PANEL UNIT ROOT TESTS WITH CROSS-SECTION DEPENDENCE: A FURTHER INVESTIGATION

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1 Econometric heory, 26, 200, doi:0.07/s PAEL UI ROO ESS WIH CROSS-SECIO DEPEDECE: A FURHER IVESIGAIO JUSHA BAI AD SEREA G Columbia University An effective way to control for cross-section correlation when conducting a panel unit root test is to remove the common factors from the data. However, there remain many ways to use the defactored residuals to construct a test. In this paper, we use the panel analysis of nonstationarity in idiosyncratic and common components (PAIC) residuals to form two new tests. One estimates the pooled autoregressive coefficient, and one simply uses a sample moment. We establish their large-sample properties using a joint limit theory. We find that when the pooled autoregressive root is estimated using data detrended by least squares, the tests have no power. his result holds regardless of how the data are defactored. All PAIC-based pooled tests have nontrivial power because of the way the linear trend is removed.. IRODUCIO Cross-section dependence can pose serious problems for testing the null hypothesis that all units in a panel are nonstationary. As first documented in O Connell (998), much of what appeared to be power gains in panel unit root tests developed under the assumption of cross-section independence over individual unit root tests is in fact the consequence of nontrivial size distortions. Many tests have been developed to relax the cross-section independence assumption. See Chang (2002), Chang and Song (2002), and Pesaran (2007), among others. An increasingly popular approach is to model the cross-section dependence using common factors. he panel analysis of nonstationarity in idiosyncratic and common components (PAIC) framework of Bai and g (2004) enables the common factors and the idiosyncratic errors to be tested separately, and Moon and Perron (2004) test the orthogonal projection of the data on the common factors. Most tests are formulated as an average of the individual statistics or their p-values. he Moon and Perron (2004) tests (henceforth MP tests) retain the spirit of the original panel unit root test of Levin, Lin, and Chu (2002), which estimates and tests the pooled We thank three referees, a co-editor, and Benoit Perron for many helpful comments and suggestions. We also acknowledge financial support from the SF (grants SES and SES ). Address correspondence to Jushan Bai, Department of Economics, Columbia University; 022 IAB, 420 West 8 th Street, ew York, Y 0027, USA; jushan.bai@columbia.edu. 088 c Cambridge University Press /0 $5.00

2 PAEL UI ROO ESS 089 first-order autoregressive parameter. As pointed out by Maddala and Wu (999), the Levin et al. (2002) type tests have good power when autoregressive roots are identical over the cross sections. On the other hand, pooling individual test statistics may be more appropriate when there is heterogeneity in the dynamic parameters. Many papers have studied the finite-sample properties of various panel unit root tests. In this paper we try to understand whether the difference in finitesample properties can be traced to how the pooled autoregressive coefficient is estimated. o this end, we first develop a set of MP type tests using the PAIC residuals, and a panel version of the modified Sargan Bhargava test (hereafter the PMSB test) that simply uses the sample moments of these residuals but does not estimate the pooled autoregressive coefficient. We then use simulations to show that autoregressive coefficient based tests have minimal power whenever ρ is constructed using data that are detrended by least squares, irrespective of how the factors are removed. We develop new PAIC-based pooled tests that do not require explicit linear detrending. he three PAIC tests have reasonable power against the trend stationary alternative because they do not involve least squares detrending. he rest of the paper is organized as follows. In Section 2, we specify the data generating process (DGP), introduce necessary notation, and discuss model assumptions. In Section 3, we consider the PAIC residual-based MP type and PMSB tests. Section 4 discusses issues related to different tests. Section 5 provides finite-sample evidence via Monte Carlo simulations. Concluding remarks are given in Section 6, and the proofs are given in the Appendix. 2. PRELIMIARIES Let D it = p j=0 δ ijt j be the deterministic component. When p = 0, D it = δ i is the individual specific fixed effect, and when p =, an individual specific time trend is also present. When there is no deterministic term, D it is null, and we will refer to this as case p =. hroughout the paper, we let M z = I z(z z) z be a matrix that projects on the orthogonal space of z. In particular, the projection matrix M 0 with z t = for all t demeans the data, and M with z t = (, t) demeans and detrends the data. rivially, M is simply an identity matrix. he DGP is X it = D it + λ i F t + e it, () ( L)F t = C(L)η t, e it = ρ i e it + ε it, where F t is an r vector of common factors that induce correlation across units, λ i is an r vector of factor loadings, e it is an idiosyncratic error, and C(L) is an r r matrix consisting of polynomials of the lag operator L, C(L) = j=0 C j L j.

3 090 JUSHA BAI AD SEREA G Let M < denote a positive constant that does not depend on or. Also let A = tr(a A) /2. We use the following assumptions based on Bai and g (2004). Assumption A. (a) If λ i is nonrandom, λ i M; ifλ i is random, E λ i 4 M; (b) i= λ iλ i p,anr r positive definite matrix. Assumption B. (a) η t iid(0, η ),E η t 4 M; (b) var( F t ) = j=0 C j η C j > 0; (c) j=0 j C j < M; and (d) C() has rank r,0 r r. Assumption C. for each i, ε it = d i (L)v it, v it iid(0,) across i and over t, E v it 8 M for all i and t; j=0 j d ij M for all i; d i () 2 c > 0 for all i and for some c > 0. Assumption D. {v is }, {η t }, and {λ j } are mutually independent. Assumption E. E F 0 M, and for every i =,...,E e i0 M. Assumptions A and B assume that there are r factors. Assumption B allows a combination of stationary and nonstationary factors. Assumption C assumes cross-sectionally independent idiosyncratic errors, which is used to invoke some of the results of Phillips and Moon (999) for joint limit theory, and for crosssectional pooling. his assumption is similar to Assumption 2 of Moon and Perron (2004). We point out that many properties of the PAIC residuals derived in the Appendix are not affected by allowing some weak cross-sectional correlations among v it. he variance of v it in the linear process ε it is normalized to be ; otherwise it can be absorbed into d i (L). Assumption D assumes that factors, factor loadings, and the idiosyncratic errors are mutually independent. Initial conditions are stated in Assumption E. For the purpose of this paper we let F t = F t + η t, where is an r r matrix. he number of nonstationary factors is determined by the number of unit roots in the polynomial matrix equation, (L) = I L = 0. Under (), X it can be nonstationary when (L) has a unit root, or ρ i =, or both. Clearly, if the common factors share a stochastic trend, X it will all be nonstationary. An important feature of the DGP given by () is that the common and the idiosyncratic components can have different orders of integration. It is only when we reject nonstationarity in both components that we can say that the data are inconsistent with unit root nonstationarity.

4 PAEL UI ROO ESS 09 Other DGPs have also been considered in the literature on panel unit root tests. he one used in Phillips and Sul (2003) is a special case of () as they only allow for one factor, and the idiosyncratic errors are independently distributed across time. Choi (2006) also assumes one factor, but the idiosyncratic errors are allowed to be serially correlated. However, the units are restricted to have a homogeneous response to F t (i.e., λ i = ). A somewhat different DGP is used in Moon and Perron (2004) and Moon, Perron, and Phillips (2007). hey let X it = D it + X 0 it, (2) Xit 0 = ρ i Xit 0 + u it, u it = λ i f t + ε it, where f t and ε it are I(0) linear processes, f t and ε it are independent, and ε it are cross-sectionally independent. otably, under (2), X it has a unit root if ρ i =. his DGP differs from () in that it essentially specifies the dynamics of the observed series (e.g., if D it = 0, then X it = Xit 0 ), whereas () specifies the dynamics of unobserved components. Assuming Xi0 0 = 0 and ρ i = ρ for all i, (2) can be written in terms of () as follows: X it = D it + λ i F t + e it, where ( ρl)f t = f t and ( ρl)e it = ε it. When ρ i = for all i, wehave F t = F t + f t and e it = e it + ε it. In this case, both F t and e it are I(). When ρ i = ρ with ρ < for all i, wehavef t = ρ F t + f t and e it = ρe it + ε it, and so both F t and e it are I(0). hus the common and idiosyncratic components in (2) are restricted to have the same order of integration. ote that when ρ i are heterogeneous, (2) cannot be expressed in terms of (). But under the null hypothesis that ρ i = for all i, () covers (2). It follows that the assumptions used for DGP () are also applicable to DGP (2). he model considered by Pesaran (2007) is identical to DGP (2) as the dynamics are expressed in terms of the observable variable X it : X it = ( ρ i L)D it + ρ i X it + u it, (3) u it = λ i f t + ε it. he construction of the test statistics based on defactored processes requires the short-run, long-run, and one-sided variance of ε it defined as ( σεi 2 ) 2 = E(ε2 it ) = dij 2, ω2 εi = ij, λ εi = (ωεi j=0 j=0d 2 σ εi 2 )/2, respectively. hroughout, ω 4 εi = (ω2 εi )2 and ω 6 εi = (ω2 εi )3, etc. As in Moon and Perron (2004) we assume that the following limits exist and the first three are

5 092 JUSHA BAI AD SEREA G strictly positive: ωε 2 = lim i= ω 2 εi, σ2 ε = lim i= σ 2 εi, φε 4 = lim i= ω 4 εi, λ ε = lim i= he subscript ε may be dropped when context is clear. For future reference, let ω 2 ε = i= ω 2 εi, σ 2 ε = λ εi. i= σ 2 εi, φ 4 ε = i= ω 4 εi, λ ε = i= be consistent estimates of ω 2 ε, σ 2 ε, φ4 ε, and λ ε, respectively. Assumptions necessary for consistent estimation of these long-run and one-sided long-run variances are given in Moon and Perron (2004). hese will not be restated here so that we can focus on the main issues we want to highlight here. λ εi (4) 3. PAIC POOLED ESS In Bai and g (2004) we showed that under () testing can still proceed even when both components are unobserved and without knowing a priori whether e it is nonstationary. he strategy is to obtain consistent estimates of the space spanned by F t (denoted by F t ) and the idiosyncratic error (denoted by ê it ). In a nutshell, we apply the method of principal components to the first differenced data and then form F t and ê it by recumulating the estimated factor components. More precisely, when D it in () is zero (p = ) or an intercept (i.e., p = 0), the first difference of the model is X it = λ i F t + e it. Denote x it = X it, f t = F t, and z it = e it. hen x it = λ i f t + z it is a pure factor model, from which we can estimate ( λ,..., λ ) and ( f 2,..., f ) and ẑ it for all i and t. Define F t = t s=2 f s and ê it = t s=2 ẑ is. When p =, we also need to remove the mean of the differenced data, which is the slope coefficient in the linear trend prior to differencing. his leads to x it = X it X i, f t = F t F, and z it = e it e i, where X i is the sample mean of X it over t and where F and e i are similarly defined.

6 PAEL UI ROO ESS 093 Bai and g (2004) provide asymptotically valid procedures for (a) determining the number of stochastic trends in F t, (b) testing if ê it are individually I() using augmented Dickey Fuller (ADF) regressions, and (c) testing if the panel is I() by pooling the p values of the individual tests. If π i is the p-value of the ADF test for the ith cross-section unit, the pooled test is Pê = 2 i= logπ i 2 4. (5) he test is asymptotically standard normal. For a two-tailed 5% test, the null hypothesis is rejected when Pê exceeds.96 in absolute value. ote that Pê does not require a pooled ordinary least squares (OLS) estimate of the AR() coefficient in the idiosyncratic errors. Pooling p values has the advantage that more heterogeneity in the units is permitted. However, a test based on a pooled estimate of ρ can be easily constructed by estimating a panel autoregression in the (cumulated) idiosyncratic errors estimated by PAIC, i.e., ê it. Specifically, for DGP with p =,0, or, pooled OLS estimation of the model ê it = ρê it + ε it yields ρ = tr(ê ê ) tr(ê ê ), where ê and ê are ( 2) matrices. he bias-corrected pooled PAIC autoregressive estimator ρ and the test statistics depend on the specification of the deterministic component D it.forp = and 0, ρ + = tr(ê ê ) λ ε tr(ê ê ), and the test statistics are ( ρ + ) P a =, (6) 2 φ ε 4/ ω4 ε P b = ( ρ + ) 2 tr(ê ê ) ω2 ε. (7) For p =, ρ + = tr(ê ê ) tr(ê ê ) + 3 σ ε 2 ω ε 2 = ρ + 3 σ ε 2 ω ε 2, φ 4 ε

7 094 JUSHA BAI AD SEREA G and the test statistics are P a = ( ρ + ) (36/5) φ ε 4 σ ε 4/ ω8 ε P b = ( ρ + ), (8) 2 tr(ê ê ) 5 6 ω 6 ε φ ε 4 σ, (9) ε 4 where λ ε, σ ε 2, ω2 ε, and φ ε 4 are defined in (4). hese nuisance parameters are estimated based on AR() residuals ε = ê ρ ê = [ ε, ε 2,..., ε ] with ε i being ( 2) for all i. HEOREM. Let ρ + be the bias-corrected pooled autoregressive coefficient for the idiosyncratic errors estimated by PAIC. Suppose the data are generated by () and Assumptions A E hold. hen under the null hypothesis that ρ i = for all i, as, with / 0, P a d (0,) and Pb d (0,). Jang and Shin (2005) studied the properties of P a,b for p = 0 by simulations. heorem provides the limiting theory for both p = 0 and p =. It shows that the t tests of the pooled autoregressive coefficient in the idiosyncratic errors are asymptotically normal. he convergence holds for and tending to infinity jointly with / 0. It is thus a joint limit in the sense of Phillips and Moon (999). he P a and P b are the analogs of t a and t b of Moon and Perron (2004), except that (a) the tests are based on PAIC residuals and (b) the method of defactoring of the data is different from the method of Moon and Perron (2004). By taking first differences of the data to estimate the factors, we also simultaneously remove the individual fixed effects. hus when p = 0, the ê it obtained from PAIC can be treated as though they come from a model with no fixed effect. It is also for this reason that in Bai and g (2004), the ADF test for ê it has a limiting distribution that depends on standard Brownian motions and not its demeaned variant. When p =, the adjustment parameters used in P a,b are also different from t a,b of Moon and Perron (2004). In this case, the PAIC residuals ê it have the property that /2 ê it converges to a Brownian bridge, and a Brownian bridge takes on the value of zero at the boundary. In consequence, the Brownian motion component in the numerator of the autoregressive estimate vanishes. he usual bias correction made to recenter the numerator of the estimator to zero is no longer appropriate. his is because the deviation of the numerator from its mean, multiplied by, is still degenerate. However, we can do bias correction to the estimator directly because ( ρ ) converges to a constant. In the present case, ( ρ ) p 3σ 2 ε /ω2 ε. his leads to ρ + as defined previously for p =. his definition of ρ + is crucial for the tests to have power in the presence of incidental trends.

8 PAEL UI ROO ESS he Pooled MSB An important feature that distinguishes stationary from nonstationary processes is that their sample moments require different rates of normalization to be bounded asymptotically. In the univariate context, a simple test based on this idea is the test of Sargan and Bhargava (983). If for a given i, e it = ε it has mean zero and unit variance and is serially uncorrelated, then Z i = 2 t= e2 it 0 W i (r) 2 dr. However, if e it is stationary, Z i = O p ( ). Stock (990) developed the modified Sargan Bhargava test (MSB test) to allow ε it = e it to be serially correlated with short- and long-run variance σεi 2 and ωεi 2, respectively. In particular, if ω2 εi is an estimate of ωεi 2 that is consistent under the null and is bounded under the alternative, 2 MSB = Z i / ω εi 2 0 Wi 2 (r)dr under the null and degenerates to zero under the alternative. hus the null is rejected when the statistic is too small. As shown in Perron and g (996) and g and Perron (200), the MSB has power similar to the ADF test of Said and Dickey (984) and the Phillips Perron test developed in Phillips and Perron (988) for the same method of detrending. An unique feature of the MSB is that it does not require estimation of ρ, which allows us to subsequently assess whether power differences across tests are due to the estimate of ρ. his motivates the following simple panel nonstationarity test for the idiosyncratic errors, denoted the panel PMSB test. Let ê be obtained from PAIC. For p =,0, the test statistic is defined as ( ( ) ) tr ê ê ω 2 PMSB = 2 ε /2, (0) φ ε 4/3 where ω ε 2/2 estimates the asymptotic mean of (/ 2 )tr(ê ê) and the denominator estimates its standard deviation. For p =, the test statistic is defined as ( ( ) ) tr ê ê ω 2 PMSB = 2 ε /6. () φ ε 4/45 he variables ω ε 2 and φ ε 4 are defined in (4), and are estimated from residuals ε = ê ρe, where ρ is the pooled least squares estimator based on ê. he null hypothesis that ρ i = for all i is rejected for small values of PMSB. We have the following result: HEOREM 2. Let PMSB be defined as in (). Under Assumptions A E, as, with / 0, we have PMSB (0,). d he convergence result again holds in the sense of a joint limit. But a sequential asymptotic argument provides the intuition for the result. For a given i, Z i = 2 t= ê2 it converges in distribution to 0 ω2 εi V i (r) 2 dr when p =, where V i is

9 096 JUSHA BAI AD SEREA G a Brownian bridge. Demeaning these random variables and averaging over the i give the stated result. Comparing the PMSB test with P a and P b tests when p = is of special interest. From Bai and g (2004), ê it = e it e i (e i e i ) (t ) + o p (), which has a time trend component with slope coefficient of O p ( /2 ). Because of the special slope coefficients, detrending is unnecessary when constructing P a and P b tests, but suitable bias correction for the autoregressive coefficient is necessary to avoid certain degeneracy (see the discussion of degeneracy following heorem ). Detrending is also unnecessary with the PMSB test because the limit of 2 ê it is simply a Brownian bridge. ot having to detrend ê it is key to having tests with good finite-sample properties when p =. 4. HE MP ESS he autoregressive coefficient ρ can also be estimated from data in levels X it = ( ρl)d it + ρ X it + u it, u it = λ i f t + ε it. As is standard in the literature on unit roots, there are three models to consider: a base case model (A) that assumes D it is null; a fixed effect model (B) that assumes D it = a i ; and an incidental trend model (C) that has D it = a i + b i t. ote that we use p =,0, to represent the DGP and use Models A C to represent how the trends are estimated. Let = (λ,...λ ) and X and X be by matrices. Based on the first step estimator ρ = tr(x M z X) tr(x M, one computes the z X ) residuals û = M z X ρ M z X, from which a factor model is estimated to obtain = ( λ,..., λ ), where M z is a projection matrix defined in Section 2. he biascorrected, defactored, pooled OLS estimator defined in Moon and Perron (2004) is ρ + = tr(x M z XM ) ψ ε tr(x M z X M ), where M = I ( ) and ψ ε is a bias correction term (given subsequently) defined on the residuals of the defactored data ε = [M z X ρ M z X ]M. he MP tests, denoted t a and t b, have the same form as P a and P b defined in (6) and (7), with X and X replacing ê and ê both in ρ + and in the tests. hat is, ( ρ + ) t a =, K a φ ε 4/ ω4 ε t b = ( ρ + ) 2 tr(x M ω ε z X )K 2 b, where M z and the parameters ψ ε, K a, and K b are defined as follows. When the data are untransformed (Model A) M z = I 2, ψ ε = λ ε, K a = 2, and K b =. φ 4 ε

10 PAEL UI ROO ESS 097 When the data are demeaned (Model B), then M z = M 0, φ ε = σ ε 2/2, K a = 3, and K b = 2. When the data are demeaned and detrended (Model C), 3 M z = M and φ ε = σ ε 2/2, K a = 5/4, and K b = 4. Model A is valid when p is or0in the DGP. here are two important differences between P a,b and t a,b. First, our tests explicitly estimate the factors and errors before testing, whereas the MP tests implicitly remove the common factors from the data and thus do not explicitly define ê it. As a result of this, the bias adjustments are also different. For p =, we obtain a bias adjustment for ρ directly, whereas Moon and Perron (2004) adjusted the bias for the numerator of ρ. Also, Moon and Perron removed the deterministic terms by the least squares estimation of the incidental parameters, whereas PAIC takes the first difference of the data. ote that in the Moon and Perron setup, Models A and B are both valid for p = 0 because under the null hypothesis that ρ =, the intercepts are identically zero. However, the finite-sample properties of the MP test are much better when A is used. Model B (demeaning) gives large size distortions, even though removing the fixed effects seems to be the natural way to proceed. It should also be remarked that Moon and Perron (2004) estimated the nuisance parameters by averaging over ω εi 2 and σ εi 2, where these latter are defined using ε = ûm. Importantly, ω εi is a function of ρ that is biased. However, the unbiased ρ + itself depends on ω εi. his problem can be remedied by iterating ρ + till convergence is achieved. his seems to improve the size of the test when p = but does not improve power. Simulations show that the MP tests are dominated by P a,b when p =. Because the MP tests are applied directly to the observable series, one might infer that t a and t b are testing the observed panel of data. It is worth reiterating that after the common factors are controlled for, one must necessarily be testing the properties of the idiosyncratic errors. his is clearly true for (2) because both the common and idiosyncratic terms have the same order of integration. Although less obvious, the statement is also true for model (), in which e it and F t are not constrained to have the same order of integration. o see this, assume no deterministic component for simplicity. he DGP defined by () can be rewritten as X it = ρ i X it + λ i F t ρ i λ i F t + ε it. (2) Because the defactored approach will remove the common factors, we can ignore them in the equations. It is then obvious that the t a,b tests (using observable data) will determine if the (weighted) average of ρ i is unity, where the ρ i are the autoregressive coefficients for the idiosyncratic error processes. he same holds for the test of Pesaran (2007), who estimates augmented autoregressions of the form (suppressing deterministic terms for simplicity and adapted to our notation) X it = (ρ i )X it + d 0 X t + d X t + e it, where X t = i= X it. Although X t is observed, it plays the same role as F t. As such, the covariate augmented Dickey Fuller (CADF), which takes an

11 098 JUSHA BAI AD SEREA G average of the t ratios on ρ i, is also a test of whether the idiosyncratic errors are nonstationary. Others control for cross-sectional correlation by adjusting the standard errors of the pooled estimate. But the method still depends on whether the factors and/or the errors are nonstationary; see Breitung and Das (2008). o make statements concerning nonstationarity for the observed series X it, researchers still have to separately test if the common factors are nonstationary. PAIC presents a framework that can establish if the components are stationary in a coherent manner. 5. FIIE-SAMPLE PROPERIES In this section, we report the finite-sample properties of Pê, PMSB, and autoregressive coefficient based tests, P a,b and t a,b.asp = is not usually a case of practical interest, we only report results for p = 0 and p =. For the t a,b tests, we follow Moon and Perron (2004) and use Model A for testing (i.e., no demeaning) instead of B (demeaning) when p = 0. o make this clear, we denote the results by ta,b A.Forp =, the t a,b tests are denoted by ta,b C (with demeaning and detrending). Jang and Shin (2005) explored the sensitivity of the MP tests to the method of demeaning but did not consider the case of incidental trends. Furthermore, they averaged the t tests of the PAIC residuals, rather than pooling the p values as in Bai and g (2004). Gengenbach, Palm, and Urbain (2009) also compared the MP tests with PAIC but also for p = 0 only. In addition, all these studies consider alternatives with little variation in the dynamic parameters. Here, we present new results by focusing on mixed I()/I(0) units and with more heterogeneity in the dynamic parameters. We report results for four models. Additional results are available on request. Models 3 are configurations of (), whereas Model 4 is based on (2). he common parameters are r =, λ i is drawn from the uniform distribution such that λ i U[,3], η t (0,), and ε it (0,). he model-specific parameters are Model. =, ρ i = for all i; Model 2. = 0.5, ρ i U[0.9,0.99]; Model 3. = 0.5, ρ i = for i =,...,/5, ρ i U[0.9,0.99] otherwise; Model 4. ρ i U[0.9,0.99], where is the autoregressive coefficient in F t = F t +η t. We consider combinations of, taking on values of 20, 50, and 00. he number of replications is 5,000. We hold the number of factors to the true value when we evaluate the adequacy of the asymptotic approximations because the theory does not incorporate sampling variability due to estimating the number of factors. However, in practice, the number of factors (r) is not known. Bai and g (2002) developed procedures that can consistently estimate r. heir simulations showed that when and are large, the number of factors can be estimated precisely. However, the number of factors can be overestimated when or is small (say, less than 20). In those

12 PAEL UI ROO ESS 099 cases, the authors recommend using BIC 3 (Bai and g, 2002, p. 202). Alternatively, classical factor analysis can be used to determine the number of factors for small or ; see Anderson (984, Ch. 4). Gengenbach et al. (2009) found that the performance of panel unit root tests can be distorted when the number of common factors is overestimated. It is possible that the BIC 3 can alleviate the problem. Regardless of which criterion to use, the finite-sample distributions of post model selection estimators typically depend on unknown model parameters in a complicated fashion. As Leeb and Potscher (2008) showed, the asymptotic distribution can be a poor approximation for the finite-sample distributions for certain DGPs. his caveat should be borne in mind. o illustrate the main point of this paper, namely, tests that explicitly detrend the data first to construct ρ have no power, we additionally report results for two and P D b. hese tests obtain ρ+ from Model B for tests that we will denote Pa D DGP with p = 0 and Model C when p =. he former is a regression of ê it on ê it, plus an individual specific constant. he latter adds a trend. he tests use the same adjustments as t a and t b of Moon and Perron (2004). Although detrending renders the numerator of ρ + nondegenerate, it also removes whatever power the tests might have, as we now see. Results are reported in able for p = 0 and able 2 for p =. he rejection rates of Model correspond to finite-sample size when the nominal size is 5%. Models 2, 3, and 4 give power. Power is not size adjusted to focus on rejection rates that one would obtain in practice. able shows that for p = 0 PMSB, P a, and P b seem to have better size properties. Apart from size discrepancies when is small, all tests have similar properties. he difference in performance is much larger when p =. able 2 shows that t a, t b, Pa D, and P b D are grossly oversized, and both tests use least-squaresdetrended data to estimate ρ. he Pa D and Pb D have no power. On the other hand, Pê, PMSB, P a, and P b are much better behaved. Importantly, these tests either do not need a pooled estimate of ρ or they do so without linearly detrending ê. Moon et al. (2007) find that the MP tests have no local power against the alternative of incidental trends. Our simulations suggest that this loss of power arises as a result of detrending the data to construct ρ. Assuming cross-section independence, Phillips and Ploberger (2002) proposed a panel unit root test in the presence of incidental trends that maximizes average power. It has some resemblance to the Sargan Bhargava test. Although optimality of the PMSB test is not shown here, the PMSB does appear to have good finitesample properties. he panel unit root null hypothesis can thus be tested without having to estimate ρ. Incidental parameters clearly create challenging problems for unit root testing using panel data, especially for tests based on an estimate of the pooled autoregressive coefficient. he question arises as to whether alternative methods of detrending might help. In unreported simulations, the finite-sample size and power of Pa D and P b D under generalized least squares detrending are still unsatisfactory. One way of resolving this problem is to avoid detrending altogether. his is the

13 00 JUSHA BAI AD SEREA G ABLE. Rejection rates when p = 0inDGP Pê PMSB P a P b P D a P D b t A a t A b Model (Size): F I(),e it I() Model 2 (Power): F I(0),e it I(0) Model 3 (Power): F I(0),e it mixed I(0), I() Model 4 (Power): F I(0),e it I(0) ote: Pê,PMSB, P a, P b, Pa D,andPD b are tests based on PAIC residuals ê. he first four are defined in (5), (), (8), and (9). he tests Pa D and P b D are constructed in the same way as t a and t b, but they estimate ρ + from an autoregression using ê with a constant; ta A and t b A are MP tests that do not demean the data.

14 ABLE 2. Rejection rates when p = indgp Pê PMSB P a P b P D a PAEL UI ROO ESS 0 P D b t C a t C b Model (Size): F I(),e it I() Model 2 (Power): F I(0),e it I(0) Model 3 (Power): F I(0),e it mixed I(0), I() Model 4 (Power): F I(0),e it I(0) ote: Pê,PMSB, P a, P b, Pa D,andPD b are tests based on PAIC residuals ê. he first four are defined in (5), (), (8), and (9). he tests Pa D and Pb D are constructed in the same way as t a and t b, but they estimate ρ + from an autoregression using ê with a constant and a linear trend. he tests ta C and tc b are MP tests that detrend the observable data.

15 02 JUSHA BAI AD SEREA G key behind the drastic difference in the properties of P a and P b on the one hand, and Pa D, P b D, t a, and t b on the other. Still, the Pê, PMSB, P a, and P b tests require that or not be too small, or else the test is oversized. Overall, tests of nonstationarity in panel data with incidental trends are quite unreliable without and being reasonably large. In a recent paper, Westerlund and Larsson (2009) provide a detailed analysis of pooled PAIC test Pê. heir justification of the procedure is more rigorous than what was given in Bai and g (2004); they also provide a small-sample bias correction. It should be stressed that Pê is not the only way to construct a pooled test in the PAIC framework. As we showed in this paper, P a, P b and PMSB are also PAIC-based pooled tests. Our simulations show that all PAIC-based pooled tests have good finite-sample properties. 6. COCLUSIO In this paper, we (a) develop a PAIC-based estimate of the pooled autoregressive coefficient and (b) develop a PMSB test that does not rely on the pooled autoregressive coefficient. Upon comparing their finite-sample properties, we find that tests based on the autoregressive coefficient have no power against incidental trends whenever linear detrending is performed before estimating the pooled autoregressive parameter. he PMSB test, the original PAIC pooled test of the p values, and the new P a and P b tests all have satisfactory properties. one of these tests require a projection ê on time trends. It is worth emphasizing that tests that control cross-section correlation only permit hypotheses concerning the idiosyncratic errors to be tested. o decide if the observed data are stationary or not, we still need the PAIC procedure to see if the factors are stationary. In fact, PAIC goes beyond unit root testing by showing that the common stochastic trends are well defined and can be consistently estimated even if e it are I() for all i. his is in contrast with a fixed spurious system in which common trends are hardly meaningful. OES. A kernel estimate based on ê it, although consistent under the null hypothesis that all units are nonstationary, is degenerate under the specific alternative that all units are stationary. Accordingly, nuisance parameters are estimated using ε it instead of ê it. 2. Estimation of ωεi 2 is discussed in Perron and g (998). 3. As long as the data are demeaned and detrended, the projection matrix M must be used even if the true DGP is given by Model A. REFERECES Anderson,.W. (984) An Introduction to Multivariate Statistical Analysis. Wiley. Bai, J. (2003) Inferential theory for factor models of large dimensions. Econometrica 7,

16 PAEL UI ROO ESS 03 Bai, J. & S. g (2002) Determining the number of factors in approximate factor models. Econometrica 70, Bai, J. & S. g (2004) A PAIC attack on unit roots and cointegration. Econometrica 72, Breitung, J. & S. Das (2008) esting for unit root in panels with a factor structure. Econometric heory 24, Chang, Y. (2002) onlinear IV unit root tests in panels with cross-section dependency. Journal of Econometrics 0, Chang, Y. & W. Song (2002) Panel Unit Root ests in the Presence of Cross-Section Heterogeneity. Manuscript, Rice University. Choi, I. (2006) Combination unit root tests for cross-sectionally correlated panels. In D. Corbae, S.. Durlauf, & B.E. Hansen (eds.), Econometric heory and Practice: Frontiers of Analysis and Applied Research: Essays in Honor of Peter C.B. Phillips, pp Cambridge University Press. Gengenbach, C., F. Palm, & J.P. Urbain (2009) Panel unit root tests in the presence of cross-sectional dependencies: Comparison and implications for modelling. Econometric Review, forthcoming. Jang, M.J. & D. Shin (2005) Comparison of panel unit root tests under cross-sectional dependence. Economics Letters 89, 2 7. Leeb, H. & B. Potscher (2008) Can one estimate the unconditional distribution of post-model-selection estimators? Econometric heory 24, Levin, A., C.F. Lin, & J. Chu (2002) Unit root tests in panel data: Asymptotic and finite sample properties. Journal of Econometrics 98, 24. Maddala, G.S. & S. Wu (999) A comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics 6, Moon, R. & B. Perron (2004) esting for a unit root in panels with dynamic factors. Journal of Econometrics 22, Moon, R., B. Perron, & P. Phillips (2007) Incidental trends and the power of panel unit root tests. Journal of Econometrics 4, g, S. & P. Perron (200) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, O Connell, P. (998) he overvaluation of purchasing power parity. Journal of International Economics 44, 9. Perron, P. & S. g (996) Useful modifications to unit root tests with dependent errors and their local asymptotic properties. Review of Economic Studies 63, Perron, P. & S. g (998) An autoregressive spectral density estimator at frequency zero for nonstationarity tests. Econometric heory 4, Pesaran, H. (2007) A simple unit root test in the presence of cross-section dependence. Journal of Applied Economics 22, Phillips, P.C.B. & R. Moon (999) Linear regression limit theory for nonstationary panel data. Econometrica 67, 057. Phillips, P.C.B. & P. Perron (988) esting for a unit root in time series regression. Biometrika 75, Phillips, P.C.B. & W. Ploberger (2002) Optimal esting for Unit Roots in Panel Data. Mimeo, University of Rochester. Phillips, P.C.B. & D. Sul (2003) Dynamic panel estimation and homogeneity testing under crosssection dependence. Econometrics Journal 6, Said, S.E. & D.A. Dickey (984) esting for unit roots in autoregressive-moving average models of unknown order. Biometrika 7, Sargan, J.D. & A. Bhargava (983) esting for residuals from least squares regression being generated by Gaussian random walk. Econometrica 5, Stock, J.H. (990) A Class of ests for Integration and Cointegration. Mimeo, Department of Economics, Harvard University. Westerlund, J. & R. Larsson (2009) A note on the pooling of individual PAIC unit root tests. Econometric heory 25,

17 04 JUSHA BAI AD SEREA G APPEDIX Assumptions A E are assumed when analyzing the properties of PAIC residuals ê it. LEMMA. Let C = min[, ]. he PAIC residuals ê it satisfy, for p =,0, 2 i= t= ê 2 it = 2 i= t= e 2 it + O p(c 2 ). Proof of Lemma. From Bai and g (2004, p. 54), ê it = e it e i + λ i H V t d i F t where V t = t s=2 v s, v t = f t Hf t, and d i = λ i H λ i. Rewrite the preceding expression as ê it = e it + A it with A it = e i +λ i H V t d i F t. hus êit 2 = e2 it +2e it A it + A 2 it. It follows that 2 i= t= ê 2 it = 2 i= + 2 t= i= e 2 it t= i= t= e it A it A 2 it = I + II+ III. (A.) Bai and g (2004, p. 63) show that 2 t= A2 it = O p(c 2 ) for each i. Averaging over i, it is still this order of magnitude. In fact, by the argument of Bai and g (2004), III = ( ( i= t= A 2 it 3 )( 2 V t 2 t= ) d i 2 i= 2 ( ) ei 2 i= ) λ i H 2 i= F t 2 t= = O p ( ) + O p ( ) + O p ( [ min[ 2, ] ) = O p (C 2 ). ext, consider II (ignoring the factor of 2), II = 2 e it e i + i= t= 2 e it λ i H V t i= t= 2 e it d i F t = a + b + c. i= t= he proof of a = O p (C 2 ) is easy and is omitted (one can even assume e i = 0). Consider b.

18 b 2 λ i e it H V t i= t= 2 /2 2 /2 ( λ i e it /2 t= i= PAEL UI ROO ESS 05 2 /2 H V t 2). t= ( By (A.4) of Bai and g (2004, p. 57), /2 2 t= H V t 2) /2 = Op ( ) = O p (C 2 ). he first expression is O p() because () /2 i= λ i e it = O p (). hus b = O p (C 2 ). Consider c: c [ t= = O p ( /2 ) i= [ t= ] 2 /2 ( e it d i i= Using equation (B.2) of Bai (2003), i.e., 2 ] 2 /2 e it d i. ) /2 F t 2 t= d i = H f s ε is + O p (C 2 ), s= (A.2) and ignoring H for simplicity, we have e it d i = i= e it i= f s ε is + /2 O p (C 2 ), s= noting that e it = /2 O p (). If we can show that for each t, E ( e it i= ) 2 f s ε is = O( ) + O( ), s= (A.3) then c = O p ( /2 )[O p ( /2 ) + O p ( /2 ) + /2 O p (C 2 )] = O p(c 2 ). But the preceding expression is proved for the case of t = subsequently (see the proof of (A.7)); the argument is exactly the same for every t. hus II = O p (C 2 ), and the lemma follows. LEMMA 2. If / 2 0, then the PAIC residuals ê it satisfy, for p =,0, n i= t= ê it ê it = i= t= e it e it + O p ( / ) + O p (C ). n

19 06 JUSHA BAI AD SEREA G Proof of Lemma 2. Using the identity t=2 êit ê it = 2 ê2 i 2 ê2 i 2 t=2 ( ê it) 2 and the corresponding identity for t=2 e it e it, then Lemma 2 is a consequence of Lemma 3, which follows. LEMMA 3. If / 2 0, then the PAIC residuals ê it satisfy, for p =,0, (i) i= (ê2 i e2 i ) = O p( / ), n (ii) (iii) i= (ê2 i e2 i ) = O p( / ) + O p (C ), i= t=2 [( ê it) 2 ( e it ) 2 ] = O p ( / ) + O p (C ). Proof of Lemma 3. Proof of (i). Because ê i is defined to be zero, it follows that the left-hand side of (i) is ( ) / )( i= e2 i = o p () if / 0. Proof of (ii). From ê i = e i + A i, it is sufficient to show that (a) i= A2 i = O p ( / ) and that (b) i= e i A i = O p ( / ) + O p (C ). Using V 2 / = O p ( ) and d i = O p (/min[, ]), it is easy to show that the expression in (a) is O p ( / ). Consider (b). e i A i = i= e i e i + i= e i λ i H V i= e i d i F. i= (A.4) he first term on the right-hand side can be shown to be O p ( /2 ). Consider the second term. Under ρ i =, e i = t= ε it, e i λ i H V V H ( i= i= = O p () V = O p (C ) t= λ i ε it ) (see Bai and g, 2004, p. 57). For the last term of (A.4), from /2 F = O p (), we need to bound ( ) ( /2 e i d i = i= e i d i ). (A.5) i= From d i = O p (min[, ] ),wehave i= e id i = O p (). hus (A.5) is o p () if / 0. But / 0 is not necessary. o see this, first use d i in (A.2) to obtain e i d i = i= e i i= f s ε is + O p ( /2 )O p (C 2 ). s= We shall show ( ) 2 E e i i= f s ε is = O( ) + O( ). s= (A.6) (A.7)

20 From e i = t= ε it, the left-hand side of (A.7) is 2 i= j= 2 PAEL UI ROO ESS 07 E( f s f k ε is ε it ε jk ε jh ). (A.8) s,k,t,h Consider i j. From cross-sectional independence and the independence of factors with the idiosyncratic errors, E( f s f k ε is ε it ε jk ε jh ) = E( f s f k )E(ε is ε it )E(ε jk ε jh ). o see the key idea, assume ε it are serially uncorrelated; then E(ε is ε it ) = E(εit 2 ) for s = t and 0 otherwise. Similarly, E(ε jk ε jh ) = E(ε 2 jk ) for h = k and 0 otherwise. From E(ε2 it ) = σi 2 for all t, terms involving i j have an upper bound (assume E(εit 2) σ i 2 under heteroskedasticity) 2 σi 2 σ j 2 i j 2 E( f s f k ) =O( ) s,k because s,k E( f s f k ) M under weak correlation for f s.ifε it is serially correlated, then the sum in (A.8) for i j is bounded by ( 2 i j 2 2 )( E( f s f k ) max s s,k t= ( E( f s f k ) )( s,k l=0 i j E(ε is ε it ) )( max k h= ) γ i (l) )( γ j (l), l=0 E(ε jk ε jh ) where γ i (l) is the autocovariance of ε it at lag l and γ j (l) is similarly defined. Replace σi 2 by l=0 γ i (l) < (and similarly for σj 2 ); the same conclusion holds. ext consider the case of i = j. Because 2 s,t,k,h E( f s f k ε is ε it ε ik ε ih ) = O(), we have 2 i= 2 s,t,k,h E( f s f k ε is ε it ε ik ε ih ) = O( ), proving (A.7). Combining (A.5) (A.7), e i d i = i= ( ) /2 [ O p ( /2 ) + O p ( /2 ) + /2 O p (C 2 ] ) = O p ( ( / ) + O p C ). his proves (b). Combining (a) and (b) yields (ii). Proof of (iii). From ê it = e it a it, where a it = λ i H v t + d i f t,wehave [( ê it ) 2 ( e it ) 2 ] = 2 i= t=2 ( e it )a it + i= t=2 ait 2. i= t=2 From Bai and g (2004, p. 58), t=2 a2 it = O p(c 2 ); thus the second term on the right-hand side of the preceding expression is bounded by O p (C 2 ). Consider the )

21 08 JUSHA BAI AD SEREA G first term ( e it )a it = i= t=2 ( e it )λ i H v t i= t=2 + ( e it )d i f t = I + II. i= t=2 By the Cauchy Schwartz inequality, ( ) I H /2 /2 ( ) /2 e it λ i t=2 i= v t 2 = O p ()O p (C ). t=2 For II, it suffices to show that II = o p () when f t is replaced by f t.ow ( ( e it )d i f ) /2 t /2 di 2 i= t=2 i= /2 2 /2 e it f t. i= t=2 he preceding expression is /2( ) /2Op i= d2 i () = O p ( / ), because i= d2 i = O p (/min( 2, )). hus (iii) is equal to O p (C 2 ) + O p(c ) + O p ( / ) = O p (C ) + O p( / ). n LEMMA 4. For p =, the PAIC residuals satisfy, with,, (i) 2 êit 2 = i= t= 2 (ẽ it ) 2 + O p (C 2 ), i= t= (ii) ê it ê it = i= t= ẽ it ẽ it + O p ( / ) + O p (C ), i= t= where ẽ it = e it e i e i e i /( )(t ). Proof of Lemma 4. his is an argument almost identical to that in the proof of Lemmas and 2. he details are omitted. ote that when p =, the PAIC residuals ê it are estimating ẽ it ; see Bai and g (2004). n Let ê τ it denote the regression residual from regressing ê it on a constant and a linear trend. We define e τ it similarly. LEMMA 5. For p =, the PAIC residuals ê τ it satisfy, (i) (ii) 2 i= (êit τ )2 = t= 2 (eit τ )2 + O p (C 2 ), i= t= êit τ êτ it = i= t= i= t= e τ it eτ it + O p( / ) + O p (C ). Proof of Lemma 5. Using the properties for Vt τ and F t τ derived in Bai and g (2004), the proof of this lemma is almost identical to that of Lemmas and 2. he details are omitted. n

22 PAEL UI ROO ESS 09 LEMMA 6. Suppose that Assumption C holds. Under ρ i = for all i, we have, as,, (i) 2 i= t= e2 p it 2 ω2 ε, (ii) with / 0, ( ) ) i= d t= e it ε it λ (0, 2 φ4 ε, ( (iii) 2 2 i= t= eit ) τ p 5 ωε 2, ( i= t= eτ it ε it + 2 σ 2 ) ( ) d 0, 60 φ4 ε, (iv) with / 0, (v) 2 i= t= (ẽ it) 2 p 6 ω2 ε, where eit τ is the demeaned and detrended version of e it and ẽ it = e it e i e i e i /( )(t ), λ = i= λ εi, σ 2 = i= σ εi 2, and λ εi, σεi 2, ω2 ε, and φε 4 are all defined in Section 2. Proof of Lemma 6. Parts (i) and (ii) are from Lemma A.2 of Moon and Perron (2004). Parts (iii), (iv), and (v) can be proved using similar arguments as in Moon and Perron (2004). hese are all joint limits. o provide intuition and justification for the parameters involved, we give a brief explanation of the preceding results using sequential argument. Consider (i). For each i, as, 2 t=2 e2 d it ωεi 2 U i, where U i = 0 W i (r) 2 dr with EU i = 2. hus, for fixed, i= 2 t=2 e2 p it 2 ω2, where ω 2 = i= ω2 εi. Part (i) is obtained by letting go to infinity. Consider (ii). For each i, t=2 e d it e it ω 2 εi Z i + λ εi, where Z i = 0 W i (r)dw i (r) and λ εi is one-sided long-run variance of ε it = e it (i.e.,λ εi = [ωεi 2 σ εi 2 ]/2). hus for fixed, ( ) i= d t= e it ε it λ i= ω2 εi Z i. Because Z i are independent and identically distributed (i.i.d.), zero mean, and var(z i ) = 2, we obtain (ii) by the central limit theorem as. Similar to (i), (iii) follows from 2 t=2 (eτ it )2 ω d εi 2 U i τ, d where Ui τ = 0 W τ (r) 2 dr with E(Ui τ ) = /5. For (iv), t=2 eτ it eτ it ωεi 2 Zτ i + λ εi, where Zi τ = 0 W τ (r)dw(r). From E(Zi τ ) = 2, we have E(ω2 εi Zτ i ) + λ εi = σεi 2 /2. hus, ( ) i= t= eτ it ε it + 2 σ 2 d i= ω2 εi Zτ i. Because var(zi τ ) = 60, (iv) is obtained by the central limit theorem when. For (v), 2 t= (ẽ it) 2 ω d εi 2 V i with V i = 0 Bi 2(r)dr, where B i is a Brownian bridge. Part (v) follows from E(V i ) = /6. he proofs of these results under joint limits are more involved; the details are omitted because of similarity to the arguments of Moon and Perron (2004). ote that joint limits in (iii) and (iv) require / 0. Also see the detailed proof for the joint limit in Lemma 8. n Proof of heorem. For p =,0, Lemmas and 2 show that pooling ê it is asymptotically the same as pooling the true errors e it. Let ρ + (resp. ρ + ) be the bias-corrected estimator based on the true idiosyncratic error matrix e and λ (resp. ê and the estimated λ ). Under the null of ρ i = for all i,wehave ) tr (e (ρ + e [ ( ) ] λ tr e e λ ) = tr(e e = ) 2 tr(e e. (A.9) )

23 0 JUSHA BAI AD SEREA G ( ) By Lemma 6(i) and (ii), (A.9) converges in distribution to 0, 2φ4 ε ωε 4 as, with / 0. his limiting distribution does not change when λ is replaced by its estimate λ because ( λ λ ) = o p (); see Moon and Perron (2004). By Lemmas and 2 the limiting distribution continues to hold when e is replaced by ê. hat is, ( ρ + ) d ( 0, 2φ4 ε ω 2 ε ). hus, P a = ( ρ + )/ 2 φ ε 4/ ω4 d ε (0,), as, with / 0. For the P b test, multiply equation (A.9) on each side by [(/ 2 )tr(e e )] /2, whose limit is (ω ε /2) /2 by part (i) of Lemma 6. We obtain ( ) (ρ + /2 ) 2 tr(e e ) = [ ) tr ]( ) /2 λ 2 tr(e e ), ( e e which converges to (0,φε 4/ω2 ε ),as, with /. It follows that P b = ) /2 ( ρ + )( (/ 2 )tr(ê ê ) ω ε 2/ φ ε 4 d (0,). For p =, recall that ê it estimates ẽ it = e it e i e i e i /( )(t ). Let ẽ be the matrix consisting of elements ẽ it. ote that ẽ i 0 for all i. And t= ẽit ẽ it = 2 ẽ2 i 2 t= ( ẽ it) 2 = 2 t= ( ẽ it) 2. But ẽ it = ε it ε i. hus t= ẽit ẽ it = 2 t= (ε it ε i ) 2 p 2 σ εi 2. hus i= t= ẽit ẽ p it 2 σε 2. ogether with Lemma 6(v), tr(ẽ ẽ ) p σ ε 2/2 2 tr(ẽ ẽ ) ωε 2/6 = 3(σ ε 2 /ω2 ε ). For given, the preceding limit is 3 σ 2 / ω2, where σ 2 = i= σ εi 2 and ω 2 = i= ω2 εi. Again, let ρ+ denote the bias-corrected estimator based on ẽ and the true parameters ( σ 2, ω2 ), i.e., ρ + = tr(ẽ ẽ ) tr(ẽ ẽ ) + 3 σ 2 ω 2. hen (ρ + ) = tr(ẽ ẽ ) (/ 2 )tr(ẽ ẽ ) + 3 σ 2 ω 2 = A B + σ 2 /2 ω 2 /6, where A = tr(ẽ ẽ) and B = (/2 )tr(ẽ ẽ ). It follows that (ρ + ) ) = (A + σ 2 B /2 + 3 B ote (A + σ 2 /2) = 2 i= t= σ 2 ω 2 (B ω 2 /6 ). [(ε it ε i ) 2 σ 2 εi ] = O p ( ).