Generalized fixed-t Panel Unit Root Tests Allowing for Structural Breaks

Transcription

1 ATHES UIVERSITY OF ECOOMICS AD BUSIESS DEPARTMET OF ECOOMICS WORKIG PAPER SERIES 08-0 Generalzed fxed-t Panel Unt Root Tests Allowng for Structural Breaks Yanns Karavas and Elas Tzavals 76 Patsson Str., Athens 04 34, Greece Tel. (++30) Fax: (++30)

2 Generalzed xed-t Panel Unt Root Tests Allowng for Structural Breaks Yanns Karavas and Elas Tzavals* Deartment of Economcs Athens Unversty of Economcs & Busness Athens 04 34, Greece July 0 Workng Paer Abstract In ths aer we suggest anel data unt root tests whch allow for seral correlaton n the dsturbance terms and structural breaks n the ndvdual e ects or lnear trends of anel data models. The lmtng dstrbutons of the tests are derved under the assumton that the tme-dmenson of the anel (T ) s xed, whle the cross-secton grows large. Thus, they are arorate for short anels, where T s small. The tests consder the cases of a known and unknown date break. For the latter case, the dstrbuton of the tests s nonstandard. The aer gves an analytc form of ths dstrbuton, based on the maxmum of T -elltcally contoured dstrbutons, whch faclates the alcaton of the tests. The aer roves the consstency of the tests. Monte Carlo evdence suggest that our tests have sze whch s very close to ts nomnal level and satsfactory ower n small-t anels. Ths s true even for cases where the degree of seral correlaton s large and negatve, where sngle tme seres unt root tests are found to be overszed. JEL class caton: C, C3 Keywords: Panel data models; unt roots; structural breaks; *The authors would lke to thank the artcants at the Conference n Honour of Professor Hashem Pesaran n Cambrdge 0 and Arellano-Valle and Genton for ther useful comments on a revous verson of ths aer. Yanns Karavas: jkarava@aueb.gr Elas Tzavals: etzavals@aueb.gr

3 Introducton A vast amount of work has been recently focused on drawng nference about unt roots based on dynamc anel data models (see, Hlouskova and Wagner (006), for a more recent survey). Snce many emrcal anel data studes rely on short anels, of artcular nterest s testng for a unt root n dynamc anel data model when the tme dmenson of the anel, denoted as T, s xed ( nte) and ts cross-secton, denoted as, grows large (see, e.g. Harrs and Tzavals (999, 004), Hadr (000), Bnder et al (005) and De Wachter et al (007)). These tests have better small-t samle erformance, comared to large-t anel unt root tests (see, e.g., Levn et al (00), Im et al (003)), gven that they assume nte T: Imlementatons of xed-t anel unt root tests nclude many nterestng alcatons n economcs such as testng for the economc convergence hyothess [see de la Fuente (997), for a survey], the urchasng ower arty hyothess under d erent economc (or exchange rate) regmes [see Culver and Paell (999), nter ala] and the e ects of lberalzaton olces on trade (see, e.g., Warzarg and Welch (004)). In ths aer, we extend the xed-t anel data unt roots test statstcs of Harrs and Tzavals (999) to allow for a common structural break n the determnstc comonents of anel data models, namely ther ndvdual e ects or lnear trends of a known and unknown date. Ths s done n a dynamc anel data framework allowng for seral correlaton of the dsturbance terms. Ths extenson s very useful gven lethora of evdence suortng the vew that the resence of unt roots n economc tme seres can be falsely attrbuted to the exstence of structural breaks n ther determnstc comonents (see, e.g., Perron (006), for a survey). On ths front, the anel data aroach o ers an nterestng and unque ersectve that t s not shared by sngle unvarate tests. The cross-sectonal dmenson of the anel can rovde useful samle nformaton whch can hel to dstngush the tye of shfts (breaks) n the determnstc comonents of the anel from the e ects of stochastc ermanent shocks. As onted out by Ba (00), ths framework can more accurately trace out structural break onts of the anel data. Allowng for seral correlaton n the unt root tests s crtcal due to ts nherent nature n economc tme seres data, as s stated n Schwert (989), Sad and Dckey (984) and Phlls (987). The aer assumes that the maxmum order of seral correlaton of the dsturbance terms of the anel data model s a functon of the tme dmenson T. Both the varance and the seral correlaton e ects n the dsturbance terms of the anel data models are allowed to be heterogenous across the ndvdual unts of the anel. Ths assumton makes the tests alcable under qute general anel data generatng rocesses, observed n realty. There are a few studes n the lterature whch suggest xed-t anel data unt root tests allowng for a common structural break n the determnstc comonents of the anel data model (see, e.g., Carron-Slvestre et al (00) and Tzavals (00), or more recently, Karavas and Tzavals (0a), and Hadr (0)). These studes however suggest unt root tests usng the smle AR anel data model as an auxlary regresson model, whch may not be oeratonal n ractce due to the assumton of no seral correlaton made for the dsturbance terms. As recently s noted by De Blander and Dhaene (0), ths may lead to erroneous nference about unt roots f there s substantal seral correlaton n the dsturbance terms. The man goal of the above studes s to ass deas how to test for unt roots n the resence of

4 breaks of known or unknown date, usng the AR anel data model as an examle. In addton to the above, there are also studes n the lterature whch suggest anel unt root tests allowng for a common structural break, but they assume that the tme-dmenson of the anel; T, s large and grows faster than ts cross-secton, denoted as (see, e.g., Carron--Slvestre et al (005), Ba and Carron--Slvestre (009), and Km (0)). These are arorate for large-t anel data sets. Alcaton of ths category of anel unt root tests to small-t anels wll lead to serous sze dstortons and crtcal ower reductons n testng the null hyothess of unt roots aganst ts statonary alternatve (see Harrs and Tzavals (999)). As shown n Karavas and Tzavals (0a), the exstence of a break n the data generatng rocess requres anel data sets wth a qute large tme-dmenson, T (e.g. T > 00), so as the large-t anel unt root tests to have satsfactory small samle roertes. The sequental anel data test statstcs suggested by the aer for the case of an unknown break are n lne to those suggested by Andrews (993), Zvot and Andrews (99), Perron (997), Perron and Vogelsang (998), nter ala), for sngle tme seres. In ths aer the lmtng dstrbuton of these test statstcs can be obtaned as the mnmum value of a nte number of correlated varates; T for the anel data model wth ndvdual e ects and T 3 for ths model allowng also for ndvdual lnear trends. Ths dstrbuton s shown analytcally based on recent results of Arellano-Valle and Genton (008), who have derved the analytc form of the robablty densty functon of the maxmum of absolutely contnuous deendent random varables. The analytc form of ths dstrbuton enables us to derve crtcal values of the suggested test statstcs wthout havng to rely on Monte Carlo analyss, as n Karavas and Tzavals (0a). Thus, t substantally facltates alcaton of the tests n ractce. The aer s organzed as follows. In Secton, we derve the lmtng dstrbutons of the anel unt root test statstcs allowng for a known, or an unknown, date break under the assumton that the dsturbance terms of anel data models are whte nose rocesses. Ths analyss hels us to better understand the testng rncles of the test statstcs. In Secton 3, we generalze the results of Secton to allow for seral correlaton n the dsturbance terms, whle n Secton 4. we extend the tests to allow also for ndvdual lnear trends. In Secton 4., we show how to carry out the tests when there s a break n the ndvdual e ects of anel data models under the null hyothess. Secton 5 conducts a Monte Carlo smulaton study to examne the small samle erformance of the tests. Secton 6 concludes the aer. All the mathematcal dervatons are rovded n the Aendx of the aer. Test statstcs and ther lmtng dstrbuton In ths secton, we resent anel unt root test statstcs for the cases that the break date s known and unknown. Ths s done under the assumton that the dsturbance terms of the AR anel data model consdered are ndeendently, dentcally normally dstrbuted ( IID). Extensons of the tests to the more general case of serally correlated and heterogenous dsturbance terms are made n the next secton. 3

5 . Known date break Consder the followng AR nonlnear dynamc anel data model: where ' ( y t = a t ( ') + 'y t + u t ; = ; ; ::;, ; ], a t = a f t T 0 and a () f t > T 0, where T 0 denotes the tme-ont of the samle, referred to as break-ont; where a common break n the ndvdual e ects of anel data model occurs, for all cross-secton unts of the anel. a and a () denote the ndvdual e ects of model before and after the break ont T 0, resectvely. Throughout the aer, we wll denote the fracton of the samle that ths break occurs as,.e. = T0 T I = T ; 3 T ; :::::; T T. Under the null hyothess of a unt root (.e. ' = ), model reduces to the ure random walk model y t = y t + u t, for all, whle, under the alternatve of statonarty (.e. ' < ), t consders a common structural break n ndvdual e ects a. The above sec caton of the null and the alternatve hyotheses s very common n sngle tme seres nference rocedures allowng for structural breaks (see, e.g., Zvot and Andrews (99), Andrews (993), Perron and Vogelsang (998). The man focus of these rocedures s to dagnose whether evdence of unt roots can be surously attrbuted to the gnorance of structural breaks n nusance arameters of the data generatng rocesses lke ndvdual e ects a. The common break assumton across all unts of the anel can be attrbuted to a monetary regme shft, whch s common across all economc unts, or to a structural economc shock whch s ndeendent of the dsturbance terms u t, lke a credt crunch or an exchange rate realgnment. As atly noted by Ba (00), even f each seres of the anel data model has ts own break ont, the common break assumton across s useful n ractce not only for ts comutatonal smlcty, but also because t allows for estmatng the mean of ossbly random break onts. Ths mean wll have an economc olcy nterest. The AR anel data model can be emloyed to carry out unt root tests allowng for a structural break n ndvdual e ects a t coe cent of ', denoted as ^'. based on the wthn grous least squares (LS) estmator of autoregressve Ths estmator s also known as least square dummy varable (LSDV) estmator (see, e.g., Baltag (995), nter ala). Under the null hyothess of ' =, t mles: " # " X # X ^' = y; 0 Q y ; y; 0 Q u, () = where y = (y ; :::; y T ) 0 s a (T X)-dmenson vector collectng the tme seres observatons of deendent varable y t of each cross-secton unt of the anel, y ; = (y 0 ; :::; y T ) 0 s vector y lagged one erod back, u = (u ; :::; u T ) s a (T X)-dmenson vector of dsturbance terms u t and Q s the (T XT ) wthn transformaton matrx of the ndvdual seres of the anel data model, y t. Let us de ne X e ; e () to be a matrx of determnstc comonents used by the LSDV estmator to demean the levels of seres y t, for all ; where e and e () are (T X)-column vectors whose elements are de ned as follows: e t = = f t T 0 and 0 otherwse, and e () t = f t > T 0 and 0 otherwse. Then, matrx Q wll be de ned as Q = I T X (X 0 X ) X 0, where I T s an dentty matrx of dmenson (T XT ). 4

6 Panel data unt root testng rocedures based on above LSDV estmator ^' have the very nterestng roerty that, under the null hyothess, are nvarant (smlar) to the ntal condtons of the anel y 0 and, after arorate sec caton of matrx X, to the ndvdual e ects of the anel data model, as wll be seen n Secton 4. The latter haens f matrx X also contans broken lnear trends. Smlarty of the tests wth resect the ntal condtons of the anel does not requre any mean or covarance statonarty condtons on the anel data rocesses y t, as assumed by generalzed method of moments, or condtonal and uncondtonal maxmum lkelhood based anel data unt root nference rocedures (see, e.g., Hsao et al (00) and Madsen (008)). These condtons may be restrctve n ractce. However, ^' s an nconsstent (asymtotc based) estmator of ' due to the wthn transformaton of the data, whch wes o ndvdual e ects a t (or ntal condtons y 0 under the null hyothess ' = ) whle allowng for a structural break under the alternatve hyothess of statonarty, ' <. Thus, the test statstcs that we suggest testng for hyothess ' = wll rely on a correcton of estmator ^' for ts nconsstency (asymtotc bas) due to the above transformaton of the data (see, e.g., Harrs and Tzavals (999, 004)). To derve the lmtng dstrbuton of these tests, we make the followng assumton about the sequence of the dsturbance terms fu t g. Assumton : (a) fu g consttutes a sequence of ndeendent dentcally dstrbuted (IID) (T X)- dmenson vectors wth means E(u ) = 0 and varance-autocovarance matrces E(u u 0 ) = ui T < + and nonzero, for all. (a) E(u t y o ) = E u t a t = E u t a () t = 0 and 8 f; ; :::; g; t f; ; :::; T g: (a3) E ut 4 < +; E(y 4 0 ) < +; E (a t )4 < +; E (a () t )4 < + and E y0 a t < +; E y0 < +: a () t Condton (a) of Assumton enables us to derve the lmtng dstrbuton of a anel data unt root test statstc based on estmator ^' under the null hyothess ' = by alyng standard asymtotc theory for IID rocesses, whle (a) and (a3) are smle regularty condtons under whch the test statstc can be roved that s consstent under the alternatve hyothess, ' <. The followng theorem rovdes the lmtng dstrbuton of such a test statstc, based on estmator ^' corrected for ts bas. For analytc convenence, ths s done under the assumton that u t s also normally dstrbuted,.e. u t IID(0; u); for all and t: Theorem Let u t IID(0; u), then, under the null hyothess ' = and known, we have Z b V =^ ^' ^b ^ d (0; ) (3) as, where ^b ^ utr( 0 Q ) P (4) ^ = y0 ; Q y ; s a consstent estmate of the asymtotc bas of ^' whch, under the null hyothess, s gven as b = u tr(0 Q ) u tr(0 Q ), ^ u s a consstent estmator of varance u under the null hyothess, whch s gven as ^ u = P = y0 y tr( ) where s the d erence oerator and s a (T XT )-dmenson matrx havng n 5

7 ts man dagonal the corresondng elements of matrx 0 Q, and zeros elsewhere, and V s a varance functon gven as V = 4 uf 0 (K T + I T )F, (5) where F = vec(q 0 ); K T s a (T XT )-dmenson commutaton matrx and I T s a (T XT )- dmenson dentty matrx. The roof s gven n the aendx. The test statstc Z, gven by Theorem, can be easly mlemented to test unt root hyothess ' = based on the tables of the standard normal dstrbuton. Theorem shows that the asymtotc bas of estmator ^' stems from the "wthn" transformaton matrx Q, whch nduces correlaton between vectors y ; and u (see, e.g. ckel (98)). Snce dsturbance terms u t are IID, the correlaton between y ; and u comes only from the man dagonal elements of the varance-autocovarance matrces of u t, de ned by Assumton as E(u u 0 ) = ui T, for all. The above bas can be estmated by the nonarametrc estmator ^b ^ and, thus, t can be subtracted from ^' to obtan a test statstc whch s normally dstrbuted and s asymtotcally net of nusance arameter e ects. To test the null hyothess ' =, ths statstc s based on the o -dagonal elements of the samle moments of varance-autocovarance matrces statstc Z as whch are equal to zero,.e. E(u t u s ) = 0 for s 6= t. Ths can be better seen by wrtng test X = u 0 ( 0 Q )u = X = h tr ( 0 Q )u u 0, (6) (see Aendx) where ( 0 Q ) s matrx wth zeros n ts man dagonal due to the subtracton of matrx from 0 Q ; mlyng tr ( 0 Q )E(u u 0 ) = 0, for all. Matrx allows us to cature the correlaton e ects between vectors y ; and u, whch are nduced by the "wthn" transformaton of the data through matrx Q and, thus, generate the bas of LSDV estmator ^'. Subtractng from 0 Q enables us to adjust ^' for ths bas. varance-autocovarance The adjusted LS estmator reles on samle moments of wth zero elements,.e. E(u t u s ) = 0, for s 6= t. These moments are weghted by the elements of matrx 0 Q. They can be consstently estmated under the null hyothess ' =. Wrtng analytcally matrx 0 Q can be easly seen that the elements of ths matrx ut more weghts to samle moments of E(u t u s ), for s 6= t, wth s and t de ned mmedately before break ont, T 0. The next theorem establshes the consstency of test statstc Z. Theorem Under condtons (a)-(a3) of Assumton, t can be roved that lm P + (Z < z a j H a ) = ; I, (7) where z a s the crtcal value of standard normal dstrbuton at sgn cance level a. The roof s gven n the aendx. ote that matrx s used to estmate u, based on estmator ^ u = P = y 0 y tr( ) where s the d erence oerator. 6

8 . Unknown break ont The results of Theorem are based on the assumton that the break ont T 0 s known. ext, we relax ths assumton and roose a test statstc of the null hyothess ' = whch, under the alternatve hyothess of statonarty, allows for a common structural break n the ndvdual e ects of model of an unknown date. As n sngle tme seres lterature (see, e.g., Zvot and Andrews (99) and Perron and Vogelsang (998)), we wll vew the selecton of the break ont as the outcome of mnmzng the standardzed test statstc Z, gven by Theorem, over all ossble break onts of the samle, after trmmng out the ntal and nal arts of the tme seres observatons of the anel data. The mnmum value of test statstcs Z, for all I; de ned as z mnz, wll gve the least favorable result of the null hyothess ' =. Let I ^ mn denote the break ont at whch the mnmum value of Z, over all I; s obtaned. Then, the null hyothess wll be rejected f we have: Z (^ mn) < c mn, (8) where c mn denotes the sze a left-tal crtcal value of the lmtng dstrbuton of mn I Z. The followng theorem enables us to tabulate the crtcal values of ths dstrbuton at any sgn cance (sze) level a. Theorem 3 Let condton (a) of Assumton hold and u t s normally dstrbuted. Then, under the null hyothess ' = and unknown, we have z mn I Z d mn (0; ) (9) I as, where [ s ] s the varance-covarance matrx of the test statstcs Z, wth elements s gven by the followng formula: s = F 0 (K T + I T )F (s) F 0 (K T + I T )F, (0) F (s)0 (K T + I T )F (s) where and s denote two d erent fractons of the samle that the break can occur. See Aendx for a roof. The result of Theorem 3 mles that crtcal values of the lmtng dstrbuton of the standardzed test statstc mn I Z, denoted c mn, can be obtaned from the dstrbuton of the mnmum value of a xed number of T correlated normal varables Z wth covarance matrx. Snce mnfz ( T ) ; Z ( 3 T ) :::; Z ( T T ) g = maxf Z ( T ) ; Z ( 3 T ) :::; Z ( T T ) g, we can use the dstrbuton of the maxmum of normal varables Z to calculate crtcal value c mn for a sgn cance level a,.e. P ( < c mn ) = P ( > c mn ) = a. The ntegral functon P ( > c mn ) = a can be calculated numercally based on the robablty densty functon (df) of. Ths densty functon has been recently derved by Arellano-Valle and Genton (008) 7

9 for the more general case of the maxmum of absolutely contnuous deendent random varables of elltcally contoured dstrbutons. For the case of normal random varables, t s gven as f (x) = X (x; ; ; )(xe T 3 ; ; ; ; ); x R, () where e T 3 s a (T 3)-column vector of untes, and are the df and cdf of the normal dstrbuton wth arguments gven as follows: ; (x) = + (x ) ; ( ; ) and ; = ; 0 ;( ; ) ; 3 where = (. ) 0 and = 4 ; ; 5 are resectvely the vector of means and the varanceautocovarance matrx of the (T )-column vector Z whch conssts of random varables Z, for ; ; I, arttoned as Z = (Z ( ).Z ) 0, where Z ( ) s a (T -3)-column vector consstng of the remanng elements of Z excludng Z : The above df of random varable, de ned as f (x), s a mxture of the normal margnal denstes (x; ; ; ) corresondng to all ossble break fractons of the samle. These denstes are weghed wth the cdf values of the (T -3)-column vector xe T 3, gven (xe T 3 ; ; (x); ; ), gvng the robablty that Z takes the largest value across all (mlyng that Z takes ts mnmum value) and the remanng varables of vector Z, collected n vector Z ( ), havng smaller values. The consstency of the test gven by Theorem 3 follows mmedately from Theorem, whch roves the consstency of Z for a known date break. Ths can be seen by notng that f, under the alternatve hyothess ' <, test statstc Z converges to mnus n nty, for I, then so does ther mnmum. 3 Generalzng the test statstcs for serally correlated and heterogenous dsturbance terms In ths secton, we generalze the test statstcs resented n the revous secton to allow for serally correlated and heterogenous dsturbance terms u t, for all. Due to the xed-t dmenson of anel data model and the allowance for a common structural break n the ndvdual e ects t, the maxmum order of seral correlaton, denoted as max, consdered by our tests wll be a functon of T. Ths wll be assumed to be the same for both samle ntervals before and after break ont T 0. Later on, we wll gve a table of values of max whch do not deend on the locaton of the break, T 0, and thus can be roved very useful for the alcaton of our tests, n ractce. To derve the lmtng dstrbuton of test statstcs based on estmator ^' under the above more general assumtons of anel data sets, we wll make the followng assumton about the sequence of the dsturbance terms fu g: Assumton : (b): fu g consttutes a sequence of ndeendent random vectors of dmenson (T X) 8

10 wth means E(u ) = 0 and varance-autocovarance matrces E(u u 0 ) = [ ;ts ], where ;ts = E(u t u s ) = 0 for s = t + max + ; :::; T and t < s: (b): The average oulaton covarance matrx P = s bounded away from zero n large samles: ;tt > 0 for some 0 > 0 and for all > 0 ; for some 0 ; and for at least one t f; :::; T g: (b3): The 4 + -th oulaton moments of y ; = ; :::;, are unformly bounded.e. for every real (T X) vector l such that l 0 l = ; E(jl 0 y j 4+ ) < B < for some B: X (b4): l 0 V ar(vec(y y 0))l > 0 for some 0 > 0; and for all > ; for some and for every real = ( T (T + )X) vector l wth l0 l = : (b5): E(u t y o ) = E u t a = E u t a () = 0 and 8 f; ; :::; g; t f; ; :::; T g: Assumton enables us to derve the lmtng dstrbuton of ^' by emloyng standard asymtotc theory under more general condtons than those of Assumton, whch consders the smle case that u t IID(0; u), for all. More sec cally, condton (b) allows the varance-autocovarance matrces of dsturbance terms u t, = E(u u 0 ), to be heterogenous across the cross-sectonal unts of the anel wth a degree of seral correlaton max less than T. The attern of seral correlaton consdered by matrces cature that mled by movng average (MA) rocesses of u t, often assumed for many economc seres (see, e.g. Schwert (989)). It can be also though of as aroxmatng that mled by AR models of u t whose autocorrelaton des out after max. to derve the lmtng dstrbuton of test statstc ^' can Condton (b) qual es alcaton of a central lmt theorem (CLT) adjusted for the asymtotc bas (nconsstency) of estmator ^' as, under the more general assumtons than condton (b). More sec cally, Condton (b) along wth condton (b4) guarantees that, the varance and the test wll be d erent than zero. Fnally, condtons (b5) and (b3) consttute weak condtons under whch the consstency of the tests can be roved. These two condtons corresond to condtons (a) and (a3) of Assumton. The lmtng dstrbuton of a normalzed anel unt root test statstc based on estmator ^' corrected for ts nconsstency under the above assumton s gven n the next secton. Ths statstc assumes that the fracton of the samle that the break occurs s known. Theorem 4 Let condtons (b) - (b5) of Assumton hold. Then, under the null hyothess ' = and known, we have as, where Z ^V = ^ ^b = ^ ^' ^b ^ d (0; ) (3) tr( ^) P (4) = y0 ; Q y ; s a consstent estmate of the asymtotc bas of ^' whch, under the null hyothess, s gven as b = tr(0 Q ) tr( 0 Q ) ; (5) where matrx s a (T XT )-dmenson matrx havng n ts man dagonal, and ts -lower and uer In sngle tme seres lterature, max s assumed to ncrease wth T wth an order of o(t = ) see Chang and Park (00). 9

11 dagonals of the man dagonal the corresondng elements of matrx 0 Q, and zero otherwse, ^ = P = (y y 0 ) s a consstent estmator of oulaton varance-autocovarance matrx varance functon gven as where F = vec(q vec(u u 0 ): The roof s gven n the aendx. and V s a V = F 0 F 0 ; (6) X 0 ) and = V ar(vec(u u 0 = )) s the varance-covarance matrx of To mlement the test statstc gven by Theorem 4, Z, to test hyothess ' =, we need consstent estmates of the varance-covarance matrx of vector vec(u u 0 ), de ned as :Ths can be done under the null hyothess based on the followng estmator: ^ = X (vec(y y)vec(y 0 y) 0 0 ). (7) = As for Z, matrx lays a crucal role n constructng test statstc Z. It adjusts LS estmator ^' for ts asymtotc bas. Ths bas now comes from two sources: the "wthn" transformaton of the data through matrx Q and the seral correlaton of dsturbance terms u t. 3 enables to adjust ^' for the above two sources of bas. Subtractng from 0 Q The adjusted LS estmator ^' enables us to test the null hyothess ' =, as t reles on samle moments of the elements of varance-autocovarance matrces, for all, whch are serally uncorrelated,.e. E(u t u s ) = 0; for s = t + max + ; :::; T and t < s. These moments are weghted by elements of matrx 0 Q, whch assgn hgher weghts to the moments whch are mmedately before the break ont T 0 than those whch are away than t. They can be consstently estmated under the null hyothess through the varance-covarance estmator ^. The weghts that matrx 0 Q assgns to the above elements of varance-autocovarance matrces obvously deend on the break ont and the maxmum order of seral correlaton max s assumed by test statstc Z. Based on the sec caton of ths matrx, Table and (8) gve values of max whch enable us to aly our tests ndeendently on the locaton of the break T 0 ; or the fracton of the samle. These values are found based on the crteron that the elements of matrx 0 Q elements of do not assgn weghts to zero whch result n a value of varance functon V whch s zero,.e. V = 0. They are useful n choosng the maxmum order of seral correlaton max, consdered by test statstc Z, esecally when the break s of an unknown date. Table : Maxmum order of seral correlaton T max max = [ T ] (8) 3 ote that, under condtons of Assumton, test statstc Z becomes dentcal to Z. 0

12 where [:] denotes the greatest nteger functon. ote that, n the case that dsturbance tests u t are normally dstrbuted, the varance functon V can be wrtten more analytcally as follows: V = F 0 (K T + I T )( )F, (9) where denotes the Kronecker roduct. 4 ts consstent estmate ^ = P = (y y 0 ). Ths form of V can be easly calculated by relacng wth Followng analogous stes to those requred by Theorem 3, test statstc Z can be easly extended to the case of an unknown break ont date, whch requres a sequental alcaton of the test. De ne ths test statstc as z mn I Z. The lmtng dstrbuton of z s gven as z mn I Z d mn(0; ), (0) I, where [ ;s ] s the varance-covarance matrx of the test statstcs Z whose elements, de ned as ;s, are gven by the followng formula: ;s = F 0 F (s) q F 0 F q F (s)0 F (s). () Crtcal values of the dstrbuton of random varable, denoted as f (x ) where x R, can be calculated by relacng the values of s n df formula () wth those of ;s. Ths also requres to obtan consstent estmates of varance-covarance matrx, n the rst ste. 4 Extenson of the tests to the case of determnstc trends In ths Secton, we wll extend the tests resented n the revous secton to allow for ndvdual (ncdental) lnear trends n the anel data generatng rocesses. We wll consder two cases of AR anel data models wth lnear trends. In the rst case, we wll assume that these trends are resent only under the alternatve hyothess of statonarty (see, e.g. Karavas and Tzavals (0a), or Zvot and Andews (99) for sngle tme seres), whle n the second that they are resent under the null hyothess of ' = ether (see, e.g. Carron--Slvestre et al (005) and Km (0). The rst of the above cases s more arorate n dstngushng between nonstatonary anel data seres whch exhbt ersstent random devatons from lnear trends, mled by the resence of ndvdual e ects under the null hyothess of unt roots, and statonary anel data seres allowng for broken ndvdual lnear trends. The second case s more sutable when consderng more exlosve anel data seres under the null hyothess of unt roots, whch can exhbt both determnstc and random ersstent shfts from ther lnear trends. 4 Ths can be easly seen usng standard results of the varance of a quadratc form for normally dstrbuted varates (see e.g. Schott(996)), whch mly V ar[vec(u u 0 )] = V ar(u u ) = (I T + K T )( ):

13 4. Broken trends under the alternatve hyothess of statonarty Consder the followng extenson of the nonlnear AR model : where t y t = t ( ') + ' + t ( ')t + 'y t + u t ; = ; :::; () are de ned by equaton and t = f t T 0 and () f t > T 0. Under the null hyothess ' =, s consttute ndvdual e ects of the anel data model, whch cature lnear trends n the level of seres y t, for all. Under the alternatve hyothess ' <, wll be gven as = f t T 0 and () f t > T 0. That s, they consttute the sloe coe cents of ndvdual lnear trends t, for all. Let us de ne matrx X = (e ; e () ; ; () ), where and () are (T X)-column vectors whose elements are gven as t = t f t T 0 ; and zero otherwse, and () t Then, the "wthn" transformaton matrx now wll be wrtten as Q = I T X = t f t > T 0, and zero otherwse. (X 0 the LSDV estmator, denoted as ^', can be wrtten under null hyothess ' = as follows: X ) X 0 and ^' = " X = y 0 ; Q y ; # " X = y 0 ; # Q u. (3) Followng analogous stes to those for the dervaton of test statstcs Z or Z, nference about unt roots can be conducted based on estmator ^', ths bas now s gven as adjusted for ts asymtotc bas. Under condtons of Assumton b = tr(0 Q ) tr( 0 Q ) (4) (see Aendx, roof of Theorem 5). However, n contrast to the case of model, the average oulaton varance-autocovarance matrx cannot be consstently estmated based on estmator ^ = P = (y y 0 ), due to the resence of ndvdual e ects under the null hyothess ' =. It can be easly seen that, under ' =, y = u + e, where e s a (T X)-vector of untes, and thus X E(y y) 0 = + J T ; (5) = where J T s a T T matrx of ones and = P = E(( ) ): The last relatonsh clearly shows that n order to rovde consstent estmates of based on ^ = P = (y y 0 ), we need to substtute out the average of squared ndvdual e ects n ^. Ths can be done wth the hel of a (T XT )-dmenson selecton matrx M, de ned as follows: M has elements m ts = 0 f ts 6= 0 and m ts = f ts = 0. That s, the elements of ths matrx corresond to the elements of matrx + J T (or P = E(y y 0 ) whch contan only. Based on matrx M, we can derve a consstent estmator of under the null hyothess, whch s gven as tr(mj T ) X ymy 0, (6) =

14 snce tr(m ) = 0, where " " sgn es convergence n robablty. As mentoned above, ths estmator enables us to substtute out ndvdual e ects n the estmator of, gven by ^. Then, a consstent estmator of the bas of the LSDV estmator ^' gven as where = ^b = ^ for model (), de ned as b tr( P = y0 ; ^ ), can be obtaned. Ths s, (7) Q y ; + tr(0 Q M) trace(mj T ) M (see Aendx, roof of Theorem 5) where s a (T XT )-dmenson matrx havng n ts man dagonal, and ts -lower and uer dagonals of the man dagonal the corresondng elements of matrx 0 Q, and zero otherwse. It can be easly seen that tr( estmator of ^b, snce tr( ( + J T )) = tr( ). ^) s a consstent Havng derved a consstent estmator of the asymtotc bas of LS estmator ^', next we derve the lmtng dstrbuton of a test statstc of hyothess ' = based on ths estmator adjusted for ts bas. Ths s done after trmmng out two tme seres observatons from the end of the samle,.e. = T0 T I = T ; T ; :::::; T T, due to the resence of ndvdual e ects and lnear trends under the alternatve hyothess of statonarty. To derve ths lmtng dstrbuton and to rove the consstency of the test, we rely the followng assumton. Assumton 3: Let all condtons of Assumton hold and we also have: E(u t ) = 0; 8 f; ; :::; g; t f; ; :::; T g; E(a t Then, the followng theorem ales. t ) = 0; 8 f; ; :::; g. Theorem 5 Let the sequence fy ;t g be generated accordng to model () and condtons (b)-(b4) of Assumton hold. Then, under the null hyothess ' = and known, we have Z ^V 0:5 ^ ^' ^b ^ d (0; ), (8) as, where V = F 0 F ; s de ned n Theorem 4, and F = vec(q of the theorem s gven n the aendx. 0 ): The roof Aart from the ntal condtons of the anel y 0, the test statstc gven by Theorem 5, Z, s smlar under the null hyothess to the ndvdual e ects of the anel, due to the allowance of broken trends n the "wthn" transformaton matrx, Q. To test the null hyothess of unt roots, test statstc Z reles on the same moments to those assumed by statstc Z, namely E(u t u s ) = 0; for s = t + max + ; :::; T and t < s. These moments now are weghted by elements of matrx 0 Q, where matrx s arorately adjusted to we o the e ects of nusance arameters on the lmtng dstrbuton of the test statstc. The maxmum order of seral correlaton of varance-autocovarance matrces test statstc Z s the same to that assumed by test statstc Z. assumed by Fnally, note that test statstc Z can be extended to the case of an unknown date break followng an analogous rocedure to that assumed for sequental tests statstcs z and z, de ned by equatons (9) and 3

15 (0), resectvely. Ths verson of the test statstc wll be denoted as z mn I Z : Its lmtng dstrbuton s gven as z mn I Z d mn I (0; ), (9) as, where [ ;s ] s the varance-covarance matrx of test statstcs Z whose elements ;s are gven by the formula: ;s = F 0 F (s) q q F 0 F F (s)0 F (s) : Crtcal values of the dstrbuton of can be derved based on the df f (x), gven by (), followng an analogous rocedure to that assumed for test statstc z. 4. Broken trends under the null hyothess of unt roots To allow for a common break n the ndvdual e ects of the anel data model under the null hyothess ' =, consder the followng extenson of AR model : y t = t ( ') + ' t + t ( ')t + 'y t + u t ; = ; :::; (30) Usng vector notaton, ths model mles that, under hyothess ' =, the rst-d erence of vector y s gven as y = e + () e () + u. As for model (), ths means that estmator ^ = P = (y y 0 ) wll not lead to consstent estmates of the average oulaton varance-autocovarance matrx the resence of ndvdual e ects and (). These mly X =, due to E(y y 0 ) = e e 0 + () e() e ()0 +, (3) where J = e e 0 and J = e () e ()0. The allowance of a break n ncdental arameters under the null hyothess requres estmaton of squared ndvdual e ects and () so as to obtan consstent estmates of matrx. To ths end, we wll follow an analogous rocedure to that ntroduced n the revous subsecton, based on selecton matrx M. selecton matrces M and M (), whch select square ndvdual e ects We wll de ne two (T XT )-dmenson block dagonal and (), resectvely. Matrx M whch has elements m ts = 0 f ts 6= 0, and m ts = f ts = 0 and, thus, t selects the elements of matrx e e 0 + () e() e ()0 + consstng only of e ects, for t; s T 0. For t or s > T 0, all elements of M are set to m ts = 0. On the other hand, Matrx M () has elements m () ts = 0 f ts 6= 0, and m ts = f ts = 0 and, thus, t selects the elements of matrx e e 0 + () e() e ()0 + only of e ects () ; for t; s > T 0. For t or s T 0, all the elements of M () are set to m () ts = 0. consstng Based on the above de ntons of selecton matrces M and M (), we can obtan the followng two consstent estmators of and () : tr(m J ) X = y 0 M y and tr(m () J ) X = y 0 M () y (), (3) resectvely, snce tr(m (j) ) = 0 for j = ; and tr(m (j) J r ) = 0 for j; r = ; and j 6= r. These estmators 4

16 can be emloyed to obtan consstent estmates of matrx, whch are net of square ndvdual e ects and () b 3 3 where. Then, a consstent estmator of the bas of the LSDV estmator ^', can be derved as 3 = + tr(0 Q M ) ^b 3 = ^ 3 tr( P = y0 ; 3 ^ ) for model (30), de ned as, (33) Q y ; trace(m J M ) + tr(0 Q M () ) trace(m () J M () ). Adjustng ^' by the above estmator of ts bas wll lead to a anel unt root test statstc whose lmtng dstrbuton wll be net of squared ndvdual e ects and () under the null hyothess. In the next theorem, we derve ths dstrbuton. Ths corresonds to the case of known date break. If the date s unknown, then t can be estmated, n a rst ste, based on the rst d erences of the ndvdual seres y t of the anel data model under the null hyothess ' =,.e. y = e + () e () + u. As shown by Ba (00), ths estmator rovdes consstent estmates of the break ont T 0, whch converges at on o( ) rate. Based on a consstent estmate of T 0, then we can aly the test statstc gven by the next theorem to conduct nference about unt roots. Theorem 6 Let the sequence fy ;t g be generated accordng to model (30) and condtons (b)-(b4) of Assumton hold. Then, under the null hyothess ' = and known, we have Z 3 ^V 0:5 3 ^ 3 ^' ^b 3 ^ 3 d (0; ), (34) as, where and F 3 = vec(q V 3 = F 0 3 F 3 (35) 0 3 ): The roof of the theorem s gven n the aendx. As test statstc Z, the test statstc gven by Theorem 6, Z 3, s smlar under the null hyothess to the ndvdual e ects and (), due to the ncluson of broken trends n the "wthn" transformaton matrx Q. Due to the resence of a break under the null hyothess ' =, the maxmum order of seral correlaton of the dsturbance terms u t, max, allowed by the test s not gven by Table. Ths s gven by a) T 3; when T s even and T 0 = T (36) b)equal or less than mnft 0 ; T T 0 g n all other cases of T or T 0 5 Based on condtons of Assumton 3, t can be roved that test statstc Z 3 s consstent, followng analogous stes to those for the roof of the consstency of test statstc Z. The test s also consstent, f 5 Agan the crteron of choosng max s the value of varance functon V 3 not to be zero. If T s even then max=mnft 0 ; T T 0 g always excet from the case that T 0 = T mnft 0 ; T T 0 g = mnf; 5g = where t becomes max= T 3: Examle: T = 0, T 0 = 3 : max = If T 0 = T = 5 : then max = T 3 = : If we used the results from (8) we would have. max = mnft 0 ; T T 0 g = mnf3; 3g = 3 whch does not aly. 5

17 the break ont s unknown and s estmated, n the rst ste, based on the rocedure mentoned above 6. 5 Smulaton Results In ths secton, we conduct a Monte Carlo study to nvestgate the small samle erformance of the tests suggested n the revous sectons. For reasons of sace, n our study, we consder only the case that the break date s unknown. We consder exerments of d erent samle szes of and T,.e. = f50; 00; 00g and T = f6; 0; 5g, whle the fractons of samle that the break occurs are assumed to be = f0:5; 0:5; 0:75g, whch facltate the choce of the break ont T 0. For all exerments, we conduct 0000 teratons. In each teraton, we assume that the data generatng rocesses are gven by models and (), resectvely, where dsturbance terms u t follow a MA rocess,.e. u t = " t + " t, wth " t ~IID(0; ); for all and t, and = f 0:5; 0:0; 0:5g. The values of the nusance arameters of the smulated models, namely the ndvdual e ects or the sloe coe cents of ndvdual lnear trends are assumed that they are drven from the followng dstrbutons: U( 0:5; 0), () U(0; 0:5); U(0; 0:05), U(0; 0:05) and () U(0:05; 0:05), where U stands for the unform dstrbuton. The small magntude of ndvdual e ects (j) or sloe coe cents (j) assumed above corresond to evdence found n the emrcal lterature, see e.g. Hall and Maresse (005). Ths small magntude also makes t harder for the tests to dstngush the null hyothess of unt root from ts alternatve of statonarty. For all smulaton exerments, we assume that the order of seral correlaton s set to =. Table (a): Sze and ower of z mn Z, for = 0: T = 0:5 ' = : ' = 0: ' = 0: = 0:5 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: If T = 5 then max=mnft 0 ; T T 0 g always (e.g. f T 0 = 7; max = mnf5; 6g = 5): 6

18 Table (b): Sze and ower of z mn Z, for = 0: T = 0:5 ' = : ' = 0: ' = 0: = 0:50 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: order. Table (c): Sze and ower of z mn Z, for = T = 0:5 ' = : ' = 0: ' = 0: = 0:50 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: Ths means that, for = 0, we assume an order of seral correlaton whch s hgher than the arorate Ths exerment wll show f our tests undererform when a hgher order of seral correlaton s assumed, whch may haen n ractce. The results of our Monte Carlo analyss for sequental test statstcs z and z, corresondng to models and (), are summarzed n Tables (a)-(c) and 3(a)-(c), resectvely, for values of f0:5; 0:5; 0:0g. These tables resent values of the sze and ower of the tests. The sze of the tests s calculated for ' = :00, whle the ower for values of ' f0:95; 0:9g. ote that, n all 7

19 exerments, the ower s calculated at the nomnal 5% sgn cance level of the dstrbuton of the tests. The results of the tables ndcate that both tests examned have sze whch s close to the nomnal level 5% consdered. Ths s true for all combnatons of and T. The sze erformance of the tests s close to ts nomnal level even f the MA arameter takes a large negatve value,.e. = 0:5. ote that, for ths case of, sngle tme seres unt root tests are crtcally overszed (see, e.g., Schwert (989)). The sze of the tests does also not deterorate, f a hgher order of seral correlaton = s assumed than the true order,.e. = 0. The sze erformance of the tests mroves as ncreases relatve to T. Ths can be attrbuted to the fact that, as ncreases relatve to T, varance-covarance matrx s more recsely estmated by estmator ^. The above results hold ndeendently on the fracton of the samle that the break occurs,. Table 3(a): Sze and ower of z mn Z ; for = 0: T = 0:5 ' = : ' = 0: ' = 0: = 0:5 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: Regardng the ower of the tests, the results of the tables ndcate that, as was exected, the tests have better ower erformance for the case of model, where there are only ndvdual e ects under the alternatve. For model (), where lnear trends are consdered ether under the alternatve hyothess ' < or the null, the ower of the tests reduces. However, the tests are not based and, consstently wth the theory, they have ower whch ncreases as the value of ' moves away from unty. The ower erformance of all three test statstcs s not a ected by the fracton of the samle that the break occurs; ; see also Karavas and Tzavals (0a). However, t deends on the value of. When s negatve (.e., = 0:5), the ower erformance of both tests deterorates. Fnally, note that the ower of the tests tends to ncrease as ncreases relatve to T. 8

20 6 Concluson Ths aer suggests anel unt root tests whch allow for a common structural break n the ndvdual e ects or lnear trends of dynamc anel data models. Common breaks n anel data models can arse n cases of a credt crunch, an ol rce shock or a change n tax olcy (see e.g. Ba (00)). The suggested tests assume that the tme-dmenson of the anel T s xed (or nte), whle the cross-secton grows large. Thus, they are arorate for short anel alcatons, where T s smaller than. They are based on the corrected for ts nconsstency (asymtotc bas) least squares dummy varable (LSDV) estmator of the autoregressve coe cent of dynamc anel data models (see Harrs and Tzavals (999)), and they allow for seral correlaton n the dsturbance terms. Ths estmator rovdes unt root tests whch have the very useful roerty of beng nvarant (smlar) under the null hyothess of a unt root n the autoregressve comonent of anel data models to the ntal condtons of the anel or the ndvdual e ects. Ths roerty does not restrct alcaton of the tests to anel data where condtons of mean or covarance statonarty of the ntal condtons or ndvdual e ects are requred. Table 3(b): Sze and ower of z mn Z, for = 0: T = 0:5 ' = : ' = 0: ' = 0: = 0:50 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: The aer derves the lmtng dstrbutons of the tests. When the break s unknown, t shows that the lmtng dstrbuton of the tests s the mnmum of a xed number of correlated normals. Ths dstrbuton s gven as a mxture of normals. It can be analytcally obtaned based on recent results of Arellano-Valle and Genton (008), for absolutely contnuous deendent varables. Knowledge of the analytc form of the lmtng dstrbuton of the tests consderably facltates calculaton of crtcal values for the mlementaton of the tests n ractce. Fnally, the aer examnes the small samle sze and ower erformance of the tests by conductng a Monte Carlo study. Ths s done for the case that the break s of an unknown date. The results of ths 9

21 exercse ndcate that our tests have the correct nomnal sze and ower whch s bgger than ther sze. As was exected, the ower of the tests s hgher for the dynamc anel data model whch consder ndvdual e ects rather than for the model whch also allows for ndvdual lnear trends. For all cases, the ower s found to ncrease f ncreases faster than T, due to the xed-t assumton of the tests. 7 Aendx Table 3(c): Sze and ower of z mn Z, for = T = 0:5 ' = : ' = 0: ' = 0: = 0:50 ' = : ' = 0: ' = 0: = 0:75 ' = : ' = 0: ' = 0: In ths aendx, we rovde roofs of the theorems or any other theoretcal results resented n the man text of the aer. Proof of Theorem : To derve the lmtng dstrbuton of the test statstc of the theorem, we wll roceed nto stages. We rst show that the LSDV estmator ^' s nconsstent, as. Then, wll construct a normalzed statstc based on ^' corrected for ts nconsstency (bas) and derve ts lmtng dstrbuton under the null hyothess of ' =, as. Decomose the vector y ; for model under hyothess ' = as y ; = ey 0 + u, (37) where the matrx s s a (T XT ) matrx de ned as r;c =, f r > c and 0 otherwse. Premultlyng (37) wth matrx Q yelds Q y ; = Q u, (38) 0

22 snce Q e = (0; 0; :::; 0) 0. Substtutng (38) nto () yelds ^' = P = y0 ; Q u P = = y0 ; Q y ; P = u0 0 Q u P. (39) = u0 0 Q u By Ktchn s Weak Law of Large umbers (KWLL), we have X u 0 0 Q u b = utr( 0 Q ) and = X u 0 0 Q u = utr( 0 Q ), (40) = where " " sgn es convergence n robablty. Usng the last results, the yet non standardzed statstc Z can be wrtten by (39) as ^ ^' ^b ^ = ^ P = y0 ; Q u ^ = ^ utr( 0 Q ) ^ X P y; 0 Q u tr( 0 Q = ) y0 y. (4) tr( ) = Snce, under the null hyothess ' =, we have u = y, the last relatonsh can be wrtten as follows: ^ = = = X = X = ^' X = ^b ^ u 0 0 Q u tr( 0 Q ) tr( ) u 0 ( 0 Q )u = W, X = X = tr u 0 u h ( 0 Q )u u 0 (4) where W consttute random varables wth mean E(W ) = E[u 0 ( 0 Q )u ] = tr[( 0 Q )E(u u 0 )] = utr( 0 Q ) = 0, for all, snce tr( 0 Q ) = tr( ) (or tr( 0 Q ) = 0) and varance V ar(w ) = V ar(u 0 ( 0 Q )u ) = V ar[f 0 vec(u u 0 )] = = F V ar[vec(u u 0 )]F 0 ; for all. The results of Theorem follows by alyng Lndeberg-Levy central lmt theorem (CLT) to the sequence of IID random varables W. Followng standard lnear algebra results (see e.g. Schott(997), varance

23 V ar[vec(u u 0 )] can be analytcally wrtten as V ar[vec(u u 0 )] = V ar(u u ) = 4 u(i T + K T ), where denotes the Kroenecker roduct. Proof of Theorem : Assume that the break ont T 0 s known. De ne vector w = (; '; ' ; :::; ' T ) 0 and matrx 0 = 0 : : : : : 0 0 : ' : : ' ' : : : : : : : : : : 0 : C A ' T ' T 3 : : ' 0 Under null hyothess ' = ; we have = : Based on the above de ntons of w and, vector y ; can be wrten as y ; = wy 0 + X + u, (43) where d = (a ( '); a () ( ')) 0 : Usng last exresson of y ;, test statstc Z can be wrtten under the alternatve hyothess ' < as follows: Z = V b =^ = ^V =^ ^' ' + ^b ^ P = y0 ; Q u P = y0 ; Q y ; ^ utr( 0 Q ) P = y0 ; Q y ; = ^V =^ (' ) + = X ^V y; 0 Q u ^ utr( 0 Q ) = ( ) n o = =^ ^V (' ) = X + ^V (y; 0 Q u y 0 y ). ext, we wll show that summand (I) dverges to (I) These two results mly that, as, test statstc Z converges to To rove the above results, we wll use the followng denttes: = (II) (44) and summand (II) s bounded n robablty., whch roves ts consstency. u = y 'y X d (45) and y = u + (' )y + X d, (46)