Unit root tests in panel data:asymptotic and nite-sample properties

Transcription

1 Journal of Econometrics 108 (2002) Unit root tests in panel data:asymptotic and nite-sample properties Andrew Levin a, Chien-Fu Lin b, Chia-Shang James Chu b; a Federal Reserve Board of Governors, Washington, DC 20551, USA b Department of Economics, National aiwan University, No. 21 Hsu Chou Road, aipei, aiwan, ROC Received 19 May 1992; revised 4 June 2001; accepted 13 July 2001 Abstract We consider pooling cross-section time series data for testing the unit root hypothesis. he degree of persistence in individual regression error, the intercept and trend coecient are allowed to vary freely across individuals. As both the cross-section and time series dimensions of the panel grow large, the pooled t-statistic has a limiting normal distribution that depends on the regression specication but is free from nuisance parameters. Monte Carlo simulations indicate that the asymptotic results provide a good approximation to the test statistics in panels of moderate size, and that the power of the panel-based unit root test is dramatically higher, compared to performing a separate unit root test for each individual time series.? 2002 Elsevier Science B.V. All rights reserved. JEL classication: C12; C23 Keywords:ADF regression; Nonstationary panel; Panel unit root test; Pooled t-statistics 1. Introduction A large body of literature during the past two decades has considered the impact of integrated time series in econometric research (cf. surveys by Diebold and Nerlove, 1990; Campbell and Perron, 1991). In univariate analysis, the Box Jenkins (Box and Jenkins, 1970) approach of studying dierence-stationary ARMA models requires a consistent and powerful test Corresponding author /02/$ - see front matter? 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 2 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 for the presence of unit roots. In general, such tests have non-standard limiting distributions; for example, the original Dickey Fuller test (Dickey and Fuller, 1979) and the subsequent augmented Dickey Fuller (ADF) test statistic (Dickey and Fuller, 1981) converge to a function of Brownian motion under quite general conditions (Said and Dickey, 1984). he critical values of the empirical distribution were rst tabulated by Dickey (cf. Fuller, 1976). Semi-parametric test procedures have also been proposed (i.e. Phillips, 1987; Phillips and Perron, 1988), with improved empirical size and power properties under certain conditions (cf. Diebold and Nerlove, 1990). In nite samples, these unit root test procedures are known to have limited power against alternative hypotheses with highly persistent deviations from equilibrium. Simulation exercises also indicate that this problem is particularly severe for small samples (see Campbell and Perron, 1991). his paper considers pooling cross-section time series data as a means of generating more powerful unit root tests. he test procedures are designed to evaluate the null hypothesis that each individual in the panel has integrated time series versus the alternative hypothesis that all individuals time series are stationary. he pooling approach yields higher test power than performing a separate unit root test for each individual. Some earlier work has analyzed the properties of panel-based unit root tests under the assumption that the data is identically distributed across individuals. Quah (1990, 1994) used random eld methods to analyze a panel with i.i.d. disturbances, and demonstrated that the Dickey Fuller test statistic has a standard normal limiting distribution as both the cross-section and time series dimensions of the panel grow arbitrarily large. Unfortunately, the random eld methodology does not allow either individual-specic eects or aggregate common factors (Quah, 1990, p. 17). Breitung and Meyer (1991) have derived the asymptotic normality of the Dickey Fuller test statistic for panel data with an arbitrarily large cross-section dimension and a small xed time series dimension (corresponding to the typical microeconomic panel data set). heir approach allows for time-specic eects and higher-order serial correlation, as long as the pattern of serial correlation is identical across individuals, but cannot be extended to panel with heterogeneous errors. More recent advances in nonstationary panel analysis include Im et al. (1995), Harris and zavalis (1996) and Phillips and Moon (1999), among others. What type of the asymptotics considered in the panel unit root test is a delicate issue. Earlier work by Anderson and Hsiao (1982) consider a stationary panel with xed time series observations while letting the cross sectional units grow arbitrarily large. Similar asymptotic method is used in nonstationary panel by Breitung and Meyer (1991) and Harris and zavalis (1996). Im et al. (1995) and Quah (1990, 1994) explore the case of joint limit in which both time series and cross sectional dimension approach innity with certain restrictions. he precise meaning regarding in what way the cross

3 A. Levin et al. / Journal of Econometrics 108 (2002) sectional and time series dimension approach innity has been clearly dened in Phillips and Moon (1999). his article is one of the earlier research that consider the joint limit asymptotics in which both N and approach innity subjecting to certain conditions such as N= 0 in some model, and N= 0 in others. he panel-based unit root test proposed in this article allows for individualspecic intercepts and time trends. Moreover, the error variance and the pattern of higher-order serial correlation are also permitted to vary freely across individuals. Our asymptotic analysis in Section 3 indicates that the proposed test statistics have an interesting mixture of the asymptotic properties of stationary panel data and the asymptotic properties of integrated time series data. In contrast to the non-standard distributions of unit root test statistics for a single time series, the panel test statistics have limiting normal distributions, as in the case of stationary panel data (cf. Hsiao, 1986). However, in contrast to the results for stationary panel data, the convergence rate of the test statistics is higher with respect to the number of time periods (referred to as super-consistency in the time series literature) than with respect to the number of individuals in the sample. Furthermore, whereas regression t-statistics for stationary panel data converge to the standard normal distribution, we nd that the asymptotic mean and variance of the unit root test statistics vary under dierent specication of the regression equation (i.e. the inclusion of individual-specic intercepts and time trends). For practical purposes, the panel based unit root tests suggested in this paper are more relevant for panels of moderate size. If the time series dimension of the panel is very large then existing unit root test procedures will generally be suciently powerful to be applied separately to each individual in the panel, though pooling a small group of individual time series can be advantageous in handling more general patterns of correlation across individuals (cf. Park, 1990; Johansen, 1991). On the other hand, if the time series dimension of the panel is very small, and the cross-section dimension is very large, then existing panel data procedures will be appropriate (cf. MaCurdy, 1982; Hsiao, 1986; Holtz-Eakin et al., 1988; Breitung and Meyer, 1991). However, panels of moderate size (say, between 10 and 250 individuals, with time series observations per individual) are frequently encountered in industry-level or cross-country econometric studies. For panels of this size, standard multivariate time series and panel data procedures may not be computationally feasible or suciently powerful, so that the unit root test procedures outlined in this paper will be particularly useful. he remainder of this paper is organized as follows:section 2 species the assumptions and outlines the panel unit root test procedure. Readers who are interested mainly in empirical application can skip the rest of the paper. Section 3 analyzes the limiting distributions of the panel test statistics. Section 4 briey discusses the Monte Carlo simulations. Concluding remarks regarding

4 4 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 the limitations of the proposed panel unit root test are oered in Section 5. All proofs are deferred to the appendix. 2. A panel unit root test 2.1. Model specications We observe the stochastic process {y it } for a panel of individuals i =1;:::;N, and each individual contains t =1;:::; time series observations. We wish to determine whether {y it } is integrated for each individual in the panel. As in the case of a single time series, the individual regression may include an intercept and time trend. We assume that all individuals in the panel have identical rst-order partial autocorrelation, but all other parameters in the error process are permitted to vary freely across individuals. Assumption 1. (a) Assume that {y it } is generated by one of the following three models: Model 1:y it = y it 1 + it. Model 2:y it = 0i + y it 1 + it. Model 3:y it = 0i + 1i t + y it 1 + it, where for i =1;:::;N. (b) he error process it is distributed independently across individuals and follows a stationary invertible ARMA process for each individual, it = ij it j + it : j=1 (c) For all i =1;:::;N and t =1;:::;, E( 4 it) ; E( 2 it) B 0; and E( 2 it)+2 E( it it j ) B : Assumption 1(a) includes three data generating processes. In Model 1, the panel unit root test procedure evaluates the null hypothesis H 0 : = 0 against the alternative H 1 : 0. he series {y it } has an individual-specic mean in Model 2, but does not contain a time trend. In this case, the panel test procedure evaluates the null hypothesis that H 0 : = 0 and 0i = 0, for all i, against H 1 : 0 and 0i R. Finally, under Model 3, the series {y it } has an individual-specic mean and time trend. In this case, the panel test procedure evaluates the null hypothesis that H 0 : = 0 and 1i = 0, for all i, against the alternative H 1 : 0 and 1i R. j=1

5 A. Levin et al. / Journal of Econometrics 108 (2002) As in the case of a single time series, if a deterministic element (e.g. an intercept or time trend) is present but not included in the regression procedure, the unit root test will be inconsistent. On the other hand, if a deterministic element is included in the regression procedure but is not present in the observed data, the statistical power of the unit root test will be reduced. See Johansen s (1992) discussion on the interactions of the unit root test and various deterministic specications. Campbell and Perron (1991) outline a method of ascertaining which deterministic elements should be included in the test procedure. For notational simplicity, d mt is used to indicate the vector of deterministic variables and m is used to indicate the corresponding vector of coecients for a particular model m =1; 2; 3. hus, d 1t = (the empty set); d 2t = {1} and d 3t = {1;t}. Assumption 1(b) is standard; individual time series may exhibit serial correlations. he nite-moment conditions of Assumption 1(c) correspond to the conditions for the weak convergence in Phillips (1987) and Phillips-Perron s (Phillips and Perron, 1988) unit root tests. In our panel unit root test we de- ne a ratio of the long-run variance to innovation variance (cf. Eq. (6)), the boundedness conditions in Assumption 1(c) ensure this ratio remains nite for every individual in the panel as the cross-section N becomes arbitrarily large est procedures Our maintain hypothesis is P i y it = y it 1 + il y it L + mi d mt + it ; m=1; 2; 3: (1) L=1 However, since p i is unknown, we therefore suggest a three-step procedure to implement our test. In step 1 we carry out separate ADF regressions for each individual in the panel, and generate two orthogonalized residuals. Step 2 requires estimating the ratio of long run to short run innovation standard deviation for each individual. In the nal step we compute the pooled t-statistics Step 1: Perform ADF regressions and generate orthogalized residuals For each individual i, we implement the ADF regression P i y it = i y it 1 + il y it L + mi d mt + it ; m=1; 2; 3: (1 ) L=1 he lag order p i is permitted to vary across individuals. Campbell and Perron (1991) recommend the method proposed by Hall (1990) for selecting the

6 6 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 appropriate lag order:for a given sample length, choose a maximum lag order p max, and then use the t-statistics of ˆ il to determine if a smaller lag order is preferred. (hese t-statistics have a standard normal distribution under the null hypothesis ( il = 0), both when i = 0 and when i 0. Having determined autoregression order p i in (1 ), we run two auxiliary regressions to generate orthogalized residuals. Regress y it and y it 1 against y it L (L =1;:::;p i ) and the appropriate deterministic variables, d mt, then save the residuals ê it and ˆv it 1 from these regressions. Specically, and P i ê it =y it ˆ il y it L ˆ mi d mt (2) L=1 P i ˆv it 1 = y it 1 il y it L mi d mt : L=1 o control for heterogeneity across individuals, we further normalize ê it and ˆv it 1 by the regression standard error form Eq. (1 ). ẽ it = ê it ˆ i ; ṽ it 1 = ˆv it 1 ˆ i ; (3) where ˆ i is the regression standard error in (1 ). Equivalently, it can also be calculated from the regression of ê it against ˆv it 1 ˆ 2 1 i = (ê it p i 1 ˆ i ˆv it 1 ) 2 : (4) t=p i Step 2: Estimate the ratio of long-run to short-run standard deviations Under the null hypothesis of a unit root, the long-run variance for Model 1 can be estimated as follows: K ˆ 2 yi = 1 1 t=2 y 2 it +2 L=1 w KL [ 1 1 t=2+l y it y it L ] : (5) For Model 2, we replace y it in (5) with y it y it, where y it is the average value of y it for individual i. If the data include a time trend (Model 3), then the trend should be removed before estimating the long-run variance. he truncation lag parameter K can be data dependent. Andrews (1991) suggests a procedure to determine K to ensure the consistency of ˆ 2 yi. he sample covariance weights wkl depend on the choice of kernel. For example, if the Bartlett kernel is used, wkl =1 L K +1 :

7 A. Levin et al. / Journal of Econometrics 108 (2002) Now for each individual i, we dene the ratio of the long-run standard deviation to the innovation standard deviation, s i = yi = i : (6) Denote its estimate by ŝ i =ˆ yi = ˆ i. Let the average standard deviation ratio be S N =(1=N) N s i, and its estimator Ŝ N =(1=N) N ŝi. his important statistic will be used to adjust the mean of the t-statistic later in step 3, see Eq. (12). Before presenting step 3, we make two remarks. First, as one referee pointed out, a natural estimate of yi 2 under the null hypothesis is ˆ 2 i=(1 pi ˆ il ) 2, and since ˆ 2 i is a consistent estimator for ˆ 2 i under the null hypothesis, ŝ i may well be estimated by 1 p i ˆ il. Secondly, it is important to emphasize that both the size and the power properties of the panel unit root test are enhanced by using rst-dierences (or demeaned rst-dierences) to estimate the long-run variance. Under the null hypothesis of a unit root, Schwert (1989) found that the long-run variance estimate based on rstdierences had much smaller bias in nite samples than the long-run variance estimate based on the residuals in level regression, and the same advantage occurs here too Step 3: Compute the panel test statistics Pool all cross sectional and time series observations to estimate ẽ it = ṽ it 1 + it ; (7) based on a total of N observations, where = p 1 is the average number of observations per individual in the panel, and p 1 N N p i is the average lag order for the individual ADF regressions. he conventional regression t-statistic for testing = 0 is given by t = ˆ SD( ˆ) ; (8) where ˆ = N t=2+p i N t=2+p i ṽ 2 it 1 SD( ˆ)= N ˆ ṽ it 1 ẽ it ; (9) t=2+p i ṽ 2 it 1 1=2 ; (10)

8 8 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 ˆ 2 = 1 N N (ẽ it ˆṽ it 1 ) 2 : (11) t=2+p i Under the hypothesis H 0 : = 0, the asymptotic results in the next section indicates that the regression t-statistic (t ) has a standard normal limiting distribution in Model 1, but diverges to negative innity for Models 2 and 3. Nevertheless, it is easy to calculate the following adjusted t-statistic: t = t N Ŝ N ˆ 2 m SD( ˆ) m ; (12) where the mean adjustment m and standard deviation adjustment m can be found in able 2 for a given deterministic specication (m =1; 2; 3) and time series dimension. (able 2 also includes quick-and-dirty choices of lag truncation parameter K for each time series dimension.) We show in Section 3 that this adjusted t-statistic t obeys the standard normal distribution, asymptotically. 3. Asymptotic properties Dene the following sample statistics for each individual: 1i = 2i = 3i = 1 2 i( p i 1) 1 2 i( p i 1) i( p i 1) t=p i+2 t=p i+2 t=p i+2 ˆv i;t 1 ê it ; (13) ˆv 2 i;t 1; (14) ê 2 it: (15) Next, dene the following two ratios for each individual: 1i = ( p i 1)i 2 ; (16) ˆ 2 i 2i = ( p i 1) 2 i 2 : (17) 2 ˆ 2 i

9 A. Levin et al. / Journal of Econometrics 108 (2002) Given the above denitions, the statistics of interest in (9), (11) and (8) can be rewritten, respectively, as ˆ = N 1 N 1i 1i N 1 N ; (18) 2i 2i ˆ 2 = 1 [ N ] 1i 3i 2 N ˆ N 1i 1i + ˆ 2 N 2i 2i ; (19) t = N 1=2 N 1i 1i ˆ [N 1 N : (20) 1=2 2i 2i ] In contrast to Harris and zavalis (1996) and Im et al. (1995), our unit root test is based on the t statistic, obtained from a pooled regression (7). Harris and zavalis (1996) work with pooled least squares estimator ˆ directly. Moreover, the asymptotic property of ˆ is investigated rather dierently in that the time series dimension is xed while the cross sectional dimension increases to innity. Im et al. (1995) also base their unit root test on the t-statistics, but it is the average of the t-statistics from individual ADF regressions rather than the pooled t-statistics considered here. It is easy to determine the limiting behaviors of 3i ; 1i and 2i. Under the null, y it = it, is an invertible ARMA process. As, and as long as the ADF lag order p i in (1) increases at an appropriate rate, ; ê it should well approximate it. Hence ( p i 1) 1 ê 2 it i 2 in probability. It follows that 3i ; 1i and 2i all converge to one in probability. For easier reference in the sequel, we state the following theorem without proof. heorem 1. Given assumption 1 and p i =0 ( ), ; for all i. Under H 0 ; 3i 1; 1i 1; and 2i 1 in probability; as. Weak convergence of 1i and 2i are well documented in the literature (e.g. Said and Dickey, 1984; Phillips and Perron, 1988). Results are summarized in the following theorem. heorem 2. Given conditions in heorem 1; 1i s i W m1i ; 2i si 2 W m2i ; where m =1; 2; 3; is the model index; and W m1i = 1 U 0 mi(r)db i (r); W m2i = 1 U 2 0 mi(r)db i (r); B i (r) is independent Brownian motion; U 1i = B i ; U 2i and U 3i are demeaned and detrended Brownian motions; respectively. Note that distributions of W m1i and W m2i depend on the deterministic specication but not on the particular values of model parameters. All moments of W m1i and W m2i exist. Specically, the mean and variance of W m1i and

10 10 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 able 1 Asymptotic moments Model (m) m1 2 m1 m2 2 m2 1. No intercepts or trends 0 1=2 1=2 1=3 2. Individual-specic intercepts 1=2 1=12 1=6 1=45 3. Individual-specic intercepts & trends 1=2 1=60 1=15 11=6300 W m2i can be computed following tedious yet straightforward algebra, and are displayed in able 1. For each of the three deterministic specications (m =1; 2; 3); the sample statistics 1i and 2i are independent across individuals i, for each time series dimension. hose interesting sample statistics dened in (18) (20) are of the form N 1 n hit hi ;h=1; 2; which is asymptotically equivalent to N 1 n hi due to heorem 1. It is natural to consider applying a LLN to each panel average N 1 n hi and a CL to the normalized sum N 1=2 n hi and then use these results to derive the limiting distributions of the panel unit root test statistics, t and t. o gain more intuitions, consider the sequence limit of ˆq = N 1=2 n 1i and Ĥ = N 1 n 2i. Under model 1, Ĥ 1=2 ˆq is a Studentized LM test for the unit root, which is asymptotically equivalent to the t statistic. Suppose that we rst let. From heorem 2, we have ˆq N 1=2 N s iw 11i. Since W 11i is a zero mean, independent sequence across individuals with E(W 11i )=0:5, we can expect to derive the asymptotic normality provided the Lindeberg condition holds. Hence, ˆq =N(0; 0:5V ), as N, where V = lim N N 1 N s2 i. A LLN is also expected to work for Ĥ. Since E(W 12i )=0:5, we have Ĥ 0:5V. Combining these results, we deduce that Ĥ 1=2 ˆq = d t N(0; 1). here are some complications when applying the above arguments to model 2, however. his is owing to the fact that E(W 21i )= 1 2 and E(W 22i)= 1 6. Hence t does not converge to a standard normal distribution. Similar problems arise in Model 3 too. As a result, some sort of adjustment is required. For this, we dene E( 1i ) 1i consistent estimator as and its ˆ 1i = ŝi m : (21) 1i hen (12) can be rewritten as t = N 1=2 N 1i ( 1i ˆ 1i ) m ˆ [N 1 N : (22) 1=2 2i 2i ] As proved later in heorem 5, this adjusted t-statistics, t converges to a N(0; 1). he mean and standard deviation adjustment, m and m

11 A. Levin et al. / Journal of Econometrics 108 (2002) used in empirical work are determined from Monte Carlo simulations, as discussed in Section 4 below. Formal denitions are given as follows. Denition. Let yt be a standard Gaussian random walk, i.e. yt = yt 1 + t, where t i.i.d. N(0; 1). Following the test procedure described in Section 2 for a given deterministic specication (m =1; 2; 3), let observation on yt be used to generate m1 as dened in (13). hen dene m E( m1 ), and as an arbitrary positive sequence which converges m1= m2 as. m By the construction of E( m1 ), the mean adjustment m converges to m1 as. We did not specify particular values of m for nite, because only the limit of the sequence is relevant for the asymptotic analysis. While the foregoing discussions help to speculate the asymptotic results, they do not handle the case in which both and N approach innity jointly. Recently, there is growing research interest in studying the asymptotics of nonstationary panel with both N and innity simultaneously, e.g. Quah (1990) and Im et al. (1995), among others. echnically, when N and goes to innity, asymptotic analysis should be based on the LLN and CL for triangular arrays. heorems 3, 4 and 5 below are proved along this line. By triangular arrays we mean that the cross sectional dimension N is an arbitrary monotonically increasing function of. We shall use the notation N instead of N to emphasize the fact that N increases with respect to. heorem 3. Suppose that (a) y it = it ; (b) assumption 1(b) and (c) hold; (c) P i =0( ); ; for all i; (d) lim N N 1 N s2 i = V exists. hen; N 1=2 N N 1i = d N 1=2 1i 1i N(0; 0:5V ); as N = 0. Suppose further that (e) lim N N 1 N s4 i exists. hen N 1 N 2i 2i p 0:5V; and t N(0; 1). heorem 3 states that the asymptotic normality holds as long as N = 0. his is particularly relevant for a typical microeconomic panel date set because the time series dimension is allowed to grow slower than the cross sectional dimension N. Other divergence speeds such as N = 0, and N = constant, are sucient, but obviously not necessary. he panel with no individual specic eects studied by Quah (1990) falls within the territory of heorem 3. Our result that N = 0 is an improvement over Quah s condition that N = constant. When individual-specic eects are introduced in Model 2 and 3, we require faster growth rate in to achieve the asymptotic normality, as the following theorem shows.

12 12 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 heorem 4. Suppose that (a) y it generated from Model 2 or 3; (b) condition (b) (d) in heorem 3 hold; under =0;N 1=2 s i m1 ) N(0;m1 2 V ); as N = 0. In addition; suppose (c) the truncation lag parameter (0; 1); then N 1=2 N = 0. N 1i ( 1i K increases at some rate ; where N 1i ( 1i ˆ 1i ) N(0;m1 2 V ); as It is useful compare heorem 4 with heorem 3. From the rst part of heorem 4, N 1=2 N 1i ( 1i s i m1 ) N(0;m1 2 V ), if there is no individual-specic eects as in heorem 3, then 11 = 0, and N 1=2 N 1i 1i N(0;m1 2 V ); as N = 0. On the other hand, since m1 0 under Models 2 and 3, mean adjustment is required and we want to make sure that such a demeaning method does not inuence the asymptotic distribution. he sucient condition is that N 1=2 N 1i (s i m1 ˆ 1i ) 0. But ˆ 1i = s i m1 +O p ( 1=2 ), thus the second term of (A.4.3) in Appendix A will not converge to zero unless N = 0. In contrast to the case of no individual specic eect (as in heorem 3), we now need faster growth rate in time series dimension in order to establish the asymptotic normality for the adjusted t-statistic. Some special cases are of interest. Assume that individual regression error it is i.i.d. for each i over, but heteroskedasticity is present across individuals, i.e. i 2 j. 2 hen we still require N = 0 to establish the asymptotic normality. In this case, even s i = 1, for all i, the convergence speed of ˆ 1i = 1 1i m continues to be dominated by 1i =1+O p ( 1=2 ), and hence ˆ 1i = m1 +O p ( 1=2 ). he reason is clearly due to the heteroskedastic error across individuals, and that ˆ i = i +O p ( 1=2 ). If we further assume away the heteroskedasticity such that i 2 = j, 2 then we have the advantage to pool all observations to estimate the common variance so ). hus, the mean adjustment term will vanish under the condition that N = 0. he foregoing discussions suggest that the relative divergence speed of N and is largely determined by whether there are individual-specic eects in the panel or not. When there is no individual-specic eect, we can establish asymptotic normality under quite general error process with slower growth rate in. On the other hand, the inclusion of individual-specic eects requires faster growth rate in even the individual regression error is a very favorable i.i.d. process. that ˆ = +O p ( 1=2 N 1=2 heorem 5. In addition to the conditions in heorem 4; if lim N N 1 N s4 i exist; then N 1 N 2i 2i p m2 V; and t N(0; 1):

13 A. Levin et al. / Journal of Econometrics 108 (2002) he inuence of extending the time dimension compared with extending the cross-section N can be further claried by examining the asymptotic distribution of ˆ. ( m1 S ˆ N 1 m2 V ; 1 2 ) m1 N 2 m2 2 V : (23) First, the Eq. (23) indicates that the variance of ˆ delta falls at rate O(N 1 2 ). he higher convergence rate with respect to is referred to as super-consistency in the time series literature. Intuitively, the presence of a unit root causes the variation of the signal y it 1 to grow arbitrarily large while the variation of the noise it remains constant. hus, an increment in the time series dimension adds more variation to the sample than an increment in the cross-section dimension N. Second, the inclusion of individual-specic intercepts and time trends (Models 2 and 3) causes a downward bias in ˆ (i.e., m1 0 for m =2; 3). he downward bias vanishes as the time series dimension grows large (so that ˆ converges to 0 asymptotically), but is not inuenced by the cross-section dimension N. hus, in contrast to the case of stationary panel data, the presence of a unit root causes the regression specication to inuence the asymptotic distribution of the panel estimate. hird, since an increment in the cross-section dimension N reduces the variance of ˆ but not its downward bias, the regression t-statistic diverges to negative innity for Models 2 and 3. In other words, adding an individual to the panel reduces the variance of ˆ around a non-zero mean, so that the regression t-statistic increasingly rejects the null hypothesis at conventional signicance levels. Consequently, it is necessary to use the estimated long run versus short run standard deviation ratios ŝ i and the adjustment factor m to demean the individual sample statistics. As the panel grows large, N= 0 ensures that the sum of individual dierences between the estimated mean ˆ 1i and the true mean 1i does not inuence the limiting distribution. he suggested test procedure requires that the data is generated independently across individuals. As in stationary panel data models, this assumption can be somewhat relaxed to allow for a limited degree of dependence via time-specic aggregate eects by subtracting the cross sectional averages y t = 1 n N y it from the observed date. (cf. Hsiao, 1986). he removal of cross-section averages from the data is equivalent to including time-specic intercepts in the regression models above, which does not aect the limiting distributions of the panel unit root test (see Levin and Lin, 1992 for further details). It is appropriate under the assumption of a single aggregate common factor which has an identical impact on all individuals in the panel, however, Quah and Sargent (1992) have recently proposed a method of identifying multiple common factors in panel data; their method may be

14 14 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 able 2 Mean and standard deviation adjustments a K : : : : : : : : : : : : : : : : : : : : : : : : : : a Notes:he adjustment factors 1 and 1 are used to adjust the mean and standard deviation of the panel unit root test statistic dened in Eq. (12) for a given regression model (m =1; 2; 3), average time series dimension, and lag truncation parameter K. In all cases, the standard error of the estimated mean adjustment is less than 0.007, and the standard error of the estimated standard deviation adjustment is less than useful for removing more complex contemporaneous correlation from the data Monte Carlo simulations In this section, we briey discuss the results of three Monte Carlo experiments. he rst set of simulations were used to determine appropriate values of the mean and standard deviation adjustments, m and m, used in the adjusted t-statistic given in Eq. (12) for a particular deterministic specication (m =1; 2; 3) and time series dimension. he lag truncation parameter K was selected according to the formula K =3:21 1=3. Due to computational limitations, the ADF lag length p i was set to the true value of zero rather than being individually estimated. At each replication, Gaussian random numbers with unit variance were used to generate 250 independent random walks of length + 1 (i.e. a panel of dimensions N = 250 and = + 1), and this data was used to construct the sample statistics given in Eqs. (7) (11) above. Based on 25,000 replications, the adjustment factor m was estimated by the mean value of t =N Ŝ N ˆ 2 SD( ˆ) and the adjustment factor m was estimated by the standard deviation of t N Ŝ N ˆ 2 SD( ˆ). he resulting estimates of m and m are given in able 2. In all cases, the standard error of the estimated mean adjustment is less than 0.007, and the standard

15 A. Levin et al. / Journal of Econometrics 108 (2002) error of the estimated standard deviation adjustment is less than As grows large, able 2 indicates that the adjustment factors converge to the asymptotic values predicted by able 1 and heorems 3 5. he second set of simulations examines the empirical size properties of the panel unit root test procedure under alternative deterministic specications and panel dimensions. For a given replication, Gaussian random numbers with unit variance were used to construct N independent random walks with time periods each; then the data was used to calculate the adjusted t-statistic using the simulated values of m, and m, and the test was performed using the critical values of the standard normal distribution at nominal sizes of 1, 5 and 10 percent. Based on 10,000 Monte Carlo replications, the empirical size for each model specication and set of panel dimensions are given in able 3. he standard error of the estimated empirical size is less than in all cases. able 3 indicates that in panels with a moderate number of individuals, the nominal size slightly underestimates the empirical size. For larger panels, the dierence between nominal and empirical size falls within the range of the Monte Carlo sampling error, consistent with the prediction of asymptotic normality given in heorems 3 5. Finally, the empirical power properties of the panel unit root test procedure were analyzed. For a given replication, the data was generated by the process y it =0:9y it 1 + it, (i.e. = 0:10) where the disturbances it are i.i.d. Gaussian random numbers with unit variance. his data was used to calculate the adjusted t-statistic as in the previous Monte Carlo experiment, and the test was performed using the critical values of the standard normal distribution at nominal sizes of 1, 5 and 10 percent. Based on 10,000 Monte Carlo replications, the empirical power for each model specication and set of panel dimensions are given in able 4. For purposes of comparison, able 4 also includes the empirical power of the Dickey Fuller test for a single time series (N = 1). he standard error of the estimated empirical power is less than 0.01 in all cases. he middle panel of able 4 illustrates the power advantages of performing unit root tests with panel data. In the absence of individual-specic eects (m = 1), the power of the standard Dickey Fuller test is quite low for short time series ( 6 50), whereas very high power can be achieved by performing a joint test for a small number of independent time series (N 10). If the model allows for individual-specic intercepts and trends, the standard Dickey Fuller test has very low power even for relatively long time series ( 100), whereas substantial power can be achieved by using the panel unit root test procedure with a panel of moderate dimensions. (i.e. N =10 and = 50, or N = 25 and = 25). Maddla and Wu (1999) have done various simulations to compare the performance of competing tests, including IPS test, LL test and the Fisher s test. Care must be taken to interpret their results. Strictly speaking,

16 16 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 able 3 Empirical size properties a Model N =10 N =25 N =50 N = 100 N = 250 m = m = m = m = m = m = m = m = m = a Note:op, middle and lower panel correspond to the ten percent, ve percent and one percent level test, respectively. comparisons between the IPS test and LL test are not valid. hough both tests have the same null hypothesis, but the alternatives are quite dierent. he alternative hypothesis in this article is that all individual series are stationary with identical rst order autoregressive coecient, while the individual rst

17 able 4 Empirical power properties a A. Levin et al. / Journal of Econometrics 108 (2002) Model N =1 N =10 N =25 N =50 N = 100 N = 250 m = m = m = m = m = m = m = m = m = a Note:op, middle and lower panel correspond to the ten percent, ve percent and one percent level test, respectively. order autoregressive coecients in IPS test are allowed to vary under the alternative. If the stationary alternative with identical AR coecients across individuals is appropriate, pooling would be more advantageous than Im et al. average t-statistics without pooling. Also note that the power simulations reported in Maddla and Wu (1999) are not size-corrected.

18 18 A. Levin et al. / Journal of Econometrics 108 (2002) Conclusion In this paper, we have developed a procedure utilizing pooled cross-section time series data to test the null hypothesis that each individual time series contains a unit root against the alternative hypothesis that each time series is stationary. As both the cross-section and time series dimensions of the panel grow large, the panel unit root test statistic has a limiting normal distribution. he Monte Carlo simulations indicate that the normal distribution provides a good approximation to the empirical distribution of the test statistic in relatively small samples, and that the panel framework can provide dramatic improvements in power compared to performing a separate unit root test for each individual time series. hus, the use of panel unit root tests may prove to be particularly useful in analyzing industry-level and cross-country data. he proposed panel based unit root test does have its limitations. First, there are some cases in which contemporaneous correlation cannot be removed by simply subtracting the cross sectional averages. he research reported in this paper depends crucially upon the independence assumption across individuals, and hence not applicable if cross sectional correlation is present. Secondly, the assumption that all individuals are identical with respect to the presence or absence of a unit root is somewhat restrictive. Readers are referred to Im et al. (1995) for a panel unit root test without the assumption of identical rst order correlation under the alternative. Acknowledgements his is a revised version of the Levin and Lin s (1992) earlier work. We would like to thank Clive Granger, Rob Engle, Neil Ericsson, Soren Johansen, Cheng Hsiao, homas MaCurdy, James McKinnon, Max Stinchcombe, C.-R. Wei, Hal White, and an anonymous referee for helpful comments and suggestions. We appreciate the assistance of Paul Homan and the sta of the UCLA supercomputer center. Appendix A. Proof of heorem 3. For each ; { 1i ;; 2;:::;N } a zero-mean, independent sequence, where 1i 1i 1i. Part one of the proof is accomplished in three steps. he following results are used in the subsequent proof. sup 1i =O p ( 1 ) i (A.3.1)

19 and A. Levin et al. / Journal of Econometrics 108 (2002) sup(i 2 0:5si 2 )=O p ( 1 );i 2 E( 2 1i ) i 2 : (A.3.2) i In step 1, we prove that N 1=2 1i obeys the central limit theorem. For this, we need to ensure the following Lindeberg condition holds (Billingsley, 1986, pp ): ( N N i 2 ) 1 2 1i I { ) N 1=2 } dp 0; as : 1i ( i 2 It is well-known that the sucient condition for the Lindeberg condition is E 1i 2+ ; for some 0 and all i =1;:::;N : (A.3.3) Straightforward but tedious algebra leads to result similar to (A.3.1) and (A.3.2), sup i [E 1i 4 (15=4)si 4 ]=O p ( 1 ). Hence there exists some such that (A.3.3) holds for all i, and the Lindeberg condition is fullled. We have from the CL that N 1=2 N 1i = N 1=2 N ( 1i 1i ) AN(0; 0:5V ); V = lim N N 1 si 2 : (A.3.4) N In step 2, we show that N 1=2 N 1i also obeys the central limit theorem. Using (A.3.1), N 1=2 N ( 1i 1i )=N 1=2 N 1i + N 1=2 N 1i : (A.3.5) From (A.3.1), he second term in (A.3.5) 6 ( N = )(1=N ) N O p( 1 ) 6 ( N = )M, since O p ( 1 ) is a bounded sequence for all i. It follows that N 1=2 N ( 1i 1i ) and N 1=2 N 1i are asymptotic equivalent as N = 0. In the nal step, we prove the asymptotic equivalence between N 1=2 N 1i and N 1=2 N 1i 1i. he strategy is to show that the dierence between them converges to zero in probability. o do so, we consider the sequence of triangular array, {( 1i 1) 1i }, and verify the condition in heorem 6:2 of Billingsley (1986, p. 81). Dene c i =E[( 1i 1) 1i ] and 2 i = var[( 1i 1) 1i ]. he required condition is that there exists a sequence v such that v 1 N 2 i 0. In fact, putting v = N 1=2 works.

20 20 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 Since 2 i 6 E[( 1i 1) 2 2 1i ], { N N 2 i 6 E[( 1i 1) 2 2 1i ] v 1 = 6 { { N 1 MN 1 N 1 N 1 E[( 1i 1) 2 2 1i ] N 1 1=2 E( 2 1i )} ; } 1=2 } 1=2 because ( 1i 1) 2 is bounded in probability. It follows that { } N 1=2 1 E( 2 1i ) MN { { MN 1 0:5MN 1 N 1 N 1 [0:5s 2 i +O p ( 1 )] s 2 i + MN 1 } 1=2 N 2 It is seen that the last term converges to zero, hence N 1=2 However, N 1=2 N N [( 1i 1) 1i ] N 1=2 c i = N 1=2 = 6 M 6 M N N 1 N N N N 1=2 O p ( )} 1 : c i p 0: E[( 1i 1) 1i ] N 1 N 1 E[ ( 1i 1) 1i ] N N N E( 1i ) O p (1= )

21 A. Levin et al. / Journal of Econometrics 108 (2002) MM N 1 1 N N O p (1= ) 6 MM N = 3=2 0: his completes proving the rst part of theorem 3. For the second part of the proof, we verify that the condition of LLN for triangular array holds (Billingsley, 1986, p. 81). Since { 2i ;; 2;:::} is an independent sequence with mean sup i [E( 2i ) 0:5si 2 ]=O p ( 1 ), and sup i [Var( 2i ) 1 3 s4 i ]=O p ( 1 ), if the required condition [ N var( 2i )] 1=2 = v 0 holds for some positive sequence {v }, then v 1 N [ 2i E( 2i )] p 0: (A.3.6) Put v = N. Assumption 3(e) ensures that [ N var( 2i )] 1=2 =v 0, hence (A.3.6) follows. But N 1 N E( 2i ) 6 N 1 N =0:5N 1 [0:5s 2 i +O p ( 1 )] N s 2 i + 1 N 1 N O p ( 1 ) 0:5V: Desired result follows from combining this result with 2i 1 in heorem 1. Finally, since ˆ 1;t N(0; 1) follows immediately. Proof of heorem 4. Since { 1i ;; 2;:::}; 1i 1i 1i, is a zero mean, independent sequence across individuals such that sup i ( 1i s i m1 )= O p ( 1 ) and sup i (i 2 s2 i m1 2 )=O p( 1 );i 2 Var( 1i ), where m =2 or 3, is the model index. Analogous to the proof in heorem 3, the Lindeberg condition holds, and the CL implies that ( N ) 1=2 N ( ) N 1=2 N 1i = N 1=2 1i N(0; 1); or i 2 N 1=2 N N 1 1i N(0; 2 m1v ): Now, N 1=2 N ( 1i 1i ) 6 N 1=2 N ( 1i s i m1 )+N 1=2 N O p( 1 ). Since N 1=2 N O p( 1 ) 0, as N = 0, we conclude that N 1=2 N ( 1i s i m1 ) N(0;m1 2 V ). Similar to the proof in heorem 3, we can also 2 i

22 22 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 prove that N 1=2 N 1i ( 1i s i m1 ) N(0; 2 m1v ): (A.4.1) For the second part of heorem 4, consider replacing m1 in (A.4.1) with its consistent estimate ˆ 1i. Write N 1=2 N 1i ( 1i ˆ 1i )=N 1=2 N 1i ( 1i s i m1 ) +N 1=2 N 1i (s i m1 ˆ 1i ): (A.4.2) We know that the rst term in (A.4.2) converges to a normal distribution. he asymptotic behavior of the second term is of interest. Note that ŝ i =ˆ yi = ˆ i is a bounded sequence for all i, due to ˆ i = i +O p ( 1=2 ) and ˆ yi = yi + O p ( 1=2 ), c.f. Andrews (1991). Since m = m1 +O p ( 1 ) by construction and 1i =1+O p ( 1=2 ); ˆ 1i =ŝ i = 1i m = s i m1 +O p ( 1=2 ). Now rewrite the second term in (A.4.2) as N 1=2 N = N 1=2 1i (s i m1 ˆ 1i ) N ( 1i 1)(s i m1 ˆ 1i )+N 1=2 N (s i m1 ˆ 1i ) 6 N 1=2 N O p ( 1=2 )O p ( 1=2 )+N 1=2 N O p ( 1=2 ): (A.4.3) he rst term in (A.4.3) 0, as N = 0. However, the second term requires that N = 0. Complete the second part of heorem 4. Proof of heorem 5. { 2i ;; 2;:::} is an independent sequence with sup i [E( 2i ) si 2 m2 ]=O p ( 1 ), and sup i [Var( 2i ) si 4 m2 2 ]=O p( 1 ). Applying the LLN as in the proof of heorem 3, N 1 N 2i N 1 N E( 2i ) p 0. But N 1 N E( 2i ) 6 N 1 N [s2 i m2 +O p ( 1 )] = m2 N 1 N s2 i + 1 N 1 N O p( 1 ) m2 V.

23 A. Levin et al. / Journal of Econometrics 108 (2002) he standard deviation adjustment m m1= m2 by construction, and ˆ 1, we have t N(0; 1). References Anderson,.W., Hsiao, C., Formulation and estimation of dynamic models using panel data. Journal of Econometrics 18, Andrews, D.W.K., Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, Billingsley, P., Probability and Measure, 2nd Edition. Wiley Press, New York. Box, G.E.P., Jenkins, G.M., ime Series Analysis:Forecasting and Control. Holden Day Press, San Francisco, CA. Breitung, J., Meyer, W., esting for unit roots in panel data:are wages on dierent bargaining levels cointegrated? Institut fur Quantitative Wirtschaftsforschung, Working paper. Campbell, J., Perron, P., Pitfalls and opportunities:what macroeconomists should know about unit roots. In:Blanchard, O., Fischer, S. (Eds.), NBER Macroeconomics Annual. MI Press, Cambridge, MA. Dickey, D.A., Fuller, W.A., Distribution of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74, Dickey, D.A., Fuller, W.A., Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, Diebold, F.X., Nerlove, M., Unit roots in economic time series:a selective survey. Advances in Econometrics 8, Fuller, W., Introduction to Statistical ime Series. John Wiley. Hall, A., esting for a Unit Root in ime Series with Pretest Data-based Model Selection. North Carolina State University, Working Paper. Harris, R., zavalis, E., Inference for unit root in dynamic panels. Unpublished manuscript. Holtz-Eakin, D., Newey, W.K., Rosen, H., Estimating vector autoregressions with panel data. Econometrica 56, Hsiao, C., Analysis of panel data, Econometric Society Monograph #11. Im, K.S., Pesaran, H., Shin, Y., 1995, esting for unit roots in heterogenous panels. Unpublished manuscript. Johansen, S., Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregression models. Econometrica 59, Johansen, S., Determination of cointegration rank in the presence of a linear trend. Oxford Bulletin of Economics and Statistics 54, Levin, A., Lin, C.-F., May Unit root tests in panel data:asymptotic and nite-sample properties. UC San Diego, Working Paper MaCurdy,.E., he use of time series processes to model the error structure of earnings in a longitudinal data analysis. Journal of Econometrics 18, Maddla, G.S., Wu, S., Comparative study of unit root tests with panel data and a new simple test. Oxford Bulletin of Economics and Statistics 61, Park, J.Y., esting for unit roots and cointegration by variable addition. Advances in Econometrics 8, Phillips, P.C.B., ime series regression with a unit root. Econometrica 55, Phillips, P.C.B., Moon, H.R., Linear regression limit theory for nonstationary panel data. Econometrica 67,

24 24 A. Levin et al. / Journal of Econometrics 108 (2002) 1 24 Phillips, P.C.B., Perron, P., esting for a unit root in time series regression. Biometrika 75, Quah, D., International patterns of growth I:persistence in cross-country disparities. MI working paper. Quah, D., Sargent,.J., A dynamic index model for large cross-country disparities. Institute for Empirical Macroeconomics DP#77. Quah, D., Exploiting cross section variation for unit root inference in dynamic data. Economic Letters 44, Said, S.E., Dickey, D.A., esting for unit roots in autoregressive-moving average processes of unknown order. Biometrika 71, Schwert, G.W., ests for unit roots:a Monte Carlo investigation. Journal of Business and Economic Statistics 7,