Testing for Panel Cointegration Using Common Correlated Effects Estimators

Spurious regression analysis in panel data when the time series are cross section dependent is analysed in the article. We show that consistent estimation of the long‐run average parameter is possible once we control for cross section dependence using cross section averages in the spirit of the common correlated effects approach proposed by Pesaran. This result is used to design a panel cointegration test statistic accounting for cross section dependence. The performance of the proposal is investigated in comparison with factor‐based methods to control for cross section dependence when strong, semi‐weak and weak cross section dependence may be present.


INTRODUCTION
During the last 20 years, the analysis of macroeconomic panels has experienced a vast and rapid development. This has been primarily due to two reasons: first, the easy availability of statistical information concerning panels of data where the time dimension is augmented by the use of cross section variation (for example, across countries, regions or industrial sectors) and second, the belief that combining these two sources of information would lead to better statistical inference.
The recent literature has seen many efforts, in particular, to design procedures aimed at estimating long-run relationships among economic variables using macro-panel data techniques. Testing for cointegration in panel data has been a particular area of focus, since it constitutes the analysis that needs to be conducted prior to estimating long-run relationships. The early articles in this area such as the ones by Kao (1999) and Pedroni (2000) assumed cross section independence among the units of the panel data, a situation that is rarely found in empirical economic analyses. Cross section dependence appears naturally when studying economic data due to, for instance, market integration processes, globalization of economic activity, offshoring processes or because of the presence of common shocks like oil price shocks. More recent articles have therefore devoted considerable attention to devising procedures relaxing the assumption of cross section independence -see, for example, Breitung and Pesaran (2008) for an overview.
There may be different sources of cross section dependence, exerting different degrees of dependence intensity. On the one hand, we may have pervasive cross section dependence due to the presence of a dominant unit in the panel data set-up, a situation that can be interpreted as if there were common factors affecting all the time series .I L/ U i;t D e i;t D H i .L/" i;t ; where D i;t denotes the deterministic part of the model that is given by either the absence of deterministic elements D i;t D 0 8i (Model 0), a vector of constant terms, D i;t D i D . i;0 ; i;1 ; : : : ; i;k / 0 (Model 1), or a vector linear time trends, D i;t D ı i .1; t / 0 , with ı i D ı 0 i;0 ; ı 0 i;1 ; : : : ; ı 0 i;k 0 , ı i;j D . i;j ; Á i;j / 0 , j D 0; 1; : : : ; k, (Model 2). The F t component denotes a .r 1/-vector of common factors and i the ..k C 1/ r/ matrix of factor loadings, and denote by K an invariant -field generated by F t so that, conditionally on K, U i;t D U y i ;t ; U 0 x i ;t 0 are independent across i . The disturbance terms v t and e i;t are assumed to be I.0/ stationary processes, i D 1; : : : ; N , t D 1; : : : ; T . Our analysis is based on the same set of assumptions as in Bai and Ng (2004) and Banerjee and Carrion-i-Silvestre (2015). Let M < 1 be a generic positive number, not depending on T and N . Further, the Euclidean norm of a generic matrix A is defined as kAk D t race .A 0 A/ 1=2 . Then Assumption B. (i) v t D C.L/w t , w t i id .0; † w /, E kw t k 4 Ä M and (ii) Var .F t / D P 1 j D0 C j † w C 0 j > 0; (iii) P 1 j D0 j kC j k < M ; and (iv) C.1/ has rank r 1 , 0 Ä r 1 Ä r.
Assumption C. (i) For each i , e i;t D H i .L/" i;t , " i;t i id 0; 2 ";i , E j" i;t j 8 Ä M , P 1 j D0 j jH i;j j < M , ! 2 i D H i .1/ 2 2 ";i > 0; (ii) E ." i;t " j;t / D i;j with P N iD1 j i;j j Ä M for all j ; and (iii) Eˇ1 p The model specification considers the case where the stochastic regressors x i;t are assumed to be either cross section independent -imposing all, but the first, rows of i to be equal to zero -or cross section dependent with dependence driven by a set of observable common factors F t . Furthermore, it is possible to assume that the set of common factors affecting the endogenous variable y i;t is different from those affecting the stochastic regressors x i;t , a situation that is covered if we define i to be a block-diagonal matrix.
Despite the presence of the operator .I L/ in equation (2), F t does not have to be I.1/. In fact, F t can be I.0/, I.1/ or a combination of both, depending on the rank of C.1/. If C.1/ D 0, then F t is I.0/. If C.1/ is of full rank, then each component of F t is I.1/. If C.1/ ¤ 0, but not full rank, then some components of F t are I.1/ and some are I.0/.
Note that although this framework is very flexible, it implies a change in the standard definition of cointegration. The usual definition of cointegration among Y i;t D .y i;t ; x 0 i;t / 0 requires F t to be I.0/, so that the observable variables capture all the common stochastic trends. However, allowing F t to be I.1/ is also relevant from an empirical point of view since F t might be accounting for effects that are not captured by Y i;t alone. In such a situation, cointegration among the elements in Y i;t up to the inclusion of I.1/ factors is both possible and desirable economically, which will imply that H i .1/ ¤ 0 but is not full rank. 2 This leads to a broader concept of cointegration, where the observable variables in Y i;t alone do not generate a cointegrating relationship. Instead, common factors are required to enter in the model to define a long-run relationship. In this regard, the framework is close to Bai et al. (2009) where cointegration is assumed allowing for the possibility of I.1/ factors.
Panel spurious regression has been tackled in Phillips and Moon (1999). Contrary to what is found at the unit level analysis -see Granger and Newbold (1974) and Phillips (1986) -pooled estimation of the parameters affecting the stochastic regressors leads to consistent estimates of the so-called long-run average coefficientˇ. For ease of exposition, let us assume that there is no deterministic component (Model 0) and specify the model that relates the dependent variable y i;t and the explanatory variables x i;t in matrix notation as where y i denotes the .T 1/ vector of the dependent variable, x i is a .T k/ matrix of explanatory variables, i is a .k 1/ vector of parameters , F is a .T r/ matrix of common factors, i is a .r 1/ vector of factor loadings and i is a .T 1/ vector collecting the idiosyncratic errors. Note that we can write (4) in terms of the elements in (1) and obtain the following: so thatˇi can be estimated as the vector of parameters that capture the relationship among the idiosyncratic component of the variables.
Let us define the projection matrix M D D I T D .D 0 D/ 1 D 0 that removes the effect of the deterministic component on the variables of the model, where D D Ã for Model 1 and D D OEÃ for Model 2 -it should be understood that M D D I T for Model 0. Then we have where the superscript ' ' indicates that the corresponding variable has been detrended and defactored. Note that at this stage, we have assumed that the common factors are observable. 3 The pooled estimator is defined as Theorem 1. Let Y i;t be a vector of .1Ck/ stochastic processes with DGP given by (1)-(3). Under the assumption that H i .1/ is positive definite almost surely for all i (spurious regression), the pooled estimator given in (6) converges as .T; N / ! 1 jointly to whereˇdenotes the long-run average regression coefficient, and Ux Ux and Ux Uy are long-run average covariance matrices of the respective idiosyncratic components defined in the Supporting Information.
The proof is provided in the Supporting Information. It is also possible to derive the limiting distribution of the estimated long-run average parameter as in Phillips and Moon (1999). However, for the purposes of testing for cointegration pursued in this article, we are only required to show consistency of the estimator as stated in Theorem 1 .

PANEL CCE COINTEGRATION TEST WITH CROSS SECTION DEPENDENCE
So far, the cross section dependence has been assumed to be driven by a set of observable common factors. However, in most cases, this situation is infeasible from an empirical point of view, and we need to devise procedures to estimate (or proxy for) the unobserved common factors.
There are two popular approaches in the literature that address this issue. First, the Ng (2002, 2004) proposal, which uses principal components to estimate the common factors and panel information criteria to choose the number of common factors. Second, we can use the cross section average method suggested in Pesaran (2006Pesaran ( , 2007 and Pesaran et al. (2013), which employs cross section averages as convenient proxies to capture the common factors without requiring the estimation of their number. This article looks in detail at this second approach and also establishes a comparison with testing procedures based on Ng (2002, 2004). Note, however, that the derivations obtained in the previous section follow if we also estimate the common factors and loadings using the approach described in Bai and Ng (2004).
Assumption F. Let us assume that rank . N / D r Ä .1 C k/ for all N as N ! 1.
If the rank condition established in Assumption F is met, we have which indicates that, for sufficiently large N , the observable averages N h t D N D t ; N Y 0 t 0 can be used to proxy the unobserved factors. Following Holly et al. (2010), let us specify the cross section augmented regression: where Ń t D N y t ; N x 0 t 0 collects the cross section averages of the dependent and the stochastic regressors of the model. To estimate theˇparameters in (7), Holly et al. (2010) use the pooled CCE estimator (PCCE) in Pesaran (2006), which is given by where x i D OEx i;1 x i;2 : : : ; x i;k , x i;j D .x i;j;1 ; x i;j;2 ; : : : ; x i;j;T / 0 denotes the .T k/ matrix of regressors of interest, y i is the .T 1/ vector of the dependent variable for the i th unit and N One interesting feature is that the PCCE estimator is easy to compute and does not require the estimation of the factors driving the cross section dependence. The main drawback is that consistency has been proved by Kapetanios et al. (2011) only under the maintained hypothesis that cointegration exists, a hypothesis that needs to be tested. Therefore, to assess the validity of the testing procedure that we apply, we need to show whether the PCCE estimator is consistent under the null hypothesis of no cointegration. This result is provided in the following Theorem.
Theorem 2. Let Y i;t be a vector of .1Ck/ stochastic processes with DGP given by (1)-(3). Under the assumption that H i .1/ is positive definite almost surely for all i (spurious regression) and rank . N / D r Ä .1 C k/ for all N as T; N ! 1, the pooled estimator given in (8) converges as .T; N / ! 1 jointly to The proof is provided in the Supporting Information.

A test for cointegration based on pooled common correlated effects estimator
Using Theorem 2 and following on from the contributions of Pesaran (2007), Holly et al. (2010) and Pesaran et al. (2013), in this section, we propose a panel cointegration test statistic that is based on the PCCE estimator. It is worth mentioning that the use of the Pesaran approach requires us to constrain the DGP that has been used so far in the sense to be described in the discussion below. Thus, Pesaran (2007) and Pesaran et al. (2013) specify the following DGP: with f t a .r 1/-vector of I.0/ common factors and i;t an I.0/ idiosyncratic disturbance term. As can be seen, equation (10) can be written as i;t D 1= .1 Â i L/ f 0 t i C i;t so that the null hypothesis of spurious regression, with F 1;t I.1/ and i;t I .1/. Under the alternative hypothesis, we have jÂ i j < 1 for i D 1; : : : N 1 ; Â i D 1 for i D N 1 C 1; : : : N , with N 1 =N ! ı, 0 < ı Ä 1 as N ! 1, so that cointegration exists for N 1 units. Further, note that under the alternative hypothesis, for the N 1 units for which 0;t I.0/ and i;t I.0/, but for the remaining N N 1 units, F 1;t I.1/ and i;t I.1/.
The Pesaran approach requires us to assume the same order of integration for all common factors and the idiosyncratic component for each specific unit of the panel, namely, I.1/ under the null hypothesis of spurious regression for all units, and I.0/ for N 1 units and I.1/ for the remaining N N 1 units under the alternative hypothesis. This is a substantial restriction of the general framework used previously for estimation, where no assumption on the order of integration properties of the common factors was needed in relation to that of the idiosyncratic components. It is worth noticing that the same stochastic processes f t that generates the common factors can play different roles under the alternative hypothesis for the different units of the panel data -that is, under the alternative hypothesis, f t can generate either F 0;t or F 1;t , depending on the specific unit of the panel.
The PCCE-based cointegration test begins by estimating the long-run average coefficient using the PCCE method. Given the consistency of the PCCE estimator under the null hypothesis of spurious regression, in the PANEL COINTEGRATION USING CCE ESTIMATORS 617 second stage, we use the PCCE-estimated parameters to define the variable: for which the following model is estimated using ordinary least squares estimation method: and the ordinary least squares residuals are then computed as The null hypothesis of no cointegration is tested analysing the order of integration of O i;t through the application of the cross section augmented Dickey-Fuller (1979) cointegration (CADF) statistic: where t Ǫ i;0 denotes the pseudo t-ratio of the estimated˛i ;0 parameter in the regression: when there is one common factor and with O A t D O t ; N x 1;t ; : : : ; N x k;t Á 0 the vector of cross section averages augmentation terms for the 1 C k multiple common factors case.
It is worth noticing that (12) and (13) define the two extreme cases, that is, the case where there is just one common factor and the case where the rank condition established at Assumption F is met with equality. For those intermediate cases where there are fewer common factors than observables -that is, r < 1 C k -the vector O A t in (13) will be defined with O t and r 1 elements of the cross section averages of the stochastic regressors. It is important to emphasize that in empirical applications, the number of common factors .r/ does not need to equal the total number of observables .1 C k/ of the model so that the intermediate cases are relevant from an empirical point of view. If analysts have knowledge, for example, based on economic theory, about the number of common factors to include in the model, one can impose the restriction of the number of common factors and use the critical values that involves .1 C k/ observable variables but r < 1 C k common factors. In this regard, we could follow the strategy in Pesaran et al. (2013) and compute the test statistic using all possible combinations of r cross section averages available in the system as a way of obtaining robust conclusions.
When the number of common factors is not known, we can follow a conservative strategy and assume that the rank condition is satisfied with equality and base inference on the estimation of (13). The price that we would pay if the true number of common factors is r < 1Ck but we impose r D 1Ck is to have a test statistic with empirical size smaller than the nominal size accompanied by loss of power. The advantage is to allow us to remain agnostic about the number of integrated stochastic trends driving the data. To derive the critical values appropriate for the PCCE-based cointegration test, note that we can substitute (7) in (11) and obtain where Q y P i;t is the unit root part of the process analysed by Pesaran (2007) -when r D 1 -and by Pesaran et al. (2013) when r > 1. It is worth noticing that testing for panel cointegration is asymptotically equivalent to testing for the panel unit root hypothesis addressed in Pesaran (2007) and Pesaran et al. (2013). Using Theorem 2, it is possible to show that as T; N ! 1 , x 0 i;t . Ǒ P C CE ˇ/ has negligible effect on the unit root test of Q y i;t so that, as T; N ! 1 , the cointegration test t Ǫ i;0 is defined by Q y P i;t and Q y P t , which are the same elements that define the limiting distributions in Pesaran (2007) -see his Theorem 3.2 for the CADF if statistic when r D 1 -and Pesaran et al. (2013) -see their Theorem 2.1 when r > 1. 5 Although the limiting distributions of the test statistics proposed in this article and the ones reported in Pesaran (2007) and Pesaran et al. (2013) are equivalent, it is the case that there are slight differences for panel data sets of small T and/or N dimensions. To save space, we only report critical values for the pooled test .CADF P /, although critical values for the individual t Ǫ i;0 test statistic can be computed using a GAUSS program available on request. Tables I and II present the critical values for the CADF P test statistic for Models 1 and 2, respectively, when there is one common factor .r D 1/ -that is, the rank condition is met with inequality -whereas Tables III and IV collect the critical values for the multiple common factor case .r > 1/ -in this case, we impose that the rank condition is met with equality. 6 The computation of the critical values is based on Pesaran et al. (2013), generating the dependent variable as y i;t D y i;t 1 C " 1;i;t and a vector of k explanatory variables x i;t D x i;t 1 C " 2;i;t , where " i;t D " 1;i;t ; " 0 2;i;t 0 i id N .0; I kC1 /, i D 1; 2; : : : ; N , t D 50; 49; : : : ; T and y i; 50 D x i; 50 D 0. Using these independent time series, we have computed the PCCE estimator and retrieved the O e i;t residuals that are used to estimate the regression equation in (12) and obtain the individual and CADF P statistics. The simulation uses 50,000 replications using different combinations of T and N . As can be seen, the critical values for the one common factor case are close to the ones computed in Pesaran (2007) when T is large, although they differ in finite samples -for example, for Model 2, compare Table II of our article with Table 1b of Pesaran (2007) when T D 200. Note also that for large T and N , the critical values do not depend on the number of regressors, since the consistency property of the Ǒ P C CE estimator implies that the CADF P statistic behaves like the Pesaran (2007) panel unit root statistic, making our critical values applicable to cases where there are more than two regressors. A similar feature is found when comparing the critical values in Pesaran et al. (2013) and the ones computed in this article for the multiple common factor case. Pesaran (2007) also proposes a truncated version of the CADF if statistic to ensure that the statistic has finite moments. In our case, the truncation takes the following form -see Pesaran (2007 p. 277 where .d 1 ; d 2 / D .6:19; 2:61/ for Model 1 and .d 1 ; d 2 / D .6:42; 1:70/ for Model 2. Note that we use the same 5 The limiting distributions are obtained using sequential and joint limits assuming that N=T ! k > 0. Since consistency only requires that N and T tend to infinity jointly, the condition that N=T ! k > 0 does not pose any difficulty. The limiting distribution of CADF if can also be derived under sequential limits provided N ! 1 before T ! 1. 6 We do not report the critical values of all possible combinations where the rank condition is satisfied with inequality, although a GAUSS program is available from the authors to compute the critical values for any desired combination. threshold values as in Pesaran (2007) given that the limiting distributions of our test statistic and that of Pesaran are the same. The unreported computations that we have carried out show that the critical values of the truncated and untruncated versions of the test statistic coincide exactly for all values of T > 15 so that, to save space, we have not presented these critical values on the article. Truncation can also be applied to the multiple common factor case, although Pesaran et al. (2013) do not provide the values of the upper and lower limits for the different number of common factors, although they can be easily obtained -a GAUSS program can be used to compute the threshold values for the truncated version of the statistic for any number of common factors.  As discussed briefly in the introduction earlier, the approach proposed in this article for testing panel cointegration differs from a common factor-based approach. An example of the latter is contained in Banerjee and Carrion-i-Silvestre (2015), who deal with the same model specification that is used in this article but where the common factors and factor loadings are estimated using principal components. In addition to accounting for cross section dependence in two different ways, the testing procedures also differ in one other crucial aspect, namely, the computation of the estimate ofˇi in the individual units of the panel. For Theorem 2 to apply,ˇneeds to be a pooled estimator, that is, the potential heterogeneity of theˇi s across the units of the panel is not taken into account in computing the CCE-based test. This is in contrast to the common factor-based test that, since it is computed after first differencing the data, allows for heterogeneity in theˇi parameter, and the test statistic for the idiosyncratic component (which is most directly comparable with the CCE-based test) is based on a mean-group test constructed by averaging across the unit-specific standardized t -statistics. In principle, this therefore adds to the flexibility of the common factor-based approach, although such flexibility is unnecessary if either strict homogeneity holds or theˇi coefficients are generated by means of a random effects-type specification. However, to counter this flexibility, there is also the disadvantage of the need to estimate more parameters to construct the corresponding test statistic.  The Monte Carlo simulations reported in the discussion below specify homogeneousˇi to present the most favourable scenario from the point of view of the use of pooled estimators while disadvantaging the factor-based tests.

Common factor model: weak and strong dependence
This section looks at the performance of the CCE-based tests for cointegration in comparison with the factor-based approach under several different specifications of cross section dependence, both strong and weak. It should be noted that under some specifications of weak or semi-strong dependence to be noted below, the factor approaches 623 are no longer optimal and do not provide consistent estimates of the factors or their loadings as typically, Assumption A(ii) is violated in such circumstances. It is nevertheless of importance to compare the results of the two approaches, since at an empirical level, it is often not clear what form the cross section dependence takes in the data. It is therefore interesting and important to note within the context of the simulation exercises the better performance of the factor-based tests despite worries about the consistency of the procedures when dependence is only weak. Many of the features of the DGP used in the succeeding discussion are influenced by the empirical examples, which help us to interpret better the results arising from the estimation of the models.

Strong Dependence
Let us first consider the DGP defined by the following: where r D ¹1; 2º, i;j N.1; 1/, i;t N.0; 1/, w j;t N 0; 2 F , j D 1; 2, and " i;t N.0; 1/ are four mutually independent groups. Under the null hypothesis of no cointegration, we specify i D 1 8i , whereas under the alternative hypothesis of cointegration, we have j i j < 1 for some i . Note that the definition of cointegration that we are testing for only focuses on the idiosyncratic component, regardless of the order of integration of the common factor. Thus, if F t I.1/ cointegration exists among .y i;t ; x i;t ; F t / but not between .y i;t ; x i;t /. It is worth noticing that the definition of the loadings implies that P N iD1 j i j D O p .N /, so that we are facing the case of strong dependence.
The simulations focus on Model 2 using the following set-up. The empirical size is analysed using i D 1, whereas the empirical power is investigated using i D ¹0:99; 0:95; 0:9º. 7 As for the common factor component, we consider one and two common factors with autoregressive parameter given by D ¹1; 0:99; 0:95º with different importance, which is modelled through the following values for the variance 2 F D ¹0:5; 1; 10º. The time dimension is set at T D ¹50; 100; 250º, and the cross section dimension is N D ¹10; 20; 50º. The nominal size is set at 5%, and the critical values tabulated in the previous section are used. The simulations are performed using GAUSS with 1000 replications. To save space, we only report the results for 2 F D 1 where the number of common factors is estimated. The results for the remaining two values of 2 F are qualitatively very similar, and the full set of tables can be found in Banerjee and Carrion-i-Silvestre (2011).
The simulations conducted in this subsection distinguish among three different situations. First, we cover the case where there is one common factor, and use the critical values that are computed for the true number of common factors -in this case, the rank condition is met with equality, that is, O r D 1. Second, we consider the case of two common factors using the critical values that are computed for the true number of common factorsin this case, the rank condition is met with equality, that is, O r D 2. Finally, we focus on the one common factor case but where we assume that there are two common factors -the rank condition is satisfied with inequality, but we use the critical values that are appropriate when it is satisfied with equality. This case is discussed to mimic the scenario of conservative inference.

A. BANERJEE AND J. L. CARRION-I-SILVESTRE
One common factor and O r D 1. Before presenting the results for the empirical size and power of the panel cointegration test statistic that is proposed in this article, we have conducted a small Monte Carlo simulation to show that the consistency property obtained in Theorem 2 gives a proper approximation in finite samples. Table 1 in the supplementary material reports the results of the mean, median and root mean square error of the Ǒ P C CE estimator with N D ¹10; 20; 50; 100º and T D ¹50; 100; 250º for the one common factor case. As can be seen, the mean and the median are close to the true value of the parameter -that is,ˇD 1 in (14) -regardless of the values of i and . This emphasizes the value of the approach since it is possible to obtain consistent estimates of when there is no cointegration ( i D 1) and when there is cointegration (j i j < 1). Moreover, because the factor is being controlled for adequately, whether the factor is integrated ( D 1) or stationary (j j < 1) does not affect the root mean square errors. Under spurious regression, root mean square error decreases with N . This result is supported by the theoretical derivations shown in Theorem 2 for the limiting distribution of Ǒ P C CE . When there is cointegration, it decreases with both N and T . Finally, for a given combination of N and T , the root mean square error is larger under the spurious regression case than when there is cointegration.
Table V presents the empirical size and power for the CADF P panel cointegration test statistic for N D ¹10; 20; 50º for the one common factor case. In each table, we also report the results for the test statistics in Banerjee and Carrion-i-Silvestre (2015) -hereafter, Z statistic -for which the number of common factors throughout this section is estimated using the panel Bayesian information criterion (BIC) in Bai and Ng (2002) with a maximum of six common factors.
As can be seen, the Z test has the correct size, regardless of the value of the autoregressive parameter of the common factor . /, except when both N and T are small. The CADF P statistic has the correct size when D 1, although we observe that the test statistic tends to be conservative (under-rejects) as moves away from 1 and T gets large. Note that this can be explained by the fact that this set-up violates the common factor restriction that is required by Pesaran's (2007) framework, namely, that i D -that is, the dynamic of the idiosyncratic component should be the same as the one driving the common factor component.
As for empirical power, we observe that the CADF P statistic does not outperform the Z statistic for any of the cases shown here. However, the empirical power of the two statistics is almost equivalent for large T which may be taken as good grounds for preferring the use of the CCE-based test when T is reasonably large. It is worth mentioning that even in those cases where the CADF P statistic becomes conservative because of the violation of the common factor restriction, it still shows good power.
So far, we have compared the panel data test statistics that are computed using the estimated idiosyncratic component. The procedure in Banerjee and Carrion-i-Silvestre (2015) also allows us to analyse the stochastic properties of the estimated common factors. The ADF statistic that is computed using the estimated common factor is reported in the columns labelled as t Q F . As can be seen, the t Q F has the correct size under the null hypothesis that D 1, with empirical power that increases, as expected, as moves away from 1 and T gets large. To sum up, for this simple scenario, the principal components-based panel cointegration test in Banerjee and Carrion-i-Silvestre (2015) shows better overall performance, with empirical size close to the nominal size and empirical power higher than those demonstrated by the CCE-based statistics. However, both approaches tend to provide the same empirical power when the time dimension is large, and the convenience of the CCE-based approach needs also to be taken into account when assessing the relative merits of these alternative testing procedures.
Finally, it could be stressed that the procedure in Banerjee and Carrion-i-Silvestre (2015) is more informative, as it allows to obtain a fuller picture of the stochastic properties of all the specified components affecting the model. As noted earlier, from an empirical point of view, assessing the stochastic properties of the common factors is particularly important since this allows us to interpret whether .y i;t ; x i;t / cointegrate alone or whether we need to consider .y i;t ; x i;t ; F t / to obtain a cointegrating relationship.

Two common factors and O
r D 2. The simulations conducted in this section are based on the DGP given by (14) to (17) using two common factors, but where instead of using (15) for the generation of the stochastic regressors x i;t , they are defined according to where x i;j N.1; 1/, j D 1; 2. Note that now, we are considering that both the dependent variable and the stochastic exogenous regressors are affected by the common factors. In this case, the rank condition is satisfied with equality provided that the number of observables .1 C k D 2/ equals the number of common factors .r D 2/, which also equals the number of cross section averages that are used in the computation of the statistics -that is, we assume that O r D 2.  Table VI reports in the columns labelled as Equality the empirical size and power for the CADF P test statistic, when N D ¹20; 50º. As can be seen, the test statistic has a liberal empirical size for N D 20, regardless of the order of integration of the common factors. However, the empirical size equals the nominal size when the number of units of the panel increases up to N D 50. It is worth noticing that in this case, the test statistic features under-rejection problems when the common factors are I.0/, a situation that violates the assumptions made in our framework. As for the empirical power, the CADF P test statistic shows decent power figures, which tends to one as T gets large, regardless of the value of N .

One common factor and O
r D 2. From an empirical point of view, it is more interesting to analyse the effects that might have on the empirical size and power of the CADF P test statistic when practitioners use more cross section averages than common factors present in the model. In this case, we have specified the DGP given by (14), (16), (17) and (18) but considering just one common factor .r D 1/. Thus, by using all cross section averages available in the system, we are covering the situation where the assumed number of common factors . O r D 2/ is larger than the true number of common factors .r D 1/. Note that now, the rank condition is met with inequality .r < 1 C k/.
Table VI reports in the columns labelled as Inequality the empirical size and power for the CADF P test statistic, when N D ¹20; 50º. As can be seen, the empirical size is close to the nominal empirical size when the common factor is I.1/, regardless of the value of N . As expected, the test statistic becomes conservative as the common factor becomes I.0/ -see the comments mentioned previously. As for the empirical power, the CADF P test statistic has good power, which tends to one as T gets large. An interesting feature is that for a given combination of . i ; /, the figures for the empirical power are smaller than the ones obtained when the correct number of cross section averages are used to capture the effects of the common factors -see the values for the empirical power offered in Table V compared with the ones in the columns labelled as Inequality of Table VI. This result is something to be expected, since we are including more regressors than needed in the regression equation in which the test statistic bases on, so that a fall in the power will be produced.

Semi-strong Dependence and Weak Dependence
Semi-strong dependence. In the previous simulation experiment, we defined a common factor model where the sum of the loadings P N iD1 j i j D O p .N /, a condition that is required to obtain a consistent estimate of the space generated by the common factors. However, it is interesting to analyse the behaviour of the test statistics when we consider departures from this specification, leading to so called semi-strong (or semi-weak) and weak cross section dependence.
For example, following Chudik et al. (2011), we may specify the loadings as Weak dependence. Alternatively we may consider the case where the loadings of the common factors are such that P N iD1 j i j D O p .1/, so that we face the case of weak dependence through the loadings. In this regard, we may also follow Chudik et al. (2011) and specify the loadings as with the rest of the parameters of the DGP as defined in the previous section. It should be noted that in this case, the factor structure is not identified, so that the application of principal components would not lead to consistent estimates of either the common factors or the factor loadings. The use of the test statistic in Banerjee and Carrion-i-Silvestre (2015) is thus strictly speaking not justified. Table VII reports the results for N D ¹10; 20; 50º when the number of common factors is estimated. As can be seen, most of the features that were outlined in the previous section are still valid. However, there are some important differences that could be noted. First, the empirical size of the CCE-based statistic is close to the nominal one even for the case where the common factor is I.0/, so that we do not see any undersize distortions in this case. This may be a reflection of the fact that the data generation processes here are better suited to the CCE approach. Second, except where we have semi-strong dependence with 2 F D 10, the panel BIC does not detect any common factor since the conditions for consistent estimation are not satisfied. We therefore report the results only for the tests on the idiosyncratic component and show that the difficulties of applying the factor approach here notwithstanding, the Z statistic remains more powerful than the CCE-based statistic even when T D 50, although they again perform equally well in terms of power as T gets large.

Spatial autocorrelation
Our final specification of the DGP follows Baltagi et al. (2007) and introduces weak cross section dependence in the panel data set-up using a spatial error model. The DGP is given by with " t D ." 1;t ; " 2;t ; : : : ; " N;t / 0 , W N is an .N N / known spatial weights matrix, # is the spatial autoregressive parameter and t is an .N 1/ error vector assumed to be distributed independently across cross section dimension with constant variance 2 . Second, it is possible to define a spatial moving average (SMA) specification: where now # is the spatial moving average parameter. Finally, we also use the spatial error component (SEC) specification: where t is an .N 1/ vector of local error components and t is an .N 1/ vector of spillover error components. The two component vectors are assumed to consist of i id terms with respective variances 2 and 2 and are uncorrelated.
Of special interest is the SEC specification since we can relate the spatial model with the common factor model that has been investigated in the previous section. We can specify the following: where now t D F t , with D 0 1 ; 0 2 ; : : : ; 0 N 0 the .N r/ matrix of loadings. Further, if we set # D 1 and W N D I N , we obtain the common factor representation used previously. This allows us to specify different models depending on the degree of weak correlation that we want to allow. For instance, if the spatial weight matrix is now V N D I N C W N with t D F t and # ¤ 0, the common factors will affect not only each unit but also their neighbours. The simulations that are reported in this section follow the set-up in Baltagi et al. (2007), who use two different values for # D ¹0:4; 0:8º and the spatial weight matrix W N given by the sparse weight matrix W .1; 1/ that defines the '1 ahead and 1 behind' matrix with the i th row .1 < i < N / of this N N matrix having non-zero elements in positions i C 1 and i 1. The W N matrix has been normalized so that the sum of the elements of each row equals one. Other sparse weight matrices W .j; j /, j D 2; 3; : : : ; 10 were used in Baltagi et al. (2007), although they claimed that qualitatively similar results were obtained. Therefore and to save space, we only use the W .1; 1/ matrix as a way to illustrate the effect of spatial dependence on the panel data cointegration tests that we consider in the article.
The simulation experiment has been conducted for N D ¹10; 20; 50º, and, in general, qualitatively similar results are obtained regardless of the number of cross section units. Consequently, in what follows, our discussion focuses on the results reported in Table VIII for N D 20, given that this panel dimension is closer to the ones used in the empirical practice -Tables 2 and 3 in the Supporting Information present the results of the empirical size and power of the panel data cointegration test statistics for N D ¹10; 50º respectively.
When the SAR specification is used, both CADF P and Z statistics show the correct size when # D 0:4. However, size distortions (over-rejection problems) are observed when # D 0:8, being the size distortions comparable for both test statistics -in some cases, size distortions are larger for the Z statistic (N D 10 and N D 20). In general, the Z statistic is more powerful, although in some cases, this might be due to the effects of the size distortions. As expected, the empirical power of both test statistics tends to one as N and/or T increase.
When the spatial dependence is driven by a SMA specification, both test statistics have the correct empirical size for # D 0:4 but show over-rejection problems when # D 0:8. In this regard, the size distortions are less important for the CADF P statistic, although the distortions almost disappear for both statistics when N D 50. As for the empirical power, we observe that the Z statistic is more powerful than the CADF P statistic in all cases.
The three SEC specifications that we have considered lead to similar qualitative results. For N D 20 and N D 50, the empirical size of the two statistics is close to the nominal one regardless of the value of T and #. Only mild over-rejection problems are found for the Z statistic when N D 10, while the CADF P statistic shows good performance. The Z statistic is more powerful than the CADF P statistic when N D 20 and N D 50, but the performance of the CADF P statistic for N D 10 is very good if one bears in mind that the empirical size is controlled. Finally, the empirical power of both statistics tends to one as T gets large.
In summary, we may conclude from the results of the simulation experiments that there is some evidence in favour of the dominance of factor-based procedures over the CCE approach. However, there may be circumstances where the factor approach is not strictly applicable (such as in the semi-strong or weak specifications, and when N is really small). Allied to the convenience of the CCE approach and equivalent performances for large T , these are good reasons to propose the use of our new test for cointegration in panels. where note that slope homogeneity is imposed. The computation of the CD test statistic in Pesaran (2015) leads us to reject the null hypothesis of no cross section correlation for the panels of the variables involved in the model, which indicates that panel cointegration test statistics that account for the presence of cross section dependence have to be used -see Table X.

House prices in the USA
For this example, we do not undertake a comparison with the test statistic in Banerjee and Carrion-i-Silvestre (2015) since the T dimension is too small relative to N for our needs (to enable consistent computation of the factors). However, it may be seen as an advantage of the CCE-based approach that a feasible test for cointegration can be constructed in the presence of cross section dependence for reasonably small N and T -see tables for size and power properties.
We have computed the individual CCE test statistics proposed in this article using up to four lags for the autoregressive correction in (12) and, as in Holly et al. (2010), considering the presence of one common factor. Table IX shows that the null hypothesis of no cointegration is rejected at the 5% level of significance in 3 (p D 0), 6 (p D 1), 8 (p D 2), 13 (p D 3) and 12 (p D 4) cases out of 49 -if the level of significance is set at the 10%, rejection happens in 5 (p D 0), 11 (p D 1), 14 (p D 2), 20 (p D 3) and 13 (p D 4) cases out of 49. The same results are obtained regardless of whether the truncated or untruncated version of the statistic is used. Therefore, even in the most favourable situation, evidence in favour of cointegration is found for only half of the units. It would be the case that pooling the individual information will lead to better statistical inference, provided that the assumption of cross section independence of O e i;t , i D 1; 2; : : : ; N , in (12) is met. The computation of the CADF P statistic gives CADF P D 1:85 .p D 0/, CADF P D 2:56 .p D 1/ and CADF P D 2:78 .p D 2/, depending on the order of the autoregressive correction that is used. As can be seen, when we compare the values of the CADF P statistic with the critical values in Table I, we conclude that, except for p D 0, the null hypothesis of no cointegration is rejected at the 5% level of significance. However, it should borne in mind that rejection of the null hypothesis does not necessarily imply that cointegration holds for all units.

Production function
The second empirical application focuses on the estimation of a production function using the data in Banerjee et al. (2010) taken from the Penn World Table database (version 6.3). We define a panel data set of developed countries that includes Australia, Austria, Belgium, Canada, Denmark, Finland, France, Greece, Ireland, Italy, Luxembourg, Netherlands, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom and the United States. The selection of these countries allows us to have a balanced panel data set covering the period between 1951 and 2007. Notice also that our data set includes almost all EU-15 countries -we have not been able to include Germany because of lack of information between 1951 and 1969 -and almost all G7 countries -the exception is Japan, for which we do not have information for the whole period. Therefore, we deal with a panel data set of dimension T D 57 and N D 19, which fits the requirement of having a panel with T larger than N . The model that is estimated is given by y i;t D˛i Cˇ1l i;t Cˇ2k i;t C u i;t ; where y i;t denotes the logarithm of the real GDP per capita, l i;t is the logarithm of the population and k i;t is the logarithm of the real capital stock per capita. As before, the CD test statistic in Pesaran (2015) rejects the null hypothesis of no cross section correlation for the panels of the variables involved in the model, which indicates that panel cointegration test statistics that account for the presence of cross section dependence have to be usedsee Table X. The CCE estimation of the slope parameters equals Ǒ D . Ǒ 1 ; Ǒ 2 / 0 D .0:8; 0:78/ 0 . Table XI presents the individual CCE t Ǫ i;0 statistics, i D 1; 2; : : : ; 19, considering that there is one common factor. As can be seen, using the untruncated version of the statistic, the null hypothesis of no cointegration can be rejected at the 5% level of significance in 2 (p D 0), 1 (p D 1), 2 ( p D 2), 1 (p D 3) and 1 (p D 4) cases out of 19 -we use the critical values for N D 20 and T D 50. If the level of significance is set at the 10% Notes: Columns 2 to 6 report the results for different lags. ** and * denote rejection of the null hypothesis of no cointegration at the 5 and 10% levels of significance respectively. a indicates that the null hypothesis is not rejected when using the truncated version of the test statistic. CCE, common correlated effects.
level, the rejection of the null hypothesis of no cointegration happens in 3 (p D 0), 3 (p D 1), 2 (p D 2), 3 (p D 3) and 3 (p D 4) cases out of 19. If we use the truncated version of the statistic, the results that are obtained are almost identical, with the marginal exception for Spain with p D 3, where now the null hypothesis of no cointegration cannot be rejected at the 10% level of significance. Thus, using the individual-based statistics, we find little evidence against the null hypothesis of no cointegration. The individual information can be combined computing the CADF P statistic, which produces CADF P D 1:68 .p D 0/, CADF P D 1:71 .p D 1/, and CADF P D 1:67 .p D 2/, depending on the order of the autoregressive correction that is used. As can be seen, when we compare the values of the CADF P statistic with the critical values in Table I for N D 20 and T D 50, we conclude that the null hypothesis of no cointegration cannot be rejected at the 5% level of significance, regardless of the order of autocorrelation that is considered. The results of the test statistic in Banerjee and Carrion-i-Silvestre (2015) with up to six common factors are reported in Table XII. We present two different sets of results depending on whether or not the variables are divided by their standard deviations when using principal components -see Banerjee and Carrion-i-Silvestre (2015) for further details. Without this transformation, the panel BIC in Bai and Ng (2002) leads to selection of the maximum number of factors that is allowed. In this case, all the estimated common factors are non-stationary. Once transformed, the panel BIC indicates that there is only one integrated common factor. However, regardless of the number of common factors or the transformation, the statistics in Banerjee and Carrion-i-Silvestre (2015) indicate that the idiosyncratic disturbance terms are stationary. It is worth mentioning that rejection of the null hypothesis of no cointegration does not necessarily mean that all cross section units are cointegrated. Therefore, we cannot conclude that the variables in the vector Y i;t D .y i;t ; l i;t ; k i;t / 0 are cointegrated, since at least one non-stationary common factor is detected. Cointegration is possible only by the inclusion of common factors in the model.

CONCLUSIONS
The article has shown that consistent estimate of the long-run average coefficient is obtained when time series in the panel data are cross section dependent, which is accounted for using a common factor model approach. The estimation procedure that is applied is based on the CCE approach in Pesaran (2006). Our result contributes to the literature of non-stationary panel data analysis, where consistent estimation of the parameters of the model is feasible in a spurious regression framework. The article conducts an extensive simulation exercise to study the finite sample performance of the statistic that has been proposed, allowing for weak and strong cross section dependence. The two empirical applications illustrate the effectiveness of the respective approaches. Where a weak dependence structure is plausible such as in the house prices example, the use of CCE-based tests provides satisfactory and confirmatory results. Where however an integrated trend may be relevant, the restriction of being unable to decompose between common and idiosyncratic components (especially to have different degrees of persistence) handicaps somewhat the CCE-based tests in relation to common factor approaches. This is especially seen in the empirical example where the cointegration possibility is found to be not among the original variables (between output, labour and income) but between the original variables and an integrated stochastic common trend.