Sequential Panel Unit Root Tests for a Mixed Panel with Nonstationary and Stationary Time Series Data

Transcription

1 Sequential Panel Unit Root et for a Mixed Panel with ontationary and Stationary ime Serie Data Donggyu Sul Department of Economic, Univerity of Auckland July, 004 Abtract he paper provide a new effective detrending method by uing recurive mean adjutment. Baed on recurive mean adjutment method, univariate unit root tet a well a panel unit root tet are propoed. More importantly, the maximum order tet under cro ectional dependence are developed to detect whether or not a panel contain nontationary unit. he cro ectional average of the econd moment of the recurive time erie mean grow over time when a panel contain nontationary unit. o reduce the ize ditortion of tet, equential panel unit root tet are propoed. he null hypothei of panel unit root (all unit are nontationary) i firt teted. When the firt null i rejected, the econd null hypothei of mixed panel unit root (ome unit are nontationary) i teted. Monte Carlo imulation how that the equential tet perform well with moderately large and. Incomplete: Pleae do not quote. he final verion of the paper will be available by the end of 004 at

2 Introduction Since Quah (990, 94) opened the door of panel unit root teting literature, everal important theoretical development have been contributed by everal reearcher. Levin, Lin and Chu (LLC, 00) generalize Quah (990, 94) panel unit root tet under the alternative of a homogenou panel. Im, Pearan and Shin (IPS, 003), Choi (00) and Maddala and Wu (MW, 999) conider panel unit root tet under the alternative of a heterogeneou panel. More recently Bai and g (B, 004a), Chang (003), Moon and Perron (MP, 004) and Phillip and Sul (PS, 003) propoe panel unit root tet under cro ectional dependence. Along with the theoretical development of panel unit root tet, their ue in empirical reearch ha grown exponentially. he mot important reaon for their popularity i that panel unit root tet reject the null hypothei of unit root more often than univariate unit root tet. hi i a natural reult becaue the goal of the panel unit root tet under the null hypothei of unit root i to emplify the power of tet through the pooling of information acro unit. However the unolved and thorny problem of panel unit root tet i that the rejection of the null hypothei of panel unit root doe not imply that all unit in the panel are tationary. Conider the following two hypothee of panel unit root. Ho 0 : ρ i = for all i HA 0 : ρ i < for all i Ho : ρ i = for all i HA : ρ i < for ome i Quah(990) and LLC tet the firt null hypothei Ho 0 againt HA 0. Some empirical reearcher think that the rejection of Ho 0 implie that all unit in a panel are tationary. Unfortunately thi i a myth. Karlon and Löthgren (000) report the even when a half of panel are tationary, the LLC tet reject Ho 0 very often. he next null hypothei Ho ha the ame problem. IPS, Choi, MW, B, MP and PS tet are deigned to reject Ho more often than LLC when there i at leat one tationary unit in a panel. Panel empirical reearcher, however, want to tet whether or not a panel conit of pure tationary unit rather than to tet whether a panel contain at leat one tationary unit. For example, teting growth convergence or long run purchaing power parity require that all unit in a panel are tationary. he rejection of Ho 0 or Ho doe not provide any meaningful anwer. he null hypothei of panel tationarity doe not have uch problem. he iue i rather whetherornotthereianefficient panel tationary tet. Hadri (000) propoe the panel KPSS tet under cro ection independence. However a Bai and g (004b) how, the finite ample performance of the panel KPSS tet i rather diappointing. Canor and Kilian (00) and Lee (996) criticize KPSS tet becaue the power of KPSS tet i almot the ame a the ize of the KPSS tet baed on Andrew (99) prewhitening HAC etimator while the ize ditortion of KPSS tet baed on ewey and Wet (987) fixed bandwith HAC etimator i eriou. Sul, Phillip and Choi (003) provide a reaon that the KPSS tet with Andrew

3 0.97 rule i not conitent under the null hypothei and propoe a new boundary rule of / where tand for the number time erie obervation. Under the new rule, the power of the KPSS tet become fairly good when compared with the KPSS tet baed on W HAC etimator, while the ize ditortion of the tet become moderate under the new rule. he problem i that the KPSS tet doe not have any power if the AR() coefficient ρ i greather than /. For example, with = 00, the KPSS tet work only when ρ 0.9. More importantly, Choi (00) report that even when the half of a panel conit of nontationary unit, the ize adjuted power of the KPSS tet baed on W fixed bandwith HAC etimator i very low. In other word, the KPSS tet doe not have enough power to detect whether or not a panel i mixed. Chang and Song (00) conider more attractive null hypothei of the mixed panel unit root and propoe a mixed panel unit root tet. Ho : ρ i = for ome i HA : ρ< for all i he null hypothei Ho i the mot deirable hypothei for empirical panel unit root tet. he rejection of Ho implie that all unit in a panel are tationary. Chang and Song (00) tet, however, uffer from a lack of power. It i worthwhile noting that teting for a mixed panel unit root might not be meaningful when the power of a tet under Ho i poor. Hence to improve the power of the mixed panel unit root tet, one hould improve the power of the panel unit root tet under the null Ho in the firt place. hi paper propoe a new mixed panel unit root tet for the cae of a contant and a linear trend under the null hypothei Ho. henewpropoedtetibaedonthefactthatthe econd moment of the recurive mean i growing over time under the null hypothei of Ho. However, the propoed tet uffer from a moderate ize ditortion epecially when there are a mall number of nontationary unit in a panel. o reduce the ize ditortion while maintaining good power of the tet, we introduce equencial panel unit root tet. he propoed teting procedure conit of two panel unit root tet. he firt null hypothei i Ho.IfthenullHo i rejected, then the econd null hypothei of mixed panel unit, Ho i teted. o increae the rejection rate of the null Ho, new panel unit root tet are propoed baed on recurive mean adjutment method. In the next ection, I introduce Shin and So (00) recurive mean adjutment for unit root tet, propoe a new efficient bia reduction method for the general autoregreive model with a linear trend and contruct a univariate unit root tet by utilizing recurive mean adjutment. In ection 3, the new efficient and powerful panel unit root tet baed on recurive mean adjutment are propoed. Since the panel recurive mean adjuted etimator produce a maller he finite ample perforamcen of the prewhitening HAC etimator i heavily depending on the mall ample bia of the AR() coefficient in the prewhitening tage. Andrew (99) ugget the 0.97 rule that the etimate of AR() coefficientireplacedby0.97whenitigreatherthan

4 variance than the leat quare dummy variable (LSDV) etimator, the pooled panel unit root tet (like the LLC type tet) i more powerful than the meta tet (like IPS, MW or Choi (00) ). In ection 4, a recurive mixed panel unit root tet i propoed and it aymptotic propertie are tudied. he cro ectional average of the econd moment of the recurive mean with a mixed panel grow very fat at order O 3 a the time erie obervation increae, while that with a pure tationary panel grow le fat at order O( ). I utilize thi information to contruct a imple recurive mixed panel unit root tet. Section 5 dicu the teting trategy under cro ection dependence and provide olution. Section 6 how the reult of Monte Carlo imulation. Section 7 provide a hort application of purchaing power parity over OECD countrie and conclude. ModelandUnitRootetwithRecuriveMeanAdjutment. Model he autoregreion model conidered in the paper fall into the following two categorie: ( y it = a i ( ρ M: (Unknown Contant) i )+ρ i y it + ε it y it = a i + x it,x it = ρ i x it + ε it ( y it = a i ( ρ)+b i ρ M: (Linear rend) i + b i ( ρ i ) t + ρ i y it + ε it y it = a i + b i t + x it,x it = ρx it + ε it where t (t =,..,) indexe the time erie obervation while the index i (i =,...,) tand for the ith cro ectional unit. he regreion error ε it i a tationary proce. In the unit root cae, the initialization of x it i taken to be x i0 = O p () andiuncorrelatedwith{ε it } t.. Univariate Unit Root et with Recurive Mean Adjutement We conider the univariate unit root tet ( =). he limiting ditribution i obtained by letting. Shin and So (999) originally introduce recurive mean adjutment in autoregreion originally to reduce the mall ample bia of leat quare etimator. Later Shin and So (00, 00) extend their recurive mean adjutment to unit root tet for the cae of unknown mean. Phillip, Park and Chang (00) tudy more extenively the propertie of recurive mean adjutment, while Sul, Phillip and Choi (003) provide the reaon why recurive mean adjutment reduce the mall ample bia and how that the recurive detrending propoed by Shin and So (00) doe not work properly. o fix idea, let conider M with iid 4

5 ε it and define recurive mean ȳ t = t t and rewrite the regreion of M in deviation from mean form: = y y t ȳ t = a ( ρ)+ρ (y t ȳ t ) ( ρ)ȳ t + ε t () From M, Plugging () to () yield ( ρ)ȳ t = a ( ρ)+( ρ) x t. () y t ȳ t = ρ (y t ȳ t )+ t (3) where t = ( ρ) x t + ε t. When ρ =, t = ε t. Since the regreion error t doe not contain overall mean of y t, the mall ample bia of ρ i ignficantly reduced. hi principle, however, cannot directly apply to the cae of linear trend. Phillip, Park and Chang (00) propoe a recurive detrending method to make the regreion error become a martingale difference equence. heir detrending etimator uffer from eriou upward bia when ρ<. Here we provide a new detrending method to reduce the mall ample bia. For M with =, oberve thi: ȳ t = t y =a + b (t ) + x t, t = y t ȳ t = a +(x t x t ) y t ȳ t = a ( ρ)+bρ + b ( ρ) t + ρ (y t ȳ t ) ( ρ)ȳ t + ε t = a ( ρ)+bρ + ρ (y t ȳ t ) ( ρ) x t + ε t he trend i eliminated but the contant i till preent. aking an overall mean adjutment yield y t ȳ (ȳ t µ) =ρ [y t ȳ (ȳ t µ)] + (u t ū) where µ = P ȳ t, ȳ = P y t and u t = ( ρ) x t + ε t. hi procedure reduce the mall ample bia ignificantly. able how the dramatic bia reduction from the recurive mean adjutment. It i worthywhile noting that the variance of the recurive mean adjuted (RMA) etimator i far le than that of the OLS etimator, epecially for the cae of the linear trend. We alo invetigated whether or not the recurive mean adjutment work with a general AR(p) pecification by mean of Monte Carlo imulation and found that the propoed new etimator work very well. In the next ection, we provide explicit bia formulae for RMA etimator. he imulation reult are not reported here to ave the pace but will be available upon requet of the author. 5

6 A able revealed, the relative variance of the RMA etimator compared to that of the OLS etimator decreae a increae. hi ueful fact can be ued for teting unit root. Shin and So (00) already propoed a univariate unit root tet baed on recurive mean adjutment for the cae of an unknown contant. Here we complete their tak by adding the cae of a linear trend. Let wt c = y t ȳ t, wt τ = y t ȳ t ȳ +µ wt c = y t ȳ t, wt τ = y t ȳ t ȳ +µ and conider the RMA etimator of ρ in the following general autorregreion w c t = ρw c t + w τ t = ρw τ t + p φ j y t j + u t j= for M p φ j ( y t j ȳ j )+u t j= for M where ȳ j itheamplemeanof y t j. he RMA ADF tet are given by t r = ˆρ e (ˆρ) where e (ˆρ) i the tandard error of ˆρ. he limiting ditribution of the tet tatitic under the local to unity alternative ρ =+c/ follow a Propoition Aume u t i a equence of i.i.d. error with E(u )=0and σ = E u <. hen For M, Z h t r = J J i Z ³ ( Z dw J J dr / ³ + cφ J J dr / where 0 Z ³ + J J µ J Φ = 0 0 Z 0 Z Jdr dr Ã! p φ i, J = J (r) = i= Z r J = J (r) =r J () d, 0 0 Z r 0 J = J ³ J J e c(r ) dw () Z 0 0 ) dr / Z J () d + 0 Here W (r) i a tandard Brownian motion on [0, ] while J (r) i the Orntein-Uhlenbeck proce. he proof of Propoition i traightforward, hence it i omitted. 3 he critical value 3 For M, Shin and So (00) provided the limiting ditribution of the tet tatitic given by t = h R 0 J J ih ½ R dw 0 J J / hr dri + cφ 0 J J i / h R dr + 0 J J R Jdrih 0 J J i ¾ / dr Jdr 6

7 of t r with c =0(for teting the unit root hypothei) are given in able. 4 able : Bia, variance and mean quare error (MSE) of RMA and OLS etimator. Contant Linear rend (A) (B) (C) (D) (A) (B) (C) (D) ρ=0.9,= ρ=0.94,= ρ=0.96,= ρ=0.98,= ρ=.00,= ρ=.00,= ρ=.00,= ρ=.00,= ρ=.00,= ote: (A) =Bia of recurive mean adjuted etimator; (B) = Bia of OLS etimator; (C) = Variance ratio of recurive mean adjuted etimator to OLS etimator; (D) = MSE ratio of recurive mean adjuted etimator to OLS etimator. able : Critical value for RMA ADF tet %.5% 5% 0% 0% Contant Linear rend he local aymptotic power comparion with the ADF tet can eaily be done. o ave pace a well a to maintainthefocuofthepaper,wedon treportthereultherebutnotethatthelocalaymptoticpowerwith RMA i much better than that with ADF tet. 7

8 3 Panel Unit Root et In thi ection, the panel unit root tet baed on the RMA etimator are propoed under cro ectional independence. wo type of the RMA panel unit root tet are conidered. he firt tet i imilar to Choi (00) and Maddala and Wu (999) Meta analyi. he Fiher tet tatitic can be deriven directly from the univariate RMA unit root tet developed in the previou ection. Let p i denote the probability value of t r tatitic of the following regreion: wit c = ρ i wit c + wit τ = ρ i wit τ + p φ ij y it j + u it for M (4) j= p φ ij ( y it j ȳ i j )+u it for M (5) j= he null hypothei of the panel unit root i given by Ho : ρ i =for all i while the alternative i H A : ρ i < for ome i he RMA Fiher tet tatitic ditribution i given by: R = ln p i χ (6) i= Here we conider only that the recurive mean adjuted Fiher tet becaue Choi (00) and Phillip and Sul (003) report that the Fiher tet perform bet among other Meta tatitic perform. he econd tet i the pooled RMA tet. Even when ρ i 6= ρ, by pooling cro ectional information, the pooled RMA unit root tet provide a greater rejection rate than the RMA Fiher tet. Further detail will be provided in Section 6. Here we tudy the aymptotic propertie of the pooled RMA etimator for the cae of ρ i =for the cae of AR(). Conider panel AR() regreion are given by wit c = ρwit c + u it for M, wit τ = ρwit τ + u it for M. Denote ˆρ pr = P P i= t= wc it wc it P P i= t= w c for M it ˆρ pr = P P i= t= wτ it wτ it P P i= t= w τ for M it 8

9 and their t tatitic are t pr = ˆρ pr e ˆρ. pr Following Harri and zavali (999), a for fixed, we have Propoition he probability limit of the pooled recurive mean adjuted etimator under the null hypothei of panel unit root i given by plim ˆρpr =0, for M and M and ˆρ pr qvar ˆρ pr L (0, ) where Var ˆρ ³ pr =ˆσ P P u i= t= w k it for k = c and τ for contant and linear trend, repectively. Appendix A provide the proof of Propotion but it i intructive to ketch it ouline here. For fixed effect cae, it i eay to ee why the pooled RMA etimator i conitent. Under the null, we have "Ã! # t plim y it u it = plim y i u it =0 t i= t= i= t= ince Ey it j u it =0for j>0a long a u it i not erially correlated. For the linear trend cae, note that Ã!Ã! plim y it u it i= t= t= Ã Ã!! Ã t! = plim y i u it t i= t= a long a u it i not erially correlated. For panel AR(p) regreion, the above reult hold a long a the enough augumented term are included in the regreion. Choi, Mark and Sul (004) find that when ρ<, the pooled RMA etimator i upward biaed and it inconitency i given by ρ ln with moderately large. 5 For the linear trend cae, alo the pooled RMA etimator i lightly upward biaed when ρ < but it bia vanihe rapidly a increae. 5 he expreion of the approximation i O( ln ) which i greater than O. However, the actual bia in abolute value i far le than that of LSDV with fixed effect. See Choi, Mark and Sul (004) for exact biae formulae. = = t= 9

10 It i worthywhile noting that the pooled RMA etimator i more efficient than the pooled mean unbiaed etimator propoed by Harri and zavali (999) and Phillip and Sul (004). Let ˆρ c be the pooled mean unbiaed etimator. hat i, ˆρ c =ˆρ + bia (,) where ˆρ i the pooled OLS etimator and bia (,) i the mean bia function provided by Harri and zavali (999) and Phillip and Sul (004). For the cae of unknown contant, bia (,)=3/ while for the cae of the linear trend, it become 7.5/ with moderately large. he variance of the mean unbiaed etimator ˆρ c i larger than the variance of ˆρ pr aymptotically. Harri and zavali (999) provide the aymptotic variance of ˆρ c under the aumption of normal error of ε it which i given by plim V (ˆρ c )= 3( ) 5 ( +) 3 5( ) ( +) 3 ( ) A, the variance ratio i given by " # ( V (ˆρ c ) lim plim V ˆρ ' pr for M for M 3. for M.87 for M For mall, the exact aymptotic variance ratio for panel AR() model are plotted in Figure. Even in mall (let ay > 5), the pooled RMA etimator i more efficient than the pooled mean unbiaed etimator for both the fixed effect and the incidental linear trend. Maddala and Wu (999) and Karlon and Löthgren (000) report that the IPS and the Fiher tet reject the null of panel unit root than the LLC tet under the alternative more often. One of the reaon i that the variance of the pooled mean unbiaed etimator i hampered by the biae. In other word, the variance of the pooled mean unbiaed etimator i alway larger than the variance of the LSDV etimator which reult in the power lo. For the pooled RMA etimator, thi i no longer true. Even when ρ i 6= ρ under the alternative, the power of the pooled RMA etimator i much better than that of the Fiher tet conidered in (6). Figure how the reult of a Monte Carlo imulation with =5of which ρ i i generated from U (0.9,.0) where U tand for the uniform ditribution. ote that the Fiher tet baed on the univariate RMA-ADF tet i uperior to the Fiher tet baed on ADF tet in term of power a well a ize. Moreover, the pooled RMA tet i more powerful than the Fiher tet baed on RMA-ADF tet. See other Monte Carlo reult later in the paper for a more detail comparion. 0

11 4 3 Fixed Effect Variance ratio Incidental rend Figure : Aymptotic variance ratio of the pooled mean unbiaed etimator to the pooled recurive mean adjuted etimator for panel AR() cae. 4 Mixed Panel Unit Root et Conider the following two null hypothee. Ho : ρ = for all i HA : ρ i < for ome i Ho : ρ i = for ome i HA : ρ i < for all i If the firt null i not rejected, then there i no need to conider the econd hypothei a long a the tet under the firt null hypothei doe not uffer from a ize ditortion. When the firt null hypothei of the panel unit root i rejected, the mixed panel become an important iue. Since the firt alternative HA implie that there are ome tationary unit in the panel, the rejection of the firt null naturally lead to tet the econd null hypothei. hi ection provide the aymptotic behavior of the pooled OLS etimator where a panel conit of nontationary and tationary unit and propoe a convienient tet for mixed panel unit root tet under the econd null hypothei. 4. Propertie of a Mixed Panel he main feature of a mixed panel i that the dominant root converge to unity a increae. Let α be the percentage of tationary unit in the panel. We further aume that the fraction

12 Pooled et for M 0.8 Meta et for M Power Pooled et or M 0. Meta et for M Figure : Power Comparion: pooled v.. U (0.9,.0) and =5. Meta tet under heterogeneity of ρ i with ρ i of tationary unit doe not depend on the ize. In other word, the number of tationary unit, α or nontationary unit, ( α) goe to infinity a. For implicity, we aume the cro ectional homogeneity: hat i ρ i = ρ for all i. he probability limit of the pooled OLS etimator with the mixed panel approache unity a increae. o fix idea, conider the cae of a known contant and let σ =for notational convinience. hen the probability limit of the pooled OLS etimator for fixed i given by plim ˆρ = α ρ ρ +( α) + α +( α) + ρ = c 0 = for moderately large ( ρ) α ρ α +( α) + ρ where ( α)(+ρ) c 0 =. α Here we aume the variance of the innovation error i identical for both tationary and nontationary cae. Otherwie, the definition of α hould be changed to account for the variance ratio of the innovation error of the tationary unit to that of nontationary unit. onethle, the probability limit of the pooled etimator ha the feature of a local to unity. A increae, the nontationary unit dominate the tationary unit, which reult in the pooled OLS e- (7)

13 timator approaching unity. Moon and Phillip (00, 004) provide the etimation method for the local unity parameter c in the cae of a homogeneou panel by utilizing generalized method of moment. If the panel i known to be mixed, then the local unity parameter c can be etimated. However it i hard to detect whether or not a panel i mixed, epecially under fixed effect or incidental linear trend cae. A ickell (987) and Phillip and Sul (004) how, the pooled OLS etimator ufferfromaerioudownwardbiabutitbiaidecreaingovertime. Let ˆρ t be the pooled OLS etimator by uing panel data upto the time t. For the cae of fixed effect, with all tationary unit, the probability limit of ˆρ t i given by while with a mixed panel, it become plim ˆρ t = ρ +ρ t + O t plim ˆρ t = c t + O t where µ µ α 5ρ + ρ c =3α α +ρ + ρ > 0 ( ρ) In practice, the true parameter ρ and c are unknown. Moreover, a more time erie obervation are available, the probability limit of the pooled OLS etimate with a pure tationary panel i increaing over time and that with a mixed panel a well. he pooled recurive mean adjuted etimator provide aymptotically clear ditinct between two panel. For the cae of pure tationary panel, the pooled recurive mean adjuted etimator i upward biaed but it bia i decreaing over t. With moderate large t, the aymptotic bia can be approximed a plim ˆρ t = ρ + ρt ln t + O t > 0 for ρ>0 In contrat, for the cae of a mixed panel, the aymptotic bia of the pooled recurive mean adjuted etimator i given by µ µ 6 α plim ˆρ t = +ρ α t + O t ln t hi oppoite relationhip hold even for the linear trend cae. With large and, it i poible to ditinguih whether or not a panel conit of pure tationary unit by utilizing thi information. However, in finite, it i rather ambiguou ince the variance of ˆρ t i too large to ditinguih the mixed panel from the pure tationary panel. 3

14 4. Maximum Order et he clear difference between nontationary and tationary panel i the different rate of growing variance. o fix idea, conider the cae of known contant a follow. y it = ρy it + ε it From the direct calculation, we have ( yit = plim i= t= σ ρ for tationary y it σ ( +) for nontationary y it (8) hat i, the econd moment of tationary y it i contant over time while that of nontationary³ y it i growing over time. ote that when ρ i 6= ρ for tationary y it, the limit become σ P i= ρ i σ ρ. Hence the order doe not change at all. Define η it = P t t = y i. he cro ectional mean of η it will be varying over time when y it i not tationary. However thi information i not of much ue when for teting panel unit root epecially with mall. With given, the approximated confidence interval of the cro ectional mean of η it even with tationary y it i given by σ / ρ ± /. With fixed, thi implie the cro ectional mean of η it can move around σ / ρ over time. Hence in order to figure it out whether or not y it i tationary, one need a very large dimenion. Alternatively, the econd moment of recurive mean provide much more helpful information. From the direct calculation, we have plim i= Ã t! t y i = = ( σ ( ρ) t + O t for tationary y it σ 3 t + O () for nontationary y it Define κ it = P t t = y i. It i worthywhile noting that the maximum order of κit doe not change even under the ARMA(p,q) cae. hi i becaue y it i O p t / and κ it i O p t when y it i tationary, while y it i O p t 3/ and κ it i O p (t) when y it i nontationary. Hence the cro ectional mean of κ it at time t i decreaing over t when κ it i tationary while it i increaing when κ it i nontationary. We utilize thi finding to develop a maximum order tet to detect whether or not a panel conit of nontationary unit. Firt conider the unknown mean cae. Fixed Effect ake another recurive mean of κ it over t and denote it a z it. ake cro ectional average and denote it a z t. A, the following propoition hold. (9) 4

15 P Propoition 3 Aume that plim i= a i = µ a < where a i i unknown contant. For every t, a, the probability limit of z t i given by plim z t = Ã! t plim y ik y i= = i k= ( = ln t + O () for all I(0) t +t +lnt + O () for all I(), 6 Appendix B provide detail proof of Propoition 3. From (9), for ARMA(p,q) cae, it i obviou that the recurive um of κ it i bounded by O p (ln t) a long a y it i tationary while that of κ it i bounded by O p t a long a y it i nontationary. Here z t i normalized by the initial value of y i to make it be independent of variance of ε it. ote that the maximum order doe not change even when ρ i 6= ρ with tationary κ it. heiuehereiratherthetime varying behavior of z t with finite, epecially when ρ i i near unity. Since z t i increaing over time either when y it i I (0) or I (), it i neceary to divide it by t β to ditinguih between tationary and nontationary unit when β. 6 Let S t = z t t β and conider it probability limit. S t = plim z t t β Figure 3 how time varying propertie of S t with β =0.9. In finite, the time varying pattern of S t can be ummarized a follow. Firtly, for fixed poitive value of β, S t ha a hump hape. Let t be the time t when S t reache at it maximum value. A ρ, t. Secondly, for fixed value of ρ, a β increae, t approache unity. Conider the following null hypothei of mixed panel unit root Ho : ρ i =for ome i HA : ρ i < for all i Under the null hypothei of a mixed panel unit root, the maximum order of S t i O t β a long a the fraction of nontationary unit ( α) > 0. eting the above null hypothei i traightforward. Conider the following imple linear regreion S t = d + d t + e t, t = t 0,..., (0) where S t i the cro ectional average of z t t β. ote that t tart from t 0 rather than the initial obervation due to hump hape of S t. Under the null hypothei of Ho,a, the ign of d mut be poitive, while under the alternative, they mut not be ignificantly poitive. Hence the null hypothei of Ho can be rewritten a H o : d 0 6 Here β i bounded by unity rather than two. he regreion in (0) provide the reaon clearly. A, if β>, then the probability limit of ˆd goe to zero. 5

16 Figure 3: Probability limit of S t with β =0.9 for AR() cae. ote that a, under the null hypothei of Ho, the t tatitic of ˆd converge to a normal ditribution when β =while plim ˆd =+ when β<. o reduce the ize ditortion of the tet, we et β =0.9 but ue the normal table to tet the null hypothei of d 0. It i neceary ince the ize ditortion could behugewithalargefractionoftationary unit α. Incidental Linear rend For the cae of linear trend, define q t a q t = t y i y iq y i i= = y q= i For fixed t, a, the probability limit of q t i given by P Propoition 4 Aume that plim i= b i = µ b < where b i i unknown trend coeifficient. For every t, a, the probability limit of q t i given by plim q t = plim = ( 5 t i= = y i y i y iq y q= i tσ ρ + O () for all I(0) 8 σ t 3 + O () for all I(). () 6

17 Probability limit of Q t Figure 4: Probability limit of Q t with γ =.8 for AR() cae See Appendix B for the detail proof of Propoition 4. Comparing to the fixed effect cae, q t i O(t) when y it i tationary. 7 Let Q t = q t t γ and conider it probability limit. Q t = plim q t t γ ote that γ mut be greater than unit but le than two. Figure 4 how time varying propertie of Q t with γ =.8. Compared to Figrue 3, the maximum time t for the linear trend cae i much greater than that for the contant cae. hi i becaue q t i O p (t) rather than O p (ln t) when y it i tationary. Of coure, an increaing value of γ reult in maller t but it alo affect the ize of the tet. he following regreion i conidered to tet the null hypothei of mixed panel unit root. Q t = f + f t + v t () Similar to (0), a, plim ˆf =+ with γ<. If γ =, then the t tatitic of ˆf follow a normal ditribution aymptotically. o ecure the ize of the tet, we et γ =.8 which i equal to β. 7 Reader may wonder why we didn t take additional recurive mean over q t. ote that (t) q t 5 κ t with ρ< while (t) q t κ t/6 with ρ =. aking additional recurive mean of q t yield 5 z t for ρ< butitbecomez t/. Hence in the finite, the performance of the recurive mean of q t become wore that that of q t. 7

18 5 Sequential Panel Unit Root et under Cro Sectional Dependence Bai and g (00), Forni, Hallin, Lippi and Reichlin (000), Moon and Perron (00), and Phillip and Sul (003) tudy parametric tructure of cro ection dependence. Phillip and Sul (004) ditinguih macro panel data from micro panel data in the term of the number of common factor. Uually macro panel data i cro ectionally aggregated data o that the number of common factor i limited wherea the micro panel data might have a number of common factor. here are two type of factor model in nontationary panel literature. Bai and g (00) conider a direct factor tructure for the data of the form K y it = λ i F t + m it. (3) = where F t i the th common factor while λ i i the factor loading coefficient for the th common factor. Uually the idioyncratic term m it i aumed to be cro ectionally independent. Bai and g (00) propoed how to etimate the number of factor in panel data, while more recently Bai and g (004) have ugget how to etimate F t from y it. Once the idioyncratic term m it i ubtracted from y it, the mixed panel unit root a well a the panel unit root tet propoed in the previou ection can be directly applied. he econd model i tudied by Forni, Lippi and Reichlin (999), Moon and Perron (00), and Phillip and Sul (003). In all thee tudie, common factor enter into the regreion error. K y it = a i + ρ i y it + u it,u it = δ i θ t + ε it, (4) where the error u it depend on K factor {θ t : =,...,K} with factor loading {δ i : =,...,K}, and ε it i aumed to be iid(0,σ i ). ote that the firt factor model i more general than the econd factor model in the ene that F t can have a different erial correlation tructure to m it. However, thee two model can be complimentary to each other when m it follow the common factor tructure like (4). For example, all panel empirical tudie for growth convergence are baed on the following imple linear pecification = y it = µ t + y o it (5) where y it i an individual per capital income of the i th economy, µ t i the common growth component and yit o i the deviation between y it and µ t. he deviation part, yit o uually ha cro ectional dependence which can be modelled by (4). hi pecification i rather like a combination between (3) and (4). hat i, m it in (3) ha the common factor tructure like 8

19 (4). onethle when the cro ectional dependence i bet modelled a (3), the mixed panel unit root tet developed in the previou ection can be directly applied to the etimated ˆm it a Bai and g (004) ugget. Otherwie, the orthogonalization method propoed by Phillip and Sul (003) i required to eliminate factor loading coefficient δ i. With orthogonalized panel data, the mixed panel unit root tet can be carried on. he latter procedure i rather complicated. Here I provide tep by tep procedure how to perform equential panel unit root tet. Step Obtain an individual RMA etimator ˆρ ri from (4) for M and (5) for M. Denote the reidual ˆε it ˆε it = y it ˆρ ri y it ˆε it = y it ˆρ ri y it p ˆφ ij y it j for M j= p ˆφ ij ( y it j ȳ i j ) j= for M If ˆρ ri, then et ˆρ ri =. ote that ˆε it ha non-zero mean for M and ha non-zero mean and linear trend for M when ρ<. Step If ˆρ ri <, demean ˆε it formordetrendˆε it for M. If ˆρ ri, then demean ˆε it for M. Contruct ample variance and covariance matrix ˆΣ ε. Step 3 Contruct the orthogonal complement matrix ˆF δ baed on ˆΣ ε. ranform the data y it by premultiplying ˆF δ and denote y it + = ˆF δ y it. ote that y it + i cro ectionally independent a. Step 4 Perform either the Fiher tet with RMA-ADF or the pooled tet where the null hypothei i ρ i =for all i. If the null cannot be rejected, then all unit in the panel are I(). If the null can be rejected, then next perform the maximum order tet. he null hypothei i ρ i =for ome i. he above 4-tep procedure provide a conitent tet either under the null hypothei or under alternative a. hi i becaue the ample (or intitial) convariance matrix i etimated under the aumption of heterogeneity of ρ i ote that the regreion error ε it 6= u it under the alternative. Hence to contruct conitent etimator of Σ ε, the ordinary demeaning or detrending i needed after obtaining ˆε it in Step. When the null hypothei of panel unit root Ho i rejected, the maximum order tet under cro ectional dependence can be carried out uing y it +. 9

20 6 Simulation Experiment We conidered a number of Monte Carlo experiment with variou data generating procee. Since the reult of Monte Carlo tudie are imilar over different data generating procee, only reult for two et of Monte Carlo experiment are reported here Deign of Data Generating Proce he data generating proce i given by or equivalently y it = a i + b i t + x it, x it = ρ i x it + u it, (6) u it = δ i θ t + ε it (7) y it = a + i + β i t + ρ i y it + u it where a + i = a i ( ρ i )+b i ρ i,β i = b i ( ρ i ) while ε it iid (0, ) over i and t, θ t iid (0, ) over t, and for (ρ i,δ i ) parameter election that are detailed below. he primary ditinction i between the homogeneou cae where ρ i = ρ for all i and the heterogeneou cae where ρ i differ acro individual i. For cro ection independence cae, δ i =0while for cro ection dependence cae, δ i U [, 3] where U i uniform ditribution. Here the lag length i aumed to be known. Panel data are generated under four pecification which differ according to their degree of the cro ectional dependence and whether or not the homogeneity retriction i impoed on ρ. hee pecification are a follow: Cae I: (AR() under o Cro-ectional Dependence) For the cae of homogeneity, we et ρ i =0.9. For the cae of heterogeneity, we et ρ i U[0.85, 0.95] for M while ρ i U[0.80, 0.90] for M. Cae II: (AR() under High Cro-ectional Dependence) We et δ i U [, 3]. Each experiment involve 5,000 replication of panel ample of (, ) obervation. Recall that α i the fraction of tationary unit. I conider five different value of α =(, 3/4, /, /4, 0). We et =30, 50 and 00 while =0, 40 and 80 o that when α =3/4, the number of nontationary unit become 5, 0 and 0 for =0, 40 and 80 repectively. It i worthy noting that aymptotically the true value of the pooled etimator with mixed panel depend on the ize of. From the formulae in (7), thee value with =30, 50 and 00 are (0.990,0.994,0.997), (0.975,0.983,0.99) and (0.975,0.983,0.99) for α = 3/4, / and /4. he initial tarting point t 0 mut be elected for teting the null hypothei of mixed panel unit root. heoretically with large, the initial tarting point t 0 could be the initial 8 Other Monte Carlo reult are available upon requet of the author. 0

21 obervation. With finite, the optimal tarting point t 0 mut be the maximum value of S t and Q t to increae the power of the mixed panel unit root tet and to reduce the ize ditortion a well. he problem i that the maximum value of S t and Q t are dependent of ρ. A it i howninfigure3and4,themaximumvaluet become the lat obervation when ρ i near unity with mall. ote that the power of the panel unit root tet with mall under Ho i alo very poor when ρ i near unity. Alo it i meaningle to tet the null hypothei of mixed panel unit root, Ho when the power of the tet under Ho i poor. With extenive numerical imulation baed on the exact aymptotic formulae of S t and Q t, we et t 0 =5. 6. Simulation Reult able 3 conit how the finite ample performance of three panel unit root tet. hey are the Fiher tet baed on the DF-GLS, the Fiher tet baed on RMA-DF and the pooled RMA tet under no cro ection dependence. he rejection rate i evaluated baed on 5% tet. All rejection rate are not ize adjuted. A it i hown in Figure, the power of the pooled RMA tet i better than thoe of the two Fiher tet for both homogeneity and heterogeneity of ρ i. More interetingly, the rejection rate increae a and increae even when α =/4. able 4 report the rejection rate of the maximum order tet and the equential mixed panel unit root tet under cro ection dependence. For the cae of the maximum order tet, the power increae a either or increae. However, the ize of the tet decreae a increae for fixed. Alo the ize of the tet increae a the propotion of tationary unit (α) increae. In contrat, the equential tet combining with the pooled RMA and maximum order tet provide reduce the ize ditortion ignificantly without much acrificing the power of the tet for the moderately large. 7 Application and Concluding Remark We apply the equential mixed panel unit root tet for long run purchaing power parity. he data ued in the paper cover the period for OECD countrie wa taken from the International Financial Statitic. he erie involved annual price indice for each country and real exchange rate calculated from the individual national price indice and the end of the period pot exchange rate. We ue BIC to determine the appropriate lag and end up with AR() pecification. able 5 report the reult. he Fiher tet baed on RMA-ADF reject the null hypothei of panel unit root for all numeraire currencie, while the pooled RMA tet reject 9 out of numeraire currencie. Hence the firt null Ho or Ho 0 i rejected fairly trongly. ext, we tet the null hypothei of mixed panel unit root. he maximum order tet cannot reject the null hypothei for all numeraire currencie that there are ome nontationary unit in the

22 panel. herefore if real exchange rate don t have any nonlinear adjutment, they are not all tationary. hi paper provide the equential mixed panel unit root tet and applie the tet to tet long run purchaing power parity. We found that long run purchaing power parity doe not hold for all bilateral currencie. Since the propoed mixed panel unit root tet i contructed baed only on linear enviroment, threhold type mixed panel unit root tet will be promiing.

23 able 3: Rejection Rate of the Fiher et baed on DF-GLS, RMA-ADF and the pooled RMA tet. AR() with no cro ection dependence Part A: Cae of Homogeneity ρ Sample Contant Linear rend α= α= 3 4 α= α= 4 α=0 α= α= 3 4 α= α= 4 α=0 Fiher et baed on ADF-GLS =30,= =30,= =30,= =50,= =50,= =50,= Fiher et baed on RMA-ADF =30,= =30,= =30,= =50,= =50,= =50,= Pooled RMA et =30,= =30,= =30,= =50,= =50,= =50,=

24 able 3:-Continue Part B: Cae of Heterogeneity ρ Sample Contant Linear rend α= α= 3 4 α= α= 4 α=0 α= α= 3 4 α= α= 4 α=0 Fiher et baed on ADF-GLS =30,= =30,= =30,= =50,= =50,= =50,= Fiher et baed on RMA-ADF =30,= =30,= =30,= =50,= =50,= =50,= Pooled RMA et =30,= =30,= =30,= =50,= =50,= =50,=

25 able 4: Rejection Rate of Sequential Mixed Panel Unit Root et AR() with high cro ection dependence Part A: Homogeneity ρ Sample Contant Linear rend Maximum Order et α= α= 3 4 α= α= 4 α=0 α= α= 3 4 α= α= 4 α=0 =30,= =30,= =30,= =50,= =50,= =50,= =00,= =00,= =00,= Sequential Panel Unit Root et: he pooled RMA & Maximum Order et =30,= =30,= =30,= =50,= =50,= =50,= =00,= =00,= =00,=

26 able 4:Continue Part A: Heterogeneity ρ Sample Contant Linear rend Maximum Order et α= α= 3 4 α= α= 4 α=0 α= α= 3 4 α= α= 4 α=0 =30,= =30,= =30,= =50,= =50,= =50,= =00,= =00,= =00,= Sequential Panel Unit Root et: he pooled RMA & Maximum Order et =30,= =30,= =30,= =50,= =50,= =50,= =00,= =00,= =00,=

27 able 5: Are Real Exchange Rate All Stationary? umeraire country Pooled RMA et Fiher et Maximum Order et Autralia -.7(0.0) 64.47(0.00) 43.67(.00) Autria -.77(0.04) 6.7(0.0) 55.0(.00) Belgium -.09(0.0) 85.95(0.00) 56.39(.00) Canada -.6(0.0) 64.54(0.00) 5.90(.00) Denmark -0.84(0.0) 64.5(0.0) 36.95(.00) Finland -3.5(0.00) 6.6(0.0) 66.0(.00) France -.0(0.0) 57.07(0.0) 55.58(.00) Germany -.8(0.0) 55.8(0.03) 43.0(.00) Greece -.5(0.0) 64.8(0.00) 55.05(.00) Ireland -.79(0.00) 57.67(0.0) 56.9(.00) Italy -.5(0.0) 55.40(0.03) 7.3(.00) Japan -.4(0.0) 54.06(0.04) 64.39(.00) etherland -0.75(0.3) 68.6(0.00) 47.46(.00) ew Zealand -.65(0.00) 69.3(0.00) 67.97(.00) orway -.36(0.0) 58.97(0.0) 53.5(.00) Portugal -.9(0.0) 54.84(0.04) 60.78(.00) Spain -.6(0.00) 56.44(0.03) 63.56(.00) Sweden -.5(0.0) 59.07(0.0) 49.49(.00) Switzerland -.8(0.03) 7.9(0.00) 63.66(.00) U.K. -.99(0.0) 67.9(0.00) 56.53(.00) U.S. -.50(0.0) 64.9(0.00) 76.98(.00) he number in parenthei i the probability value. 7

28 8 Appendix We retate model here firt and take the following aumption. ( y it = a i ( ρ M: (Unknown Cotant) i )+ρ i y it + ε it y it = a i + x it,x it = ρ i x it + ε it ( y it = a i ( ρ)+b i ρ M: (Linear rend) i + b i ( ρ i ) t + ρ i y it + ε it y it = a i + b i t + x it,x it = ρx it + ε it Aumption he ε it have zero mean, finite +ν moment for ome ν>0, are independent over i and t with E(ε it )=σ i for all t, and lim P i= σ i = σ. P Aumption lim i= a P i = µ a <, lim i= b P i = µ b <, lim i= a i = µ P a < and lim i= b i = µ b <. he folllowing lemma are ueful to prove Propotion and 3. Denote P E i =plim i=, then we have Lemma (Stationary x t ) P - E i t= x itx it =( )ρσ x ³ P ³ - E i t= x it = σ x + ρ ρ P k= ( ρk ) P ³ -3 E P t i t= t = x P ³ ³ i = σ ρ x t= t t ρ ρt ( ρ) ρ ³ P -4 E i t= t P t = x i P ³ -5 E P i t= x t it t = x i -6 E i P t= ³ P x t it t = x i ³ P -7 E i t= it ³ x P t= + σ x ³ P k= -8 E i ³ P t= x it ³ P t= σ x ³ P k= = σ x P t j= ³ P =j = σ x P t= ³ t = σ xρ P ³ t= t P t = x i h P k j= ρ k j P =j t P t = x i h P k j= ρ k j P =j P +σ P ³ x k= ρk k P ³ j= =j P =j+(k ) ³ ρ t ρ ³ ρ t ρ = σ x ( ) + σ x i = σ x ( ) + σ x i σ x ³ P h P k k= j= ρ j P i =k+j ³ P h P k k= ³ P P j= ρj =j i =k+j + ³ P j= ρ j P =j j= ρ j P + σ x 8

29 Lemma ( Unit Root x t ): - E i P t= x itx it = σ P ³ P - E i t= x it = σ t= t n P P t t= j= j + P t= t P o j=t+ P ³ -3 E P t i t= t = x i = σ P ³ 6 t= t t + 3 t3 t ³ P -4 E i t= t P t = x i = σ P ³ j= j P =j P + P ³ k k= j= j P ³ =j P =j+(k ) -5 E i P t= -6 E i P t= ³ P x t it t = x i ³ P x t it t = x i ³ P -7 E i t= it ³ x P t= t P t = x i -8 E i ³ P t= x it ³ P t= + σ P j= j P =j t = σ P ³ Pt t= t = ( ) = σ P ³ Pt t= t = ( ) P t = x i = σ ( ) σ P P k k= j= j P =k = σ ( ) σ P P k k= j= j P =k σ ( ) hrough imulation, each formulae i varified. 8. Appendix A: Proof of Propoition 8.. he cae of unknown contant It i traightforward to ee there i no aymptotic bia. o obtain the aymptotic variance of ˆρ pr, we conider the following limit. plim = plim i= t= (y it ȳ it ) " Ã x it!ã x it i= t= t= = σ µ + σ ( +) k k= j= t= j t!# t x i = =k 9

30 ote that the Hamonic number H = P = can be approximated to be ln + γ where γ i Euler contant. Similarly =k Z = k d + γ =ln ln k + γ k k= j= j =k ' Z hi approximation alo provide µz k j (ln ln k) dj dk = ( k) k ln ( k) k ln k +( k) k (k ln k ln k + k), k k= j= plim j =k i= t= and ' 36 3 (ln ) ln Finally, the aymptotic limit i given by ½ (y it ȳ it ) = σ 36 7 ¾ ln + O () Meanwhile the probability limit of the quare of the regreion reidual i given by plim i= t= û it = plim = plim i= i= w c it ˆρwit c t= ε it = σ Hence the aymptotic variance of ˆρ pr for the cae of unknown contant i given by plim P i= P t= û it P P i= t= (y it ȳ it ) = 36 7 ln + O () 8.. For the cae of linear trend: he bia of ˆρ pr i given by t= plim ˆρpr = plim C D where D = (y it y i [ȳ it ȳ i ]) i= t= 30

31 and C = (y it y i [ȳ it ȳ i ]) (u it u i ) i= t= We conider the denominator term firt. From Lemma - through -, we have plim (x it x i ) = σ ( ), 6 i= t= = (y it y i [ȳ it ȳ i ]) t= (y it y i ) +4 t= plim [ȳ it ȳ i ] 4 t= i= t= (y it y i )[ȳ it ȳ i ] t= (x it x i ) = σ 6 ( ), plim = plim = [ȳ it ȳ i ] µ Ã t t i= t= i= t= µ µ t 6 t + 3 t3 t t= j + plim = plim j= =j i= t= = k k= j= y i! j =j (y it y i )[ȳ it ȳ i ] ( Ã! t y it y i t t= = k j i= = plim k= i= t= j= =k = A +4B 4C = 9 + O ( ) " t= =j+(k ) µ Ã!# t y i t = Ã! t ) y i y it t t= = Hence (y it y i [ȳ it ȳ i ]) t= 3

32 where A = 6 ( ) = 6 + O ( ) µ µ B = t 6 t + 3 t3 t j t= j= =j k j (8) k= j= = O ( ) =j C = = 8 + O ( ) t= k= j= Since =j+(k ) k= k j= µ µ t 6 t + 3 t3 t j k j= j =j k= =j =j+(k ) k j= j =k j =k = 6 + O ( ) = 4 + O () = 8 + O ( ) = 36 + O( ) he nominator term i alo imilarly obtained by plim C = plim plim plim i= t= (y it y i )(u it u i ) i= t= From the direct calculation, i= t= [ȳ it ȳ i ](u it u i ) (x it x i )(u it u i ) = + 3

33 plim i= t= = plim = k= j=k =j plim = Eu " = =... " +Eu i= x = x j= =j k= j=k =j while [ȳ it ȳ i ](u it u i ) ÃÃ t t i= ince ÃÃ t= + x = = t= t t = = + + x + x + + x = After numerical calculation, we have plim C =0 Hence we have!ã!! x i u it t=!ã!! x i u it t= = plim ˆρpr = plim C D =0 # = o obtain the aymptotic variance, recall that u it u i = ε it ε i and ˆρ pr = ρ + O p. Oberve thi. plim i= (u it u i ) = σ O t= Hence the aymptotic variance of ˆρ pr for the cae of unknown contant i given by P i= P t= plim (u it u i ) P P i= t= (y it y i [ȳ it ȳ i ]) = 9 + O 3 # 33

34 8..3 Aymptotic ormality Following Harri and zavali (999), we aume x i0 =0. For fixed, a, the aymptotic normality hold due to the Linderberg-Lecy CL. 8. Appendix B: Proof of Propotion 3 and Proof of Propoition 3 Let σ =for notational convinience. ote that when ρ<, from lemma -3 Ã! t x µ ik a + σ ln t + O () ( ρ) plim = (9) x i= t= i µ k= a + σ ( ρ) = ln(t)+o () (0) plim ince µ a i finite and i independent of t. When ρ =, from the direct calculation (ee lemma -3) Ã t x ik x i i= t= 8.. Proof of Propoition 4 t= = t k= Expanding q t for every t give t y i y iq = = q= Ã t! t y i +4 = =! = µ a + σ 6 t + 3 t + 6 ln t + O () µ a + σ () q= = t +t +lnt + O () () 6 Ã t! y iq 4 y i = t = q= y iq (3) ake probability limit a. he limit of the firt term i given in (9) and (). he limit of the econd term i rather complicated. o find the maximum order of the econd term, conider P tp x i = x i + (x i + x i )+ + = " x i = + x i + + x i = (x i + + x i ) = # 34