A Simple Panel Stationarity Test in the Presence of Cross-Sectional Dependence

Transcription

1 A Simle Panel Stationarity est in the Presence of Cross-Sectional Deendence Kaddour Hadri Eiji Kurozumi 2 Queen s University Management School Deartment of Economics Queen s University Hitotsubashi University ovember 4, 28 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 (27) 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. We also develo a Lagrange multilier (LM) test allowing for cross-sectional deendence and, under restrictive assumtions, comare our augmented KPSS test with the extended LM test under the null of stationarity, under the local alternative and under the xed alternative, and discuss the di erences between these two tests. 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 augmented KPSS test. JEL classi cation: C2, C33 Key words: Panel data; stationarity; KPSS test; cross-sectional deendence; LM test; locally best test. Corresonding author: Kaddour Hadri, 25 University Square, Queen s University Management School, Queen s University, Belfast, B7, UK. el +44 () Fax +44 () k.hadri@qub.ac.uk 2 Kurozumi s research was artially suorted by the Ministry of Education, Culture, Sorts, Science and echnology under Grants-in-Aid o

2 . Introduction Since the beginning of the 9 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 (2), Hadri (2), Hadri and Larsson (25), Im, Pesaran and Shin (23), Levin, Lin and Chu (22), Maddala and Wu (999), Phillis and Moon (999), Quah (994) and Shin and Snell (26). On the alication side, the early contributions were the work of O Connell (998), Oh (996), Paell (997, 22), Wu (996) and Wu and Wu (2), 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 (25)). 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 (2), Baltagi (2) 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 (2, 24) 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 (24), Phillis and Sul (23), Bai and g (24), Moon and Perron (24), Choi and Chue (27) and Pesaran (27) for unit root anel tests. For 2

3 anel stationarity tests, the only contributions so far are by Bai and g (25) and Harris, Leybourne and McCabe (25), both of which corrected for cross-sectional deendence by using the rincial comonent analysis roosed by Bai and g (24). Choi and Chue (27) utilize subsamling technique to tackle cross-sectional deendence. Phillis and Sul (23), Bai and g (24), Moon and Perron (24) and Pesaran (27) emloy factor models to allow for cross-sectional correlation (cf. to de Silva, Hadri and remayne (27) for the comarison of the three last tests). Pesaran (27) 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 (24) who exloited the exibility of the bootstra method to deal with the ervasive roblem of cross-sectional deendence of general form. Breitung and Pesaran (25) 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 3

4 seed of and going to in nity. In this aer, we adat Pesaran (27) aroach to the anel stationarity test of Hadri (2) due to its concetual simlicity. Our test is basically the same as the Kwiatkowski et al. (992) test (KPSS test hereafter) with the regression augmented by cross-sectional average of the observations. We show that the limiting null distribution is the same as the test suggested by Hadri (2), which is an Lagrange multilier (LM) test without crosssectional deendence. We also extend Hadri s test and develo the LM test allowing for cross-sectional deendence. We comare our augmented KPSS test with the extended LM test under the null of stationarity, under the local alternative and under the xed alternative, and discuss the di erence between these two tests. We then extend our test to the case of the serially correlated shocks, and use Monte Carlo simulations to examine the nite samle roerties of the augmented KPSS test. he aer is organized as follows. Section 2 sets u the model and assumtions, and de ne the augmented test statistic. We also develo the LM test allowing for cross-sectional deendence. Section 3 is devoted to the comarison of our augmented KPSS test under restrictive assumtions with the extended LM test under the null of stationarity, under the local alternative and under the xed alternative. We show that the limiting null distribution of the augmented KPSS test is the same as that of Hadri s (2) test. We also examine whether our theoretical result is valid in nite samles via simle Monte Carlo simulations. In Section 4, we relax Assumtion in order to allow for serial correlation in the error term and roose a modi cation of the 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 A) A for a full column rank matrix A. he symbols ;! and =) signify convergence in robability and weak convergence resectively as! with xed, while ;! and =) means convergence in robability and weak convergence resectively when!. We denote sequential 4

5 convergence in robability and sequential weak convergence by ;;! and ; =) resectively as! followed by!. We denote [ r] element of vector A by [A] [ r] where [ r] is the largest integer less than r. 2. Model and est Statistics 2.. Model and assumtions Let us consider the following model: y it = zt 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 i = 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] : In this aer, we consider these two cases. Accordingly, we de ne i = i when z = and i = [ i ; i ] when z = [; t]. In model (), zt i is the individual e ect while f t is one dimensional unobserved common factor and " it is the individual-seci c (idiosyncratic) error. By stacking y it with resect to t, model () can be exressed as y i z r i f y i2 7 6 z2 7 6 r i2 7 6 f y i 7 5 = 6 4. z 7 5 i r i f 7 5 i " i " i2. " i ; or r i r i2. r i = v i v i2. v i y i = Z i + r i + f i + " i (2) = Z i + Lv i + f i + " i ; 5

6 where Z = [ ; d] with = [; ; ; ] and d = [; 2; ; ] being vectors. Further, we have y Z y = 6 4 or y Z... Z L L... L v v 2. v f f f y = (I Z) + r + ( f) + " (3) = (I Z) + (I L)v + ( f) + ": " " 2. " 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:(; 2 "); f t i:i:d:(; 2 f ); v it i:i:d:(; 2 v); and 2 ", 2 f and 2 v are known. (ii) here exist real numbers M, M and M such that j i j < M < for all i and < M < jj < M < for all, where = P i= i. Assumtion (i) is restrictive and not ractical. he assumtion of normality with homoskedasticity is required to derive the LM test and to discuss the otimal roerty of the tests. he variances of the innovations are assumed to be known in order to make the theoretical investigation as simle as ossible. In fact, if the variances are unknown, we need to estimate them and the asymtotic roerty of the tests deends on those estimators. Our urose in this section is to examine the theoretical e ect of augmentation, which is exlained below, on stationarity tests. Assumtion (i) will be relaxed in Section 4 to a more ractical one. 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 and above both in nite samles and in asymtotics. he latter roerty is imortant in 6

7 order to eliminate the common factor e ect from the regression. A similar assumtion is also entertained in Pesaran (27). 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 : 2 v 2 " = 8i v.s. H : (4) where = 2 v= 2 " is a signal-to-noise ratio. Under H, r it becomes equal to zero for all i so that y it is stationary whereas some or all of the cross-sectional units have a unit root under the alternative A simle stationarity test A anel stationarity test has already been roosed by Hadri (2) and Shin and Snell (26) for cross-sectionally indeendent data and we extend Hadri s test to the crosssectionally deendent case. Hadri (2) 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 (992) 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 (2) 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 (27), which develos anel unit root tests with cross-sectional deendence. As in Pesaran (27), we rst take a cross-sectional average of the model: y t = z t + r t + f t + " t ; (5) where y t = P i= y it, t = P i= i, r t = P i= r it, t = P i= i and " t = P i= " it. Since 6= by assumtion, we can solve equation (5) with resect to f t 7

8 as follows: f t = y t z t r t " t : By inserting this solution of f t into model () we obtain the following augmented regression model: y it = zt ~ 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 (2). hat is, (S ) Z A = (7) where and 8 < : S = X i= ote that S i can also be exressed as S i with S i = 2 " 2 y im w L LM w y i = = 6 ; 2 = 2 = 45 when z t = z t = ; = = 5 ; 2 = 2 = 63 when z t = z t = [; t] : S i = 2 " 2 X (Sit w ) 2 where Sit w = t= with ^ it obtained for each i by regressing y it on w t = [z t; y t ] 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 it has the limiting distribution. We call Z A the augmented KPSS test statistic. tx s= ^ is 2.3. An LM test for anel stationarity Although the augmented KPSS test is easy to imlement, we do not know whether it has an otimal roerty. ote that if y t were deterministic, we could see that Z A is equivalent to the LM test statistic and, because the LM test is a locally best invariant test under the 8

9 assumtion of normality, the augmented KPSS test would also be locally otimal. One might think that this local otimality holds because y t converges in robability to its mean, z t +ft, as! and f t is indeendent of " it. In order to investigate the local otimality of the augmented KPSS test, we consider the LM test and comare the two tests. Under the assumtion of normality, the log-likelihood function of y, denoted by `, is exressed as ` = const 2 log jj 2 fy (I Z)g fy (I Z)g; where V ar(y) = 2 "I LL + A I with A = 2 f + 2 "I : he artial derivative of ` with resect to is given by = const + 2 fy fy (I Z)g: (8) j H = A I ; and the maximum likelihood (ML) estimator of under H is = 2 "I LL ; (9) H ^ = h(i Z ) H (I Z)i (I Z ) H y = (A Z Z) (A Z )y = [I (Z Z) Z ]y: () hus, the maximum likelihood estimator of under H is the same as the OLS estimator. By evaluating (8) under the null hyothesis using (9) and (), the LM test statistic is given by LM = = 2 fy (I Z)^g (A I )( 2 "I LL )(A I )fy (I Z)^g 2 y ( 2 "A 2 M z LL M z )y: 9

10 By normalizing LM, the extended LM test statistic for anel stationarity is given by LM Z LM = () where and are the same as in Z A. ote that LM can also be exressed as LM = X X i= j= 2 "a ij 2 LM ij with LM ij = 2 X SitS z jt z and Sit z = t= tx ^u is ; s= where a ij 2 being the (i; j) element of A 2 and ^u it for each i is obtained by regressing y it on z t. 3. heoretical Proerty of the Augmented KPSS and LM ests 3.. he limiting distributions of the test statistics In this section we comare the augmented KPSS test with the extended LM test. ote that the LM test is known to be a locally best invariant test under Assumtion. Because there is no one-to-one transformation between Z A and Z LM, we can see that the augmented KPSS test does not have the local otimality in nite samles. hen, our interest moves on to whether it is asymtotically locally otimal or not. In order to investigate the asymtotic local otimality of the augmented KPSS test, we comare it with the LM test statistic under the null hyothesis, under the local alternative and under the xed alternative. We rst give the limiting distributions of the two test statistics under the null hyothesis. heorem Assume that Assumtion holds. Under H, as! followed by!, the augmented KPSS and LM test statistics have a limiting standard normal distribution for both cases of z t = and z t = [; t], Z A ; Z LM ; =) (; ): test. ote that the rejection region of Z A and Z LM is the right hand tail as in Hadri s (2)

11 Remark Although we derived only the sequential weak limit of the test statistics, it is not di cult to see that the sequential limit is the same as the joint limit where and go to in nity simultaneously if we additionally assume the rate condition =!. his is because all the innovations are i.i.d. normal and the variances are known (see Shin and Snell (26)). In articular, the assumtion of known variances is helful to establish the joint limit because it is su cient to consider the joint limit of the numerator of the test statistic in this case. According to Phillis and Moon (999), this rate condition indicates that the joint limit theory holds when the cross-sectional dimension is moderate while is relatively large. However, we will relax the assumtion of normality and consider the deendent shocks in a later section. In this case, it would be di cult to establish the equivalence between the sequential and joint limits, and we will rely only on the sequential limit technique. In order to see if the sequential limit theory can successfully aroximate the nite samle behavior of the test statistics, we will conduct Monte Carlo simulations in a later section. heorem shows that Pesaran s (27) method works well in order to eliminate crosssectional deendence even for testing the null hyothesis of stationarity and the augmented KPSS test is asymtotically equivalent to the LM test statistic under the null hyothesis. We next investigate the asymtotic roerty of the test statistics under the local alternative, which is exressed as H` : = c2 2 where c is some constant: ote that for a single time series analysis, the local alternative is given by = c 2 = 2. Since the sum of S i is normalized by as in Z A, the local alternative for anel stationarity tests becomes = c 2 =( 2 ). heorem 2 Assume that Assumtion holds. Under H`, as! followed by!, the augmented KPSS and LM test statistics have the same limiting distribution given by Z ; Z A ; Z LM =) (; ) + c2 E Fi v (r) 2 dr

12 where Fi v(r) = R r Bv i (t)dt R r R R z(t) dt z(t)z(t) dt z(t)bv i (t)dt with Bv i (r) are indeendent Brownian motions, which are indeendent of B " i (r), z(r) = when z t = and z(r) = [; r] when z t = [; t]. his result imlies that both the augmented KPSS and extended LM test statistics have the same asymtotic local distribution. Since the LM test is locally best invariant, we can see that the augmented KPSS test has the same asymtotic local otimality. We nally investigate the asymtotic roerty of the test statistics under the xed alternative H. he following theorem gives the di erence of owers between the two tests when the alternative is not local but far away from =. heorem 3 Assume that Assumtion holds. Under H, as! followed by!, Z Z ; A =) 2 2 E vi G v i (r) 2 dr Z Z ; LM =) 2 2 E Fi v (r) 2 dr ; where G v i (r) = R r Bv i (t)dt R r z 2 (t)dt(r z 2(t)z 2 (t)) R z 2(t)B v i (t)dt with z 2(t) = [z (t); B v (t)] and B v (t) is a standard Brownian motion indeendent of B v i (t) and E vi denotes the exectation oerator with resect to B v i (r). ote that since G v i (r) deends on Bv i (r) and Bv (r) and they are indeendent, we can see that E vi [ R Gv i (r)2 dr] still deends on B v () and then it is stochastic, while E[ R F i v (r) 2 dr] is deterministic. his is an interesting result because, when the asymtotic local owers are the same for two tests, it is often the case that they also have the same limiting distribution under the xed alternative. For anel stationarity tests, the two tests have the same local asymtotic ower from heorem 2 but the owers are di erent under the xed alternative from heorem 3. his imlies that although the two tests are locally otimal, they are not equivalent in a wide range under the alternative Finite samle roerty under restrictive assumtions 2

13 In this subsection we investigate how accurately does the asymtotic theory aroximate the nite samle behavior of the augmented KPSS and LM tests. We consider the following data generating rocess for nite samle simulations: y it = z t i + r it + f t i + " it ; f t i:i:d:(; ); " it i:i:d:(; ) r it = r it + v it ; v it i:i:d:(; ); H : = H : = :; :; : where i = i for the constant case while i = [ i ; i ] for the trend case with i and i being drawn from indeendent U(; :2), i are drawn from +U(; 4) for strong cross-sectional correlation case (SCC) and from U(; :2) for weak cross-sectional correlation case (WCC), and i, i and i are xed throughout the iterations. We consider all the airs of =, 2, 3, 5, and =, 2, 3, 5,, 2. he level of signi cance is.5 and the number of relications is, in all exeriments. able shows the sizes of the tests. We can observe that the emirical size of the augmented KPSS test is close to the nominal one when is equal to or greater than 5 for SCC case while it is slightly undersized for WCC case. On the other hand, the size of the LM test is close to the nominal one irresective of and but it is slightly undersized for SCC case while it is slightly oversized for WCC case. Overall, the null distributions of the two tests seem to be well aroximated by a standard normal distribution suggested by heorem in view of the size of the tests. able 2 reorts the owers of the tests. For given and, the uer, middle and lower entries are the owers of the tests for = :,. and., resectively. From the table the owers of the tests become higher for larger and, although the tests have low ower when is small. We can also observe that the owers become higher for larger. For examle, the size of the augmented KPSS test for = 5, SCC and constant case is relatively close to.5 for all the values of while the emirical ower when = : is.45,.22,.254,.342 and.539 for =, 2, 3, 5 and, resectively. able 2 imlies that the tests are consistent as roved by heorem 3. In order to see if the augmented KPSS test can be seen as the asymtotically locally 3

14 best test indicated by heorem 2, we calculated the size adjusted ower of the tests. Figure draws the ower curves for selected cases. From the gure we observe that the ower of the augmented KPSS test is almost the same as that of the LM test for the constant case. When a linear trend is included, the augmented KPSS test is as owerful as the LM test when is small while the former is slightly less owerful than the latter for the trend case. As a whole, the nite samle behavior of the augmented KPSS and LM tests is well aroximated by the asymtotic theory established in the revious section when and are of moderate size. 4. Extension to general case 4.. Modi cation of the augmented KPSS test So far, we have investigated the theoretical roerty of the augmented KPSS test under restrictive assumtions. In this section we relax Assumtions (i) and consider a more ractical situation. Because the LM test statistic will deend on nuisance arameters in a comlicated way under general assumtions and it would be di cult to correct Z LM so that it becomes free of nuisance arameters, we concentrate on the modi cation of the augmented KPSS test. 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: y it = z t i + f t i + " it ; " it = i " it + + i " it + it : (2) he lag length may change deending on the cross-sectional units but we suress the deendence of on i for notational convenience. Assumtion 2 (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. he stochastic rocess it are indeendent of f t and i:i:d:(; 2 i ) across i and t with a nite fourth moments. 4 (ii)

15 his assumtion allows the common factor to be stationary but still assumes that it is indeendent of the idiosyncratic errors, which are nite order AR rocesses with the i.i.d. innovations. We assume Assumtions (ii) and 2 in the rest of this section. Since our interest is whether y it are (trend) stationary or unit root rocesses, the testing roblem is given by H : i () 6= 8i v.s. H : i () = 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 t; y t ; y t ; ; y t ] because " it are AR() rocesses and construct S w it using this regression residuals. Along the same line as (2) in the roof of heorem it is not di cult to see that the numerator of each S i weakly converges to 2 X (Sit w ) 2 =) 2 i t= Z where 2 i = 2 i =( i i ) 2 and V " i (r) = B" i (r) V " i (r) + O 2 dr R r R R z(t) dt z(t)z(t) dt 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 2 i order to correct for serial correlation. Several consistent estimators of the long-run variance for arametric model have been roosed in the literature for a 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 (26). 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 (25). heir method is originally develoed for the rewhitening method, but it is in 5

16 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 t^ i + ^ i y it + + ^ i y it + ^ iy t + + ^ iy t + ^ it ; and construct the estimator of the long-run variance by 8 9 ^ 2 ^ 2 < i isp C = where ^i = min ( ^i ) 2 : X = ; ^ ij : ; and ^2 i = j= X t= ^ 2 it: We then roose to construct the test statistic (7) using S i = We denote this test statistic as Z SP C A. ^ 2 isp C 2 X (Sit w ) 2 : t= he other method we consider is the lag-augmented method roosed by Choi (993) 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 = zt ~ i + ~ i y it + + ~ i y it + ~ i+ y it + ~ iy t + + ~ iy t + ~ it ; and construct the test statistic using S i = ^ 2 ila 2 X (Sit w ) 2 where ^ 2 ila = t= ^ 2 i ( ~ i ~ i ) 2 : We denote this test statistic as Z LA A. he consistency of ^ 2 isp C and ^2 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 xed alternative. Unfortunately, it is tedious to derive the joint limit of Z SP C A or Z LA A 6 under general assumtions and we do not

17 ursue 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 behavior of these tests Finite samle roerty under general assumtions In this section we conduct Monte Carlo simulations to investigate the nite samle roerties of the 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 i + f t i + " it ; " it = i " it + it ; where f t i:i:d:(; ), it i:i:d:(; ), f t and it are indeendent of each other, i, i and i are set as in Subsection 3.2., the i are drawn from : + U(; :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 (25) (HLM hereafter). According to HLM, we rst estimate the idiosyncratic errors " it by the rincial comonent method roosed by Bai and g (24) and next aly the stationarity test roosed by Harris, McCabe and Leybourne (23) to the estimated series of " t ; " 2t ; ; " 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 (25). able 2 reorts the sizes of the tests. here are no entries for HLM test when = because the time dimension is too short to calculate their test statistic. When only a constant is included in the model, the 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 = and = 2. For WCC (weak cross-correlation) case Z SP C A is undersized excet 7

18 for the case of =. similar roerty as Z SP C A he augmented KPSS test corrected by the LA method has 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 2. able 4 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 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.9,.22 and.78 when = and = 3 for the constant case with SCC, while the owers of these tests are.437,.262 and.28. On the other hand, the sizes of these tests are.58,.76 and.54 when = and = for the constant case with WCC while the owers are.878,.82 and.. 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 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 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 (2) test to correct for cross-sectional deendence à la Pesaran (27). We showed that the limiting null distribution of this augmented KPSS test is the same as the original Hadri s test that is the LM test without cross-sectional deendence. We also extended Hadri s test by develoing an LM test correcting for cross- 8

19 sectional deendence. hen, we comared our augmented KPSS test with the extended LM test. We found that the augmented KPSS test is asymtotically locally otimal but it is not asymtotically equivalent to the LM test under the xed alternative. he Monte Carlo simulations indicated that we should carefully use the anel stationarity tests because they are undersized in some cases but su er from over rejection in other cases. 9

20 References [] Bai, J. and S. g (24). A anic attack on unit roots and cointegration. Econometrica 72, [2] Bai, J. and S. g (25). A new look at anel testing of stationarity and the PPP hyothesis, D. W. K. Andrews and J. H. Stock, ed., Identi cation and Inference for Econometric Models. Essays in Honor of homas Rothenberg. Cambridge University Press, Cambridge. [3] Baltagi, B. H. (2). Econometric Analysis of Panel Data. Chichester, Wiley. [4] Baltagi, B. H. and C. Kao (2). onstationary anels, cointegration in anels and dynamic anels: A survey. Advances in Econometrics 5, 7-5. [5] Banerjee, A. (999). Panel data unit roots and cointegration: An overview. Oxford Bulletin of Economics and Statistics Secial issue, [6] Banerjee, A., M. Marcellino and C. Osbat (2). esting for PPP: Should we use anel methods? Emirical Economics 3, [7] Banerjee, A., M. Marcellino and C. Osbat (24). Some cautions on the use of anel methods for integrated series of macroeconomic data. Econometrics Journal 7, [8] Breitung, J. and W. Meyer (994). esting for unit roots in anel data: Are wages on di erent bargaining levels cointegrated? Alied Economics 26, [9] Breitung, J. and Pesaran, M. H. (28). Unit roots and cointegration in anels. In: L. Matyas and P. Sevestre (eds): he Econometrics of Panel Data: Fundamentals and Recent Develoments in heory and Practice. Dordrecht: Kluwer Academic Publishers, 3rd edition, Chat. 9, [] Chang, Y. (24). Bootstra unit root tests in anels with cross-sectional deendency. Journal of Econometrics,

21 [] Choi, I. (993). Asymtotic normality of the least-squares estimates for higher order autoregressive integrated rocesses with some alications. Econometric heory 9, [2] Choi, I. (2). Unit root tests for anel data. Journal of International Money and Banking 2, [3] Choi, I. and. K. Chue (27). Subsamling hyothesis tests for nonstationary anels with alications to exchange rates and stock rices. Journal of alied econometrics 22, [4] Culver, S. E. and D. H. Paell (997). Is there a unit root in the in ation rate? Evidence from sequential break and anel data models, Journal of Alied Econometrics 2, 4, [5] de Silva, S., K. Hadri and A. R. remayne (27). Panel unit root tests in the resence of cross-sectional deendence: Finite samle erformance and an alication. Unublished manuscrit, University of Sydney. [6] Kwiatkowski, D., P. C. B. Phillis, P. Schmidt and Y. Shin (992). esting the null hyothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, [7] Hadri, K. (2). esting for stationarity in heterogeneous anel data. Econometrics Journal 3, [8] Hadri, K. and Larsson, R. (25). esting for stationarity in heterogeneous anel data where the time dimension is nite. Econometrics Journal 8, [9] Harris, D., B. McCabe and S. Leybourne (23). Some limit theory for autocovariances whose order deends on samle size. Econometric heory 9,

22 [2] Harris, D., S. Leybourne and B. McCabe (25). Panel stationarity tests for urchasing ower arity with cross-sectional deendence. Journal of Business and Economic Statistics 23, [2] Im, K., M. H. Pesaran and Y. Shin (23). esting for unit roots in heterogeneous anels. Journal of Econometrics 5, [22] Levin, A., C. F. Lin and C. S. J. Chu (22). Unit root tests in anel data: Asymtotics and nite-samle roerties. Journal of Econometrics 8, -24. [23] Maddala, G. S. and S. Wu (999). A comarative study of unit root tests with anel data and a new simle test. Oxford Bulletin of Economics and Statistics 6, [24] Moon, R. and B. Perron (24). esting for unit roots in anels with dynamic factors. Journal of Econometrics 22, 8-26 [25] Pesaran, M. H. (27). A simle anel unit root test in the resence of cross-section deendence. Journal of Alied Econometrics 22, [26] Phillis, P. C. B. and H. R. Moon (999). Linear regression limit theory for nonstationary anel data. Econometrica 67, 57-. [27] Phillis, P. C, B. and D. Sul (23). Dynamic anel estimation and homogeneity testing under cross section deendence. Econometrics Journal 6, [28] O Connell, P. G. J. (998). he overvaluation of urchasing ower arity. Journal of International Economics 44, -9. [29] Oh, K. Y. (996). Purchasing ower arity and unit root tests using anel data. Journal of International Money and Finance 5, [3] Paell, D. H. (997). Searching for stationarity: Purchasing ower arity under the current oat. Journal of International Economics 43,

23 [3] Paell, D. H. (22). he great areciation, the great dereciation and the urchasing ower arity hyothesis. Journal of International Economics 57, [32] Quah, D. (994). Exloiting cross section variation for unit root inference in dynamic data. Economics Letters 44, 9-9. [33] Shin, Y. and A. Snell (26). Mean grou tests for stationarity in heterogeneous anels. Econometrics Journal 9, [34] Sul, D., P. C. B. Phillis and C. Y. Choi (25). Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics 67, [35] oda, H. Y., and. Yamamoto (995). Statistical inference in vector autoregressions with ossibly integrated rocesses. Journal of Econometrics 66, [36] Wu, S. and J. L. Wu (2). Is urchasing ower arity overvalued? Journal of Money, Credit and Banking 33, [37] Wu, Y. (996). Are real exchange rates nonstationary? Evidence from a anel data test. Journal of Money, Credit and Banking 28,

24 Aendix In this aendix, 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. Proof of heorem We rst exress y t in matrix form. Since y t = z t + r t + f t + " t ; we have 2 or 6 4 y y 2. y = 6 4 z z 2. z r r 2. r f f 2. f " " 2. " y = Z + r + f + ": (3) Since 6=, we have f = (y Z r ")=. By inserting this into (2), the model becomes where ~ i = i =. y i = Z( i ~ i ) + ~i y + (r i ~ i r) + (" i ~ i ") (4) Let W = [ ; d; y] = [Z; y]. Under the null hyothesis, S i can be exressed in matrix form as S i = from (4) because (r i ~ i r) disaears under H. Let Q = I2 = 2 " 2 y im w L LM w y i 2 " 2 (" i ~ i ") M w L LM w (" i ~ i ") (5) ; D = diagfd ; g and D 2 = diagfd ; g where D = diagf ; g and de ne W W Q = [Z; y ] where y = y Z = f + ", so that we have the equality M w = M w and thus we can relace M w in (5) with M w. 24

25 Lemma Under H for r, (i) (ii) L(" i ~ i ") LW D [ r] [ r] (iii) D2 W W D (iv) D 2 W (" i ~ i ") =) " B ~ " i (r) = " B i " (r) + O ; =) = =) = =) = Z r Z r z(t) dt; f B f (r) + " B " (r) ; z(t) dt; f B f (r) + O ; " R z(t)z(t) dt R z(t) f db f (t) + " d B " (t) 2 2 f + 2 " " R z(t)z(t) dt f R z(t)dbf (t) + O 2 2 f + O R " z(t)d B ~ i "(r) " R " z(t)db" i (r) + O where z(r) = [; r], B " i (r) for i = ; and Bf (r) are indeendent standard Brownian motions, B " (r) = P i= B" i (r) and ~ B " i (r) = B" i (r) ~ i B " (r). # ; # # ; ; Proof of Lemma : Weak convergences in (i) is established by the functional central limit theorem (FCL) and the continuous maing theorem (CM). he second equality holds because j~ i j < M =M < for all i from Assumtion (ii) and B " (r) = O (= ) by the weak low of large numbers (WLL). (ii) Since y = f + " under H, we have, by the FCL, 2 LW D = 4 [ r] [ r] X ztd X ; f t + X!3 " it 5 [ r] t= t= i= Z r =) z(t) dt; f B f (r) + " B " (r) : (6) (iii) he result of the (; ) block is easily obtained. For the (; 2) block,, we can see by 25

26 the FCL that y = d y = X t= f t + X X t= i= X tf t + t= while for the (3; 3) element, t= " it =) f B f () + " B(); (7) X X t i= " it =) Z t f db f (t) + " db(t) (8) y y = 2 X t= f 2 t + 2 X t= X i= " it! X X f t " it t= i= ;! 2 2 f + 2 " ; by the WLL. (iv) he result of the rst two element is obtained by the FCL. For the last element, we have, by the WLL, y " i = ;! 2 " ; X t= f t " it + X t= " 2 it + X X " it t= j6=i " jt and similarly, We then have (iv).2 y " ;! X i= 2 " = 2 " : Since LM w (" i ~ i ") = L(" i ~ i ") LW D D2 W W D D 2 W (" i ~ i "); 26

27 we have, by Lemma and the CM, so that LM w (" i =) " (B " i (r) ~ i ") [ r] Z r Z z(t) dt Z ) z(t)z(t) dt z(t)db i " (t) + O " V i " (r) + O ; (9) his imlies that S i =) Z V " i (r) + O 2 dr: (2) Z S ;;! E V i " (r) 2 dr : (2) By aroriately normalizing S, we obtain the result for Z A. In order to derive the limiting distribution of the LM test statistic, we rst note that under H, (I M z )y = ( M z f) + (I M z )" ; ( 2 f M z ) + ( 2 "I M z ) = (; A M z ) = (A =2 M z ) where = [ ; ; ] (; I I ). hen, LM can be exressed as LM = = 2 (A =2 M z )( 2 "A 2 LL )(A =2 M z ) 2 (I M z L)( 2 "A I )(I L M z ): (22) We rst investigate the matrix A. ote that 2 "A can be exressed as 2 "A = 2 "( 2 f + 2 "I ) 2 f + 2 f 2 A : " 2 " 27

28 Since rk( ) = and ( ) = ( ), the ( ) eigenvalues of are and the non-zero eigenvalue is, for which the corresonding eigenvector is. hen, there exists an orthonormal matrix P such that P P = P P = I and P P = diagf ; ; ; g : his imlies that P 2 "A P = I = f 2 " 2 " 2 "+ 2 f 2 f 2 f " A : (23) By inserting (23) into (22), we obtain LM = = = 2 (I M z L)(P P 2 "A P P I )(I L M z ) 2 (I M z L)( A I )(I L M z ) 2 " 2 " + 2 f 2 M z LL M z + X 2 i M z LL M z i ; i=2 where = [ ; ; ] = (P I ) (; I I ). ote that the rst term converges to zero in robability as! followed by!, whereas for the second term, we have, by the FCL and the CM, 2 i M z LL M z i ( =) Z Z B i (r) V i (r)2 dr Z r Z Z ) 2 z(t) dt z(t)z(t) dt z(t)db i (t) dr where B i for i = 2; ; are indeendent standard Brownian motions. See also Hadri (2). hen, by aroriately normalizing LM, we obtain the result. 28

29 Proof of heorem 2 We rst give the following lemma. Lemma 2 Under H` (i) L(r i ~ i r) [ r] (ii) LW D for r, [ r] (iii) D2 W W D (iv) D 2 W (r i ~ i r) (v) D 2 W (" i ~ i ") =) c Z r " =4 Z c r " = =) = =) = =) = =) = =4 Z r Z r ~B v i (t)dt B v i (t)dt + O ; z(t) dt; f B f (r) + " B " (r) + c " =4 z(t) dt; f B f (r) + O ; R z(t)z(t) dt Z r R z(t) f db f (t) + " d B " (t) + c" =4 R z(t) B v (t)dt 2 2 f + 2 " " R z(t)z(t) dt f R z(t)dbf (t) + O 2 2 f + O # " c" R =4 z(t) B ~ i v(t)dt " c" R =4 z(t)bv i (t)dt + O R " z(t)d B ~ i "(r) " R " z(t)db" i (r) + O Proof of Lemma 2: (i) Since v it i:i:d:(; c 2 2 "=( 2 )), we have [ r] v it ri[ r] = X t= # : # ; B v (t)dt ; =) c " =4 Bv i (r) (24) by the FCL. We then obtain the weak convergence in (i) by the CM. # ; ; 29

30 (ii) he rst element has already been obtained in heorem. For the second element, since y t = r t + f t + " t under H`, we have [ r] X t= y t = =) c " =4 [ r] X X rit + [ r] X f t + i= t= Z r t= [ r] X X i= B v (t)dt + f B f (r) + " B " (r): (iii) Similarly to the roof of (ii), we can see that the (,2) block of D2 W W D becomes D Z y = =) while the (2,2) block is given by (iv) Since y y = X y t (r it ~ i r t ) = t= t= " P t= (r # t + f t + " t ) P t= t(r t + f t + " t ) " c " # B v () + f B f () + =4 " B " () c " R =4 td B v (t) + R t(f db f (t) + " db " ; (t)) = 2 X t= r 2 t + 2 f 2 t + " 2 t + 2r t f t + 2r t " t + 2f t " t X ft 2 + t= =) 2 2 f + 2 " : 2 ;! ; X t= X t= " 2 t + O ( r t ) (r it ~ i r t ) + " it X (f t + " t ) (r it ~ i r t ) t= we have D 2 W (r i ~ i r) = =) " D P t= z t (rit r t ) P t= y t (r it r t ) # " c" =4 R z(t) ~ B v i (t)dt : # 3

31 (v) he rst element has already been obtained in heorem. For the second element, note that y (" i ~ i ") = X rt (" it t= ~ i " t ) + X (f t + " t )(" it ~ i " t ): he rst term on the right hand side is O (= ) while the second term converges in robability to zero as roved in Lemma (iv). hen, we obtain (v).2 In order to derive the limiting distribution of Z A under H`, note that LM wy i = LM w (" i ~ i ") + LM w (r i ~ i r). hen, we can see from Lemma (i) and Lemma 2 that where LM w y i Using this, we have S i [ r] Z r Fi v (r) = Bi v (t)dt =) Z =) " V i " (r) + O + c " =4 Fi v (r) + O Z r t= Z Z z(t) dt z(t)z(t) dt z(t)bi v (t)dt: V i " (r) + O + c 2 =4 Fi v (r) + O dr: Since B i "(r) and Bv i (r) are indeendent, we can see that the sum of the cross roduct between V i " (r) and F v i (r) is O (= ), so that Z A =) X Z i= =) (; ) + c2 E V " i (r) 2 dr Z Fi v (r) 2 dr : + c2 X i= Z Fi v (r) 2 dr + O In order to derive the limiting distribution of the LM test statistic, note that (I M z )y = (I M z )r + (A =2 M z ): hen, the denominator of the LM test statistic can be exressed as (LM ) = ( 2 2 "A 2 M z LL M z ) (25) + 2 r ( 2 "A 2 M z LL M z )r ( 2 "A 2 M z LL M z )": 3

32 he rst term on the right hand side of (25) converges in distribution to a standard normal distribution as! followed by! as roved in heorem. Since A 2 = P 2 A P, the second term on the right hand side of (25) is exressed as 2 r ( 2 "A 2 M z LL M z )r = 2 r (P I )( 2 " 2 A M zll M z )(P I )r: ote that (P I )r = (P I )(I L)v = (I L)v = r where v (P I )v (; 2 "(I I )) and r (I L)v. Since v has the same distribution as v, we can see that r has the same distributional roerty as r, so that we can aly the limit theorem used for the derivation of the limiting distribution of the augmented KPSS test statistic. hat is, 2 r ( 2 "A 2 M z LL M z )r = 2 r ( 2 " 2 A M zll M z )r = ; =) c 2 E 4 " ( 2 " + 2 f 2 Z 2 r M z LL M z r + 2 Fi v (r) 2 dr ; X i=2 r i M z LL M z r i where F v i induced by v it. is de ned as F v i with Bi v relaced by Bi v, which are standard Brownian motions Since v it is indeendent of " it, the third term of (25) can be shown to converge in robability to as! followed by!. We then obtain the theorem. Proof of heorem 3 We rst give the following lemma, which can be roved by the FCL and the CM and then we omit the roof. 32

33 Lemma 3 Under H f for r, Z r (i) L(r i ~ i r) =) " ~B i v (t)dt; [ r] Z r Z r (ii) LW D2 =) z(t) dt; " B v (t)dt ; [ r] " R R (iii) D2 W W D2 =) z(t)z(t) dt " z(t) B v (t)dt R " z(t) B v (t)dt 2 2 R " B v (t) 2 dt " R (iv) D 2 W (r i ~ i r) =) " z(t) B ~ # i v(t)dt 2 2 R " B v (t) B ~ i v(t)dt : # ; then, It is not di cult to see that [LM w (r i ~ i r)] [ r] dominates [LM w (" i ~ i ")] [ r] and 2 S i = 2 " 4 (r i ~ i r)m w L LM w (r i ~ i r) + o (): Since LM w (r i ~ i r) = L(r i ~ i r) W D 2 (D 2 W W D 2 ) D 2 (r i ~ i r), we have, using Lemma 3, 2 S i Z =) 2 Z r ~B v i (t)dt " R z(t)z(t) dt R z(t) B v (t)dt Z Z r = 2 Bi v (t)dt Z r Z Z 2 (t) dt Z r z(t) dt; R R Z r B v (t)dt z(t) B # " v R (t)dt B v (t) 2 z(t) ~ # 9 B R i v(t)dt = dt B v (t) B ~ i v(t)dt ; Z ) 2 Z 2 (t)z 2 (t) dt Z 2 (t)bi v (t)dt dr + O ; 2 dr 33

34 where Z 2 (r) = [z(t) ; B v (r)]. Hence, we have 2 Z A = =) X i= X i= Z X Z 2 (S i ) Z r 2 Bi (t)dt v dr ( Z r i= Z Z r X i= +O : Z Z 2 Z 2 (t) dt Z 2 (t)z 2 (t) dt Z 2 (t)bi (t)dt) v dr Z Z Z r Z 2 (t) dt Z 2 (t)z 2 (t) dt Z 2 (t)bi v (t)dt B v i (t)dtdr he rst term of (26) converges in robability to ( 2 =)E[ R (R r Bv i (t)dt)2 dr] by the WLL. For the second term, we note that X Z Z ((Bi v (s)bi v (t) min(s; t)) Z 2 (s)z 2 (t) dsdt ;! (27) i= because " # X E ((Bi v (s)bi v (t) min(s; t))z 2 (s)z 2 (t) i= ( " #) h =2 E Z 2 (s)z 2 (t) 2i X E (Bi v (s)bi v (t) min(s; t)) 2 i= C where C is some constant. Since B v (t) =) B v (t), which is a standard Brownian motion indeendent of Bi v (t), we have (26) Z 2 (r) =) [z(r) ; B v 2(r)] z 2 (r): (28) Hence, we observe from (27) and (28) that the second term of (26) weakly converges to 2 Z ( Z r Z H 2 z 2 (t) dt z 2 (t)z 2 (t) dt Z Z Z Z ) r min(s; t)z 2 (s)z 2 (t) dsdt z 2 (t)z 2 (t) dt z 2 (t)dt dr (29) 34

35 as!. In exactly the same manner, we can see that as! the third term of (26) weakly converges to 2 H 2 2 Z ( Z r Z Z z 2 (t) dt z 2 (t)z 2 (t) dt Z r min(s; t)z 2 (t)dsdt From (26), (29) and (3), we conclude that ( " Z Z # ) r 2 Z ; A =) 2 E B v 2 i (t)dt dr + H 2H 2 : Since the term in braces is the same as E v [ R Gv i (r)2 dr], we have the result for Z A. ) dr: (3) he limiting distribution of the LM test statistic can be derived in the same way using the FCL and the CM. 35

36 able. Size of the tests: base case constant case trend case SCC WCC SCC WCC Z A Z LM Z A Z LM Z A Z LM Z A Z LM

37 able 2. Power of the tests: base case constant case trend case SCC WCC SCC WCC Z A Z LM Z A Z LM Z A Z LM Z A Z LM