A Simple Panel Unit Root Test in the Presence of Cross Section Dependence

Transcription

1 A Smple Panel Unt Root est n the Presence of Cross Secton Dependence M. Hashem Pesaran Cambrdge Unversty & USC September 3, Revsed January 5 Abstract A number of panel unt root tests that allow for cross secton dependence have been proposed n the lterature that use orthogonalzaton type procedures to asymptotcally elmnate the cross dependence of the seres before standard panel unt root tests are appled to the transformed seres. In ths paper we propose a smple alternatve where the standard ADF regressons are augmented wth the cross secton averages of lagged levels and frst-dfferences of the ndvdual seres. ew asymptotc results are obtaned both for the ndvdual CADF statstcs, and ther smple averages. It s shown that the ndvdual CADF statstcs are asymptotcally smlar and do not depend on the factor loadngs. he lmt dstrbuton of the average CADF statstc s shown to exst and ts crtcal values are tabulated. Small sample propertes of the proposed test are nvestgated by Monte Carlo experments. he proposed test s appled to a panel of 7 OECD real exchange rate seres as well as to log real earnngs of households n the PSID data. JEL Classfcaton: C, C5, C, C3. Key Words: Panel Unt Root ests, Cross Secton Dependence, Monte Carlo Results, Purchasng Power Party, Real Earnng Dynamcs. I would lke to thank Soren Johansen and Chrs Rogers for helpful dscussons wth respect to the analyss of exchangeable processes, and Jörg Bretung, Roger Moon and Ron Smth for useful general comments. I am also partcularly grateful to Yongcheol Shn for carryng out some prelmnary computatons that were nstrumental n helpng me form some of the deas developed formally n the paper. Excellent research assstance by Mutta Akusuwan s also gratefully acknowledged who carred out all the Monte Carlo computatons reported n ths paper. Computatons of the emprcal applcatons n Secton 7 were carred out by Vanessa Smth and akash Yamagata.

2 Introducton Over the past decade the problem of testng for unt roots n heterogeneous panels has attracted a great of deal attenton. See, for example, Bowman (999), Cho (), Hadr (), Im, Pesaran and Shn (995, 3), Levn, Lee, and Chu (), Maddala and Wu (999), and Shn and Snell (). Baltag and Kao () provde an early revew. hs lterature, however, assumed that the ndvdual tme seres n the panel were cross-sectonally ndependently dstrbuted. Whle t was recognzed that ths was a rather restrctve assumpton, partcularly n the context of cross country (regon) regressons, t was thought that cross-sectonally de-meanng the seres before applcaton of the panel unt root test could partly deal wth the problem. (see Im, Pesaran and Shn (995)). However, t was clear that cross-secton de-meanng could not work n general where par-wse cross-secton covarances of the error terms dffered across the ndvdual seres. Recognzng ths defcency new panel unt root tests have been proposed n the lterature by Chang (), Cho (), Phllps and Sul (3), Ba and g (4), Bretung and Das (4), Cho and Chue (4), Moon and Perron (4), and Smth, Leybourne, Km and ewbold (4). Chang () proposes a non-lnear nstrumental varable approach to deal wth the cross secton dependence of a general form and establshes that ndvdual Dckey-Fuller (DF) or the Augmented DF (ADF) statstcs are asymptotcally ndependent when an ntegrable functon of the lagged dependent varables are used as nstruments. From ths she concludes that her test s vald for both (the tme seres dmenson) and (the cross secton dmenson) are large. However, as shown by Im and Pesaran (3), her test s vald only f s fxed as. Usng Monte Carlo technques, Im and Pesaran show that Chang s test s grossly over-szed for moderate degrees of cross secton dependence, even for relatvely small values of. Cho () models the cross dependence usng a two-way error-component model whch mposes the same par-wse error covarances across the dfferent cross secton unts. hs provdes a generalzaton of the cross-secton de-meanng procedure proposed n Im, Pesaran and Shn (995) but t can stll be restrctve n the context of heterogeneous panels. Cho and Chue (4) and Smth, Leybourne, Km and ewbold (4) use bootstrap technques to deal wth cross secton dependence. Bretung and Das (4) employ feasble generalzed least-squares estmates that are applcable n cases where. Harrs, Leybourne, and McCabe (4) propose a test of jont statonarty (as opposed to unt roots) n panels under cross secton dependence usng the sum of lag-k sample autocovarances where k s taken to be an ncreasng functon of. Ba and g (4), Moon and Perron (4), and Phllps and Sul (3) avod the restrctve nature of the cross-secton de-meanng procedure by allowng the common factors to have dfferental effects on dfferent cross secton unts. In the context of a resdual onefactor model Phllps and Sul (3) show that n the presence of cross secton dependence the standard panel unt root tests are no longer asymptotcally smlar, and propose an Followng Maddala and Wu (999) bootstrap technques have also been utlzed to deal wth cross secton dependence n panel unt root tests. See, for example, Smth, Leybourne, Km and ewbold (3), and Chang (3). hese procedures are also vald f s fxed as. []

3 orthogonalzaton procedure whch n effect asymptotcally elmnates the common factors before precedng to the applcaton of standard panel unt root tests. Sequental asymptotc results are provded n the case where, and then. Independently, smlar orthogonalzaton procedures are used by Ba and g (4) and Moon and Perron (4) n a more general set up. Moon and Perron (4) propose a pooled panel unt root test based on de-factored observatons and suggest estmatng the factor loadngs that enter ther proposed statstc by the prncpal component method. hey derve asymptotc propertes of ther test under the unt root null and local alternatves, assumng n partcular that /, as and. hey show that ther proposed test has good asymptotc power propertes f the model does not contan determnstc (ncdental) trends. In a related paper, Moon, Perron and Phllps (3) propose a pont optmal nvarant panel unt root test whch s shown to have local power even n the presence of determnstc trends. Ba and g (4) consder a more general set up and allow for the possblty of unt roots (and contegraton) n the common factors, but contnue to assume that /, as and. o deal wth such a possblty they apply the prncpal component procedure to the frst-dfferenced verson of the model, and estmate the factor loadngs and the frst dfferences of the common factors. Standard unt root tests are then appled to the factors and the ndvdual de-factored seres, both computed as partal sums of the estmated frst dfferences. In ths paper we adopt a dfferent approach to dealng wth the problem of cross secton dependence. Instead of basng the unt root tests on devatons from the estmated factors, we augment the standard DF (or ADF) regressons wth the cross secton averages of lagged levels and frst-dfferences of the ndvdual seres. Standard panel unt root tests can now be based on the smple averages of the ndvdual cross sectonally augmented ADF statstcs (denoted by CADF), or sutable transformatons of the assocated rejecton probabltes. he ndvdual CADF statstcs or the rejecton probabltes can then be used to develop modfed versons of the t-bar test proposed by Im, Pesaran and Shn (IPS), the nverse ch-squared test (or the P test) proposed by Maddala and Wu (999), and the nverse normal test (or the Z test) suggested by Cho (). A truncated verson of the test s also consdered where the ndvdual CADF statstcs are sutably truncated to avod undue nfluences of extreme outcomes that could arse when s small (n the regon of ). ew asymptotc results are obtaned both for the ndvdual CADF statstcs, and ther smple averages, referred to as the cross-sectonally augmented IPS (CIPS) test. he asymptotc null dstrbuton of the ndvdual CADF and the assocated CIPS = P = CADF statstcs are nvestgated as followed wth,as well as jontly when and tendngtonfnty such that / k, wherek s a fxed fnte non-zero constant. It s shown that the CADF statstcs are asymptotcally smlar and do not depend on the factor loadngs. But they are asymptotcally correlated due to ther dependence on Clearly, t s also possble to construct the CADF test based on recent modfcatons of the ADF test proposed n the lterature, for example, the ADF-GLS test of Ellott et al. (996), the weghted symmetrc ADF (WS-ADF) test of Fuller and Park (995) and Fuller (996, Secton..3), or the Max-ADF test of Leybourne (995). he use of the latter two modfcatons of ADF statstcs n IPS panel unt root test have been recently consdered by Smth, Leybourne, Km and ewbold (4) who report sgnfcant gan n power as compared to the IPS test based on standard ADF statstcs. []

4 the common factor. As a result the standard central lmt theorems do not apply to the CIPS statstc (or the other combnaton or meta type tests proposed by Maddala and Wu, and Cho). However, t s shown that the lmt dstrbuton of the truncated verson of the CIPS statstc (denoted by CIPS ) exsts and s free of nusance parameters. he crtcal values of CIPS and CIPS statstcs are tabulated for the three man specfcatons of the determnstcs, namely n the case of models wthout ntercepts or trends, models wth ndvdual-specfc ntercepts, and models wth ncdental lnear trends. 3 he small sample propertes of the proposed tests are nvestgated by Monte Carlo experments, for a varety of models wth ncdental determnstcs (ntercepts as well as lnear trends), cross dependence (low and hgh) and ndvdual specfc resdual seral correlaton (postve and negatve), and sample szes, and =,, 3, 5,. he smulatons show that the cross sectonally augmented panel unt root tests have satsfactory sze and power even for relatvely small values of and. hs s partcularly true of the truncated verson of the CIPS test and the cross sectonally augmented verson of Cho s nverse normal combnaton test. hese tests show satsfactory sze propertes even for very small sample szes, namely when = =, and there s a hgh degree of cross secton dependence wth a moderate degree of resdual seral correlaton. Perhaps not surprsngly, the power of the tests crtcally depends on the sample szes and, and on whether the model contans lnear tme trends. In the case of models wth lnear tme trends power starts to rse wth only f s 3 or more. For >3 the power rses qute rapdly wth both and.in ther respectve smulatons Ba and g (4) report Monte Carlo results for = and =,, Moon and Perron (4) for =, 3, =,, and Phllps and Sul (3) for =5,, and =,, 3. All these studes consder experments where s much larger than, and hence are dffcult to evaluate n relaton to our smulaton results where could be small relatve to and vce versa. 4 he plan of the paper s as follows. Secton sets out the basc model. Secton 3 ntroduces the cross sectonally augmented regressons for the ndvdual seres for models wthout resdual seral correlatons, and shows that n ths case the CADF statstc does not depend on nusance parameters as for any fxed >3. he null dstrbuton of the CADF statstc s derved under sequental and jont asymptotcs and t s shown that the CADF statstcs for dfferent seres are asymptotcally correlated and form an exchangeable sequence. Asymptotc crtcal values for the CADF dstrbuton are provded n Secton 3., together wth smulated values of ts moments, as well as the asymptotc correlaton coeffcent for any par of CADF statstcs. All the three man specfcatons of the determnstcs (no ntercept or trend, ntercept only, and a lnear trend) are covered. he varous CADF based panel unt root tests (the cross sectonally augmented versons of the IPS, P and the Z tests) are dscussed n Secton 4. Secton 5 extends the results to the case where the ndvdual specfc errors are serally correlated. It s shown that the ndvdual CADF statstcs have the same asymptotc dstrbuton as n the serally uncorrelated case, so long 3 Crtcal values for the cross sectonally augmented combnaton (or meta) tests are avalable from the author on request. 4 A comparatve study of the small sample propertes of the panel unt root tests proposed n ths paper wth those advanced by Moon and Perron (4) and Ba and g (4) s provded n Gengenbach, Palm and Urban (4). [3]

5 as the CADF regressons are further augmented wth the lagged changes of the ndvdual seres as well as the lagged changes of the cross secton averages. Small sample performance of the proposed tests are nvestgated n Secton 6 usng Monte Carlo experments. Secton 7 provdes two emprcal applcatons. In one applcaton the proposed test s appled to quarterly real exchange rates from 7 OECD countres, and the results are compared to the bootstrap tests recently appled to the same data set by Smth et al. (4). 5 hs applcaton clearly demonstrates the mportance of allowng for cross secton dependence n panel unt root tests. A second applcaton tests for unt roots n real earnngs of households usng the Panel Study of Income Dynamcs (PSID) data recently analyzed by Meghr and Pstaferr (4). he results show no evdence of unt roots n log real earnngs for the sample as a whole, although the test outcomes are less clear cut for some of the sub-samples once cross secton dependence s taken nto account. hs applcaton hghlghts the mportance of frst testng for the presence of cross secton dependence before testng for unt roots n panels, n order to avod unnecessary loss of power. he paper ends wth some concludng remarks n Secton 8. otatons: a n = O(b n ) states the determnstc sequence {a n } s at most of order b n, x n = O p (y n ) states the vector of random varables, x n, s at most of order y n n probablty,and x n = o p (y n ) s of smaller order n probablty than y n. denotes convergence n quadratc mean (q.m.) or mean square errors and = convergence n dstrbuton. All asymptotcs are carred out under, ether wth a fxed, sequentally, or jontly wth. In partcular, = ( ) denotes convergence n dstrbuton (q.m.) wth fxed as, = ( ) denotes convergence n dstrbuton (q.m.) for fxed (or when there s no - dependence) as, =, denotes sequental convergence wth frst followed by (smlarly = ),, (,) j = denotes jont convergence wth, jontly such that / k, wherek s a fxed fnte non-zero constant. v denotes asymptotc equvalence n dstrbuton, wth v, v,, v,, v,and (,) j v, smlarly defned as =, =, etc. A Smple Dynamc Panel wth Cross-Secton Dependence Let y t be the observaton on the th cross-secton unt at tme t and suppose that t s generated accordng to the followng smple dynamc lnear heterogeneous panel data model y t =( φ ) µ + φ y,t + u t,=,..., ; t =,...,, (.) where ntal value, y, s gven, and the error term, u t, has the one-factor structure u t = γ f t + ε t, (.) n whch f t s the unobserved common effect, and ε t s the ndvdual-specfc (dosyncratc) error. 5 hs applcaton also corrects an error dentfed by Vanessa Smth (one of the authors of the Smth et al. paper) n the process of replcatng the test results for ncluson n the current paper. [4]

6 It s convenent to wrte (.) and (.) y t = α + β y,t + γ f t + ε t, (.3) where α =( φ ) µ, β = ( φ )and y t = y t y,t. he unt root hypothess of nterest, φ =, can now be expressed as aganst the possbly heterogeneous alternatves, H : β = for all, (.4) H : β <, =,,...,, β =,= +, +,...,. (.5) We shall assume that /, the fracton of the ndvdual processes that are statonary, s non-zero and tends to the fxed value δ such that < δ as. As noted n Im, Pesaran and Shn (3) ths condton s necessary for the consstency of the panel unt root tests. We shall make the followng assumptons: Assumpton : he dosyncratc shocks, ε t, =,,...,, t =,,...,, are ndependently dstrbuted both across and t, have mean zero, varance σ,andfnte fourth-order moment. Assumpton : he common factor, f t, s serally uncorrelated wth mean zero and a constant varance, σf,andfnte fourth-order moment. Wthout loss of generalty σ f wll be set equal to unty. Assumpton 3: ε t, f t,andγ are ndependently dstrbuted for all. he cross-secton ndependence of ε t (across ) s standard n one factor models, although ts valdty n more general settngs may requre specfcaton of more than one common factor n (.). Assumptons and together mply that the composte error, u t,sserally uncorrelated. hs restrcton can be relaxed by consderng statonary error processes of the type px u t = ρ j u,t j + γ f t + ε t. A smple example of ths generalzaton wll be consdered n Secton 5. j= 3 Unt Root ests for One-Factor Resdual Models wth Serally Uncorrelated Errors Let γ = P j= γ j and suppose that γ 6= for a fxed and as. hen followng the lne of reasonng n Pesaran (4), the common factor f t can be proxed by the cross secton mean of y t,namelyȳ t = P j= y jt, and ts lagged value(s), ȳ t, ȳ t,... for suffcently large. In the smple case where u t s serally uncorrelated, t turns out that ȳ t and ȳ t (or equvalently ȳ t and ȳ t )are suffcent for asymptotcally flterng out the effects of the unobserved common factor, f t. We shall therefore base our test of the unt root [5]

7 hypothess, (.4), on the t-rato of the OLS estmate of b (ˆb ) n the followng cross-sectonally augmented DF (CADF) regresson 6 Denotng ths t-rato by t (,) wehave where y t = a + b y,t + c ȳ t + d ȳ t + e t. (3.6) t (,) = y M w y, ˆσ y, M w y, /, (3.7) y =( y, y,..., y ), y, =(y,y,...,y, ) (3.8) M w = I W W W W, W = τ, ȳ, ȳ, (3.9) τ =(,,...,), ȳ =( ȳ, ȳ,..., ȳ ), ȳ =(ȳ, ȳ,...,ȳ ), (3.) ˆσ = y M,w y, (3.) 4 M,w = I G (G G ) G, and G = y,, W. (3.) In computng the t-rato of ˆb estmator of σ, t s more convenent to use the followng alternatve σ = y M w y, (3.3) 3 As we shall see under the null hypothess ˆσ and σ are both consstent for σ,as and tend to nfnty. But for nvestgatng the lmtng propertes of the proposed test, the use of σ smplfes the analyss consderably. he t-rato assocated wth σ s gven by 3 y t (,) = M w y, y / M w y y /. (3.4), M w y, Under β =wehave y t = γ f t + ε t and hence y = γ f + ε, (3.5) 6 he dea of augmentng ADF regressons wth covarates has been consdered by Hansen (995), where statonary covarates are added to the ADF regresson wth the am of ncreasng the power of the unt root test. In our applcaton the covarates are non-statonary as well as beng endogenous for a fnte. [6]

8 y, = y τ + γ s f, + s,, (3.6) ȳ = γf + ε, (3.7) ȳ =ȳ τ + γs f, + s, (3.8) where ε =(ε, ε,..., ε ), f =(f,f,..., f ), ε =( ε, ε,..., ε ), ε t = P j= ε jt, y s a gven ntal value (fxed or random), ȳ = P j= y j, s, =(,s,...,s, ), s f, = (s f,s f,..., s f, ), s =(, s,..., s )wths t = P t j= ε j, and s t = P j= s jt, for t =,,..,ands ft = P t j= f j. Usng (3.7) to elmnate f from (3.5), and notng that by assumpton γ 6=,wehave where herefore, where ξ =(ξ, ξ,...,ξ ) (,ω I ), µ ω = σ δ and σ = P j= σ j <. herefore, y = δ ȳ+ξ t, (3.9) δ = γ / γ, andξ t = ε t δ ε t. (3.) M w y = M w ξ, and M,w y = M,w ξ, (3.) µ + δ σ = σ + O, (3.) M w y = ω M w υ, and M,w y = ω M,w υ, (3.3) where υ = ξ /ω (, I ). Smlarly, usng (3.8) to elmnate s f, from (3.6) we obtan and hence y, =(y δ ȳ ) τ + δ ȳ + s, δ s, (3.4) M w y, = ω M w s,, (3.5) where s, =(s, δ s ) /ω. It s easly seen that s s the random walk assocated wth υ. Usng (3.), (3.3) and (3.5) n (3.7) we have 4υ t (,) = M w s, (υ M,wυ ) / s M /. (3.6), w s, [7]

9 Smlarly usng (3.3) and (3.5) n (3.4) we have 3υ t (,) = M w s, υ / M w υ s /. (3.7), M w s, Hence, the exact null dstrbuton of t (,) or t (,) wll depend on the nusance parameters only through ther effects on M w and M,w. But, as shown n the Appendx ths dependence vanshes as, rrespectve of whether s fxed or tends to nfnty jontly wth. In the case where s fxed, to ensure that the CADF statstcs, t (,) (or t (,)), do not depend on the nusance parameters the effect of the ntal cross-secton mean, ȳ, must also be elmnated. hs can be acheved by applyng the test to the devatons y t ȳ. he followng theorems provde a formal statement of these results. heorem 3. Suppose the seres y t,for =,,..., and t =,,...,, are generated under (.4) accordng to (.3) and by constructon ȳ (the cross-secton mean of the ntal observatons) s set to zero. hen under Assumptons and the null dstrbuton of t (,) gven by (3.4), wll be free of nusance parameters as for any fxed > 3. In partcular, we have (n quadratc mean) t (,) µ ε ε q Ψ σ ( 3) 3 ε s, σ f q q Ψ / ³ s, s, f h σ h Ψ f h /, (3.8) where Ψ f = f f f τ τ f s f, f s f, τ 3/ 3/ f s f, 3/ τ s f, 3/ s f, s f,, q = f ε σ τ ε σ s f, ε σ, h = f s, σ 3/ τ s, σ 3/ s f, s, σ, ε /σ and f are ndependently dstrbuted as (, I ), s = s, + ε,ands f = s f, + f. he crtcal values of the CADF test can be computed by stochastc smulaton for any fxed >3, and for gven dstrbutonal assumptons for the random varables (ε,f). heorem 3. Let y t be defned by (.3) and consder the statstcs t (,) and t (,) defned by (3.7) and (3.4), respectvely. Suppose that assumptons -3 hold and γ tends to a fnte non-zero lmt as, then under (.4) and as and h,t (,) and t (,) have the same sequental (, ) and jont (,) j lmt dstrbutons, referred to as Cross-sectonally Augmented Dckey-Fuller (CADF) dstrbuton gven by CADF f = R W (r) dw (r) ψ fλ f κ f ³ R, (3.9) / W (r) dr κ f Λ f κ f [8]

10 where Ã Λ f = R! W f (r) dr R W f (r) dr R W, (3.3) f (r) dr and µ à R! W () ψ f = R W, κ f = W (r) dr R f (r) dw (r) W, (3.3) f (r) W (r) dr wth W (r) and W f (r) beng ndependent standard Brownan motons. For the jont lmt dstrbuton to hold t s also requred that as (,) j,/ k, wherek s a non-zero, fnte constant. Remark 3. he crtcal values of CADF f can be computed by stochastc smulaton assumng that µ εt v (, I f + ),fort =,,...,. t Remark 3. From (3.9) t s clear that CADF f = G (W,W f ), (3.3) where G(.) s a general non-lnear functon common for all. herefore, CADF f and CADF jf are dependently dstrbuted, wth the same degree of dependence for all 6= j. Remark 3.3 he random varables CADF f,cadf f,..., CADF f form an exchangeable sequence. hs follows from the fact that under Assumpton 3, condtonal on W f the random varables {CADF f } are dentcally and ndependently dstrbuted. 7 Remark 3.4 he dstrbuton of CADF f reduces to the standard DF dstrbuton under f t =. he sngularlty of Λ f n ths case can be dealt wth by use of generalzed nverse. It s easly seen that lm f CADF f = DF = R W (r) dw (r) W () R W (r) dr R ³ R W (r) dr W (r) dr /, as requred. herefore, erroneous use of CADF n cases where a smple DF statstc would have suffced, although neffcent, wll not be nvald. Remark 3.5 he CADF test can be appled to test the unt root hypothess n the case of a sngle tme seres when nformaton on the cross-secton average, ȳ t, s avalable. For example, when testng for a unt root n the UK output one could use OECD output as a proxy for a possble common technologcal effect n output seres across countres, and apply the CADF test nstead of the standard DF test. Of course, dfferent crtcal values would now apply, and must be computed usng the CADF dstrbuton gven by (3.9). 7 See, for example, heorem.. n aylor, Daffer and Patterson (985, p.3). [9]

11 Remark 3.6 In the above analyss we have opted for smple averages (.e. ȳ) ndealng wth the cross dependence problem. But t s clear that weghted averages could also be used nstead. For example, ȳ t n the th CADF regresson can be replaced wth yt = P j= w jy jt where w j, j =,..., are weghts specfc totheseres. However, n the case where the factor loadngs (γ ) are ndependent random draws from a common dstrbuton the choce of w =(w,w,..., w ) has no effect on the asymptotc propertes of the proposed tests so long as for each, P j= w j as. here would be some small sample dfferences acrossthetestsusngdfferent weghtng schemes, but ths s unlkely to be mportant. Remark 3.7 Our assumpton that γ 6=, wth a non-zero lmt as mght be vewed by some as restrctve. But t s nstructve to recall from (3.7) that n the case where γ =, we have ȳ t = ε t, and therefore ȳ t wouldtendtozeroas,andȳ t to a fxed constant for all t. In most economc and fnancal panels of nterest ths does not seem to be a very lkely outcome. However, the case where γ, as could be of theoretcal nterest and s worth further consderaton. 3. Crtcal Values of the Indvdual CADF est Fgure dsplays the smulated cumulatve dstrbuton functon of the CADF statstc under the null hypothess usng 5, replcatons for = and = 5. For comparson the smulated cumulatve dstrbuton functon of the standard DF statstc s also provded. 8 We expect the smulated dstrbuton to be very close to theoretcal dstrbuton gven by (3.9). Perhaps not surprsngly the CADF dstrbuton s more skewed to the left as compared to the standard DF dstrbuton. hs s clearly reflected n the crtcal values of the two dstrbutons summarzed n able A for the three standard cases consdered n the lterature: no ntercept, ntercept only, ntercept and a lnear trend. 8 he seres y t = y,t +f t +ε t,for =,,...,, and t = 5, 49,...,,,..., 5 were frst generated from y, 5 =,wthf t and ε t as d (, ). hen 5, CADF regressons of y t on an ntercept, y,t,ȳ t and ȳ t were computed over the sample t =,,..., 5. Fgure plots the ordered values of the OLS t-ratos of y,t n these regressons. []

12 able A: Crtcal Values of the CADF and DF Dstrbutons (=,=5, 5, replcatons) %.5% 5% % o ntercept DF CADF Intercept DF CADF Lnear rend DF CADF Crtcal values of the ndvdual CADF dstrbuton for values of and n the range of to for the three standard cases (of no ntercept and no trend, ntercept only, and ntercept and trend) are gven n ables a to c, respectvely. Another nterestng aspect of the CADF dstrbuton, whch becomes mportant when the test s used n a panel data context, s the par-wse dependence of the CADF f statstcs across, mentoned above. he smulated values of the smple par-wse correlaton coeffcent, Corr(CADF f,cadf f ), together wth smulated mean and standard devaton of the CADF dstrbuton for dfferent values of and are gven n ables a to c for the three standard cases. hese smulated moments are remarkably stable for dfferent values of and n excess of. he smulated estmate of the correlaton coeffcent s around.3 for the ntercept case and n the range.-. for the lnear trend case, both qute small but non-zero. 3. ormal Approxmaton to the Dstrbuton of CADF he CADF dstrbuton, lke the standard DF dstrbuton, departs from normalty n two mportant respects: It has a substantally negatve mean and ts standard devaton s less than unty, although not by a large amount. However, the dstrbuton of a standardzed verson of the CADF statstc, defned by [t (,) E (CADF f )] / p Var(CADF f ), looks remarkably lke a standard normal dstrbuton, where E (CADF f )and p Var(CADF f ) are gven n ables a to c. he smulated densty functons of the the standardzed CADF for the ntercept and the lnear trend cases, computed wth =, = 5, and 5, replcatons are dsplayed n Fgures and 3, respectvely. he skewness and Kurtoss 3 coeffcents of the standardzed CADF dstrbutons are.7 and.7 for the ntercept case, and.6 and.9 for the lnear trend case. hey are qute small, although statstcally hghly sgnfcant. evertheless, the closeness of the approxmaton partcularly for the left tal of []

13 the dstrbuton suggests a relatvely smple ormal test, once the mean and the standard devaton of CADF dstrbuton s computed. 4 CADF Panel Unt Root ests Gven that the null dstrbuton of the ndvdual CADF statstcs are asymptotcally ndependent of the nusance parameters, the varous panel unt root tests developed n the lterature for the case of the cross-sectonally ndependent errors can also be appled to the present more general case. Here we focus on a generalzaton of the t-bar test proposed by IPS and consder a cross-sectonally augmented verson of the IPS test based on CIPS(,) = X = t (,), (4.33) where t (,) s the cross-sectonally augmented Dckey-Fuller statstc for the th cross secton unt gven by the t-rato of the coeffcent of y,t n the CADF regresson defned by (3.6). One could also consder combnng the p-values of the ndvdual tests as proposed by Maddala and Wu (999) and Cho (). Examples are the nverse ch-squared (or Fsher) test defned by P (,) = X ln (p ), (4.34) where p s the p-value correspondng to the unt root test of the th ndvdual cross secton unt. Another possblty would be to use the nverse normal test defned by = Z(,) = X = Φ (p ). (4.35) Here we focus on the t-bar verson of the panel unt root test, (4.33), and consder the mean devatons D(,) = X = [t (,) CADF f ], where CADF f s the stochastc lmt of t (,) as and tend to nfnty such that / k ( <k< ). See (3.9). It seems reasonable to expect that D(,) =o p () for and suffcently large. hs conjecture would clearly hold n the case where t (,)have fnte moments for all and above some gven threshold values, say,and.however, such moment condtons are dffcult to establsh even under cross-secton ndependence. (see IPS) One possble method of dealng wth these techncal dffcultes would be to base the t-bar test on a sutably truncated verson of the CADF statstcs. he smulatons reported n Secton 3. suggest that the standardzed verson of these statstcs are very close to beng []

14 standard ormal wth fnte frst and second order moments. herefore, for the purpose of the panel unt root test t would be equally vald to base the test on an average of the truncated versons of t (,), say t (,), where t (,) =t (,), f K <t (,) <K, t (,) = K,f t (,) K, (4.36) t (,) =K,f t (,) K. where K and K are postve constants that are suffcently large so that Pr[ K <t (,) <K ] s suffcently large, say n excess of Usng the normal approxmaton of t (,) as a crude benchmark we would have q K = E (CADF f ) Φ (ε/) Var(CADF f ), and q K = E (CADF f )+Φ ( ε/) Var(CADF f ), where ε s a suffcently small postve constant. For example, settng ε = 6, n the case of models wthout an ntercept or trend (usng the mean and the standard devatons n able a) we would have K = (.5) = 6., and K = (.5) = 4.6. Smlarly, for models wth an ntercept we have K =6. 9, and K =.6, and fnally for models wth a lnear trend we obtan K =6.4, and K =.7 he assocated truncated panel unt root test s now gven by CIPS (,) = X t = (,). (4.37) Snce, by constructon all moments of t (,) exst t then follows that CIPS (,) = X = CADF f + o p (), (4.38) where CADF f s gven by CADFf = CADF f, f K <CADF f <K, CADFf = K,f CADF f K, CADFf = K,f CADF f K, (4.39) and CADF f s defned by (3.9) n heorem 3.. he dstrbutons of the average CADF statstc or ts truncated counterpart, CADF = P = CADF f, are non-standard even for suffcently large. hs s due to the dependence of the ndvdual CADF f varates on the common process W f whch nvaldates the applcaton of the standard central lmt theorems to CADF or CADF, and s n contrast to the results obtaned by IPS under cross-secton ndependence where a standardzed verson of CADF = P = CADF f, was shown to be normally dstrbuted for suffcently large. evertheless, t s possble [3]

15 to show that CADF converges n dstrbuton as, wthout any need for further normalzaton. Recall that CADF f = G(W,W f ),=,,...,, wherew,w,..., W and W f are..d. Brownan motons. Smlarly, CADFf defned by (4.39) wll be a nonlnear functon of W and W f and hence condtonal on W f, CADFf wll be ndependently dstrbuted across. herefore, snce by constructon t follows that E CADF f <, X = CADF f a.s. π K π K + E (CADF f W f, K <CADF f <K ), (4.4) where π =Pr(CADF f K W f )andπ =Pr(CADF f K W f ). hs result smplfes further f we could also establsh that E CADF f <, a property that we conjecture to be true. By lettng K and K, and notng that n ths case π K π K, we have X = CADF f a.s. E (CADF f W f ). he above results establsh that the CADF converges almost surely to a dstrbuton whch depends on K,K and W f. hs dstrbuton does not seem analytcally tractable, but can be readly smulated usng (4.37). We smulated the dstrbuton of CIPS settng =, = 5, and usng 5, replcatons under the followng cases:. Models wthout ntercepts or trends (I), wth K =6., and K =4.6,. Models wth ntercept only (II), wth K =6. 9, and K =.6, 3. Models wth a lnear trend (III), wth K =6.4, and K =.7. he smulated densty functons for these three cases are dsplayed n Fgures 4-6. All the three denstes show marked departures from normalty, although the extent of the departure depends on the nature of the determnstc ncluded n the model. he densty n the case of the model wthout any determnstcs show the greatest degree of departure from normalty and s n fact bmodal. he densty for the other two models are un-modal but are hghly skewed. he densty for the model wth a lnear trend s closest to beng normal. hs pattern of departures from normalty s n accordance wth the estmates of par-wse correlaton coeffcents of the ndvdual CADF statstcs reported n ables a-c. he larger the value of ths correlaton coeffcent, the greater one would expect the densty to depart from normalty. Recall that the asymptotc correlaton coeffcents of the ndvdual CADF statstcs are.,.3 and. for the models I to III, respectvely. We carred out the same analyss for the non-truncated verson, usng CIPS defned by (4.33), and obtaned dentcal results. he fnte sample dstrbutons of CIPS (,) and CIPS (,) dffer only for very small values of and are ndstngushable for >. he [4]

16 comparatve small sample performances of the CIPS and the CIPS tests wll be consdered n Secton 6. he %, 5% and % crtcal values of CIPS and CIPS tests are gven n ables 3a-3c for models I-III, respectvely. In most cases the crtcal values for the two versons of the CIPS test are dentcal and only one value s reported. In cases where the two crtcal values dffer the truncated verson s ncluded n brackets. Smlar arguments also apply to the other forms of the panel unt root tests gven by (4.34) and (4.35). he cross sectonally augmented versons of these statstcs, where the rejecton probabltes, p, are computed usng the CADF regressons, (3.6), wll be denoted by CP(,) andcz(,). ote that n the presence of cross secton dependence these statstcs are no longer asymptotcally normally dstrbuted and ther crtcal values must be obtaned by stochastc smulatons. he %, 5% and % crtcal values of CP(,) and CZ(,) are computed by Mutta Akusuwan for all pars of, =, 5,, 3, 5, 7,,, and are avalable from the author on request. 5 Case of Serally Correlated Errors he CIPS testng procedure can be readly extended to the case where n addton to the cross dependence, the ndvdual-specfc error terms are also serally correlated. 5. Alternatve Resdual Seral Correlaton Models wth Cross Dependence he resdual seral correlaton can be modelled n a number of dfferent ways, drectly va the dosyncractc components, through the common effects, or a mxture of the two. o smplfy the exposton we shall confne our analyss to statonary frst-order autoregressve processes and consder three general types of specfcatons. 9 Inthecasewhereonlythe dosyncratc components are serally correlated we have where u t = γ f t + v t, (5.4) v t = ρ v,t + ε t, ρ < (5.4) and ε t..d.(, σ ). In conjuncton wth (.3) ths would yeld the followng augmented Dckey-Fuller regresson y t = µ β ( ρ )+β y,t + ρ ( + β ) y,t + γ (f t ρ f t )+ε t, (5.43) wth cross-sectonally dependent errors. 9 he analyss can be readly extended to hgher order processes. [5]

17 where In the case where the resdual seral correlaton s confned to the common effects we have u t = γ f t + ε t, (5.44) f t = λf t + ξ t, λ <, (5.45) and ξ t are serally uncorrelated wth mean zero and a constant varance. he seral correlaton n the common effects nduces movng average errors n the ndvdual ADF regressons, and we have y t = µ β ( λ)+β ( λ) y,t + λ( + β ) y,t + γ ξ t + ε t λε,t. (5.46) In ths case the cross-secton dependence s characterzed through the resdual common effects, ξ t. A thrd possblty would be to model the resdual seral correlaton frst as u t = ρ u,t + η t, ρ <, for =,,...,, (5.47) and then allow for the cross secton dependence by assumng a one-factor model for the resduals Under ths specfcaton we have η t = γ f t + ε t. (5.48) y t = µ β ( ρ )+β ( ρ )y,t + ρ ( + β ) y,t + γ f t + ε t. (5.49) 5. Indvdual-Specfc CADF Statstcs for the Serally Correlated Case All three models yeld the same ADF regressons, but wth dfferent error specfcatons and parameter heterogenety. he asymptotc theory to be developed n ths secton can be adapted to deal wth all three specfcatons, but to save space here we focus on the thrd specfcaton gven by (5.49). We shall also confne our attenton to the case where the autoregressve coeffcents, ρ are homogeneous across, but shall consder the mplcatons of relaxng ths assumpton usng Monte Carlo smulatons. he mathematcal detals become much more complcated f ρ s allowed to dffer across. o deal wth the unobserved common effects, f t,wefrst note that n ths case under the unt root hypothess we have (usng (5.49) wth β =andρ = ρ) y t = ρ y,t + γ f t + ε t, he case of non-statonary common effects wll not be consdered here. [6]

18 and f t = γ ( ȳ t ρ ȳ t ) γ ε t. Hence, for suffcently large, and under our assumpton that γ tends to a non-zero lmt as, the common effects can be proxed by a lnear combnatons of ȳ t and ȳ t.in addton the DF regressons must be augmented for resdual seral correlaton and the lagged levels of the cross secton means of the processes, namely y,t and ȳ t. Accordngly, we propose runnng the followng CADF regressons whch are augmented to asymptotcally flter out the effects of both cross secton and tme dependence patterns n the resduals: y = b y, + W c + e, where W = y,, ȳ, ȳ, τ, ȳ s a 5 matrx of observatons defnednsecton 3. he ndvdual CADF statstcs are gven by t (,) = y M y, ˆσ y, M y, /, (5.5) where ˆσ = y M,w y, 6 M = I W W W W, M,w = I G (G G ) G,andG = y,, W. Smlarly, we have the Lagrange multpler verson defned 5 y t (,) = M y, y / M y y /. (5.5), M y, As wth the serally uncorrelated case both versons of the CADF tests are asymptotcally equvalent. o establsh the asymptotc nvarance of the above CADF statstcs to the coeffcents of the common effects, γ,wefrst note that under β = or and where y t = ρ y,t + δ ( ȳ t ρ ȳ t )+(ε t δ ε t ), (5.5) y t = δ ȳ t +z t (ρ) δ z t (ρ), (5.53) y t =(y δ ȳ )+δ ȳ t + s z,t δ s zt, (5.54) z t (ρ) =( ρl) ε t, (5.55) [7]

19 s z,t = tx z j (ρ), s zt = j= X = s z,t, (5.56) and L s a one-perod lag operator. Usng these results we now have the followng generalzatons of (3.9) and (3.4): and y = ρ y, + δ ( ȳ ρ ȳ )+(ε δ ε), (5.57) y, =(y δ ȳ ) τ + δ ȳ +(s z, δ s z, ), (5.58) whch f used n (5.5) yelds (under β =) t (,) = υ M s z, ³ υ / ³ M υ s z, M s z, 5 where as before υ =(ε δ ε) /ω (, I ), ω s defned by (3.), and s z, =(s z, δ s z, ) /ω. /, (5.59) he elements of s z, and s z, are defned n (5.56). he exact sample dstrbuton of t (,) depends on δ, γ and ρ, but as stated n the followng theorem ths dependence vanshes for and,suchthat/ k, where k s a fnte, non-zero constant. heorem 5. Let y t be defned by (5.49) wth ρ = ρ <, and consder the statstcs t (,) and t (,) defned by (5.5) and (5.5), respectvely. Suppose that Assumptons -3 hold and γ tends to a fnte non-zero lmt as, then under (.4) and as and h,t (,) and t (,) have the same sequental (, ) and jont (,) j lmt dstrbutons gven by (3.9), obtaned under ρ =. For a proof see Secton A.3 n the Appendx. hs theorem establshes that Augmented Dckey-Fuller regresson results n pure tme seres contexts also apples to cross sectonally augmented regressons. Although, our proof assumes a frst-order error process, the approach readly extends to hgher order processes. For example, for an AR(p) error specfcaton the relevant ndvdual CADF statstcs wll be gven by the OLS t-rato of b n the followng p th order cross-secton/tme-seres augmented regresson: y t = a + b y,t + c ȳ t + px d j ȳ t j + j= px δ j y,t j + e t. (5.6) hs testng procedure also readly extend to models contanng lnear trends. Clearly, the same crtcal values reported n ables 3a-3c wll also be applcable here. It s worth notng that heorem 5. also holds under (5.43) and (5.46), so long as λ <. Detals of the proof are very smlar and can be obtaned from the author on request. [8] j=

20 5.3 Panel Unt Root ests for Panels wth Serally Correlated Errors It s now relatvely easy to construct panel unt root tests that smultaneously take account of cross-secton dependence and resdual seral correlaton. Once agan we focus on the truncated verson of the CIPS statstc gven by (4.37), wth t (,) computed usng the cross-secton/tme seres augmented regresson, (5.6), subject to the truncaton scheme defned by (4.36). Usng theorem 5. and notng that the result of the theorem apples equally to the truncated verson of the CADF statstcs we have Hence t (,) =CADF f + o p (). CIPS (,) j v X = CADF f, and CIPS n the case of serally correlated errors has the same lmt dstrbuton as (4.4) obtaned under ρ = and the crtcal values reported n ables 3a-3c also apples equally to the serally correlated case. 6 Small Sample Performance: Monte Carlo Evdence In ths secton we consder the small sample performance of the cross sectonally augmented unt root tests proposed n the paper usng Monte Carlo technques. Intally we shall consder dynamc panels wth fxed effects and cross secton dependence, but wthout resdual seral correlaton or lnear trends. he data generatng process (DGP) n ths case s gven by y t =( φ ) µ + φ y,t + u t,=,,...,, t = 5, 5,...,,,..., ; (6.6) where u t = γ f t + ε t, f t d(, ), (6.6) and ε t..d.(, σ ), wth σ du [.5,.5]. (6.63) We shall consder two levels of cross secton dependence where we generate γ du [,.] as an example of low cross secton dependence, and γ du [, 3] to represent the case of hgh cross secton dependence. he average par-wse cross correlaton coeffcent of u t and u jt under these two scenaros are % and 5%, respectvely, and cover a wde range of values applcable n practce. he order of augmentaton, p, can be estmated usng model selecton crtera such as Akake or Schwartz appled to the underlyng tme seres specfcaton, namely (5.6). [9]

21 o examne the mpact of the resdual seral correlaton on the proposed tests we consdered a number of experments where the errors ε t were generated as ε t = ρ ε,t + e t, e t..d.(, σ ), σ du [.5,.5], (6.64) wth ρ du [.,.4], as an example of postve resdual seral correlatons, and ρ du [.4,.], as an example of negatve resdual seral correlatons. hs yelds the augmented ADF model gven by (5.43). hese experments were also carred out under low and hgh cross secton dependence scenaros. hs DGP dffers from model (5.49) that underles the theoretcal dervatons n Secton 5, and s ntended to check the robustness of our analyss to alternatve resdual seral correlaton models. It also allows the resdual seral correlaton coeffcents, ρ,todffer across ; thus provdng an opportunty to check the robustness of our results to such heterogenetes. In a thrd set of experments we allow for determnstc trends n the DGP and the CADF regressons. For ths case y t were generated as follows: y t = µ +( φ ) δ t + φ y,t + u t, wth µ du [.,.] and δ du [.,.]. hs ensures that y t has the same average trend propertes under the null and the alternatve hypotheses. he errors, u t,were generated accordng to (6.6), (6.63) and (6.64) for dfferent values of ρ as set out above. Sze and power of the tests were computed under the null φ = for all, andtheheterogeneous alternatves φ du [.85,.95], usng, replcatons per experment. 3 he tests were one-sded wth the nomnal sze set at 5%, and were conducted for all combnatons of and =,, 3, 5,. All the parameters, µ, δ, φ, ρ, σ, and γ were generated ndependently of the errors, e t (ε t )andf t ;wthf t also generated ndependently of e t (ε t ). 6. Sze Dstorton of the Standard Panel Unt Root ests Before reportng the results for the proposed cross sectonally augmented tests, t would be helpful frst to examne the extent to whch the sze of the standard panel unt root tests (that assume cross secton ndependence) are dstorted n the presence of cross secton dependence. able 4 reports the emprcal szes of the IPS, truncated IPS, the nverse ch-squared (P ), and the nverse normal (Z) tests when the DGP s subject to cross secton dependence wth serally uncorrelated errors as defned by (6.6) and (6.6). 4 All these tests are based on smple DF regressons and utlze the ndvdual DF statstcs, or the assocated rejecton probabltes. 5 he IPS statstc s the famlar standardzed t bar statstc defned by (see IPS (3)) IPS(,) = {t-bar E [t β =]} p Var[t β =], = (, ). (6.65) 3 Under the alternatve hypothess µ are drawn as µ du[,.]. 4 In calculaton of P and Z statstcs the rejecton probabltes, p, are truncated to le n the range [., ], n order to avod very extreme values affectng these test statstcs. hs s n effect a knd of truncaton, smlar to the truncated verson of the IPS statstcs. 5 See ( 4.34), ( 4.35), and the notes to able 4 for further detals. []

22 where t-bar = P = t,andt s the t-rato of the estmated coeffcent of y,t n the OLS regresson of y t on an ntercept and y,t. 6 he truncated verson of the IPS test uses the same formula as above but replaces t wth the ndvdually truncated statstc, t, defned by (4.36) wth K =6.9, and K =.6. InthecaseofP and Z tests we report two sets of results: one set based on normal approxmatons as orgnally proposed by Maddala and Wu (999), and Cho (), and another set based on emprcal crtcal values obtaned from the smulated dstrbuton of these statstcs under the null hypothess. We refer to the latter versons of these tests as P and the Z tests. 7 As to be expected the extent of over-rejecton of the tests very much depends on the degree of cross secton dependence. Under the low cross secton dependence the dfferent verson of the IPS and the Z tests perform reasonably well. he standard P test tends to over-reject for small values of, but the normal approxmaton begns to work as s ncreased. Overall, when the cross secton dependence s low all tests (possbly except for the P test) have the correct sze, whch s n lne wth the results n the lterature, reported, for example, by Cho (). But n the case of hgh cross secton scenaro all the tests tend to over-reject, often by a substantal amount. Clearly, the standard panel unt root tests that do not allow for cross secton dependence can be serously based f the the degree of cross secton dependence s suffcently large. It would now be nterestng to see f the cross sectonally augmented versons of these tests can resolve ther sze dstortons under the hgh cross secton dependence scenaro. 6. Sze and Power n the Case of Models wth Serally Uncorrelated Errors he tests to be consdered are the cross sectonally augmented IPS test, CIPS(, ), and ts truncated verson, CIPS (,), and the cross sectonally augmented versons of the nverse ch-squared and the nverse normal tests, denoted by CP(,) andcz(,), respectvely. he CIPS and CIPS statstcs are defned by (4.33) and (4.37), respectvely. he computaton of thecp and CZ statstcs requre the estmaton of ndvdual-specfc rejecton probabltes by stochastc smulatons. In partcular, the cross-sectonally augmented nverse ch-squared test statstc s gven by CP (,) = X ln [ˆp (,)], (6.66) = where the rejecton probabltes are computed as bp (,) = S SX I ht (,) CDF (s) s= (6.67) 6 he IPS and other panel unt root tests can be readly adapted for use wth unbalanced panels where the avalable tme perods dffer across. In the case of standard IPS test ths generalzaton s consdered n Im, Pesaran and Shn (3). 7 he crtcal values of the P and Z tests are avalable from the author on request. []

23 CDF (s) s the s th random draw from the dstrbuton of CDF, and S s the number of replcatons used to compute bp (,), whch we also set equal to 5,. I [A] s the ndcator functon that takes the value of when A>, and otherwse. Smlarly, CZ(,) = Φ [ˆp (,)]. (6.68) = o avod very extreme values the rejecton probabltes were truncated to le n the range [., ]. he sze and power characterstcs of these tests are summarzed n ables 5a and 5b, respectvely. here are no evdence of sze dstortons n the case of the CIPS, CIPS, and the versons of the CP,andCZ tests (denoted by CP,and g CZ) g that use the correct crtcal values. ot surprsngly, the use of normal approxmatons for the CP, andcz tests does not work here snce due to the cross secton dependence these test statstcs are not normally dstrbuted even for suffcently large and. herefore, t s only vald to consder a power comparson of CIPS, CIPS, CP, g andcz g tests, as summarzed n able 5b. It s clear that CP g test s generally domnated by the other three tests whch are very smlar ndeed. one of the tests exhbt much power when =, rrespectve of the sze of. Only when s ncreased to and beyond one can begn to see the beneft ofncreasng on the power of the tests. Fnally, n the present smple case of serally uncorrelated resduals lttle seems to be ganed by the truncaton procedure. 6.3 Sze and Power n the Case of Models wth Serally Correlated Errors In the case of models wth serally correlated errors, the cross sectonally augmented tests (CIPS, CIPS, g CP,and g CZ ) are computed both for the basc CADF regressons wthout tme seres augmentaton (whch we denote by CADF()), and the CADF regressons are augmented wth lagged changes of y t and ȳ t, as n (5.6), whch we refer to by CADF(p), where p s the order of the tme seres augmentaton. We computed the tests for p =,, and focussed on the hgh cross secton scenaro. he sze and power results for the experments wth postve resdual seral correlaton are summarzed n ables 6a and 6b, and the ones for negatve resdual seral correlaton are gven n ables 7a and 7b. As to be expected sgnfcant sze dstortons wll be present f CADF regressons are not augmented to account for the tme seres dependence. here are substantal under rejectons for postve resdual seral correlaton, and substantal over-rejectons n the case of negatve resdual seral correlatons. But the test szes stablze at around 5% when the CADF regressons are augmented wth y,t. Recall that ȳ t s already ncluded n the CADF regressons. Irrespectve of whether the resdual seral correlatons are postve or negatve, the tests based on CADF() regressons tend to have the correct sze. here s, however, some evdence that for small (less than n these experments) the g CP test, and to a lesser extent, the CIPS test are over-szed. But the truncated verson of the CIPS does not seem to suffer from ths problem even for as small as. he truncaton of the extreme ndvdual CADF statstcs seem to have pad out n the present applcaton where s very small relatve to the number of parameters of the underlyng CADF() regressons. X []

24 Focussng on CIPS and g CZ we note from ables 6b and 7b that both tests have very smlar power propertes. ether of the tests seem to have any power for = or less, and as n the serally uncorrelated case, the power does not rse wth f s too small. However, wth = or hgher the power of both tests begn to rse qute rapdly wth. 8 he tests tend to show hgher power for negatve as compared to postve resdual seral correlatons. Fnally, there s very lttle to choose between the two tests, although as noted earler the CIPS statstc s much smpler to compute. 6.4 Sze and Power n the Case of Models wth Lnear rends and Serally Correlated Errors Sze and power of CIPS, CIPS, g CP, and g CZ tests n the case of models wth lnear determnstc trends are summarzed n ables 8-. ables 8a and 8b gve the results for models wthout resdual seral correlaton and show that all the varous test contnue to have szes very close to the nomnal value of 5%. However, as to be expected the ncluson of lnear trends n the CADF regressons come at the cost of a lower power. We now need to be 3 or more before power begns to ncrease wth. For example, when =thepower of the tests stays around 7% rrespectve of the value of. Butwhen =5thepowerof the CIPS test rses from 8% to 6% as s ncreased from to. Once agan the g CP test s domnated by the other three tests whch have very smlar power characterstcs. he results for the lnear trend case combned wth resdual seral correlaton are presented n ables 9 and. able 9a and 9b gve the results for postve resdual correlatons, and ables a and b summarze the results for negatve resdual seral correlatons. he szes of the CIPS and g CZ tests contnue to be satsfactory, even for = once the augmentaton for the resdual seral correlaton s mplemented (see the results under ACDF()). In contrast, the CIPS and g CP tests are grossly over-szed when =. ote that n the case of CADF() regressons wth lnear trends the number of parameters beng estmated s 7, and wth only 3 degrees of freedom remanng the non-truncated ndvdual CADF () statstcs mght not have moments, whch could be the reason why the CIPS test breaks down. he applcaton of the truncaton procedure fxes the lack of the moment problem and renders the truncated CIPS test vald even when the degrees of freedom of the underlyng CADF regressons s as low as 3. Smlarly, the g CZ statstc overcome the problem of extreme values by usng the nverse probablty transformaton and by the fact that the rejecton probabltes used n the constructon of g CZ are truncated to avod very extreme values. See (6.68). 7 Emprcal Applcatons In ths secton we apply our proposed panel unt root test to two dfferent types of panel data: () an nternatonal macro data set composed of 7 real exchange rate seres, and () 8 hese Monte Carlo results are n lne wth the theoretcal fndngs of Moon, Perron and Phllps (3) who show that local power of panel unt root tests s n / neghborhood of the null n the case of models that contan ntercepts only and /4 for models wth lnear trends. [3]

25 a mcro panel data on real earnngs of households from the PSID data. 7. Real Exchange Rates Panel unt root tests have been used n the lterature prmarly to test the purchasng power party (PPP) hypothess. 9 hese applcatons have been partcularly mportant consderng the relatve lack of power of unt root tests appled to sngle seres. Relance on long tme seres coverng sxty or more years of data n order to enhance the power of sngle-seres unt root tests have also been problematc due to changes n exchange rate regmes and the ncdence of structural breaks wthn the same regme. However, as orgnally emphaszed by O Connel (998) panel unt root tests can also lead to spurous results (spurously favourng the PPP hypothess) f there are sgnfcant degrees of error cross secton dependence and ths s gnored by the panel unt root tests. hs s confrmed by the Monte Carlo results summarzed n able 4, but only f the degree of error cross secton dependence s suffcently hgh. Applcaton of panel unt root tests that allow for cross secton dependence s, therefore, desrable once t s establshed that the panel s subject to a sgnfcant degree of error cross secton dependence. In cases where cross secton dependence s not suffcently hgh, loss of power mght result f panel unt root tests that allow for cross secton dependence are used. herefore, before an approprate choce of a panel unt root test s made t s mportant that the degree of cross secton dependence s frst tested. In ths sub-secton we consder two panels of quarterly real exchange rates from 7 OECD countres. he frst panel covers the perod 974Q to 998Q4 ( = ), and the second shorter panel covers the perod 988Q-998Q4 ( = 44) whch has been recently analyzed by Smth et al. (4), on the grounds that the latter s less lkely to be subject to structural breaks. Log real exchange rates are computed as y t = s t + p ust p t, where s t s the log of the nomnal exchange rate of country th currency n terms of U.S. dollar, p ust and p t are logarthms of consumer prce ndces n the U.S. and country, respectvely. Here we also take the opportunty of correctng an error n the computaton of the real exchange rate used n the panel unt root tests reported n Smth et al. (4, p.65). hese authors base ther tests on s t p ust + p t, whch would have been correct (apart from a sgn) f the nomnal exchange rate had been defned as U.S. dollars per unt of country th currency. But as the authors state and the data deposted on the JAE Archve establshes the nomnal exchange rates used are unts (fractons) of foregn currency n one U.S. dollar. 9 Panel unt root tests have also been appled to test the convergence of log output per capta across countres. But as argued n Pesaran (4b), unt root tests appled to per capta output seres ether ndvdually or n ther panel forms are not nformatve about wthn or cross country convergence. Further detals together wth the data are avalable from Journal of Appled Econometrcs Data Archve. ( hs error was dentfed by Vanessa Smth (one of the authors of the Smth et al. paper) n the process of replcatng and extendng the test results for ncluson n the current paper. [4]

26 As the frst stage n our analyss we estmated ndvdual ADF(p) regressons (wthout cross secton augmentatons) for p =,, 3 and 4, for the two sample perods 974Q to 998Q4 and 988Q-998Q4, and computed par-wse cross secton correlaton coeffcents of the resduals from these regressons (namely ˆρ j ). he smple average of these correlaton coeffcents across all the (7 8) / = 53 pars, ˆρ, together wth the assocated cross secton dependence (CD) test statstcs proposed n Pesaran (4c) are gven n able. he average cross secton error correlaton coeffcents s around.6 whch s qute large. he CD statstcs are also hghly sgnfcant and leave lttle room for doubtng that the errors of real exchange rate equatons are hghly correlated across countres. hs concluson s qute robust to the choce of p and the sample perod and s n lne wth the fndngs of O Connel (998) and others, but s set wthn a more formal statstcal framework. able : Cross Secton Correlatons of the Errors n the ADF(p) Regressons of Real Exchange Rates Across Countres ( =7) Panel A 974Q-998Q4 ( = ) p = p = p =3 p =4 ˆρ CD Panel B 988Q-998Q4( = 44) p = p = p =3 p =4 ˆρ CD hepaneluntrootteststatstcsbasedonadf(p) andcadf(p) regressons are summarzednable. heips statstc s the standardzed t bar test of Im, Pesaran and Shn (3) defned by (6.65), and CIPS statstc s the cross secton average of the t rato of the OLS coeffcent of y,t n the CADF regressons (5.6). Under the unt root hypothess and no cross secton dependence IP S s asymptotcally dstrbuted as (, ), and therefore on the bass of the IPS statstcs n able t would be concluded that the unt root hypothess s rejected for p, wth the probablty of rejecton beng partcularly hgh for p =4. 3 But, due to the large and sgnfcant degree of cross secton dependence n real exchange rates documented n able, ths concluson mght not be safe and we should be consderng the CIPS test that allows for cross secton dependence. At the 5% sgnfcance level the crtcal value of the CIPS statstc for =7and n the range of 3 s around -.. (see able 3b). herefore, accordng to the CIPS test the null of unt root can not be rejected at the 5% level rrespectve of the value of p. herefore, the apparent support obtaned for the PPP hypothess usng the IPS test could be spurous. ³ Specfcally, ˆρ = ( ) X X = j=+ bρ j, and CD = h / ( ) ˆρ. Under the null hypothess of zero cross secton dependence CD s asymptotcally dstrbuted as (, ). 3 he same concluson would be reached f p s selected for each seres by model selecton crtera such as AIC. [5]

27 able : IPS and CIPS est Statstcs for the 7 OECD Quarterly Real Exchange Rates Panel A 974Q-998Q4 ( = ) ests p = p = p =3 p =4 IPS CIPS Panel B 988Q-998Q4 ( =44) ests p = p = p =3 p =4 IPS CIPS ote: All statstcs are based on unvarate AR(p) specfcatons n the level of the varables wth p 4 ncludng an ntercept term only and the underlyng ADF and CADF regressons are estmated on the same sample perod, namely, 974Q-998Q4 and 988Q-998Q4 for panels A and B respectvely. he above concluson s n lne wth the test results reported by Harrs, Leybourne and McCabe (4) for a smlar data set usng an altogether dfferent procedure that tests the null hypothess of the real exchange rates beng statonary. A smlar concluson s also reached by Cho and Chue (4) for the G7 countres usng subsamplng technques. urnng to the bootstrap procedure advanced n Smth et al. (4), n able 3 we report thep-valuesofthreeoutofthefve tests consdered by these authors, namely t, Max,and WS whch are the bootstrap versons of the ADF test, the Max test of Leybourne (995), and the weghted symmetrc (WS) test of Pantula et al. (994) that allow for cross secton dependence usng bootstarp blocks of m =3or. 4 hese test results and the concluson one obtans from them concernng the valdty of the PPP hypothess crtcally depends on the ADF verson of the test used n the bootstrap procedure. he t verson of the test (whch s based on the same underlyng statstc used n the CIPS test) does not reject the unt root hypothess for values of p 3. For p =4thet test rejects the null at 7% and 5% levels for the sample perods 974Q-998Q4 and 988Q-998Q4, respectvely. In contrast, the Max andthewstestsrejecttheuntroothypothessat6%orlesssolongasp. Overall, the test results are nconclusve and further analyss of the small sample propertes of the unt root tests based on the Max and the WS varants of the ADF statstcs would seem desrable. It would also be nterestng to nvestgate the small sample propertes of CIPS type tests based on Max and WS statstcs computed usng CADF regressons. 4 For detals of the bootstrap procedure used see Smth et al. (4). I am grateful to Vanessa Smth for carryng out the computatons. [6]

28 able 3: P-values for Bootstrap Panel Data Unt Root ests Appled to 7 OECD Quarterly Real Exchange Rates Panel A 974Q-998Q4 ( = ) p = p = p =3 p =4 Block Sze m m m m Statstc t Max WS Panel B 988Q-998Q4 ( = 44) p = p = p =3 p =4 Block Sze m m m m Statstc t Max WS ote: All statstcs are based on unvarate AR(p) specfcatons n the level of the varables wth p 4 ncludng an ntercept term only and the regressons are estmated on the same sample perod, namely, 974Q-998Q4 and 988Q-998Q4 for panels A and B respectvely. Estmates were obtaned usng 5 replcatons. 7. Real Earnngs In ther analyss of varance dynamcs of real earnngs Meghr and Pstaferr (4) mpose a unt root on log real earnngs of households n the PSID. hey consder households wth male heads aged 5 to 55 wth at least nne years of usable earnngs data. In our applcaton we further restrct the set of households to those wth twenty two years of usable earnngs so that we end wth = when estmatng ADF (p) orcadf(p) regressons wth p =. Asour Monte Carlo results show the CIPS test tends to lack power when s smaller than. In fact for models wth lnear trends one would even need a larger for the CIPS test to have reasonable power even for reasonably large. We also follow Meghr and Pstaferr (4) and group the households by ther educatonal backgrounds nto Hgh School Dropouts (HSD, those wth less than grades of schoolng), Hgh School Graduates (HSG, those wth at least a hgh school dploma, but no college degree), and College Graduates (CLG, those wth a college degree or more). Although our proposed test allows for heterogenety of ncome dynamcs across households t s not necessarly true that all household groupngs would be subject to the same degree of cross secton dependence. he CD statstcs for all the households ( = 8), and the three sub-groups ( HSD = 36, HSG = 87, and CLG = 58), together wth the assocated IPS and CIPS statstcs are gven n able 4. he CD test s statstcally sgnfcant for the sample as a whole, and for two of the three educatonal groups. It s not statstcally sgnfcant n the case of the hgh school drop outs. hs outcome s robust to the choce of p and does not depend on whether a lnear trend s ncluded n the earnngs equatons. Based on the IPS test, the unt [7]

29 root hypothess s rejected for the whole sample and n the case of all the three sub-groups, rrespectve of whether a lnear trend s ncluded n the ndvdual earnng equatons. But once the CIPS test, that allows for error cross secton dependence, s consdered the test outcomes are mxed. he unt root hypothess s rejected for the sample as a whole, but not for all the sub-groups. 5 hs could be partly due to lack of power of the CIPS test n small samples as compared to the IPS test. It s, therefore, mportant that the CIPS test s used when there are sgnfcant evdence of cross secton dependence. hs s readly seen n the case of the test results for the HSD group, where the unt root hypothess s rejected by IPS but not f CIPS s used n the case of models wth lnear trends. able 4: Panel Unt Root ests Appled to Log Real Earnngs of Households n PSID Data ( = ) Panel A Wth Intercept Household Groupngs All HSD HSG CLG = 8 =36 =87 =58 p = CD IPS CIPS p = CD IPS CIPS Panel B Wth Intercept and rend Household Groupngs All HSD HSG CLG = 8 =36 =87 =58 p = CD IPS CIPS p = CD IPS CIPS Concludng Remarks hs paper presents a new and smple procedure for testng unt roots n dynamc panels subject to (possbly) cross sectonally dependent as well as serally correlated errors. he 5 he crtcal values for the CIPS tests are gven n ables 3a-3c. For example, for =and = 8 the 5% crtcal value of the CIPS test n the case of models wth an ntercept s -.4, and for models wth an ntercept and a lnear tme trend t s [8]

30 procedure nvolves augmentng the standard ADF regressons for the ndvdual seres wth current and lagged cross secton averages of all the seres n the panel. hs s a natural extenson of the DF approach to dealng wth resdual seral correlaton where lagged changes of the seres are used to flter out the tme seres dependence when s suffcently large. Here we propose to use cross secton averages to perform a smlar task n dealng wth the cross dependence problem. Our approach should be seen as provdng a smple alternatve to the orthogonalzaton type procedures advanced n the lterature by Ba and g (4), Moon and Perron (4), and Phllps and Sul (3). Although we have provded extensve smulaton results n support of our proposed tests, further smulaton experments are needed to shed lght on the relatve merts of the varous panel unt roots that are now avalable n the lterature. Our analyss and testng approach can also be extended n a number of drectons. One obvous generalzaton s to allow for a rcher pattern of cross dependence by ncludng addtonal common factors n the model. hs s lkely to pose addtonal techncal dffcultes, but can be dealt wth by augmentng the ndvdual ADF regressons wth addtonal cross secton averages formed over sub-groups, such as regons, sectors or ndustres. Another worthwhle extenson would be to consder cross secton augmented versons of unt root tests due to Ellott et al. (996), Fuller and Park (995), and Leybourne (995). Such tests are lkely to have better small sample power propertes. In ther analyss Ba and g (4) also consder the possblty of unt root n the common factors. However, under ther set up the unt root propertes of the common factor(s) and the dosyncractc component of the ndvdual seres are unrelated. As a result they are able to carry out separate unt root tests n the common and the dosyncractc components. he specfcaton used by Ba and g s gven by the statc factor model (assumng one factor for ease of comparson) y t = α + α t + γ f t + v t, where f t s the common factor, γ the assocated factor loadngs, and v t the dosyncractc component assumed ndependently dstrbuted of f t. he unt root propertes of y t s determned by the maxmum order of ntegraton of the two seres f t and v t. Hence, y t wll be I() f ether v t and/or f t contan a unt root. Averagng across and lettng, q.m. q.m. for each t, v t, f v t s statonary, and v t c, wherec s a fxed constant f v t s I(). herefore, a unt root n f t may be tested by testng the presence of a unt root n ȳ t ndependently of whether the dosyncractc components are I() or I(). By contrast, n our specfcatons (see (5.43), (5.46), and (5.49)), the common factor s ntroduced to model cross secton dependence of the statonary components. As a result when testng φ =,the order of ntegraton of y t changes from beng I() f f t s statonary, to I() f f t s I(). herefore, n our set up t makes sense not to allow f t to have a unt root. he models advanced here and the statc factor model used by Ba and g serve dfferent purposes. [9]

31 Appendx A: Mathematcal Proofs A. Some Prelmnary Order Results Recall from (3.) and (3.) that υ = ξ /ω =(ε δ ε) /ω,whereω = σ + O (s, δ s ) /ω where s = s, + υ.. Also s, = (A.) Hence τ υ = τ ε δ (τ ε), ω τ s, 3/ = τ s, δ (τ s ) ω 3/, (A.) ε υ = ε ε δ ( ε ε) ω, ε s, 3/ = ε s, δ ( ε s ) ω 3/, (A.3) f υ = f ε δ (f ε) ω, f s, 3/ = f s, δ (f s ) ω 3/, (A.4) s s, = s s, δ s s ω, s f, s s, f, s, δ ³s f, s = ω, (A.5) s f, υ = s f, ε δ ³ s f, ε ω, s υ = s ε δ s ε, (A.6) ω υ υ = ε ε + δ ( ε ε) δ (ε ε) ω, (A.7) s, s, = s, s, + δ s s δ s, s ω, (A.8) ow usng results n Pesaran (4, Appendx) t s easly seen that µ τ ε µ τ ε µ E =,Var = O, (A.9) µ f ε µ f E ε µ =,Var = O, (A.) µ Ã! µ µ ε ε ε ε E = O,Var = O, (A.) µ ε ε σ E = µ ε µ,var ε = O. (A.) [A.]

32 Also usng results n Fuller (996, p. 547) and carryng out smlar dervatons we also have µ µ s E ε µ s =,Var ε = O, (A.3) 3/ 3/ µ s E ε µ s =,Var ε µ s E τ µ s =,Var τ 3/ 3/ µ s E s µ s E s, µ = O µ = O µ s,var s µ = O, (A.4) µ = O, (A.5) µ s,var s, µ s E s f, µ s =,Var s f, µ s µ f, ε s f, ε E =,Var E µ = O, (A.6) µ = O, (A.7) µ = O, (A.8) µ = O, (A.9) µ f ε µ f E ε µ =,Var = O, (A.) µ f s µ f s µ 3/ =,Var 3/ = O, (A.) A. Asymptotc Dstrbuton of t (,) - Serally Uncorrelated Case he t (,) statstc defnedby(3.7)maybewrttenas ³q ³ υ t (,) = ³ υ M wυ 3 M w s, / ³ s, M ws,. / (A.) where υ (, I ), and s, s defned by (A.) whch s the standardzed random walk assocated to υ. Frst consder the numerator of t (,), and note that υ M w s, = υ s, υ WD ³ Ã! D W D W s, WD, (A.3) where D = / /. (A.4) [A.]

33 Also D W υ = ȳ υ τ υ ȳ υ, D W s, = ȳ s, 3/ τ s, 3/ ȳ s,, (A.5) Usng (3.7) and (3.8) we have D W WD = ȳ υ ȳ ȳ ȳ τ τ ȳ ȳ ȳ 3/ ȳ τ 3/ ȳ ȳ 3/ τ ȳ 3/ ȳ ȳ. (A.6) µ f υ µ ε = γ υ +, (A.7) ȳ υ µ τ υ =ȳ + γ µ s f, υ + µ s υ, (A.8) ȳ s, 3/ = γ µ f s, 3/ + µ ε s, 3/, (A.9) ȳ s, =ȳ µ τ s, + γ µ s f, s, + µ s s,, (A.3) τ ȳ = γ µ τ f + µ τ ε, (A.3) τ µ ȳ =ȳ 3/ + γ µ τ s f, + 3/ µ τ s, (A.3) 3/ ȳ ȳ µ f = γ f + γ µ f ε + µ ε ε, (A.33) ȳ µ ȳ f µ τ ε µ τ f = γȳ 3/ +ȳ 3/ + γ s f, 3/ µ 3/ f s µ ε s f, µ ε s + γ 3/ + γ 3/ + 3/ (A.34) µ s f, s f, ȳ ȳ = ȳ + γ µ τ s f, +ȳ γ + µ s s +ȳ µ τ s + γ µ s f, s. (A.35) Smlarly, apart from ³s, s, and (υ υ ), the remanng terms n the denomnator of t (,) may also be wrtten n terms of the above expressons. [A.3]

34 A.. Fxed and In ths case Var( ε t )= σ,var( s t)= t σ, where σ = P j= σ j <. Hence, for a fxed those elements n t (,) thatnvolve ε and s wll converge to zero n mean square errors as. Assumng also that the seres, y t,arentheformof devatons from the cross-sectonal mean of the ntal observatons so that ȳ =, usng the above results for a fxed and as we have (n mean square errors) D W WD Γ Ψ Γ, D W υ Γ q, D W s, Γ h s, s, s, s, σ, (υ υ ) ε ε σ, where γ s the lmt of γ as,and υ s, Γ = γ, Ψ f = γ q = f ε σ τ ε σ s f, ε σ ε s, σ,, h = f f f τ τ f s f, f s f, τ 3/ 3/ f s, σ 3/ τ s, σ 3/ s f, s, σ f s f, 3/ τ s f, 3/ s f, s f,., Usng these results n (A.) we fnally obtan: t (,) µ ε ε σ ( 3) q Ψ 3 ε s, σ f q q Ψ / ³ s, s, f h σ h Ψ f h /, (A.36) whch s free of nusance parameters and ts probablty dstrbuton can be smulated for any gven value of >3. Recall that f t and ε t /σ are ndependently dstrbuted as d(, ). A.. Sequental Asymptotc : then Frst, usng famlar results from the unt root lterature as, we have (See, for example, Hamlton (994, p.486)) τ s, σ 3/ = Z W (r) dr, s, s, σ = Z W (r) dr, τ ε σ = W (), ε s, σ τ s f, 3/ = = Z Z W (r) dw (r) =(/) W (), W f (r) dr, s f, s f, = Z W f (r) dr [A.4]

35 where W (r) andw f (r) are ndependently dstrbuted standard Brownan motons defnedon[, ]. Smlarly, s f, s, σ = Z W f (r) W (r) dr, f ε σ = W f (), s f, ε σ = Z W f (r) dw (r), where W f (r) andw f (r) are also dstrbuted as standard Brownan motons. Fnally, t s easly seen that f f, τ f, f s, σ 3/, ε ε q Ψ f q, σ ( 3). 3 Usng these results n (A.36) as, we obtan the followng sequental lmt dstrbuton R t (,) =, W (r) dw (r) ψ f Λ f κ f ³, (A.37) R / W (r) dr κ f Λ f κ f where Ã Λ f = R W f (r) dr R! W f (r) dr R W f (r) dr, (A.38) µ ψ f = W () R W f (r) dw (r) à R, κ f = W (r) dr R W f (r) W (r) dr!. (A.39) A..3 Jont Asymptotcs Usng the order results (A.9) to )A.) t s easly seen that all the terms n (A.) to (A.8) that contan the cross-secton means, ε and s, converge n quadratc mean to zero as,, jontly so long as /. hs latter condton s satsfed f / k, wherewherek s a fxed fnte non-zero constant. It therefore follows that the asymptotc result, (A.37) also holds under jont asymptotc and so long as / wehave t (,) (,) j = R W (r) dw (r) ψ f Λ f κ f ³. (A.4) R / W (r) dr κ f Λ f κ f It s also easly seen that wth / ˆσ = y M,w y 4 (,) j σ, and hence we also have t (,) (,) j = R W (r) dw (r) ψ f Λ ³ R W (r) dr κ f Λ f f κ f κ f /, (A.4) where t (,) sdefned by (3.7). [A.5]

36 A.3 Asymptotc Dstrbuton of t (, ) - Serally Correlated Case Consder frst the numerator of (5.59) and note that υ M s z, = υ s z, Ã υ W ³ D D W W D W D s z,!, where µ D = / I 4, W = y,, ȳ, ȳ, τ, ȳ, s z, = (s z, δ s z, ) /ω. he elements of s z, and s z, are defned by (5.55) and (5.56), and can be wrtten n terms of general frst-dfference statonary processes. Recall also that ω = σ + O. Usng the results set out above, together wth the famlar results on statonary frst-dfference processes summarzed, for example, n Proposton 7.3 of Hamlton (994), the followng lmts can now be establshed under jont asymptotcs (wth and, such that / k, >k>) D W (,) j υ = D W s z, D W W D (,) j (,) j = γ ρ υ s z, q γ +σ ρ W () γ ρ W () γ W ρ () W () R W (r) dw (r) γ ρ µ V 3, 3 Γ ρ Λ f Γ ρ R W (r) dr = ( ρ) R W f (r) W (r) dr (,) j = ρ Z q γ +σ ρ W () γ ρ W () γ W ρ () Γ ρ ψ f µ = W (r) dw (r), 3 ρ Γ ρκ f where Λ f, ψ f,and κ f,aredefned by (A.38) and (A.39), γ 6= s the lmt of γ as, W (r) and W f (r) are ndependent standard Brownan motons, and V = γ + σ ργ γ γ γ ρ ργ γ γ ργ γ γ ργ γ µ, Γ ρ = γ ρ,. Smlarly, s z, M s z, = s z, s z, Ã! Ã s z, W D ³ D W W D D W! s z,, υ M υ 5 = υ υ 5 5 υ W ³ D D W W D D W υ, [A.6]

37 where t s also easly seen that s z, s z, (,) j = ( ρ) Z W f (r) dr, and υ M υ 5 (,) j v Usng the above results n (5.59) we now have ( ρ < ) υ M,w υ (,) j. 6 t (,) (,) j v t (,) (,) j = n R ρ W (r) dw (r) ψ f Γ ρ (Γ ρ Λ f Γ ρ ) ρ Γ ρκ f ( ρ) R W f (r) dr ρ κ f Γ ρ (Γ ρ Λ f Γ ρ ) ρ Γ ρκ f o /, whch reduces to the desred result: R W (r) dw (r) ψ f Λ ³ R W (r) dr κ f Λ f f κ f, / κ f the jont asymptotc lmt dstrbuton of the CADF obtaned n the case of serally uncorrelated errors gven by (A.4) or (A.4). [A.7]

38 References [] Ba, J. and S. g, (4), A PAIC Attack on Unt Roots and Contegraton, Econometrca, 7, [] Baltag, B. H., Kao, C., (), onstatonary Panels, Contegraton n Panels and Dynamc Panels: A Survey. Advances n Econometrcs 5, 7-5. [3] Bowman, D., (999), EffcentestsforAutoregressveUntRootsnPanelData. Unpublshed manuscrpt, Board of Governors of the Federal Reserve System, Washngton, D.C. [4] Bretung, J. and Das, S., (4), Panel Unt Root ests Under Cross Sectonal Dependence, manuscrpt, Insttute of Econometrcs, Unversty of Bonn. [5] Chang, Y., (), onlnear IV Unt Root ests n Panels wth Cross-Sectonal Dependence, Journal of Econometrcs, 6-9. [6] Chang, Y., (3), Bootstrap Unt Root est n Panels wth Cross-Sectonal Dependency Journal of Econometrcs, (forthcomng). [7] Cho, I., (), Unt Root ests for Panel Data, Journal of Internatonal Money and Bankng, [8] Cho, I. (), Combnaton Unt Root ests for Cross-Sectonally Correlated Panels, Mmeo, Hong Kong Unversty of Scence and echnology. [9] Ellott, G.,.J. Rothenberg, and J.H. Stock, (996), Effcent ests for an Autoregressve Unt Root, Econometrca, 64, 4, [] Fuller, W.A, (996), Introducton to Statstcal me Seres, Second edton, John Wley &Sons,ewYork. [] Fuller, W.A. and H.J. Park (995), Alternatve estmators and unt root tests for the autoregressve process, Journal of me Seres Analyss, 6, [] Gengenback, C., F.C. Palm, and J-P. Urban, (4), Panel Unt Root ests n the Presence of Cross Sectonal Dependence: Comparsons and Implcatons for Modellng, MEEOR Research Memorandum 439, Department of Quanttatve Economcs, Unverstet Maastrcht, he etherlands. [3] Hadr, K., (), estng for Statonarty n Heterogeneous Panel Data. Econometrcs Journal, 3, [4] Harrs, D., Leybourne, S. and McCabe, B., (4), Panel Statonarty ests for Purchasng Power Party wth Cross-Sectonal Dependenc, manuscrpt, Unversty of ottngham, August. [5] Hamlton, J.D., (994), me Seres Analyss. Prnceton Unversty Press, Prnceton. [R.]

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40 [3] Phllps, P.C.B. and P. Perron, (988), estng for a unt root n tme seres regresson, Bometrka, 75, [3] Phllps, P.C.B. and D. Sul, (3), Dynamc Panel Estmaton and Homogenety estng Under Cross Secton Dependence, Econometrcs Journal, 6, [33] Shn, Y. and A. Snell, (), Mean Group ests for Statonarty n Heterogeneous Panels, Manuscrpt, Department of Economcs, Unversty of Ednburgh. [34] Smth, L.V., S. Leybourne, -H Km, and P. ewbold, (4), More Powerful Panel Data Unt Root ests wth an Applcaton to Mean Reverson n Real Exchange Rates, Journal of Appled Econometrcs, 9, [35] aylor, R.L., Daffer, P.Z., and Patterson, R.F.,(985), Lmt heorems for Sums of Exchangeable Random Varables, Rowman and Allanheld,.J. [R.3]

41 Fgure : Cumulatve Dstrbuton Functon of DF and Cross-Sectonally Augmented DF Statstcs (he Intercept Case) Fgure : Smulated Densty Functon of the Standardzed CADF Dstrbuton as Compared to the ormal Densty (he Intercept Case)

42 Fgure 3: Smulated Densty Functon of the Standardzed CADF Dstrbuton as Compared to the ormal Densty (he Lnear rend Case) Fgure 4: Smulated Densty of the CIPS* Statstc - Case of no Intercept or rend

43 Fgure 5: Smulated Densty of the CIPS* Statstc - he Intercept Case Fgure 6: Smulated Densty of the CIPS* Statstc - he Lnear rend Case