <, = ; :::; notng that the null hyothess holds fo c = 8.

4 Remak. Unde Assumtons, the ode of the ntal condtons s gven by = O( =4 T = ). We consde seuental asymtotc theoy, whee T! followed by!. All oofs ae gven n the Aendx. The followng lemma gves the dstbuton of the IPS statstc as T! Lemma. Unde Assumtons ( V (t )Z T! ) = = A E ( ) + c + = + 48 = c + =4 + = A = A 4 A 5 c = ) A c A A c = 4 ( c ) = A A 4 = + O ( =4 ) A A 4 c 5 =4 = ( c ) = A 5 whee eesents the lmt dstbuton of the DF statstc wth ntecet and A = R W ()dw (); A = R n R W R R o t (s)ds W (s)dsdt dw (); A = R n R W () W R R o t (s)ds W (s)dsdt d; A 4 = R ( )W ()d; A 5 = R ( )dw (); = R W () d; and W () s a standad ownan moton ocess. W () = W () R W (s)ds The behavou as! of the consttuent tems n the lmt n Lemma s gven by the followng sees of lemmas Lemma. = ( = A E ( ) )! ) (; V ( )): Lemma. Let c = lm! P = c. Then () () = =! c ) ce A A c! ) ce (v) A A = A A 4 c 5 ; () = ; (v)! ) ce A! A c ) ce ; = c A A 4 5 A :! A ) ce ;

5 Lemma 4. () = ( c ) = A A 4 () = O ( = ); () ( c ) = A 5 = O ( = ); = ( c ) A 4A 5 = = O ( = ): The local asymtotc lmt of the IPS test as T! followed by! s gven by Theoem. Unde Assumtons, wth c = lm! P = c Z T!;! ) (; ) + c E + E A + c 48 E A 4 E Ths follows dectly fom combnng the esults of Lemmas -. E A A V ( ) A A 4 V ( ) 5 Remak. Settng = n Theoem gves the local asymtotc dstbuton of the IPS test fo asymtotcally neglgble ntal condtons; that s, wheneve = o( =4 T = ). ote that ths coects the esult stated n etung and Pesaan (8), whee the o set tem n c contans only E( ), theeby ncoectly omttng the othe two exectatons. Values fo the moments nvolved n Theoem can be obtaned usng Monte Calo smulaton. We obtaned the values by dect smulaton of the dstbutons, aoxmatng the Wene ocesses usng..d. (; ) andom vaates, and wth the ntegals aoxmated by nomalzed sums of stes. Hee and thoughout the est of the ae, smulatons wee ogammed n Gauss 7. usng 5, elcatons. Substtuton of these moments nto the lmt exesson n Theoem gves se to the followng Coallay. Unde the condtons of Theoem Z T!;! ) (; ) + (:8 :5 )c: () Coollay shows that fo a zeo (common) ntal condton, the lmt dstbuton of Z s (; ) + :8c and hence the owe of the IPS test (snce t s left taled) s monotoncally nceasng n c <. Howeve, once we allow fo (common) non-zeo ntal condtons, fo a gven c <, the owe s monotoncally deceasng n jj. In fact, t also follows fom () that fo the IPS test conducted at any chosen sgn cance level, asymtotcally, ts owe wll fall below nomnal sze once :8 :5 < ; that s, once jj > :445. These asymtotc oetes of the IPS test ae demonstated gahcally n Fgue, whee the nomnal sgnfance level s 5%, c f ; 4; 6; 8; g and f; :; :::; 4:g. Fo comason, also shown s the local asymtotc owe of the unvaate DF test (fo = + c=t ), as evously studed by Mülle and Ellott () and Havey and Leyboune (5), when c =. The contast n behavou s comletely evdent, hghlghtng the fact that the behavou of anel unt oot tests cannot necessaly be nfeed fom the coesondng behavou of the unvaate counteats. Whle the unvaate DF test has owe that s monotoncally nceasng n, the 4

6 IPS test dslays the ooste behavou, wth owe adly deceasng as the magntude of nceases. y the tme =, when the ntal values ae one standad devaton away fom the sees mean, owe has oughly halved elatve to the = case; fo > :445, the above esult that IPS owe falls below sze s clealy obseved fo all values of c. Fnally, t s motant to assess the extent to whch the asymtotc behavou of the IPS test manfests tself when (and T ) s nte. In Fgue we eot nte samle owe smulatons fo f; ; ; 5; g when T =, wth f; :; :::; 4:g. These ae based on smulaton of Z wth data geneated fom the model () wth = and the v t geneated as..d. (; ) ndeendently acoss. We set = fo all such that = =( ) and, to make the comasons moe staghtfowad, fo each value of, s selected such that nte samle owe s eual to 5% when =. Results fo the unvaate DF test ae agan eoted fo comason. We see that as nceases the owe cuve of the IPS test less and less esembles that of the sng-n- unvaate case and mgates towads that of the decayng-n- lage case descbed above. In vey boad tems, owe s nceasng n when < and deceasng when >. Moeove, Z can ossess extemely low nte samle owe when = 5 o moe and not close to zeo. Thus, ou lage asymtotcs do ndeed aea to be a decent edcto of what mght occu n nte samles when s not small. We would theefoe suggest that ou ndngs seve a note of cauton to those alyng the IPS test when thee exsts uncetanty egadng the magntude of the ntal condtons. 4 Refeences etung, J. and Pesaan, M. H. (8). Unt oots and contegaton n anels. In Matyas, L. and Seveste, P. (eds.), The Econometcs of Panel Data, d edn., Kluwe Academc Publshes, fothcomng. Evans, G..A. and Savn,.E. (98). Testng fo Unt Roots:. Econometca 49, Havey, D. I. and Leyboune, S. J. (5). On testng fo unt oots and the ntal obsevaton. Econometcs Jounal 8, 97. Im, K. S., Pesaan, M. H. and Shn, Y. (). Testng fo unt oots n heteogeneous anels. Jounal of Econometcs 5, Mülle, U. K. and Ellott, G. (). Tests fo unt oots and the ntal condton. Econometca 7, Phlls, P. C.. (987). Towads a un ed asymtotc theoy fo autoegesson. ometka 74, Tanaka, K. (996). Tme Sees Analyss: onstatonay and onnvetble Dstbuton Theoy. John Wley, ew Yok. Aendx Poof of Lemma Usng esults fom Mülle and Ellott () and Phlls (987), we have that as T! t R T! R ) c = K ;c () d + K ;c ()dw () R K ;c () d 5

7 whee K ;c = K ;c () R K ;c (s)ds and K ;c () = (e c = )( c = ) = + W () + c =R e( s)c =W (s)ds: ext, va a Taylo sees exanson of the fom we may wte K ;c () = c ( e xc = = + xc = + O( ) c ) = =4 + W () + c =R W (s)ds + O ( =4 ) K ;c = ( )c ( c ) = =4 + W () + c = n R W (s)ds R R t W (s)dsdto + O ( =4 ) whee W = W () R W (s)ds. Substtutng and eaangng, we nd t T! ) c = + O ( =4 ) + A + c = A + c ( c ) = =4 A 5 + O ( =4 ) + c ( c ) = = + c = A + c ( c ) = =4 A 4 + O ( =4 ) () whee the A j ; j = ; :::; 5 and ae as de ned n the man text. Wtng the second tem as F ( + G ) =, whee F eesents the numeato and G = c ( c ) = = + c = A + c ( c ) = =4 A 4 + O ( =4 ) then a Taylo sees exanson aound G = gves ( + G ) = = = G + G O ( =4 ) c ( c ) = = c ( c ) = =4 A 4 c = A + c ( c ) = A 4 + O 5 ( =4 ): (4) On combnng () and (4), togethe wth + O ( =4 ) = + O ( =4 ), we have t T! ) A + c = + c = A c = A A + 48 c = A 4 c = A A ( c ) = 4 A A 4 ( + c = A 4A 5 + O ( =4 ) c ) = =4 A 5 and the esult of the lemma then follows by consdeng V (t )Z = = f P = t E(t )g. Poof of Lemma Ths follows fom alcaton of a standad cental lmt theoem n fo..d. andom vaables wth bounded vaance. 6

8 Poof of Lemma () Wte = c = c E( ) + = = ce( ) + O ( = )! ) c(e ) = c f E( )g usng a standad weak law of lage numbes n fo..d. andom vaables, whch ales snce E ( ) <. Results ()-(v) follow smlaly. Poof of Lemma 4 We wll show (), snce () and () follow smlaly. Ths nvolves showng and A A 4 A = ; (5) A E A 4 < : (6) Then by ndeendence we wll have E 4@ ( c ) = A A 4 A 5 A = jc j E A 4 = = O = hence ovng (). To show (5), the numeato can be wtten A A 4 = R W () d R W () dw () = R = R = W () d n R W () dw () R W () dr n R o dw () s s dw (s) R s s dw (s) R dw () R 4 : o dw () R ( ) dw () R dw () dw () 4 ow,, ae jontly nomal wth mean zeo, and we nd that E() = R E ( ) = R s s s That R s s dw (s) E() = R s s ds = ; E( ) = ; d = ; E ( ) = R s s ds = 6 ; ds = ; E ( ) = R d = : A Snce s ndeendent of both and, t follows that = =6 =6 = E ( ) = E ( ) E ( ) = AA : 7

9 and hence E 4 = so that A A 4 has zeo mean. ow consde the skewness of A A 4 n E 4 o = 8 E( ) 6 E( ) + E( ) 64 E( ): The st and thd tems ae obvously zeo snce s ndeendent of both and, and s symmetc due to nomalty. The fouth tem s also zeo due to the nomalty of. The second tem can be wtten E( ) = E( )E( ): ext consde a oulaton egesson of on. Snce E( )=E( ) = 5 we can wte = 5 + whee and ae ndeendent and jontly nomal wth mean zeo. Substtutng = 5 + nto E( ) gves E( ) = Ef( )g = 5E() 5 + E( ) + 5E( 4 ) = 5E() 5 + E()E( ) + 5E()E( 4 ) = : Thus n E 4 o = and so the dstbuton of A A 4 s symmetc about zeo. ow t follows that s! A A 4 A A = E sgn (A A 4 ) 4 s! = E (sgn (A A 4 )) E A A 4 = ovded (6) holds. To show (6) we aly the Cauchy Schwaz neualty A E A 4 E A 4 A 4 = 4 E 6 = and snce t s clea that A ; and A 4; have moments of all odes, we just need to vefy E 6 <. Fom euaton (5.7) of Evans and Savn (98), we can check the exstence of the ght hand sde of Z E = t E e t dt; > : () Fom euaton (4.) of Tanaka (996) we can deduce the mgf of to be E e t = (t) = sn = t and hence fo t > t follows that E e t = (t) = snh =. t y the change of vaable u = t we ave at the ntegal E = Z () 8 u = du = (snh u)

10 whch exsts fo any >. If t s needed to vefy ths exstence, we can wte Z u = du = = (snh u) In the st tem we use snh u u to wte Z u = du + = (snh u) Z u = du: = (snh u) Z u = du = (snh u) Z u d whch exsts and s eual to () fo >. In the second tem we use snh (u) = eu e u eu e on [; ), so Z u = du = (snh u) e Z u = e u= du + e < : 9