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1 A Simle Panel Stationarity itle Cross-Sectional Deendence est in Author(s) Hadri, Kaddour; Kurozumi, Eiji Citation Issue Date ye echnical Reort ext Version ublisher URL htt://hdl.handle.net/0086/8605 Right Hitotsubashi University Reository

2 Global COE Hi-Stat Discussion Paer Series 06 Research Unit for Statistical and Emirical Analysis in Social Sciences (Hi-Stat) A Simle Panel Stationarity est in the Presence of Cross-Sectional Deendence Kaddour Hadri Eiji Kurozumi Hi-Stat Discussion Paer December 008 (Revised: June 00) Hi-Stat Institute of Economic Research Hitotsubashi University - aka, Kunitatchi okyo, Jaan htt://gcoe.ier.hit-u.ac.j

3 A Simle Panel Stationarity est in the Presence of Cross-Sectional Deendence Kaddour Hadri Eiji Kurozumi Queen s University Management School Deartment of Economics Queen s University Belfast Hitotsubashi University Revised December 009 Abstract his aer develos a simle test for the null hyothesis of stationarity in heterogeneous anel data with cross-sectional deendence in the form of a common factor in the disturbance. We do not estimate the common factor but mo-u its e ect by emloying the same method as the one roosed in Pesaran (007) in the unit root testing context. Our test is basically the same as the KPSS test but the regression is augmented by cross-sectional average of the observations. he limiting distribution under the null is shown to be a standard normal. he latter result is derived using the joint asymtotic limits where and! simultaneously (under the additional condition that =! 0). We also extend our test to the more realistic case where the shocks are serially correlated. We use Monte Carlo simulations to examine the nite samle roerty of the anel augmented KPSS test. JEL classi cation: C, C Key words: Panel data; stationarity; KPSS test; cross-sectional deendence; joint asymtotic. We would like to thank the editor, Pierre Perron, and three anonymous referees for their comments, suggestions and constructive criticisms which heled to greatly imrove the aer. Evidently, any remaining errors are ours. Corresonding author: Kaddour Hadri, 5 University Square, Queen s University Management School, Queen s University, Belfast, B7, UK. el +44 (0) Fax +44 (0) k.hadri@qub.ac.uk Kurozumi s research was artially suorted by the Ministry of Education, Culture, Sorts, Science and echnology under Grants-in-Aid o.8704.

4 . Introduction Since the beginning of the 90 s, the theoretical and emirical econometrics literature witnessed a formidable outut on testing unit root and stationarity in anel data with large (time dimension) and (cross-section dimension). he main motive for alying unit root and stationarity tests to anel data is to imrove the ower of the tests relative to their univariate counterarts. his was suorted by the ensuing alications and simulations. he early theoretical contributions are by Breitung and Meyer (994), Choi (00), Hadri (000), Hadri and Larsson (005), Im, Pesaran and Shin (00), Levin, Lin and Chu (00), Maddala and Wu (999), Phillis and Moon (999), Quah (994) and Shin and Snell (006). On the alication side, the early contributions were the work of O Connell (998), Oh (996), Paell (997, 00), Wu (996) and Wu and Wu (00), who focused on testing the existence of urchasing ower arity. Culver and Paell (997) alied anel unit root tests to the in ation rate for a subset of OECD countries. hey have also been emloyed in testing outut convergence and more recently in the analysis of business cycle synchronization, house rice convergence, regional migration and household income dynamics (cf. Breitung and Pesaran (008)). All these " rst generation" anel tests are based on the incredible assumtion that the cross-sectional units are indeendent or at least not cross-sectionally correlated. Banerjee (999), Baltagi and Kao (000), Baltagi (00) rovide comrehensive surveys on the rst generation anel tests. However, in most emirical alications this assumtion is erroneous. O Connell (998) was the rst to show via simulation that the anel tests are considerably distorted when the indeendence assumtion is violated, whether the null hyothesis is a unit root or stationarity. Banerjee, Marcellino and Osbat (00, 004) argued against the use of anel unit root tests due to this roblem. herefore, it became imerative that in alications using anel tests to account for the ossibility of crosssectional deendence. his led, recently, to a urry of aers accounting for cross-sectional deendence of di erent forms or second generation anel unit root tests. he most noticeable roosals in this area are by Chang (004), Phillis and Sul (00), Bai and g (004), Moon and Perron (004), Choi and Chue (007) and Pesaran (007) for unit root anel tests. For

5 anel stationarity tests, the only contributions so far are by Bai and g (005) and Harris, Leybourne and McCabe (005), both of which corrected for cross-sectional deendence by using the rincial comonent analysis roosed by Bai and g (004). Choi and Chue (007) utilize subsamling technique to tackle cross-sectional deendence. Phillis and Sul (00), Bai and g (004), Moon and Perron (004) and Pesaran (007) emloy factor models to allow for cross-sectional correlation (cf. de Silva, Hadri and remayne (009) for the comarison of the three last tests). Pesaran (007) considers only one factor and instead of estimating it, he augments the ADF regressions with the crosssectional averages of lagged levels and rst-di erences of the individual series to account for the cross-sectional deendence generated by this one factor. Other contributions are by Maddala and Wu (999) and Chang (004) who exloited the exibility of the bootstra method to deal with the ervasive roblem of cross-sectional deendence of general form. Breitung and Pesaran (008) give an excellent survey of the rst and second generation anel tests. he transfer of testing for unit root and stationarity from univariate time series to large anel data contributed to a signi cant increase of the ower of those tests. However, this transfer led to a number of di culties besides the roblem of cross-sectional deendence. In articular, the asymtotic theory is by far more intricate due to the resence of two indices: the time dimension and the number of cross-sections. he limit theory for this class of anel data has been develoed in a seminal aer by Phillis and Moon (999). In their aer they study inter alia the limit theory that allows for both sequential limits, wherein! followed by! ; and joint limits where ;! simultaneously. hey also mention, in the same aer, the diagonal ath limit theory in which the assage to in nity is done along a seci c diagonal ath. he drawback of sequential limits is that in certain cases, they can give asymtotic results which are misleading. he downside of diagonal ath limit theory is that the assumed exansion ath ( (); )! may not rovide an aroriate aroximation for a given (; ) situation. Finally, the joint limit theory requires, generally, a moment condition as well as a rate condition on the relative

6 seed of and going to in nity. he main contribution of this aer 4 is the derivation under the null and under the local alternative of the limiting distribution of the Kwiatkowski et al. (99) test (KPSS test hereafter) corrected for cross-sectional deendency generated by a one factor error structure. For each unit, our test is basically the KPSS test with the regression augmented by the crosssectional average of the observations. In the anel data context, this amounts to adating Pesaran (007) aroach accounting for cross-sectional deendence to the anel stationarity test of Hadri (000). he choice of the Pesaran (007) aroach is due essentially to its concetual simlicity. We show that the limiting null distribution of our anel augmented KPSS test is the same as the anel stationary test roosed by Hadri (000), which is an Lagrange multilier (LM) test without cross-sectional deendence. Our theoretical result are obtained via the joint asymtotic theory where ;! concurrently (under the added condition that =! 0) and the limiting distributions under the null are shown to be standard normal :We then extend our anel augmented KPSS test to the more realistic and useful case of the serially correlated shocks. he test is shown to have a standard normal distribution as a limiting null distribution emloying the sequential asymtotic theory where! followed by!. We conjecture that the sequential limit is the same as the joint limit under the additional condition that =! 0: he test is very easy to imlement. We use Monte Carlo simulations to examine the nite samle roerties of the anel augmented KPSS test allowing for serial correlation. he aer is organized as follows. Section sets u the model and assumtions, and de ne the augmented anel test statistic. In section, we show that the limiting null distribution of the anel augmented KPSS test is the same as that of Hadri s (000) test. We also examine whether our theoretical results are valid in nite samles via simle Monte Carlo simulations. In Section 4, we relax Assumtion in order to allow for serial correlation 4 In the original version of the aer, we derived a Lagrange multilier (LM) test, which is a locally best invariant test under the assumtion of normality, allowing for cross-sectional deendence. We also comared a anel augmented KPSS test with the extended LM test under the null of stationarity, under the local alternative and under the xed alternative and discussed the di erences between the two tests. he latter results were derived using the joint asymtotic limits where and! jointly. 4

7 in the error term and roose a modi cation of the anel augmented KPSS test statistic to correct for the resence of this serial correlation. Once again, we examine the nite samle roerties of the roosed test statistic via Monte Carlo simulations. Section 5 concludes the aer. All the roofs are resented in the Aendix. A summary word on notation. We de ne M A = I A(A 0 A) A 0 for a full column rank matrix A. he symbols (; )! and (; ) =) mean joint convergence in robability and joint weak convergence resectively when both and go to in nity simultaneously, while or =) signi es weak convergence when only or goes to in nity. =). Model and est Statistics.. Model and assumtions Let us consider the following model: y it = zt 0 i + r it + u it ; r it = r it + v it ; u it = f t i + " it () for i = ; ; and t = ; ; where z t is deterministic and r i0 = 0 for all i. he commonly used seci cation of z t in the literature is either z t = z t = or z t = zt = [; t] 0 : In this aer, we consider these two cases. Accordingly, we de ne i = i when z = and i = [ i ; i ] 0 when z = [; t] 0. In model (), zt 0 i is the individual e ect while f t is one dimensional unobserved common factor, i is the loading factor and " it is the individualseci c (idiosyncratic) error. By stacking y it with resect to t, model () can be exressed as y i z 0 r i f y i 7 6 z r i 7 6 f y i 7 5 = z 0 r i r i. r i 7 5 i r i f = i v i v i. v i 7 5 " i " i. " i 7 5 ; 5

8 or y i = Z i + r i + f i + " i () = Z i + Bv i + f i + " i ; where Z = [ ; d] with = [; ; ; ] 0, d = [; ; ; ] 0 being vectors and B is a matrix with ones on the main diagonal and everywhere below it. anaka (996) calls matrix B, the random walk generating matrix. Further, we have y Z B v f y = Z B v f y Z B v f or y = (I Z) + r + ( f) + " () = (I Z) + (I B)v + ( f) + ": " ". " 7 5 In this section, we assume the following simle assumtion: Assumtion (i) he stochastic rocesses f" it g, ff t g and fv it g are indeendent, " it i:i:d:(0; "); f t i:i:d:(0; f ); v it i:i:d:(0; v); for simlicity, " is assumed to be known. (ii) here exist real numbers M, M and M such that j i j < M < for all i and 0 < M < jj < M < for all, where = P i= i. Assumtion (i) is too restrictive and not ractical. It will be relaxed in Section 4 to a more realistic one. he assumtion of normality with homoskedasticity is adoted here to simlify the derivation of the theoretical results. he cross-sectional indeendence of " it across i is standard in one-factor models. he indeendence over t will be relaxed in Section 4 to allow for serial correlation. In section 4 the common factor f t will be allowed 6

9 to follow a general stationary rocess. he indeendence of v it across i and t is standard in KPSS models. Instead of considering " known, we can also estimate it emloying the estimator b " = P i= P t= b" it=; where b" it is the residual from the augmented regression. It can be shown that this estimator jointly converges in robability to " under both the null hyothesis and the local alternative. hen, we could relace " in the de nition of S i by its estimator b ": Assumtion (ii) is concerned with the weights of the common factor f t. his assumtion imlies that each individual is ossibly a ected by the common factor with the nite weight i and that the absolute value of the average of i is bounded away from 0 and above both in nite samles and in asymtotics. he latter roerty is imortant in order to eliminate the common factor e ect from the regression. A similar assumtion is also entertained in Pesaran (007). he main urose of this section is to examine the theoretical e ect of augmentation, which is exlained below, on stationarity tests. We consider a test for the null hyothesis of (trend) stationarity against the alternative of a unit root for model (). Since all the innovations are homoskedastic, the testing roblem is given by H 0 : v = 0 v.s. H : > 0 (4) " where = v= " is a signal-to-noise ratio. Under H 0, r it becomes equal to zero for all i so that y it is stationary whereas all of the cross-sectional units have a unit root under the alternative. Remark Allowing for only one factor could be considered as too restrictive. Bai and g (005) examined a more general case than here in which they allowed for more than one factor. However, they were disaointed by their results. In the case of more than one factor, it is not su cient to augment the regression by y t in order to eliminate the e ect of the cross-sectional deendence from the test statistic. In such a case, the limiting distribution of our test statistic would deend on nuisance arameters. o correct for multifactor error structure, we may be able to adat the method roosed by Pesaran, Smith and Yamagata (009) in the unit root context to our case. Pesaran et al. (009) method can be considered as a generalisation of Pesaran (007) emloyed here. But, the alication of the new roosal 7

10 to our test is not straightforward and is therefore left for future research. he downside of allowing for multifactor error structure is that the number of factors has to be estimated emloying information criteria. Simulations carried out by de Silva et al. (009) to comare anel unit root tests allowing for multifactor error structure, show that the number of factors estimated is often greater than the true number of factors in small samles and this may be resonsible for the uward size distorsions observed in their simulations... A simle stationarity test A anel stationarity test has already been roosed by Hadri (000) and Shin and Snell (006) for cross-sectionally indeendent data. Here, we extend Hadri s test to the cross-sectionally deendent case. Hadri (000) showed that if there is no cross-sectional deendence in a model, we can construct the LM test using the regression residuals of y it on z t in the same way as KPSS (99) and that the limiting distribution of the standardized LM test statistic is standard normal under the null hyothesis. However, it can be shown that Hadri s (000) test deends on nuisance arameters even asymtotically if there exits crosssectional deendence; we then need to develo a stationarity test that takes into account cross-sectional deendence. In order to eliminate the e ect of the common factor from the test statistic, we make use of the simle method roosed by Pesaran (007), which develos anel unit root tests with cross-sectional deendence. As in Pesaran (007), we rst take a cross-sectional average of the model: where y t y t = z 0 t + r t + f t + " t ; (5) = P i= y it, = P i= i, r t = P i= r it, = P i= i and " t = P i= " it. Since 6= 0 by assumtion, we can solve equation (5) with resect to f t as follows: f t = y t z 0 t r t " t : By inserting this solution of f t into model () we obtain the following augmented regression 8

11 model: y it = z 0 t ~ i + ~ i y t + it ; (6) where ~ i = i ~ i, ~i = i = and it = r it ~ i r t + " it ~ i " t. Based on (6) we roose to regress y it on z t and y t for each i and construct the test statistic in the same way as Hadri (000). hat is, (S ) Z A = (7) where S = X i= S i with S i = " y0 im w B 0 BM w y i 8 < = = 6 ; = = 45 when z t = z t = ; and : = = 5 ; = = 600 when z t = zt = [; t] 0 : ote that S i can also be exressed as S i = " X (Sit w ) where Sit w = t= with ^ it obtained for each i by regressing y it on w t = [z 0 t; y t ] 0 for t = ; ;. From (7) we can see that S is the average of the KPSS test statistic across i and Z A is normalized so that we obtain a limiting distribution. We call Z A the anel augmented KPSS test statistic. tx s= ^ is. heoretical Proerty of the anel Augmented KPSS test In this section we derive the joint limit distribution, where ;! simultaneously, of the anel augmented KPSS test under the null hyothesis and under the local alternative. heorem Assume that Assumtion holds. Under H 0, as and go to in nity simultaneously with =! 0, the anel augmented KPSS test statistic has a limiting standard normal distribution for both cases of z t = and z t = [; t] 0, Z A (; ) =) (0; ): 9

12 Proof. see Aendix. his shows that the sequential asymtotic is equivalent to the joint asymtotic only under the condition that =! 0 as and go to in nity jointly. he use of the sequential asymtotic on its own will not uncover this condition. ote that the rejection region of Z A is the right hand tail as in Hadri s (000) test. he condition that =! 0 as and! jointly, means that the test is suitable for anels with larger than : heorem shows that Pesaran s (007) method works well in order to eliminate crosssectional deendence for testing the null hyothesis of stationarity. We next investigate the asymtotic roerty of the test statistics under the local alternative, which is exressed as H` : = c where c is some constant: ote that for a single time series analysis, the local alternative is given by = c =. Since the sum of S i is normalized by as in Z A, the local alternative for anel stationarity tests becomes = c =( ). heorem Assume that Assumtion holds. Under H`, as and go to in nity simultaneously with =! 0, the anel augmented KPSS test statistics has a limiting distribution given by Z (; ) Z A =) (0; ) + c E Fi v (r) dr 0 where Fi v(r) = R r 0 Bv i (s)ds R r R R 0 z(s)0 ds 0 z(s)z(s)0 ds 0 z(s)bv i (s)ds and Bv i (r) are indeendent Brownian motions across i, z(r) = and E[ R 0 F i v(r) dr]= = 45=90 when z t = and z(r) = [; r] 0 and E[ R 0 F v i (r) dr]= = 600=(=600) when z t = [; t] 0. Proof see Aendix. heorem shows that the test is more owerful when only a constant is included in the regression than the trending case as is the univariate KPSS test, because 45=90 > 600=(=600). 0

13 We investigated via Monte Carlo simulations how accurately does the asymtotic theory aroximate the nite samle behavior of the anel augmented KPSS. As a whole, we found that the nite samle behaviour of the anel augmented KPSS test is well aroximated by the asymtotic theory established in this section when and are of moderate size. Due to sace constraint the results are not reorted here but can be requested from the authors. 4. Extension to the case of serially correlated errors 4.. Modi cation of the anel augmented KPSS test So far, we have investigated the theoretical roerty of the anel augmented KPSS test under restrictive assumtions. In this section we relax Assumtions (i) and consider a more ractical and useful situation in which we allow for serial correlation. Since it is often the case that the observed rocess can be aroximated by an autoregressive (AR) model, we do not consider the error comonent model () but an AR() model instead in this section 5 : y it = z 0 t i + f t i + " it ; " it = i " it + + i " it + it : (8) he lag length may change deending on the cross-sectional units but we suress the deendence of on i for notational convenience. Assumtion (i) he stochastic rocess f t is stationary with a nite fourth moment and the functional central limit theorem (FCL) holds for the artial sum rocess of f t. (ii) he stochastic rocess it is indeendent of f t and i:i:d:(0; i ) across i and t with a nite fourth moments. his assumtion allows the common factor to be stationary but still resumes that it is indeendent of the idiosyncratic errors, which are nite order AR rocesses with i.i.d. innovations. We assume Assumtions (ii) and in the rest of this section. 5 We do not consider a general linear rocess instead of an AR() model because in the case of a general linear rocess the long-run variance estimator based on oda and Yamamoto (995), used here, will diverge to in nity at a rate under the alternative when the rocess is AR(): As a result, our test based on the lag-augmented method becomes inconsistent.

14 Since our interest is whether y it are (trend) stationary or unit root rocesses, the testing roblem is given by H 0 0 : i () 6= 0 8i v.s. H 0 : i () = 0 for some i; where i (L) = i L i L. In this case we need to modify the original KPSS test statistic for serial correlation as well as cross-sectional deendence. For the correction of cross-sectional deendence, we regress y it on w t = [z 0 t; y t ; y t ; ; y t ] because " it are AR() rocesses and construct S w it using these regression residuals. In this case it is not di cult to see that the numerator of each S i weakly converges to X (Sit w ) =) i t= Z where i = i =( i i ) and V " i (r) = B" i (r) 0 V " i (r) + O dr R r R R 0 z(t)0 dt 0 z(t)z(t)0 dt 0 z(t)db" i (t) with B i " (t) are indeendent standard Brownian motions. his suggests that we should divide the numerator of each S i by a consistent estimator of the long-run variance i order to correct for serial correlation. Several consistent estimators of the long-run variance 6 for arametric model have been roosed in the literature for univariate time series. For examle, Leybourne and McCabe (994) roose to correct the stationarity test for serial correlation by estimating the AR coe cients based on the ML method for the ARIMA model. heir method is also alied to anel data with no cross-sectional deendence by Shin and Snell (006). However, our reliminary simulation shows that this method does not work well in nite samles and we do not use this method in this aer. We next consider to make use of the new truncation rule roosed by Sul, Phillis and Choi (005). heir method is originally develoed for the rewhitening method, but it is 6 We cannot use here the estimator of the long-run variance roosed in Perron and g (998), desite its good roerties in nite samles, because it is consistent under the null of a unit root but not under the null of stationarity which we are considering in this aer. in

15 also alicable to arametric model. We rst estimate the AR() model augmented by the lags of y t for each i by the least squares method y it = z 0 t^ i + ^ i y it + + ^ i y it + ^ i0y t + + ^ iy t + ^ it ; and construct the estimator of the long-run variance by 8 9 ^ ^ < i isp C = where ^i = min ( ^i ) : X = ; ^ ij : ; and ^ i = j= X t= ^ it: We then roose to construct the test statistic (7) using S SP C i = ^ isp C X (Sit w ) : t= We denote this test statistic as Z SP C A. he other method we consider is the lag-augmented method roosed by Choi (99) and oda and Yamamoto (995). According to these aers, we intentionally add an additional lag of y t and estimate an AR( + ) model instead of an AR() model: y it = z 0 t ~ i + ~ i y it + + ~ i y it + ~ i+ y it + ~ i0y t + + ~ iy t + ~ it ; and construct the test statistic using S LA i = ^ ila X (Sit w ) where ^ ila = t= ^ i ( ~ i ~ i ) : We denote this test statistic as Z LA A. he consistency of ^ isp C and ^ ila under the null hyothesis is established in the standard way and we omit here the details. On the other hand, they are shown to diverge to in nity at a rate of under the alternative, so that S i can be seen as a consistent stationarity test for univariate time series. It is also shown by using the sequential limit that the null distributions of Z SP C A and Z LA A are asymtotically standard normal in the same way as heorem while they diverge to in nity under the alternative. Unfortunately, it is tedious to derive the joint limit of Z SP C A or Z LA A under general assumtions and we do not ursue

16 it. Instead, we shall conduct Monte Carlo simulations in the next section in order to see whether or not the sequential limit theory can aroximate the nite samle behaviour of these tests. However, we conjecture that the sequential limit is the same as the joint limit under the additional condition that =! 0: 4.. Finite samle roerty under general assumtions In this section we conduct Monte Carlo simulations to investigate the nite samle roerties of the anel augmented KPSS test using the long-run variance estimated by the SPC or the LA methods in order to correct for serial correlation in the innovations. he data generating rocess in this subsection is given as follows: y it = zt 0 i + f t i + " it ; " it = i " it + it ; where f t i:i:d:(0; ), it i:i:d:(0; ), f t and it are indeendent of each other, i = i for the constant case while i = [ i ; i ] 0 for the trend case with i and i being drawn from indeendent U(0; 0:0), i are drawn from +U(0; 4) for strong cross-sectional correlation case (SCC) and from U(0; 0:0) for weak cross-sectional correlation case (WCC), and i, i and i are xed throughout the iterations. he i are drawn from 0: + U(0; 0:8) under the null hyothesis and they remain xed throughout the iterations. On the other hand, the i are set to be equal to for all i under the alternative. For the urose of comarison, we also calculate the test statistic roosed by Harris, Leybourne and McCabe (005) (HLM hereafter). According to HLM, we rst estimate the idiosyncratic errors " it by the rincial comonent method roosed by Bai and g (004) and next aly the stationarity test roosed by Harris, McCabe and Leybourne (00) to the estimated series of " t ; " t ; ; " t. HLM method requires to redetermine the order of the autocovariance and the bandwidth arameter for the kernel estimate of the long-run variance; we set these arameters as recommended in HLM (005). able reorts the sizes of the tests. here are no entries for HLM test when = 0 because the time dimension is too short to calculate their test statistic. When only a 4

17 constant is included in the model, the anel augmented KPSS test corrected by the SPC method tends to be undersized for moderate size of for SCC (strong cross-correlation) case while it is oversized for small or large size of, although the over-rejection is not so severe when = 00 and = 00. For WCC (weak cross-correlation) case Z SP C A is undersized 7 excet for the case of = 0. he augmented KPSS test corrected by the LA method has a similar roerty as Z SP C A for SCC case while the size of the test is relatively well controlled for WCC case. On the other hand, the size of HLM test seems to be better controlled for moderate or large size of, although the test becomes undersized for large size of and small or moderate size of. When both a constant and a linear trend are included in the model, the overall roerty of Z SP C A and Z LA A is reserved while HLM test tends to be undersized for larger than 0. able shows the nominal owers of the tests. Because of the size distortion of the tests it is not easy to comare the owers of these tests but we observe that all the tests are less owerful for the moderate size of due to the undersize roerty of the tests. In some cases the anel augmented KPSS test aarently dominates HLM test but the reversed relation is observed in other cases. For examle, the emirical sizes of ZA SP C, ZA LA and HLM test are 0.009, 0.0 and when = 0 and = 0 for the constant case with SCC, while the owers of these tests are 0.47, 0.6 and 0.8. On the other hand, the sizes of these tests are 0.058, and when = 0 and = 00 for the constant case with WCC while the owers are 0.878, 0.8 and.00. Although our simulations are limited, it is di cult to recommend one of these tests because none of them dominates the others. It seems that HLM test tends to work relatively 7 It seems that the long-run variance is well estimated by the method roosed by Sul et al. (005). But it is well known that the numerator of the KPSS statistic has a downward bias (cf inter alia Shin and Snell (006) and Kurozumi and anaka (009)). As a result, each test statistic S i is downward biased. hese downward biases accumulate as increases leading to the undersize of the tests based on ZA SP C : Another roblem raised by one of the referees is that the centering and scaling constants are derived asymtotically,! : When is nite these constants may be inaroriate. We did some additional simulations emloying the bias corrections roosed by Kurozumi and anaka (009) and the centering and scaling constants for xed suggested by Hadri and Larsson (005). We found that the results after corrections are very similar to those before corrections excet for the case where = 0. In the latter case, the nite samle corrections are e ective in reducing the severe over-size distorsions. 5

18 well in the constant case because the size of the test is more or less controlled in many cases and it has moderate ower, whereas the anel augmented KPSS test with SPC correction seems to erform best in many cases corresonding to the trend case (all the other tests tend to be undersized in this case) and is most owerful in many cases. 5. Conclusion In this aer we extended Hadri s (000) test to correct for cross-sectional deendence à la Pesaran (007). We showed that the limiting null distribution of this anel augmented KPSS test is the same as the original Hadri s test that is the LM test without cross-sectional deendence. All the theoretical results are derived via the joint asymtotic theory and the limiting distributions under the null are shown to be standard normal. We also roose a more ractical anel augmented KPSS under more realistic assumtions and allowing for serial correlation in the error disturbances. he Monte Carlo simulations indicated that we should use the anel stationarity tests with care because they are undersized in some cases but su er from over rejection in other cases. 6

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22 [] Moon, R. and B. Perron (004). esting for unit roots in anels with dynamic factors. Journal of Econometrics, 8-6. [] O Connell, P. G. J. (998). he overvaluation of urchasing ower arity. Journal of International Economics 44, -9. [] Oh, K. Y. (996). Purchasing ower arity and unit root tests using anel data. Journal of International Money and Finance 5, [4] Paell, D. H. (997). Searching for stationarity: Purchasing ower arity under the current oat. Journal of International Economics 4, -. [5] Paell, D. H. (00). he great areciation, the great dereciation and the urchasing ower arity hyothesis. Journal of International Economics 57, 5-8. [6] Perron, P. and S. g (998). An autoregressive sectral density estimator at frequency zero for non-stationary tests. Econometric heory, Vol. 4(05), [7] Pesaran, M. H. (007). A simle anel unit root test in the resence of cross-section deendence. Journal of Alied Econometrics, 65-. [8] Pesaran, M. H., L. V. Smith and. Yamagata (009). A anel unit root test in the resence of a multifactor error structure. working aer, University of Cambridge. [9] Phillis, P. C. B. and H. R. Moon (999). Linear regression limit theory for nonstationary anel data. Econometrica 67, [40] Phillis, P. C, B. and D. Sul (00). Dynamic anel estimation and homogeneity testing under cross section deendence. Econometrics Journal 6, 7-5. [4] Quah, D. (994). Exloiting cross section variation for unit root inference in dynamic data. Economics Letters 44, 9-9. [4] Shin, Y. and A. Snell (006). Mean grou tests for stationarity in heterogeneous anels. Econometrics Journal 9,

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24 Aendix In this aendix, we denote some constant that is indeendent of, and the subscrits i and t as C, C, C,. We rove the theorems only for the case where z t = [; t]. he roof for the level case with z t = roceeds in exactly the same way and thus we omit it. We also assume " = in this aendix without loss of generality because we know " under Assumtion (i). We rst exress y t in matrix form. Since y t = zt 0 + r t + f t + " t ; we have or 6 4 y y. y 7 5 = 6 4 z 0 z 0. z r r. r f f. f " ". " y = Z + r + f + ": (9) 7 5 Since 6= 0, we have f = (y Z r ")=. By inserting this into (), the model becomes where ~ i = i =. y i = Z( i ~ i ) + ~i y + (r i ~ i r) + (" i ~ i ") (0) Let W = [ ; d; y] = [Z; y] and W = W Q = [Z; y ] where y = y Z = r + f + ", I D 0 Q = ; D = 0 and D 0 0 = 0 : Because M w = M w, S i in the augmented KPSS test statistic can be exressed in matrix form as S i = y0 im w B 0 BMy i : Before roceeding with the roof of theorems, we state two lemmas, which will be used in the roof reeatedly. he rst lemma states various moments related to r it. Since these can be obtained by direct calculation, we omit the roof.

25 Lemma Let v it i:i:d:(0; v) for i = ; ; and t = ; ;, r it = P t s= v is and r t = P i= r it. hen, E 4 E 4 E 4 E 4 E [r is r it ] = v min(s; t) ()! tx r is 5 = v t(t + )(t + ) () 6 s=! tx sr is 5 = v 0 t(t + )(t + )(t + t + ) () s= E [r s r t ] = v min(s; t) (4)! tx r s 5 = v t(t + )(t + ) (5) 6 s=! tx sr s 5 = v 0 t(t + )(t + )(t + t + ); (6) s= " E " E " E! tx r is s=! tx r is s=! X r it t=!# X r it t=!# X tr it t=!# X tr it t= = v t(t + )( t + ) (7) 6 = v 4 t(t + )(6 + 6 t t + ) (8) = v 4 ( + )( ); (9) E [r is r it r iu r iv ] = 4 v(st + su) for s t u v: (0) he next lemma gives the su cient condition on the equivalence of the sequential limit to the joint limit. otice that when the statistic S i weakly converges to S i as!, we can construct the robability sace on which both S i and S i exist, as discussed in Phillis and Moon (999). Lemma Let S i and S i are i.i.d. sequences across i (i = ; ; ) on the same robability sace. Assume that S i does not deend on, S i is indeendent of S j for i 6= j

26 and S i =) S i as!. If (a) E[S i ]! E[S i ] < as both and go to in nity, and (b) su ; E[Si ] <, then, X i= S i (; )! : Proof of Lemma : Since S i is an i.i.d. sequence, we have, for any arbitrary " > 0,! X P S i E[S i ] > " " E[(S i E[S i ]) ] " su E[Si ]! 0 i= by condition (b) as both and go to in nity. Because E[S i ]! by condition (a), we can see that P i= S i Proof of heorem (; )!. ; Because r i and r disaear under the null hyothesis, S i can be exressed in matrix form under H 0 as S i = y0 im w B 0 BM w y i = (" i ~ i ") 0 M w B 0 BM w (" i ~ i ") = "0 im w B 0 BM w " i = S i ~ i S i + ~ i S i ; say: ~ i "0 M w B 0 BM w " i + ~ i "0 M w B 0 BM w " Let S 0 i = " 0 i M zb 0 BM z " i. Since Shin and Snell (006) showed that it is su cient to rove that P i= (S 0 i ) X i= (S i S 0 i) X i= X i= (; ) =) (0; ); (; )! 0; () S i (; )! 0; () S i (; )! 0: () 4

27 Let J 0i = B" i, [J ; J ] = BW D = [ BZD ; = By ], [L 0 i ; L0 i ] = D W 0 " i = [(D Z 0 " i ) 0 ; ( = y 0 " i ) 0 ] 0, K = [[K ij ]] = D W 0 W D and K = [[K ij ]] for i; j = ;. hen, we have BM w " i = B" i BW D D W 0 W D D W 0 " i = (J 0i J K L i ) fj K L i + (J K + J K )L i g: (4) Similarly, by letting J 0 = P i= J 0i, L = P i= L i and L = P i= L i, we can see that BM w " i = J 0 (J K + J K ) L (J K + J K ) L : (5) We rst rove (). Using exression (4), S i can be decomosed into S i = (J 0i J K L i ) 0 (J 0i J K L i ) (J 0i J K L i ) 0 fj K L i + (J K + J K )L i g +fj K L i + (J K + J K )L i g 0 fj K L i + (J K + J K )L i g = S a i + S b i + S c i; say: (6) In order to evaluate each term, we use the following lemma. Lemma Under the null hyothesis, as both and go to in nity simultaneously, (i) EkJ 0i k C, P i= kj 0ik = O ( ) and k J 0 k = O ( ), (ii) kj k = O(), (iii) EkJ k C and kj k = O ( ), (iv) K = O(), K = K 0 = O ( ), K = O (), K = K + O ( ), K = K 0 = O ( ) and K = O (), (v) EkL i k C, P i= kl ik = O ( ) and k L k = O ( ), (vi) EkL i k C, P i= kl ik = O ( ) and L = O ( ), (vii) P i= kj 0ikkL`i k = O ( ) for `; m = ; and P i= kl ikkl i k = O ( ). Proof of Lemma : (i) Since " it is i.i.d.(0; ), we have! EkJ 0i k = E 4 X tx " is 5 = t= s= X t C: t= 5

28 Since C does not deend on i and J 0i is indeendent of J 0j for i 6= j, we can see that E[ P i= kj 0ik ] C and E[k J 0 k ] C=, which imly (i). (ii) It is easy to see that " JJ 0 P = t= t P 4 t= t P t s= s P 4 t= t P t s= s P 5 t= (P t s= s) # = O : (iii) Since y t = f t + " t under H 0, f t and " t are indeendent and " t is i:i:d:(0; =), ( EkJ k = X tx ) E 4 (f s + " s ) 5 and thus we obtain (iii). = = t= s= 8 X < : E 4 t= X t= tx s= f t + t! f s 5 + E C 4! 9 tx = " s 5 ; (iv) he result of K is obvious, while K and K are exressed as " K = P D Z 0 y = t= (f t + " t ) P t= t(f t + " t ) # and s= K = X (f t + " t ) : We can see that each comonent of K has mean zero and variance bounded above by C= while the exectation of K is bounded above by C. On the other hand, the orders of K ij for i; j = ; are obtained by using the inversion formula of a artitioned matrix. For examle, K = K + K K (K K K K ) K K = K + O he orders of K = K 0 and K are obtained similarly. he rst and second assertions of (v) and (vi) are obtained by noting that EkL i k = E 4!! X " it 5 + E 4 X t" it 5 C; t= t= t= : (7) 6

29 and ( EkL i k = E 4 CE 4 X t= ) (f t + " t ) " it 5 X t= f t " it! + X t=! " t " it 5 C: he third assertion of (v) is established from! EkL k = X E 4 " t 5 + E 4 t= X t=! t" t 5 C : For L, note that L = X t= f t " t + he rst term is easily shown to be O (= ) because f t is indeendent of " t, whereas the exectation of the second term equals =. Since = dominates =, which follows from condition =! 0, we have k L k = O ( =). (vii) Since EkJ 0i kkl`i k (EkJ 0i k EkL`i k ) = and EkL i kkl i k (EkL i k EkL i k ) = by Cauchy-Schwarz inequality, we obtain the result by using (i), (v) and (vi). Since S 0 i = (J 0i J K L i) 0 (J 0i J K L i), we have S a i S 0 i = fj (K K )L ig 0 fj (K K )L ig Using Lemma (ii), (iv) and (v), we can see that X i= X t= " t : (J 0i J K L i) 0 fj (K K )L ig: (8) kfj (K K )L ig0fj (K K )L igk kj k kk K k X i= kl i k = O! : Similarly, the second term on the right hand side of (8) becomes O ( = ). Hence, the right hand side of (8) converges to 0 in robability when both and go to in nity and =! 0. 7

30 In exactly the same manner, we have X i= ks b ik C X i= (kj 0i k + kj kkk kkl i k) (kj kkk kkl i k + kj kkk kkl i k + kj kkk kkl i k)! = O ; X ksik c i= X i= = O fkj kkk kkl i k + (kj kkk k + kj kkk k)kl i kg! : herefore, we obtained (). In order to rove () and (), notice that X S i C "0 M w B 0 BM w i= C "0 M w B 0 BM w "; X S i C "0 M w B 0 BM w ": i= X i= " i hen, it is su cient to show that "0 M w B 0 BM w " (; )! 0; which can be roved by noting that BM w " = O from exression (5) and Lemma. We thus obtain the result for the augmented KPSS test statistic. Proof of heorem 8

31 We rst show that Lemma still holds under the local alternative H`. Since y t = r t + f t +" t under H`, it is su cient to show that Lemma (iii), (iv), (vi) and (vii) hold because only J, K, K and L i are related to y t. Using Lemma with v = " = c =( ), (iii) is rove by noting that! X EkJ k = E 4 tx y s 5 t= s=! ( X E 4 tx r s + t= X C t= t 5 s= o show (iv), note that " P # K = t= (r t + f t + " t ) P t= t(r t + f t + " t ) + C C : and K = ) tx (f s + " s ) 5 s= X (r t + f t + " t ) : As shown in the roof of Lemma (iv) the variances of the second and third terms in each row of K are bounded by C=, whereas E[( P t= r t) ] C=( ) and E[( P t= tr t) ] C=( ) from (5) and (6). his shows that K = O (= ). Similarly, we can see that K = O () using Lemma and noting that E[ P t= r t ] C=( ) from (4). (vi) is obtained by noting that L i = X t= r t " it + X t= t= (f t + " t )" it ; and E[( = P t= r t" it ) ] C=( ),which is roved using (4) and the indeendence between r t and " it. ext, we give a lemma similar to Lemma. Let J r 0i = Br i, [L r0 i ; Lr0 i ]0 = D W 0 r i = [(D Z 0 r i ) 0 ; ( = y 0 r i ) 0 ] 0, J 0 r = P i= J 0i r, L r = P i= Lr i and L r = P i= Lr i. Lemma 4 Under the local alternative H`, as both and go to in nity simultaneously, (i) EkJ r 0i k C P, i= kj 0i r k = O () and kj 0 rk = O ( ), (ii) EkL r =4 i k C, 9

32 P i= klr i k = O () and kl k = O ( ), (iii) EkL r =4 i k C P, i= klr i k = O ( ) and k L k = O ( =4 ), (iv) P i= kj 0i r kklr i k = O P (), i= kj 0i r kklr i k = O ( ) and P i= klr i kklr i k = O ( ). Proof of Lemma 4: (i) Since kj r 0i k = P t= ( P t s= r is), we have EkJ r 0i k C= using (). We can also see that Ek J r 0 k = X E 4 t= tx s=! r s 5 C using (5) and v = c =( ). his imlies k J r 0 k = O (= =4 ). (ii) is obtained using (), (), (5) and (6) by noting that EkL r ik = E 4!! X X r it + tr it 5 t= t= EkL r k = E 4!! X X r t + tr t 5 : t= (iii) From the de nition of L r i, we can see that 8 < kl r ik C : + t= t= t= 0 X rit! X X r it r jt A t= t= j6=i X!! 9 X = f t r it + " t r it ; : (9) Since E[ris r it ] 4 v from (0) and 4 v = c =( 4 ), it is observed that! E 4 X rit 5 X X E[r isrit] t= 4 v s= t= X s= t= X C : (0) Similarly, since E[r is r it r js r kt ] = 0 for j; k 6= i and j 6= k and E[r is r it ] vs from () we 0

33 have 0 E 4@ E 4 E 4 X X r it r jt A 5 = t= j6=i X t= X t=! f t r it 5 =! " t r it 5 = 4 vc X X X E[r is r it ]E[r js r jt ] s= t= j6=i X s= t= X s X E ft E r it t= X t= E " t E r it From (9) () the rst and second assertions of (iii) hold. Regarding to L r, we can see using (4) that L r = = O X t= r t + X X f t r t + " t r t t= t= + O + O =4 5=4 (iv) is roved in exactly the same manner as Lemma (vii). C : () C : () C : () = O =4 Using Lemmas and 4, we can distinguish between the main term that weakly converges and the negligible term in the test statistic. Under the local alternative, : BM w y i = BM w " i ~ i BM w " + BM w r i i BM w r: Using Lemmas and 4, it is shown that BM w " = O ; BM w r = O =4 ; and thus, since j~ i j C, we can see that X ~ i " 0 im w B 0 BLM w r C BM w " BM w r = O i= X ~ i r 0 im w B 0 BM w " C BM w " BM w r = O i= =4 =4 ; :

34 herefore, the cross roducts between the terms related with " i and r, r i and ", and " and r converge to zero in robability as both and go to in nity. In addition, using exression (4) and Lemmas and 4, it is observed that BM w " i (J 0i J K L i) kj kkk K kkl ik + kj K L i + (J K + J K )L i k = O ; BM w r i (J0i r J K Lr i) kj kkk K kklr ik + kj K L r i + (J K + J K )L r ik = O =4 which imly we have only to consider (J 0i J K L i) and (J0i r J K Lr i ) in the limit. Moreover, the cross roduct between these two terms is shown to be negligible by noting that E 4 E 4 E 4 E 4 X i= X i= X i= X J0iJ 0 0i r i= J 0 0iJ K Lr i J r0 0iJ K L i L 0 ik J 0 J K Lr i!!!! 5 C 5 C 5 C 5 C X EkJ 0i k EkJ0ik r i= X EkJ 0i k EkL r ik i= X EkJ0ik r EkL i k i= X EkL i k EkL r ik i= C ; C ; C ; C ; where we used the fact that the deterministic term kj K k is bounded above by a constant and " it is indeendent of r it. Using these results, the augmented KPSS test statistic becomes ; Z A = + X n i= X i= J 0i J K L 0 i J 0i J K L i o J0i r J K 0 Lr i J 0i J K i Lr + o ():

35 he rst term weakly converges to a standard normal distribution as roved in heorem, whereas the second term can be exressed as X i= = J r 0i J K Lr i 0 J r 0i J K Lr i X ( X i= t= where we used the fact that tx s= r is t(4 t ) X r it + t= t(4 t ) the t-th row of J K = ; 6t( t ) 4 6t( t ) : ) X tr it ; he joint robability limit of (4) is obtained by Lemma. Using Lemma it can be shown that h E J r 0i J K 0 Lr i J0i r J K i i Lr = v = 600 c + O t= O() (4) ; (5) while the second moment is bounded above uniformly over and using (0). For examle, since t(4 t )= 4 for all t and, (! X E 4 tx r is t= C 4 X C 4 v 4 tx s= X X ux t= s= t 0 = u= v= u 0 = X On the other hand, since =4 r [ r] J r 0i J K Z =) c 0 = c Z 0 Lr i tx 0 J0i J K X X t(4 t ) ux t= s= t 0 = u= v= u 0 = X t0= X E[r is r it 0r iv r iu 0] X C:!) r it0 5 =) cbi v (r), we can see using exression (4) that Lr i Z Z r Bi v (s)ds r(4 r) 0 F v i (r) dr; 0 B v i (s)ds + 6r( r) Z 0 sb v i (s)ds dr

36 whose moment is c =600 by direct calculation. By alying Lemma, we have X i= Z J0i r J K 0 Lr i J0i r J K (; ) Lr i! c E Fi v (r) dr = c : When z t =, the above robability limit can be shown to be c =(90) in exactly the same manner. 4