Testing for Unit Roots in Small Panels with Short-run and Long-run Cross-sectional Dependencies 1

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1 Testing for Unit Roots in Small Panels with Short-run and Long-run Cross-sectional Dependencies 1 Yoosoon Chang Department of Economics Texas A&M University Wonho Song Korea Institute for International Economic Policy Abstract An IV approach, using as instruments nonlinear transformations of the lagged levels, is explored to test for unit roots in panels with general dependency and heterogeneity across cross-sectional units. We allow not only for the cross-sectional dependencies of innovations, but also for the presence of cointegration across cross-sectional levels. Unbalanced panels and panels with differing individual short-run dynamics and cross-sectionally related dynamics are also permitted. We also more carefully formulate the unit root hypotheses in panels. In particular, using order statistics we make it possible to test for and against the presence of unit roots in some of the individual units for a given panel. The individual IV t-ratios, which are the bases of our tests, are asymptotically normally distributed and cross-sectionally independent. Therefore, the critical values of the order statistics as well as the usual average statistic can be easily obtained from simple elementary probability computations. We show via a set of simulations that our tests work well, while other existing tests fail to perform properly. As an illustration, we apply our tests to the panels of real exchange rates, and find no evidence for the purchasing power parity hypothesis, which is in sharp contrast with the previous studies. This version: August 9, 28. JEL Classification: C12, C15, C32, C33. Key words and phrases: Panels with cross-sectional dependency and heterogeneity, unit root test, cointegration, covariate, nonlinear instrument, order statistics. 1 We would like to thank the editor and two anonymous referees for many helpful comments, which have greatly improved an earlier version of this paper. Earlier versions of the paper were circulated under the title Panel unit root tests in the presence of cross-sectional dependency and heterogeneity and Unit root tests for panels in the presence of short-run and long-run dependencies: Nonlinear IV approach with fixed N and large T, and presented at the 22 Winter Meetings of the Econometric Society at Atlanta, GA, the 1th International Panel Data Conference at The Academy of Science at Berlin, Germany, and at the International Conference on Current Advances and Trends in Nonparametric Statistics at Crete, Greece. We are grateful to H.R. Moon, J.Y. Park, P. Pedroni, B. Perron and seminar participants at University of Maryland, Graduate School of Business at University of Chicago, IZA at University of Bonn, European Central Bank, University of Central Florida for their helpful comments. Chang thanks the NSF for research support under Grants SES and SES Make correspondence to: Yoosoon Chang, Department of Economics, Texas A&M University, College Station, TX 77843, yoosoon@tamu.edu.

2 1. Introduction Panel unit root tests have been one of the most active research areas for the past several years. This is largely due to the availability of panel data with long time span, and the growing use of cross-country and cross-region data over time to test for many important economic inter-relationships, especially those involving convergencies/divergencies of various economic variables. The notable contributors in theoretical research on the subject include Levin, Lin and Chu (22), Im, Pesaran and Shin (23), Maddala and Wu (1999), Choi (21) and Chang (22, 24). There have been numerous related empirical researches as well. Examples include MacDonald (1996), Oh (1996) and Papell (1997), just to name a few. The papers by Banerjee (1999), Phillips and Moon (2) and Baltagi and Kao (2) provide extensive surveys on the recent developments on the testing for unit roots in panels. See also Choi (26) and Phillips and Sul (23) for some related work in this line of research. In this paper, we propose a new approach to test for unit roots in small panels. Our approach has several novel aspects. First, we allow for the presence of cointegration across cross-sectional units, as well as for the cross-sectional dependencies of innovations. The presence of cointegration introduces long-run cross-sectional dependencies in levels, while the cross-sectional dependencies in innovations are of a short-run nature. We therefore permit cross-sectional dependencies both in the short-run and in the long-run. It appears that there is a high potential for such possibilities in many panels of practical interests. However, none of the existing tests is applicable for such panels. In particular, they all require the absence of cross-sectional cointegrating relationships, and become invalid in the presence of cointegration among cross-sectional levels. Second, our tests rely on the models augmented by cross-sectional dynamics and other covariates. One of the main motivations to use panels to test for unit roots is to increase the power. An important possibility, however, has been overlooked in the literature, i.e., the possibility of using covariates. The idea of using covariates to test for a unit root was first suggested by Hansen (1995), and its implementation using bootstrap was studied later by Chang, Sickles and Song (21). They made it clear that there is a huge potential gain in power if covariates are appropriately chosen. Of course, the choice of proper covariates may be difficult in practical applications. In the panel context, however, some of potential covariates to account for the inter-relatedness of cross-sectional dynamics naturally come up front. For instance, we may include the lagged differences of other cross sections to account for interactions in the short-run dynamics. Finally, we formulate the panel unit root hypotheses more carefully. Obviously, the power increase is not the only reason to test for unit roots in panels. We are often interested in testing for unit roots collectively for cross-sectional units included in a certain panel. In particular, we may want to test for and against the existence of unit roots in not all, but only a fraction of cross-sectional units. Such formulation seems more appropriate for the investigation of important hypotheses such as the purchasing power parity and the growth convergence theories, among many others. The hypotheses can be tested more effectively using order statistics such as maximum and minimum of individual tests rather than their average as in the majority of the existing tests. 1

3 Our approach is based on the IV approach by Chang (22), which uses as instruments nonlinear transformations of the lagged levels to deal with cross-sectional dependencies of unknown form. We show in the paper that the nonlinear IV methodology can still be used in cointegrated panels. Indeed, the presence of cointegration can be allowed simply by using an orthogonal set of functions as instrument generating functions. The tests by Chang (22) are invalid in the presence of cross-sectional cointegration, since they rely on the instruments generated by a single function for all cross-sectional units. If a set of orthogonal instrument generating functions are used, the resulting IV t-ratios become asymptotically normal and independent, in the presence of cointegration as well as the cross-correlation of the error terms. The distributions of their various functionals including maximum and minimum are therefore nuisance parameter free and can easily be obtained. We conduct a set of simulations to evaluate the finite sample performances of our tests. Largely, our tests perform well and are preferred to other existing tests. In particular, our tests perform significantly better than other tests when there are cointegrating relations in the panel. The other tests suffer from severe size distortions in such cases. The performances of our order statistics in small samples, however, are mixed and are dependent upon the sample size T in time series dimension: The minimum statistic performs quite well even with moderate T, whereas the maximum statistic requires a large T for reliable performance. For the purpose of illustration, we apply our tests to the analysis of the purchasing power parity (PPP) hypothesis. Our tests do not provide any evidence in favor of the PPP hypothesis for real exchange rates, which is quite in contrast to the results in the previous literature. See Chang (22) and Wu and Wu (21) for some recent examples. All of the previous results were, however, obtained using the tests that assume either cross-sectional independence or no cointegration. The rest of the paper is organized as follows. Section 2 provides background and heuristics for our approach. In Section 2, we introduce all important aspects of the methodology developed in the paper. The discussions in the section are nontechnical, and the results are presented and interpreted intuitively. The rigorous and mathematical developments of our theory are included in Section 3. Section 3 introduces assumptions and specifically defines the models and test statistics for general panel models, and develops the necessary asymptotics. The results from simulations and empirical applications are summarized in Sections 4 and 5, respectively, and the concluding remarks follow in Section 6. The mathematical proofs are collected in Appendix. 2. Background and Heuristics We consider a panel model generated by y it = α i y i,t 1 + u it, i = 1,..., N; t = 1,..., T, (1) where (u it ) will be specified later. As usual, the index i denotes individual cross-sectional units, such as individuals, households, industries or countries, and the index t denotes time periods. We will make various simplifying assumptions throughout this section, to 2

4 effectively focus on the main issues. These assumptions are made for the expositional purpose, and by far stronger than necessary. They will be relaxed in later sections. We will, however, allow (u it ) to be correlated cross-sectionally across i, as well as serially over time t. Moreover, the number of time series observations T i for each individual i may differ across cross-sectional units, which is necessary to analyze unbalanced panels. 2.1 Unit Root Hypotheses We are interested in testing unit root null hypotheses for the panel given in (1). More precisely, we consider the following sets of hypotheses. Hypotheses (A) H : α i = 1 for all i versus H 1 : α i < 1 for all i. Hypotheses (B) H : α i = 1 for all i versus H 1 : α i < 1 for some i. Hypotheses (C) H : α i = 1 for some i versus H 1 : α i < 1 for all i. Hypotheses (A) and (B) both include the same null hypothesis, which implies that the unit root is present in all individual units. However, their null hypotheses compete with different alternative hypotheses. It is tested in Hypotheses (A) against the hypothesis that all individual units are stationary, while in Hypotheses (B) the alternative is that there are some stationary individual units. On the contrary, the null hypothesis in Hypotheses (C) holds as long as the unit root exists in at least one individual unit, and is tested against the alternative hypothesis that all individual units are stationary. The alternative hypotheses in both Hypotheses (B) and (C) are the negations of their null hypotheses. This is not the case for Hypotheses (A). Virtually all the existing literature on panel unit root tests effectively looks at Hypotheses (A). Some recent works, including Im, Pesaran and Shin (23) and Chang (22), allow for heterogeneous panels, and formulate the null and alternative hypotheses as in Hypotheses (B). However, strictly speaking, their use of average t-ratios can only be justified for the test of Hypotheses (A). To properly test Hypotheses (B), the minimum, instead of the average, of individual t-ratios might have been used. It is indeed not difficult to see that the tests based on the minimum would dominate those relying on the averages in terms of power for the test of Hypotheses (B). In our simulations, the minimum statistic actually yields much higher power than the average statistic, especially when only a small fraction of individual units are stationary. Hypotheses (C) have never been considered in the literature, though they seem to be more relevant in many interesting empirical applications such as tests for the purchasing power parity and the growth convergence theories. Note that the rejection of H in favor of H 1 in Hypotheses (C) directly implies that all (y it ) s are stationary, and therefore, purchasing power parities or growth convergences hold if we let (y it ) s be real exchange rates or differences in growth rates, respectively. No test, however, is available to deal with Hypotheses (C) appropriately. Here we propose to use the maximum of individual t-ratios for the test of Hypotheses (C). 3

5 2.2 Use of Covariates For general panel models, we will let (u it ) in (1) be specified as u it = β ix it + ε it, (2) where (x it ) are some stationary time series and (ε it ) are independent and identically distributed innovations with mean zero and finite variance. If (x it ) are observed, then the test of the hypothesis α i = 1 may be based on the regression y it = α i y i,t 1 + β ix it + ε it. (3) In fact, virtually all the existing panel unit root tests rely on the regression (3) with x it = ( y i,t 1,..., y i,t Pi ) for some P i 1. Of course, the lagged differences are included to account for the serial correlation in (u it ) and whiten the errors in regression. In the paper, we also use other covariates in regression (3) in addition to the lagged differences. The unit root regression with covariates was first considered in Hansen (1995) and studied subsequently by Chang, Sickles and Song (21). It was referred to by them as covariates augmented Dickey-Fuller (CADF) regression. Both Hansen (1995) and Chang, Sickles and Song (21) show that using covariates offers a great potential in power gain for the test of a unit root. We may indeed readily show that the test of the hypothesis α i = 1 in regression (3) is asymptotically identical to that in regression y it = α i y i,t 1 + ε it. (4) This is due to the asymptotic orthogonality between the stationary and integrated regressors, i.e., (x it ) and (y i,t 1 ). Note that the variance of the error term (ε it ) in (4) is smaller than that of (u it ) in (1), as long as (x it ) are correlated with (u it ). In general, the error variance will be reduced if the unit root regression is augmented with any relevant stationary covariates. Clearly, stationary covariates having higher correlations with (u it ) would more effectively reduce the error variance in the unit root regression. In many panels of interest, we naturally expect to have short-run dynamics that are inter-related across different cross-sectional units, which would make it necessary to include the dynamics of other cross-sectional units to properly model own dynamics. 2 Therefore, we include in (x it ) ( y jt, y j,t 1,..., y j,t Qi ) for some j s not equal to i and Q i 1, as well as the own lagged differences. In the PPP application reported in the paper, we choose as covariates the current and lagged differences of other cross-sectional units that have highest correlations with the errors from the regression only on its own lagged differences. On the other hand, in the simulations, we simply include the current differences of all other cross-sectional units with equal weights. This is mainly to reduce the required computing time. 2 We may even need to take into consideration the long-run trends of other cross-sectional units, since in the presence of cointegration error correction mechanism comes into play and the cross-sectional long-run disequilibrium errors would interfere with own short-run dynamics. 4

6 2.3 Nonlinear IV Estimation and Its Asymptotics Our methodology developed in the paper relies on a particular type of IV estimation. To focus on basic ideas, we will consider the simple unit root regression given by (1) throughout this section. The actual tests developed in the paper are based on the unit root regression augmented with covariates as in (3) and applicable for much more general models. For our IV estimation of regression (1), we use the instrument generated by a nonlinear function F i F i (y i,t 1 ) for the lagged level y i,t 1 of each cross-sectional unit i = 1,..., N. The idea of using this type of IV estimation was explored earlier by Chang (22) to deal with the crosssectional dependency of unknown form. The transformations (F i ) will be referred to as the instrument generating functions (IGF). We assume that Assumption 2.1 Let (F i ) be regularly integrable and satisfy (a) xf i(x)dx for all i and (b) F i(x)f j (x)dx = for all i j. The class of regularly integrable transformations was first introduced in Park and Phillips (1999), to which the reader is referred for details. They are just transformations on R satisfying some mild technical regularity conditions. Assumption 2.1 (a) needs to hold, since otherwise we would have instrument failure and the resulting IV estimator becomes inconsistent. It is analogous to the non-orthogonality (between the instruments and the regressors) requirement for the validity of IV estimation in standard stationary regressions. See Chang (22) for more detailed discussions. Assumption 2.1 (b) is newly introduced here and necessary to allow for the presence of cointegration. If cointegration is present, the procedure in Chang (22) relying on the same IGF for all cross-sectional units becomes invalid. This will be explained in detail in what follows. To understand the asymptotics of our statistics, we need to introduce some basic notions. Momentarily, we assume for simplicity that (u it ) are mean zero and independently and identically distributed across t with finite variance, and define a stochastic process U T for u t = (u 1t,..., u Nt ) as [T r] U T (r) = T 1/2 on [, 1], where [s] denotes the largest integer not exceeding s. invariance principle that U T d U u t Then it follows from as T, where U = (U 1,..., U N ) is an N-dimensional vector Brownian motion. The invariance principle plays an essential role in the development of our asymptotics. Our asymptotic theory also involves the local time of Brownian motion, which we will introduce briefly below. The reader is referred to Park and Phillips (1999, 21), Chang, 5

7 Park and Phillips (21), and the references cited there for the concept of local time and its use in the asymptotics for nonlinear models with integrated time series. The local time L i of the Brownian motion U i, for i = 1,..., N, is defined by 1 L i (t, s) = lim ϵ 2ϵ t 1{ U i (r) s < ϵ} dr. Roughly, the local time L i measures the time that the Brownian motion U i spends in the neighborhood of s, up to time t. It is well known that L i is continuous in both t and s. For any locally integrable function G on R, we have an important formula t G(U i (r)) dr = G(s)L i (t, s) ds, (5) which is called the occupation times formula. Our main statistical results are obtained by the sample mean asymptotics and sample covariance asymptotics for nonlinear nonstationary models developed in Park and Phillips (1999, 21) and Chang, Park and Phillips (21). Let G be any bounded locally integrable function on R. For the sample mean asymptotics, we have 1 T T 1 G(y i,t 1 ) d T G( T U i (r))dr = T = G(s)L i (1, s/ T )ds = L i (1, ) G( T s)l i (1, s)ds G(s)ds + o a.s. (1) (6) by the successive applications of the occupation times formula (5), change of variables and the continuity of L i (1, ). For the sample covariance asymptotics, we have 1 4 T T G(y i,t 1 )u it d whose limit distribution is determined by its quadratic variation 4 T 1 T G 2 ( T U i (r))dr = L i (1, ) 1 G( T U i )du i, (7) G 2 (s)ds + o a.s. (1), (8) and its quadratic covariation with U i (r) = r du i 1 4 T G( T U i (r))dr = 1 ) 4 (L i (1, ) G(s)ds + o a.s. (1) T a.s.. (9) We may derive the asymptotics in (8) and (9) similarly as above in (6). Now it follows from (9) that the sample covariance is asymptotically independent of U i, and therefore, independent also of its local time L i. Consequently, we may deduce that the sample covariance in (7) has asymptotic mixed normal distribution 1 4 T T ( ) G(y i,t 1 )u it d MN, L i (1, ) G 2 (s)ds (1) 6

8 with mixing variate given by the limit quadratic variation in (8). Now we let τ i be the IV t-ratio for testing α i = 1 for the i-th cross-sectional unit. For simplicity, we assume that the variance of (u it ) is identical for all i and known to be unity. Unknown error variances can be consistently estimated and do not affect our main asymptotic results. Let IGF F i be regularly integrable. Under the null hypothesis of unit root, we have ( 1 τ i = T T F 2 i (y i,t 1 ) ) 1/2 ( 1 4 T T F i (y i,t 1 )u it ) and therefore, we may readily deduce from the above mean and covariance asymptotics respectively in (6) and (1) that τ i d N(, 1) (11) as T. Under Assumption 2.1, we may indeed extend the individual asymptotic normality in (11) and show that τ i s are asymptotically independent, as well as normal, across all cross-sectional indexes i s for which α i = 1. As in (7), we have 1 4 T 1 4 T T T F i (y i,t 1 )u it d F j (y j,t 1 )u jt d 4 T 4 T 1 1 F i ( T U i )du i, F j ( T U j )du j, which become independent if and only if their quadratic covariation 1 σ ij T F i ( T U i (r))f j ( T U j (r))dr a.s. (12) as T, where σ ij denotes the covariance between (u it ) and (u jt ). However, it is well known that 1 F i ( T U i (r))f j ( T U j (r))dr = O a.s. (log T/T ) (13) for any Brownian motions U i and U j so long as they are not degenerate, and this implies that the condition (12) holds even when σ ij. Chang (22) uses this result to develop the unit root tests for panels with crosssectionally correlated innovations. However, (12) does not hold in the presence of cointegration between (y it ) and (y jt ). In this case, their limiting Brownian motions U i and U j become degenerate. If the cointegrating relationship is given by the unit coefficient, for instance, then we would have U i = U j, and therefore, 1 T F i ( T U i (r))f j ( T U j (r))dr = L i (1, ) (F i F j )(s)ds + o a.s. (1), as in (6). Consequently, the asymptotic independence of τ i and τ j generally breaks down, and holds only when F i and F j are orthogonal. This is the reason why the method by, 7

9 Chang (22) becomes invalid in the presence of cointegration, and explains why we need to use an orthogonal set of IGF s to preserve the asymptotic independence for cointegrated cross-sectional units. 2.4 Unit Root Tests in Panels For the tests of Hypotheses (A) (C), we define S = 1 N N i=1 S min = min i, 1 i N S max = max i, 1 i N where τ i as earlier is the IV t-ratio for the i-th cross-sectional unit. The average statistic S is proposed for the test of Hypotheses (A), and comparable to other existing tests. The minimum statistic S min is more appropriate for the test of Hypotheses (B). To test for Hypotheses (B), the average statistic S can also be used, but the test based on S min would be preferable as discussed previously. The maximum statistic S max can be used to test Hypotheses (C). Obviously, the average statistic S and the minimum statistic S min cannot be used to test for Hypotheses (C), since they would have incorrect sizes. The limit distributions of the statistics S, S min and S max can be easily derived from that τ i s are asymptotically independent and standard normal. We now let Φ be the distribution function for the standard normal distribution, and let λ be the size of the tests. The critical values c(λ) of the average test S for Hypotheses (A) are defined as usual from Φ(c(λ)) = λ. For a given size λ, we define c M max(λ) and c min (λ) by τ i, Φ(c M max(λ)) M = λ, (1 Φ(c min (λ))) N = 1 λ where M is the number of cross-sectional units having unit roots. These provide the critical values of the statistics S min and S max for the tests of Hypotheses (B) and (C). The following table shows the tests and critical values that should be used to test each of Hypotheses (A) (C). Hypotheses Test Statistics Critical Values Hypotheses (A) S c(λ) Hypotheses (B) S min c min (λ) Hypotheses (C) S max c M max(λ) The critical values c M max(λ) and c min (λ) for sizes λ = 1%, 5% and 1% are tabulated in Table 1 for some cases of M, N up to 1. 8

10 3. Models, Test Statistics and Statistical Theory In this section, we explicitly define model and test statistics, and establish the limit distribution theories of the tests. The models and asymptotic theories are truly general. In particular, they allow for the cross-sectional dependency in both the long-run and the short-run. We permit not only the cross-correlation of the innovations and/or crosssectional dynamics in the short-run, but also the comovements of the stochastic trends in the long-run. The models with deterministic trends are also considered. Moreover, we let each cross-sectional unit have different number of time series observations T i to accommodate unbalanced panels. 3.1 Models We now generalize the data generating process for our model introduced in (1). The initial values (y 1,..., y N ) of (y 1t,..., y Nt ) do not affect our subsequent asymptotic analysis as long as they are stochastically bounded, and therefore we set them at zero for expositional brevity. We let y t = (y 1t,..., y Nt ) and assume that there are N M cointegrating relationships in the unit root process (y t ), which are represented by the cointegrating vectors (c j ), j = 1,..., N M. 3 The usual vector autoregression and error correction representation allow us to specify the short-run dynamics of (y t ) as P N ij N M y it = a ij y j,t k + b ij c jy t 1 + ε it (14) j=1 k=1 for each cross-sectional unit, where (ε it ) are white noise, i = 1,..., N, and is the difference operator. To ensure our representation in (14), we assume that Assumption 3.1 Let (y t ) permit a finite order VAR representation and has N M linearly independent cointegrating relationships. Moreover, if we let u t = y t, then we may write u t = Π(L)ε t, where ε t = (ε 1t,..., ε Nt ), L is the lag operator, and Π(z) = k= Π kz k with Π = I and k= k Π k <. As is well known, we may deduce from the Granger representation theorem that (y t ) can be written as in (14), and that rank Π(1) = M. It follows from the Beveridge-Nelson decomposition that j=1 u t = Π(1)ε t + (ũ t 1 ũ t ), where ũ t = Π(L)ε t 3 Therefore, M is the number of unit roots in an N-dimensional time series (y t ), which we assumed to be unknown. When we formally present our statistical theory in Section 3.3, we will relate M with M the number of cross-sections having unit roots defined earlier in Section

11 with Π(z) = k= Π k z k and Π k = j=k+1 Π j. Consequently, y t = Π(1) t ε k + (ũ ũ t ). k=1 Note that ( Π k ) is absolutely summable due to the 1-summability of (Π k ) given in Assumption 3.1, and therefore, (ũ t ) is well defined and stationary. Moreover, if M < N and if C is an N (N M ) matrix such that C Π(1) =, then each column (c j ), j = 1,..., N M, of C represents a cointegrating vector for (y t ). The data generating process for the innovations (ε t ) is assumed to satisfy the following assumption. Assumption 3.2 (ε t ) is an iid (, Σ) sequence of random variables with E ε t l < for some l > 4, and its distribution is absolutely continuous with respect to Lebesgue measure and has characteristic function φ such that lim s s r φ(s) =, for some r >. Assumption 3.2 lays out the technical conditions that are required to invoke the asymptotic theories for the nonstationary nonlinear models developed by Park and Phillips (1999). Our unit root tests at individual levels will be based on the regression P i y it = α i y i,t 1 + α i,k y i,t k + β i,k w i,t k + ε it (15) k=1 for i = 1,..., N, where we interpret (w it ) as covariates added to the augmented Dickey- Fuller (ADF) regression for the i-th cross-sectional unit. As we explained earlier, we mainly consider as covariates ( y jt ) for some j s not equal to i. We may use a single unit or multiple units, possibly with fixed weights, of other cross-sections. It is important to note that the vector autoregression and error correction formulation of the cointegrated unit root panels in (14) suggests that we use such covariates. We may obviously rewrite (1) and (14) as (15) with the lagged differences of other cross sections and linear combinations of the lagged levels of all cross sections as covariates. In the subsequent development of our theory, we will assume that the data generating process is given by (14) under Assumptions 3.1 and 3.2, and that (y it ) is generated as in (15) with an appropriate selection of covariates (w it ). Though not explicitly dealt with for our development of asymptotics in the paper, we may let P i increase appropriately with the sample size and allow for a broader class of models that are not representable as in (15) with fixed P i. Under the setting that P i increases with the sample size, the inclusion of covariates (w it ) in (15) is irrelevant from the perspective of asymptotic theory. Our tests will indeed be valid with no or any arbitrary set of covariates as long as P i at an appropriate rate. This is well known in the unit root test literature, and can be readily shown along the lines of, e.g., Chang and Park (22). 4 However, the choice of covariates can be very important for the finite 4 The proof for the asymptotic validity of our tests with P i is essentially identical to that of ADF Q i k= 1

12 sample performances of the tests. For instance, without a proper collection of covariates we may have to include excessively many differenced lag terms of its own to fit the data generated by (14). This would certainly have adverse effects on the finite sample powers of the tests. 3.2 Test Statistics Our tests are based on the nonlinear IV estimation of regression (15). Naturally, we use the variables themselves as instruments for the augmented regressors x it = ( y i,t 1,..., y i,t Pi ; w it, w i,t 1,..., w i,t Q i ). (16) Hence, for the entire regressors (y i,t 1, x it ), we use the instruments given by (F i (y i,t 1 ), x it), similarly as in Chang (22). For IGF s, we use the Hermite functions of odd orders k = 2i 1, i = 1,..., N. They satisfy all the conditions in Assumption 2.1, and therefore, can be used as a proper set of IGF s. The Hermite function G k of order k, k =, 1, 2,..., is defined as G k (x) = (2 k k! π) 1/2 H k (x)e x2 /2, (17) where H k is the Hermite polynomial of order k given by dk H k (x) = ( 1) k e x2 dx k e x2. It is well known that the class of Hermite functions introduced above forms an orthonormal basis for L 2 (R), i.e., the Hilbert space of square integrable functions on R. We thus have G j (x)g k (x)dx = δ jk for all j and k, where δ jk is the Kronecker delta. Therefore, we may define the IGF s (F i ) by F i = G 2i 1 for i = 1,..., N. The orthogonality of the IGF s yields the orthogonality of the IV t-ratios only when (y it ) is asymptotically of the same scale across i = 1,..., N. Note that the orthogonality between functions is not preserved under arbitrary rescaling of their arguments. This will be seen more clearly in the proof of Lemma 3.1. Roughly, we have for i = 1,..., N and for r [, 1] y i[ti r] = y i[ti r] T i d Ti U i (r), Ti test, and therefore, we do not provide the details in the paper. Such asymptotics based on the scheme letting P i justify the common use of ADF test for a broad class of general unit root processes, even though the test is applied on a finite order autoregression. 11

13 where U i is the limit Brownian motion introduced in Section 2.3. Therefore, (y it ) s are asymptotically of the same scale if U i s have the same variance and T i s are identical for all cross-sectional units i = 1,..., N. The variance of U i, i.e., the long-run variance of ( y it ) can be easily estimated consistently. Therefore, we may assume without loss of generality that ( y it ) s have the same long-run variance for i = 1,..., N, since if necessary we may always normalize them using their estimated long-run variances, so that they all have the unit long-run variance. Unless stated otherwise, this convention will be made throughout the paper. The use of different IGF s, such as Hermite functions of distinctive orders, across crosssectional units makes it necessary to order cross-sectional units. Theoretically, it does not matter how to order them and the IGF s can be applied to cross-sectional units in any order. The actual test values, however, are dependent in finite samples upon particular orders of cross-sectional units. Therefore, it seems desirable to have a fixed rule in ordering cross-sectional units and avoid the arbitrariness caused by different ordering schemes. In our simulation and PPP application, we order cross-sectional units according to their estimated long-run variances. 5 Consequently, i = 1 denotes the individual unit with the smallest estimated long-run variance, and i = N the unit with the largest estimated longrun variance. We found by simulation that this ordering provides more desirable finite sample performances of our tests compared to other ordering schemes such as those based on descending order of the estimated long-run variances. For the unbalanced panels, T i s are different across i = 1,..., N. In this case, we set an arbitrary cross-sectional unit, say, the first unit, to be the scale numeraire, and let yit T1 = y it. (18) Ti Note that y i[t i r] = T 1 y i[ti r] Ti d T1 U i (r). for i = 1,..., N and for r [, 1]. Therefore, given our convention that U i s all have the same variance, (yit ) s are asymptotically of the same scale for i = 1,..., N even in unbalanced panels. For expositional brevity, we assume in the subsequent presentation of our theories that the scale adjustment is already done for all (y it ) s, and continue to use (y it ) in the place of (yit ) for i = 1,..., N. This should cause no confusion. Given all the conventions introduced above, we now define individual IV t-ratios explicitly. Let y i = y i,1. y i,ti, y li = y i,. y i,ti 1, X i = x i,1. x i,t i, ε i = 5 We may use the long-run variance estimates standardized by the short-run variance estimates to order the cross-sectional units. The standardized long-run variances are independent of the scale, and may be appropriate when the observations across cross-sectional units differ in scale. This idea was suggested by one of the referees. ε i,1. ε i,ti, 12

14 where (x it ) is defined in (16). 6 Then the covariates augmented autoregression (15) can be written in matrix form as y i = y li α i + X i β i + ε i = Y i δ i + ε i, (19) where β i = (α i,1,..., α i,pi ; β i,, β i,1,..., β i,q i ), Y i = (y li, X i ), and δ i = (α i, β i ). For the regression (19), we consider the estimator ˆδ i of δ i given by ˆδ i = (ˆαi ˆβ i ) = (Z iy i ) 1 Z iy i = ( Fi (y li ) y li F i (y li ) ) 1 ( X i Fi (y li ) y i X i y li X i X i X i y i ), (2) where Z i = (F i (y li ), X i ) with F i (y li ) = (F i (y i, ),..., F i (y i,ti 1)). The estimator ˆδ i is thus defined to be the IV estimator constructed from the instruments Z i. The IV estimator ˆα i for the AR coefficient α i corresponds to the first element of ˆδ i given in (2). Under the null, we have where A Ti = F i (y li ) ε i F i (y li ) X i (X ix i ) 1 X iε i = T i T i F i (y i,t 1 )ε it F i (y i,t 1 )x it ˆα i 1 = B 1 T i A Ti, (21) ( Ti B Ti = F i (y li ) y li F i (y li ) X i (X ix i ) 1 X iy li = T i and the variance of A Ti T i F i (y i,t 1 )y i,t 1 F i (y i,t 1 )x it is given by x it x it ( Ti x it x it ) 1 Ti x it ε it, ) 1 Ti x it y i,t 1, under Assumption 2.2, where σ 2 i EC Ti C Ti = F i (y li ) F i (y li ) F i (y li ) X i (X ix i ) 1 X if i (y li ) = T i T i F i (y i,t 1 ) 2 F i (y i,t 1 )x it ( Ti x it x it ) 1 Ti x it F i (y i,t 1 ). For testing the unit root hypothesis α i = 1 for each i = 1,..., N, we construct the t-ratio statistic from the nonlinear IV estimator ˆα i defined in (21). More specifically, we construct such an IV t-ratio for testing for a unit root in (1) or (15) as τ i = ˆα i 1 s(ˆα i ), (22) 6 The regressor (x it) includes the lagged differenced terms y i,t 1,..., y i,t Pi of y it. By convention, we assume that y it is observed for t = P i,..., T i and set the range of time index to be t = 1,..., T i. This convention will be made throughout the paper. 13

15 where s(ˆα i ) is the standard error of the IV estimator ˆα i given by The ˆσ i 2 is the usual variance estimator given by Ti 1 residual from the augmented regression (15), viz., s(ˆα i ) 2 = ˆσ 2 i B 2 T i C Ti. (23) Ti ˆε2 it, where (ˆε it) is the fitted P i ˆε it = y it ˆα i y i,t 1 ˆα i,k y i,t k ˆβ i,k w i,t k = y it ˆα i y i,t 1 x ˆβ it i. k=1 It is natural in our context to use the IV estimate (ˆα i, ˆβ i ) given in (2) to get the fitted residual ˆε it. However, we may obviously use any other estimator of (α i, β i ) as long as it yields a consistent estimate for the residual error variance. 3.3 Statistical Theory Now we let Q i k= α i = 1, 1 i M and α i < 1, M + 1 i N, so M is the number of cross-sectional units having unit roots. 7 By convention, we set M = if α i < 1 for all 1 i N. To derive the joint asymptotics for all τ i, i = 1,..., M, we define T min = min T i, T max = max T i 1 i M 1 i M and assume Assumption 3.3 Assume T min, T max /T 2 min, which will simply be signified by T in our subsequent asymptotics. Assumption 3.3 gives the premier for our asymptotics. Our asymptotics are based on T -asymptotics and require that the time spans for all cross-sectional units be large for our asymptotics to work. However, we allow for unbalanced panels and they only need to be balanced asymptotically. Our conditions here are fairly weak, and we may therefore expect them to hold widely. The conditions in Assumption 3.3 are stronger than those in Assumption 4.1 of Chang (22), which require T min and T max (log T max ) 4 /Tmin 3. This is because we allow for the presence of cointegration. Even though it can be effectively dealt with by using a set of orthogonal IGF s, we need a slightly more stringent assumption on the balancedness of the underlying panels. We have 7 We defined earlier M to be the number of independent unit roots, net of the number of cointegration relationships. In general, we have M M, with the inequality becomes strict if there exists cointegrating relationships in (y t). Due to the orthogonality of the set of IGF s, our asymptotics are not affected by the presence of cointegration, and they do not involve M. 14

16 Lemma 3.1 Under Assumptions 2.1 and , we have τ i d N(, 1) as T, jointly for all i = 1,..., M and independently across i = 1,..., M. Lemma 3.1 extends the asymptotic results for the IV t-ratios derived earlier by Chang (22) for the panel unit root models with cross-sectional dependencies in the errors. In particular, our lemma here establishes the asymptotic individual normality and crosssectional independence of the IV t-ratios for the general panel unit root models with the presence of covariates and cross-sectional cointegration. The asymptotic normality of the individual IV t-ratio τ i is derived easily from the asymptotics for nonlinear transformations of integrated processes established in Park and Phillips (1999, 21) and Chang, Park and Phillips (21), as we explained in Section 2.3. Under Assumptions 2.1 and , it is indeed straightforward to show that the asymptotic normality holds for our general model with covariates as T i, i = 1,..., M. On the other hand, the asymptotic independence of the individual IV t-ratios τ i s follows from the orthogonality of the IGF s even in the presence of cointegration. This was also discussed in detail in Section 2.3. The asymptotic independence of τ i s is crucial for the subsequent development of our theory. Note that here we allow for the presence of cointegration as well as unknown form of cross-sectional dependencies in the innovations. Our limit theory here is thus fundamentally different from the unit root asymptotics that apply to the usual t-ratios. As is well known, the usual t-ratios have nonnormal and cross-sectionally dependent limit distributions, for the unit root panels with crosssectional dependencies in the errors. For our general unit root panels with the presence of covariates and cross-sectional cointegration, the asymptotics for the usual t-ratios diverge even further from our results for the IV t-ratios. As shown in Hansen (1995) and Chang, Sickles and Song (21), the usual t-ratios in the unit root models with covariates have limit distributions that involve nuisance parameters. Moreover, in the presence of cross-sectional cointegration, it is not difficult to see that the usual t-ratios are strongly dependent and far from being independent cross-sectionally even when the cross-sectional dimension becomes large. The following lemma summarizes the asymptotic behaviors of S, S min and S max. Theorem 3.2 Let Assumptions 2.1 and hold. If M = N, If 1 M N, then lim P{S c(λ)} T = λ, lim min c min (λ)} T = λ. lim P{S max c M max(λ)} = λ, T lim P{S max c 1 max(λ) = c(λ)} λ. T On the other hand, S, S max p if M =, and S min p if M < N. 15

17 Theorem 3.2 implies that all our tests have the prescribed asymptotic sizes. The tests using statistics S and S min with critical values c(λ) and c min (λ), respectively, have the exact size λ asymptotically under the null hypotheses in Hypotheses (A) and (B). However, the null hypothesis in Hypotheses (C) is composite, and the rejection probabilities of the test relying on S max with critical values c(λ) may not be exactly λ even asymptotically. The size λ in this case is the maximum rejection probabilities that may result in under the null hypothesis. Theorem 3.2 also shows that all our tests are consistent for Hypotheses (A) (C). Our unit root test based on the IV t-ratio statistic is consistent. Under the alternative of stationarity, the IV t-ratio τ i given in (22) indeed diverges at the T i -rate. This can be shown using the same argument as the one in Chang (22, p.27), to which the interested reader is referred. Consequently, the IV t-ratio τ i diverges at the same rate as the usual OLS-based t-type unit root tests such as the augmented Dickey-Fuller test, under the alternative of stationarity. 3.4 Models with Deterministic Components The models with deterministic components can be analyzed similarly using properly demeaned or detrended data. As argued and demonstrated in Chang (22), a proper demeaning or detrending scheme required here must be able to remove the nonzero mean and time trend successfully, while preserving the predictability of the instruments and ultimately the Gaussian limit theory for the nonlinear IV unit root tests. We now introduce our demeaning and detrending schemes. The methods are basically similar to those given in Chang (22), but with additional attention paid to the covariates. If the time series (z it ) with a nonzero mean is given by z it = µ i + y it, (24) where the stochastic component (y it ) is generated as in (1), then we may test for the presence of a unit root in (y it ) from the covariates augmented regression (15) defined with the demeaned series y µ it, yµ i,t 1, yµ i,t k and wµ i,t k of z it, z i,t 1, z i,t k and w i,t k, viz., P i y µ it = α iy µ i,t 1 + Q i α i,k y µ i,t k + β i,k wµ i,t k + e it. (25) k=1 As in Chang (22), the demeaned series y µ it and yµ i,t 1 are constructed by subtracting respectively from y it and y i,t 1 the mean of the partial sample up to time (t 1), i.e., (t 1) 1 t 1 k=1 z ik, which is the least squares estimator of µ i in (24) computed using the sample up to (t 1) only. 8 The (t 1)-adaptive demeaning is used to maintain the martingale property and thus the Gaussian limit theory of our nonlinear IV t-ratios. The terms y µ i,t k are simply the differences of the original data, i.e., z i,t k, for k = 1, 2,..., P i, 8 Note that Ly µ it yµ i,t 1. Our procedure requires that the same partial sample mean up to time (t 1) should be subtracted from both y it and y i,t 1 to define y µ it and yµ i,t 1. See Chang (22) for more detailed discussion on this. k= 16

18 and the (e it ) are regression errors. For the demeaned covariate series w µ i,t k, we may use z j,t k for k =, 1,..., Q i and j i. We may then construct the nonlinear IV t-ratio statistic τ µ i for the process (z it ) in (24) with a nonzero mean based on the nonlinear IV estimator for α i computed from the covariates augmented regression (25), just as in (22). Similarly as in the models with nonzero means, we may also test for unit roots in the models with deterministic time trends using the nonlinear IV unit root test constructed from the properly detrended data. More explicitly, for the time series with a linear time trend z it = µ i + δ i t + y it (26) where (y it ) is generated as in (1), we may test for a unit root in (y it ) from the regression (15) defined with the properly detrended series yit τ, yτ i,t 1, yτ i,t k and wτ i,t k of the given data z it, z i,t 1, z i,t k and w i,t k, viz., P i Q i yit τ = α i yi,t 1 τ + α i,k yi,t k τ + β i,k wτ i,t k + e it. (27) k=1 The detrended data yit τ and yτ i,t 1 are constructed following the adaptive detrending scheme introduced in Chang (22) which uses the least squares estimators of the drift and trend coefficients, µ i and δ i, from the model (26) using again the observations up to time (t 1) only. The adaptive detrending preserves the predictability of our instrument F (yi,t 1 τ ). The lagged differences are also detrended just as in Chang (22), viz., yi,t k τ = z i,t k z iti /T i, where the grand sample mean of z it, i.e., Ti 1 Ti k=1 z ik, is used to eliminate the nonzero mean of z i,t k, for k = 1,..., P i. Clearly, we may use z j,t k z jtj /T j for k =, 1,..., Q i and j i as covariates. The grand sample mean z jtj /T j of z jt is used here also to eliminate the nonzero mean of z j,t k, for k =, 1,..., Q i. The nonlinear IV t-ratio τi τ for testing unit roots in the model (26) with linear time trend is then defined as in (22) using the nonlinear IV estimator for α i computed from the regression (27) based on the adaptively detrended data. With the adaptive demeaning or adaptive detrending, the predictability of our nonlinear instrument F i (y µ i,t 1 ) or F i(yi,t 1 τ ) is retained, and consequently our previous results continue to apply, including the normal distribution theory for the IV t-ratio statistic. We may now derive the limit theories of the statistics τ µ i and τi τ for the models with nonzero means and deterministic trends in the similar manner as we did to establish the limit theory given in Lemma 3.1. Corollary 3.3 The results in Lemma 3.1 hold also for both τ µ i and τ τ i. It follows that the standard normal limit theory of the covariates augmented nonlinear IV t-ratio statistics continues to hold for the models with deterministic components. k= 17

19 4. Simulations In this section, we conduct a set of simulations to investigate the finite sample performances of the newly proposed panel unit root tests for testing the three hypotheses formulated in Section 2. For the simulations, we consider the model (24) with a nonzero mean and the stochastic component (y it ) specified as in (1) with the innovations (u it ) generated by the following three DGP s: DGP1 : u it = β i u i,t 1 + η it, DGP2 : u it = β i u i,t 1 + ν i ξ t + η it, DGP3 : u it = β i u i,t 1 + ν i ξ t + η it, for i = 1,..., N; t = 1,..., T, where ξ t is the scalar common stochastic trend and η t = (η 1t,..., η Nt ) an N-dimensional innovation vector with cross-sectional dependence. Note that DGP1 is the same model analyzed in Chang (22), DGP2 is a version of the widely used dynamic factor model, and DGP3 is new and introduced here to allow for cointegration across cross-sectional units. By exploring these distinct forms of crosscorrelations, we aim to see how our tests perform relative to the existing tests in each situation. DGP1 generates cross-sectional correlations from dependent innovations (η it ) with the covariance matrix, say V, which is unrestricted except for being symmetric and nonsingular. The innovations in DGP2 and DGP3 also have the same error covariance V. However, DGP2 and DGP3 have another level of cross-correlations coming from the presence of the common stochastic trend (ξ t ). Using this common factor, DGP2 and DGP3 can generate stronger cross-sectional dependencies compared to those generated by DGP1. In DGP2, the generated series (y it ) contain both nonstationary common factors and nonstationary individual errors under the unit root null hypothesis, hence there is no cointegrating relationship among cross sections. In DGP3, however, cross-sectional cointegration is present, and they are generated by the nonstationary common stochastic trend coupled with stationary individual errors. Thus, there exists a cointegrating relationship between any pair of (y it ) and (y jt ), with (N 1) linearly independent cointegrating relations among N individual units. The parameters in our DGP s are generated as follows. The AR coefficient β i is drawn randomly from Uniform[.2,.4]. The parameter ν i, so called factor loadings, that controls the relative importance of common versus idiosyncratic shocks is also drawn randomly from Uniform[.5, 3]. The processes (ξ t ) and (η t ) are independent and drawn from iid N(, 1) and iid N(, V ), respectively. The parameters of the (N N) covariance matrix V of the innovations (η t ) are also drawn randomly. To ensure that V is a symmetric positive definite matrix and to avoid the near-singularity problem, we generate V following the steps outlined in Chang (22). The steps are presented here for convenience: (1) Generate an (N N) matrix R from Uniform[,1]. (2) Construct from R an orthogonal matrix H = R(R R) 1/2. (3) Generate a set of N eigenvalues, λ 1,..., λ N. Let λ 1 = r > and λ N = 1 and draw λ 2,..., λ N 1 from Uniform[r,1]. 18

20 (4) Form a diagonal matrix Λ with (λ 1,..., λ N ) on the diagonal. (5) Construct the covariance matrix V using the spectral representation V = HΛH. Constructed as such, the covariance matrix V will be symmetric and nonsingular with eigenvalues ranging from r to 1. The ratio r of the minimum eigenvalue to the maximum provides a measure for the degree of correlations and heterogeneity in the error covariance matrix V. The covariance matrix V becomes singular as r tends to zero and becomes spherical as r approaches 1. For the simulations, we set r =.1 as in Chang (22), and use the correlation matrix obtained from the covariance matrix V in the usual manner. We set the AR coefficient α i in (1) at α i = 1 for the cross sections that have unit root, and generate α i randomly from Uniform[.8,1] for the stationary cross sections. Since we are using randomly drawn parameter values, we simulate 1 times and report the average of the finite sample performances of the tests. Each simulation run is carried out with 3, iterations. We assume that there exist nonzero means in the data, and thus we use the adaptively demeaned series as described in Section 3.4 for our nonlinear IV tests. The panels with the cross-sectional dimensions N = 1, 2, 5 and the time series dimensions T = 1, 3 are considered for the 5% nominal test size. In our simulations, we consider the following tests: Nonlinear IV Unit Root Tests S C, Smin C, SC max S H, Smin H S A, Smin A ave, min and max tests with single IGF and no covariate ave and min tests with orthogonal IGF s and no covariate ave and min tests with orthogonal IGF s and covariate Other Existing Tests IPS MP test by Im, Pesaran and Shin test by Moon and Perron The test S C, based on a single integrable IGF, is developed in Chang (22) for Hypotheses (A). It is considered here for the comparison with other IV tests. The tests Smin C and Smax C are, respectively, the minimum and maximum counterparts of the average test S C for Hypotheses (B) and (C). The tests S C, Smin C and SC max are valid only for DGP1 and DGP2. The presence of cross-sectional cointegration in DGP3 invalidates these tests. Considered subsequently are our new tests S H and S A, which are the average tests based on the orthogonal IGF s, respectively without and with covariate. In the simulations, due to computing time, we simply use as covariate for each cross-section i the cross-section average of current differences ( y jt ) of the remaining N 1 other cross-sections, which may not necessarily be the best choice. 9 The tests Smin H and SA min are their minimum 9 Though we do not report the details here, we have observed that we may improve the performance of S A by selecting the best set of covariates from a pool of all potential covariates described in Section 3.1. This can be done by choosing the ones that have the highest correlations with the error process. For more 19

21 counterparts. We may compare the performance of (S H, Smin H ) and (SA, Smin A ) to analyze the effect of including covariate. All tests based on orthogonal IGF s are applicable for all DGP s. They are, however, intended to effectively deal with the presence of crosssectional cointegration in DGP3. The maximum tests with the orthogonal IGF s are not examined. The cross-sectional cointegration presumes the presence of unit roots, and therefore, Hypotheses (C) are not particularly interesting in this situation. We also consider two other existing tests IPS and MP, by Im, Pesaran and Shin (23) and Moon and Perron (24), respectively. The IPS test is based on the average of the individual t-ratios computed from the usual sample ADF regressions with mean and variance modifications. More explicitly, the IPS test is defined as N( t N N 1 N i=1 IPS = E(t i)) N 1 N i=1 var(t i) where t i is the t-statistic for testing α i = 1 for the i-th sample ADF regression, and t N = N 1 N i=1 t i. The values of the expectation and variance, E(t i ) and var(t i ), for each individual t i depend on T i and the lag order P i, and are computed via simulations from independent normal samples. See Table 3 in Im, Pesaran, and Shin (23). The IPS test assumes cross-sectional independence and hence it is not valid under our DGP s. The MP test models the cross-sectional dependence using an approximate dynamic linear factor model. They model the error process y it = u it as u it = δ i ξ t + ε it, where (ε it ) are cross-sectionally independent idiosyncratic shocks, δ i factor loadings for the i-th unit, and ξ t an unknown number of unobservable dynamic factors that are common to all individual units. Then, it follows under the null of α i = 1 for each i = 1,..., N, t t y it = y i,t 1 + u it = y i + δ i ξ k + ε ik. Under this setup, first the data is demeaned, and the cross-correlations generated by the nonstationary common factors t k=1 ξ k are removed by projecting the panel data to the space orthogonal to the factor loadings δ i. Then, they calculate the t-statistic to test for unit roots existing in such de-factored panel data, say ỹ, based on the modified pooled OLS estimator α pool of α obtained from the pooled regression k=1 k=1 ỹ = αỹ 1 + ε, (28) where ỹ = (ỹ 1,..., ỹ N ) and ε = ( ε 1,..., ε N ). 1 More explicitly, the MP test is defined as NT ( αpool 1) MP =, 3 ϕ 4 ε/ ω ε 4 discussions on this, see Hansen (1995) and Chang, Sickles, and Song (21). This ensures that the effective error has smallest variance, which will in turn lead to the largest power gains from the inclusion of those covariates. For the PPP application in the following section, we propose to choose as the covariates for the i-th regression the current and lagged differences of j-th series for some j i, which has the highest correlation with the regression error in the i-th regression. 1 Note that in (28) the regression errors are now cross-sectionally independent since the de-factored data, ey, used in the estimation, are free of the common factors that generate the correlations across cross sections. 2

22 where ω 2 ε = 1 N N i=1 ω2 ε,i, ϕ 4 ε = 1 N N i=1 ω4 ε,i and ω2 ε,i are the long-run variance of (ε it). 11 Here, we note that the MP test may be used to test for unit roots in the models generated by DGP1 or DGP2 only when η it s are independent across i. It is invalid under DGP3 even with the cross-sectional independence of η it s. For the construction of the nonlinear IV test S C with single IGF, we use the same IGF suggested in Chang (22). For the tests S H and S A with orthogonal IGF s, we use a set of N Hermite functions defined in (17). We normalize the data as suggested in Section 3.2 by dividing y it s, i = 1,..., N, by their estimated long-run variances. We then scale the data by multiplying the scale constant c chosen by the following scheme c = K T 1/2, (29) for all i = 1,..., N. 12 For the tests S C, Smin C and SC max with single IGF, we set the constant at K =4. On the other hand, we use the constant K =3.5 and 2.5 respectively for the tests S H and S A, and K =2 for the minimum tests Smin H and SA min. Note that we use a smaller value of K for the average test with orthogonal IGF s when the covariate is augmented. This is because in this case we are effectively controlling the cross-correlations twice, by using the orthogonal IGF s and by including the covariate. The included covariate would pick up some cross-correlations, and thus the IGF s need not be as integrable as those used for the tests without the covariate. The cross-correlations are controlled only by the orthogonal IGF s for the tests without the covariate. The simulation results are reported in Figures We first discuss the performances of the nonlinear IV average tests along with IPS and MP, and subsequently present the results for the nonlinear IV minimum and maximum tests. As expected, the IPS and MP tests fail in all DGP s we consider here, as can be seen from the considerable size distortions. Their size distortions are large especially in DGP3 with the strongest crosscorrelations. The distortions of IPS test get larger as N increases for a given T and they do not seem to be attenuated as T increases. The MP test seems to have better size property as T gets large, although they are still not very satisfactory. Among the three DGP s, the IPS suffer least from the size distortions in DGP1, while the MP test performs best in DGP2. The simple average IV test S C performs well for DGP1, but it fails in DGP2 and DGP3. That S C does not work in DGP3 with cross-cointegration is not surprising. However, the poor performance of S C in DGP2 is rather unexpected since it is still valid in DGP2. It appears that the simple average IV test does not work properly in the presence of strong cross-correlations induced by common factors. As expected, our new nonlinear IV tests, S H and S A, based on the orthogonal IGF s perform reasonably well in DGP3. The S H test has good sizes for all T s and N s considered. On the other hand, the covariate augmented test S A has good sizes for the modest 11 The maximum number of factors for the MP test is set at 8 as in Moon and Perron (24). 12 This scaling scheme is essentially the same as the one used in Chang (22). Here we set the scaling constant c as in (29) only for small T, so that it does not affect our asymptotic results in the paper. As T gets large, the choice of c indeed becomes unimportant and does not affect the performances of our tests. More formally, we may set c = (K T 1/2 )1{T T } + (K T 1/2 )1{T > T } for some fixed T. The choice of T = 5 seems to give reasonable results for both small and large T s. 13 Detailed results of the simulation are available from the authors upon request. 21

23 size N, but starts to under-reject for the larger N. In DGP1 and DGP2, both tests suffer from downward size distortions, and the problem is worse for the larger N. Their size problem, however, tends to improve as T gets large, and the sizes of S H and S A do seem to become reasonably good for the modest size N and the larger T even in DGP1 and DGP2. We note that the use of orthogonal IGF s is not necessary for DGP1 and DGP2. As mentioned above, the size performance of the covariate augmented test S A may not be as good as that of the test S H, indicating that the use of the covariate may deteriorate the size property of our test. However, S A test performs noticeably better than S H in terms of power, so there is a substantial gain in using the covariate. The results on the finite sample performances of the minimum tests are reported also in Figures 1-6 along with those of their average counterparts we just discussed above. The minimum tests are expected to perform better under the alternative hypothesis that only some fractions of the panel are stationary. To specify such alternatives, we set I, the number of I() series in the panel, to be 1%, 2%, 3%,, 1% of each of the given N. Overall, all of the minimum tests, Smin C, SH min and SA min, work reasonably well in all DGP s with stable sizes and large discriminatory powers compared to their average test counterparts. The Smin C test works very well for DGP1. The finite sample sizes of SC min are indeed very close to the nominal test size for all DGP s and for all combinations of N and T, even though the test is valid only for DGP1 and DGP2, and not for DGP3. The Smin H and SA min tests tend to under-reject for large N for all DGP s considered. Notice that when only a fraction, not all, of the panel are stationary, the powers of the minimum tests are significantly larger than their average counterparts. This is as expected, since the minimum would obviously provide more discriminatory power than the average against such alternatives where only some cross sections are stationary. Finally, we discuss the performances of the maximum tests which are constructed for Hypotheses (C) with the composite null hypothesis where only some fractions of the panel are nonstationary. For the null hypotheses, we set I 1, the number of I(1) series in the panel, to be 1%, 2%, 3%,, 1% of the given N, as in the formulations for the alternatives of Hypotheses (B). The simulation results on the sizes and powers of the maximum test Smax C are provided in Figures 7-8. The test Smax C yields reasonable sizes and powers for DGP1. In DGP1, Smax C has stable sizes that are quite close to the nominal test size in most of the N and T combinations we consider here, except when I 1 is very small. The finite sample powers of Smax C are also quite good in DGP1. Not surprisingly, the test Smax C does not perform well in DGP2 with strong cross-correlations driven by the common factor. Our simulation results therefore seem to indicate that the maximum test Smax C relying on a single IGF is reliable only for the panels with mild cross-correlations such as those generated by DGP1. In this sense, the usefulness of the maximum tests seems somewhat limited. 5. Empirical Illustrations In this section, we illustrate the usefulness of our new panel unit root tests by applying them to the long standing empirical problem of testing for the purchasing power parity (PPP) hypothesis. Although considerable amount of intellectual efforts have been put out 22

24 to investigate this problem, it is widely agreed that the question of whether or not the PPP holds has not been settled yet. We revisit this PPP problem with our new tests to examine whether the PPP holds for the post-1973 period of floating exchange rates. As noted by Murray and Papell (22), the data covering only the recent float have some advantages over the long horizon data. For instance, it does not mix observations from different nominal exchange rate regimes and the data are available for more countries. The data we use are the quarterly and monthly end-of-period real exchange rates for twenty industrialized countries, 14 obtained from the International Monetary Fund s International Financial Statistics. For monthly real exchange rates, we consider seventeen out of twenty countries since no monthly data is available for Australia, Ireland, and New Zealand. Our data cover the entire recent float period, , and include 14 and 312 observations for quarterly and monthly data, respectively. All exchanges rates are in natural logarithms and constructed using U.S. dollar as numeraire currency and CPI s as deflators. Dealing with the presence of cross-sectional dependency has been one of the main econometric issues in the panel unit root testing. In the previous studies, it is commonly assumed away by imposing cross-sectional independence, 15 which can be quite unrealistic in many economic applications including the studies on the exchanges rates. Due to the strong links across economies, the real exchange rates usually exhibit high correlations across cross sections. Indeed this is seen clearly from the estimated correlation matrix of the first differences of the quarterly real exchange rates presented in Table 2. They are highly correlated especially among European countries. Pairwise correlations are particularly strong (over.9) among Austria, Belgium, Denmark, France, Germany, and the Netherlands. The pairwise correlations among the other countries are also significant and most of them are over.6. These numbers suggest that the overall cross-correlations among the real exchange rates are quite substantial and therefore should not be ignored. 16 Moreover, such high correlations among the exchange rates from European countries indicate that there may exist some cointegrating relations among them. Note that our tests allow for dependent and possibly cointegrated panels. We carry out panel unit root testing for the full panels and also for the smaller panels which exclude the exchange rates from Australia, Canada, Greece, Japan, and Portugal. The smaller panels are considered to see if the observation made by Papell (22) also holds in our study. Papell (22) observes that the exchange rates from the five aforementioned countries follow different behavioral patterns, and that removal of these countries from the panel enables one to reject the unit root null hypothesis. The results are provided in Tables 3 and 4 for the quarterly and monthly data, respectively. In each table, the first line reports the results obtained from using the full panel, while the second line provides the results from using the smaller panel. The order of the lagged differences is selected for each cross section by the BIC criterion with the maximum lag order set at The covariates 14 The countries considered include Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, and the United Kingdom. 15 O Connell (1998) controls for cross-sectional dependence by using the orthogonalized data which are obtained by the GLS transformation based on the estimated covariance matrix of the data. 16 Similar observations were made for monthly real exchange rates in Moon and Perron (24). 17 The BIC selected 1 for the lag order for most of the cross sections. The results are not very sensitive 23

25 used for each cross-section are chosen from the pool of current and lagged differences of other cross-sections, according to the selection rule that ranks highest the one with the highest correlation with the regression error. For both quarterly and monthly data, our tests, S H and S A, do not reject the unit root null hypothesis. This is the case regardless of whether we exclude those five countries from the full panel or not. That the tests S H and S A, constructed with and without the covariate, produced the same results in all cases indicates that the use of the covariate does not affect the results. We note in particular that all of the minimum tests, Smin C, Smin H and SA min, do not reject the null hypothesis for all cases considered. Recall from the simulation results reported in the previous section that all of the minimum tests show stable sizes for the sample sizes of our data, and have more discriminatory power over any average tests especially when only a fraction of the panel are stationary. Therefore, the results from the minimum tests strongly indicate that there is no stationary series in the panels of the real exchange rates. In sharp contrast, the S C and IPS tests provide strong evidence against the unit root null hypothesis. When the full panels are used, the S C test rejects the null for both quarterly and monthly data, and the IPS test for quarterly data. If we exclude the five aforementioned countries from the panel, these results get strengthened and the IPS test now reject the null also for the monthly data. The test results of S C and IPS, however, may be spurious due to the likely existence of cointegrating relations among the cross sections in our data. These tests do not allow for cross-sectional cointegration and thus suffer from serious upward size distortions, as we saw from our simulation experiments with cointegrated panels. On the other hand, the MP test does not reject the null hypothesis in most of the cases considered. It rejects only for the quarterly data with the smaller panel. 18 However, these results may also be misleading, again due to the presence of cointegration, which the test is not designed to deal with. We also saw from our simulations that the MP test may have either upward or downward size distortions depending upon the time series and cross-sectional dimensions. That our new average tests, S H and S A, and all the minimum tests, Smin C, SH min and Smin A, do not provide any evidence in favor of the PPP hypothesis is in sharp contrast with most of the results in the previous literature. See Chang (22) and Wu and Wu (21) for some recent examples. All of the previous results were, however, obtained using the tests that assume cross-sectional independence and/or no cointegration. As our simulation experiments demonstrated, such tests suffer greatly from size distortions when there are strong cross-correlations induced by common stochastic trends and therefore are not suitable for cointegrated panels. On the other hand, our tests are designed to handle the cointegration among cross sections as well as cross-correlated regression errors, and hence our results against the PPP hypothesis seem reliable and appealing. to the lag order used. 18 These results are consistent with those of Moon and Perron (24), which are obtained using the period-average real exchange rates for the period

26 6. Conclusions This paper extends the existing methodologies for panel unit root tests in three important directions. First, we allow for dependencies across individual cross sections at both short-run and long-run levels. We allow for inter-relatedness of cross-sectional short-run dynamics and the presence of long-run relationships in cross-sectional levels. Many panels of practical interest seem to have such complicated cross-sectional dependencies. Second, our theory permits the use of covariates to increase the power. Covariates may naturally include the terms to account for cross-sectional dependencies as well as the ones to control idiosyncrasies of individual cross-sectional units. If properly chosen, the inclusion of covariates would substantially improve the power of the test, as demonstrated earlier by several authors. Third, we re-examine the formulation of the unit root hypothesis in panels, and propose to analyze the null and alternative hypotheses that only a fraction of cross-sectional units have unit roots. Such formulations are more appropriate for some of the most commonly investigated panel models such as purchasing power parity and growth convergence. The tests developed in the paper are valid for very general panels. They allow not only for unknown forms of cross-sectional dependencies at several different levels, but also for various kinds of heterogeneities such as unbalancedness, differing dynamics and other idiosyncratic characteristics for individual units. These indeed appear to be the common characteristics of many panels used in empirical studies. Nevertheless, none of the currently available tests are applicable for such general panels. In addition to their applicability, our tests are easy to implement. The relevant statistical theories are quite straightforward and all Gaussian, and the critical values are given by either the standard normal or its simple functionals. Appendix: Mathematical Proofs Proof of Lemma 3.1 The proof for the asymptotic normality in (11) is similar to Chang (22). The asymptotics of the following sample moments involving integrable transformations of a unit root process follow directly from Park and Phillips (1999, 21) as T 1/4 i T 1/2 i T i T i ( ) F i (y i,t 1 )ε it d MN, σi 2 L i (1, ) F i (s) 2 ds, F i (y i,t 1 ) 2 d L i (1, ) F i (s) 2 ds. Note that our asymptotic results here are different, up to a scalar factor, from Chang (22) that represents the asymptotics in terms of the local time of the standard Brownian motion. Since y i,t k, k = 1,..., P i, and w i,t j, j =,..., Q i, are stationary, we also 25

27 have T 3/4 i T i T 3/4 T i i F i (y i,t 1 ) y i,t k p, for all k = 1,..., P i, F i (y i,t 1 )w i,t j p, for all j =,..., Q i, due to the asymptotic orthogonality between stationary variables and integrable transformations of integrated processes established in Lemma 5 (e) of Chang, Park and Phillips (21). Using (21) and (23), we may write τ i defined in (22) as τ i = B 1 T i A Ti (ˆσ 2 i B 2 T i C Ti ) 1/2 = A Ti ˆσ i C 1/2 T i = ˆσ i ( T 1/4 i T 1/2 i T i T i F i (y i,t 1 )ε it F i (y i,t 1 ) 2 ) 1/2 + o p (1). Therefore, the asymptotic normality in (11) follows immediately. Now we prove that τ 1,..., τ M are asymptotically independent, which follows due to asymptotic normality if we show that τ i and τ j are asymptotically orthogonal for all i, j = 1,..., M. The proof goes exactly the same as that in Chang (22), except for the pairs τ i and τ j for which the corresponding cross-sectional units (y it ) and (y jt ) are cointegrated. Note that Assumption 4.1 in Chang (22) holds under our Assumption 3.2. Therefore, we assume (y it ) and (y jt ) are cointegrated. To establish the asymptotic orthogonality of τ i and τ j, it suffices to show that 1 4 Ti T j F i ( T i U iti (r))f j ( T j U jtj (r))dr p. (3) See Chang (22) for details. As shown in Chang, Park and Phillips (21), we have T 1 log T 1 = 1 T 1 log T 1 F i ( T 1 U iti (r))f j ( T 1 U jtj (r))dr 1 F i ( T 1 U i (r))f j ( T 1 U j (r))dr + o p (1). However, due to our convention in (18) made on scale adjustment, we may assume that and that 1 = F i ( T i U iti (r))f j ( T j U jtj (r))dr 1 F i ( T 1 U iti (r))f j ( T 1 U jtj (r))dr U i = U j. 26

28 Moreover, we have 1 (F i F j )( T 1 U i (r))dr = O p (T 3/4 1 ), since F i and F j are assumed to be orthogonal and (F if j )(s)ds =. deduced from T 1/4 1 T1 (F i F j )(U i (r))dr = O p (1), which is shown in, e.g., Revuz and Yor (1994, Proposition 2.8, p.528). Notice that T 1/4 1 T1 (F i F j )(U i (r))dr = d T 3/4 1 1 (F i F j )( T 1 U i (r))dr, This can be which follows immediately from the change-of-variable formula and the fact that U i (T 1 r) = d T1 U i (r) for every r. We may assume without loss of generality that the numeraire unit has the largest number of observations, i.e., T 1 T i for all i. Then we have 1 4 Ti T j F i ( T i U iti (r))f j ( T j U jtj (r))dr = 4 1 T i T j F i ( T 1 U iti (r))f j ( T 1 U jtj (r))dr = 4 T i T j 1 ( ) = O p T 1/4 j /T 1/2 i ( ) = O p Tmax/T 1/4 1/2 min (F i F j )( ( 1/4 T i T 1/4 ) j log T 1 T 1 U i (r))dr + o p T 1 since T 1 is set at T max. Now (3) follows immediately given Assumption 3.3. Proof of Theorem 3.2 We first consider the case M = N. The statistic S has standard normal limiting distribution and, therefore, the stated result follows immediately. For the statistic S min, we note that { } lim P {S min x} = lim P T T = 1 since τ i s are asymptotically independent normals. i=1 min τ i x 1 i N N lim P{τ i > x} T i = 1 (1 Φ(x)) N 27

29 For the case 1 M N, we have Note that for i = 1,..., M, and for i = M + 1,..., N, lim P {S max x} = lim P T T = i=1 { } max τ i x 1 i N N lim P{τ i x} T i = Φ(x) M. lim P{τ i x} = Φ(x) T i lim P{τ i x} = 1 T i since τ i p as T i in this case. Therefore, lim P { S max c M max(λ) } = Φ(c M max(λ)) M = λ T and lim P {S max c(λ)} = Φ(c(λ)) M = λφ(c(λ)) M 1 λ T as was to be shown. The consistency of the tests then follows immediately from the result in Lemma 3.1. Proof of Corollary 3.3 The proof of this corollary is essentially the same as that of Corollary 5.1 in Chang (22). What is new is just how we handle the covariates with deterministic components, especially in the covariance asymptotics. Therefore, the details will be omitted. In the proof, we will also assume for simplicity that the panels are balanced, i.e., T i = T for all i. This is to avoid notational complexity, which is unnecessary and not essential. The proofs for the general unbalanced panels are virtually identical. For the demeaned series, note that the covariate is written as y µ j,t k = z j,t k = y j,t k, for k =, 1,..., Q i. Then it follows that the additional sample moment introduced by the included covariate vanishes in the limit T 3/4 T F i (y µ i,t 1 ) yµ j,t k = T 3/4 T F i (y µ i,t 1 ) y j,t k p, for k =, 1,..., Q i, since y j,t k, k =, 1,..., Q i, are stationary, due to the asymptotic orthogonality between stationary variables and integrable transformations of integrated processes established in Lemma 5 (e) of Chang, Park and Phillips (21). This implies that the leading term 1 4 T T F i (y µ i,t 1 )ε it 28

30 is dominant, and consequently the regression equation (25) becomes the legitimate equation for the test of unit roots in the stochastic component (y it ) just as in the case without the covariates considered in Chang (22). For the detrended series, we have However, we may easily deduce that T 3/4 T y τ j,t k = z j,t k 1 T z jt = y j,t k 1 T y jt. F i (y τ i,t 1)(T 1 y jt ) = T 3/4 (T 1/2 y jt ) and consequently, we have T 3/4 T F i (y τ i,t 1) y τ j,t k = T 3/4 T ( T 1/2 T F i (y τ i,t 1) ) = O p (T 3/4 ), F i (y τ i,t 1) y j,t k + O p (T 3/4 ) p, for k =, 1,..., Q i, similarly as above. Therefore, the leading term 1 4 T T F i (y τ i,t 1)ε it becomes dominant, and we may easily see that the regression equation (27) is valid for the test of unit roots in the stochastic component (y it ) of the time series (z it ) in (26) with a linear time series. References Baltagi, B.H., and C. Kao (2). Nonstationary panels, cointegration in panels and dynamic panels: A survey, in Nonstationary panels, panel cointegration, and dynamic panels, pp. 7-51, Advances in Econometrics, vol. 15. Amsterdam; New York and Tokyo: Elsevier Science, JAI Banerjee, A. (1999). Panel data unit roots and cointegration: An overview, Oxford Bulletin of Economics and Statistics, 61, Chang, Y. (22). Nonlinear IV unit root tests in panels with cross-sectional dependency, Journal of Econometrics, 11, Chang, Y. (24). Bootstrap unit root tests in panels with cross-sectional dependency, Journal of Econometrics, 12, Chang, Y. and J.Y. Park (22). On the asymptotics of ADF tests for unit roots, Econometric Reviews, 21,

31 Chang, Y., J.Y. Park and P.C.B. Phillips (21). Nonlinear econometric models with cointegrated and deterministically trending regressors, Econometrics Journal, 4, Chang, Y., R. Sickles, and W. Song (21), Bootstrapping unit root tests with covariates, Rice University, mimeographed. Choi, I. (21). Unit root tests for panel data, Finance, 2, Journal of International Money and Choi, I. (26). Combination Unit root tests for cross-sectionally correlated panels, in Frontiers of Analysis and Applied Research: Essays in Honor of Peter C.B. Phillips, D. Corbae, S. Durlauf and B. Hansen, Eds, Econometric Theory and Practice Series, Cambridge University Press, Cambridge. Hansen, B.E. (1995). Rethinking the univariate approach to unit root testing: Using covariates to increase power, Econometric Theory, 11, Im, K., H. Pesaran and Y. Shin (23). Testing for unit roots in heterogeneous panels, Journal of Econometrics 115, pp Levin, A., C. Lin and C.J. Chu (22). Unit root tests in panel data: Asymptotic and finite-sample properties, Journal of Econometrics 18, MacDonald, R. (1996). Letters, 5, Panel unit root tests and real exchange rates, Economics Maddala, G.S. and S. Wu (1999). A comparative study of unit root tests with panel data and a new simple test, Oxford Bulletin of Economics and Statistics, 61, Moon, H.R. and B. Perron (24). factors, mimeographed. Testing for a unit root in panels with dynamic Murray, C., and D. H. Papell (22), The Purchasing Power Parity Persistence Paradigm, Journal of International Economics, 56, O Connell, P.G.J. (1998). The overvaluation of purchasing power parity, Journal of International Economics, 44, Oh, K.Y. (1996). Purchasing power parity and unit root tests using panel data, Journal of International Money and Finance, 15, Papell, D.H. (1997). Searching for stationarity: Purchasing power parity under the current float, Journal of International Economics, 43, Papell, D.H. (22). The great appreciation, the great depreciation, and the purchasing power parity hypothesis, Journal of International Economics 57, Park, J.Y. and P.C.B. Phillips (1999). Asymptotics for nonlinear transformations of integrated time series, Econometric Theory, 15,

32 Park, J.Y. and P.C.B. Phillips (21). Nonlinear regressions with integrated time series, Econometrica, 69, Phillips, P.C.B. and H.R. Moon (2). Nonstationary panel data analysis: An overview of some recent developments, Econometric Reviews, 19, Phillips, P.C.B. and D. Sul (23). Dynamic panel estimation and homogeneity testing under cross section dependence, Econometrics Journal, 6, Revuz, D. and M. Yor (1994). Continuous Martingale and Brownian Motion, New Your: Springer-Verlag. Wu, J.L. and S. Wu (21). Is purchasing power parity overvalued?, Journal of Money, Credit and Banking, 33,

33 Table 1: Critical Values for S max and S min S max S min M 1% 5% 1% N 1% 5% 1%

34 Table 2. Correlation Matrix of First Differenced Quarterly Log Real Exchange Rates Aust. Belg. Denm. Finl. Fran. Germ. Gree. Irel. Ital. Neth. Norw. Port. Spai. Swed. Swit. U.K. Aust. Cana. Japa. Austria Belgium.965 Denmark Finland France Germany Greece Ireland Italy Netherl Norway Portugal Spain Sweden Switzer U.K Australia Canada Japan New Zeal

35 Table 3. Test Results for Quarterly Real Exchange Rates N T S C Smin C S H Smin H S A Smin A IPS MP * * * * -2.12* Notes: 1. The results in the second line are obtained excluding Australia, Canada, Greece, Japan, and Portugal. 2. * indicates significance at the 5% level. Table 4. Test Results for Monthly Real Exchange Rates N T S C Smin C S H Smin H S A Smin A IPS MP * * * Notes: 1. The results in the second line are obtained excluding Canada, Greece, Japan, and Portugal. 2. * indicates significance at the 5% level. 34

36 Figure 1. Size and Size-adjusted Powers for DGP1: N=2, T=1 Figure 2. Size and Size-adjusted Powers for DGP1: N=2, T=3 35

37 Figure 3. Size and Size-adjusted Powers for DGP2: N=2, T=1 Figure 4. Size and Size-adjusted Powers for DGP2: N=2, T=3 36