Homogenous Panel Unit Root Tests under Cross ... - Semantic Scholar

Homogenous Panel Unit Root Tests under Cross Sectional Dependence: Finite Sample Modifications and the Wild Bootstrap

Helmut Herwartz and Florian Siedenburg

Christian-Albrechts-University of Kiel

This draft: February 2007

Abstract We investigate the performance of some homogenous first and second generation panel unit root tests under alternative forms of cross sectional dependence. We formalize contemporaneous correlation through factor models, spatial autoregressive error models and combinations thereof. Our findings confirm that while the first generation test of Levin, Lin, and Chu (2002) suffers from substantial size biases in dependent panels, the pooled test procedure of J¨onsson (2005) and Breitung and Das (2005) is robust in large samples. We propose modifications of the latter test with improved finite sample size properties while retaining (size adjusted) power features. We show that the wild bootstrap is a feasible and efficient means to immunize the considered homogenous panel unit root tests against cross sectional dependence.

JEL Classification: C23, C12 Keywords: Panel unit root tests, spatial dependence, wild bootstrap. Remark: Financial support of Deutsche Forschungsgemeinschaft (DFG) (HE 2188/1-1) is gratefully acknowledged. Corresponding author: F. Siedenburg, Institute for Statistics and Econometrics, Christian-Albrechts-University of Kiel, Olshausenstr. 40-60, D–24118 Kiel, Germany. Email: [email protected]. Phone: +49-431-8802225.

1

Introduction

Testing for unit roots in panel data has been a very active research topic in econometrics for more than a decade now. In typical macroeconomic applications with annual data, available time series are rather short, leading to low power of univariate unit root tests, especially under nearly integrated alternatives. Panel unit root tests (PURTs) have been proposed to overcome this problem. Apart from offering power improvements, PURTs are also a valuable tool in directly testing economic hypotheses which imply stationarity of specific variables. Probably the most prominent example for testing economic hypothesis by means of PURTs is the purchasing power parity (PPP) hypothesis (see Taylor and Taylor (2004) for a recent survey). Further examples, among others, include the Fisher hypothesis (Fisher, 1930) which implies stationarity of real interest rates (Herwartz and Reimers, 2006) and the permanent income hypothesis postulating random walk behavior of private consumption (J¨onsson, 2005). Making use of the cross sectional dimension, however, introduces some additional difficulties. First of all, one has to formulate some specific form of the alternative hypothesis. On the one hand, for homogeneous PURTs one assumes that under the alternative the process is stationary for all cross section members. Heterogeneous tests, on the other hand, are designed to reject the null hypothesis if there is at least one cross section member for which the process is stationary. While imposing less restrictive assumptions on the data generating process (DGP), the latter category of tests poses the problem to interprete a rejection of the null hypothesis. J¨onsson (2005) points out that when testing economic hypotheses by means of PURTs, homogeneous tests should be applied since economic theory is supposed to either hold or not to hold simultaneously for a group of cross sectional units. Another critical point in panel unit root testing are the assumptions with regard to the prevalence of contemporaneous error term correlation. Breitung and Pesaran (2006) distinguish two generations of PURTs. First generation PURTs (e.g. Levin, Lin, and Chu, 2002, Im, Pesaran, and Shin, 2003) are characterized by the underlying assumption

1

of cross sectional independence. Since the work of O’Connell (1998), however, it is understood that neglecting cross sectional dependence leads to severe size distortions of first generation PURTs. Under second generation tests, therefore, this assumption is relaxed and cross sectional error term correlation is allowed. Hurlin and Mignon (2004) and Breitung and Pesaran (2006) provide recent surveys over this rapidly expanding literature. Two general directions of second generation PURTs can be identified: First, approaches proceeding from the assumption that the cross sectional dependence is induced by one or more common factors that vary along the time dimension but are invariant across panel units (Bai and Ng, 2004, Moon and Perron, 2004 or Pesaran, 2005). The second main direction allows cross sectional correlation of the SUR type originally developed by Zellner (1962). This latter class of tests can be traced back to the GLS approach of O’Connell (1998) and includes instrumental variable and bootstrap methods (Chang, 2002, 2004), as well as tests based on robust covariance estimators (J¨onsson, 2005, Breitung and Das, 2005). In a large scale Monte Carlo study, Baltagi, Bresson, and Pirotte (2005) analyze the performance of various PURTs under spatially dependent error terms. Spatial dependence is a particular persuasive concept of cross sectional dependence as it implies that contemporaneous correlation is stronger between neighboring entities than between entities far away from each other. The concept of spatial dependence is widely used in regional and urban economics and has a long tradition in spatial econometrics (e.g. Anselin, 1988). To our knowledge, however, it has only quite recently been formally introduced into the area of panel econometrics (Elhorst, 2003). The rare consideration of spatial dependence in panel unit root testing is surprising since already O’Connell (1998) noted that [...] ”[A]ny EC-wide shock that influences prices or exchange rates will cause [them] these exchange rates to move together. Or [...] shocks which originate in Germany may propagate to France but not to the U.S.” The results of Baltagi et al. (2005) show that all analyzed tests are to some extent sensitive to spatial autocorrelation. The rejection frequencies are generally upward biased under H0 and the size of the bias depends positively on the strength of 2

the spatial correlation. These findings indicate that there is scope to develop test procedures which are robust under general forms of cross sectional dependence, including spatial correlation. It is the purpose of this paper to evaluate complementary avenues to improve the finite sample behavior of particular homogeneous PURTs namely the test statistics advocated by Levin, Lin, and Chu (2002) and J¨onsson (2005) and Breitung and Das (2005). We focus throughout on the impact of (neglected) cross sectional correlation. Consequently, we consider the most simple testing problem, namely the case of distinguishing the panel unit root against a stationary AR(1) alternative excluding any deterministic components. In the spirit of White (1980), we first consider a robust test statistic employing a covariance estimator that avoids to estimate a covariance matrix formalized via the SUR approach. Secondly, we draw upon former work of and MacKinnon and White (1985) and Davidson and Flachaire (2001) and build covariance estimators from modified pooled regression residuals. Thirdly, noting from Herwartz and Neumann (2005) and Herwartz (2006) that the wild bootstrap is capable to immunize test statistics against nuisance parameters invoked by SUR type disturbances we design a scheme for resampling homogeneous PURTs. For the basic statistic introduced in Levin, Lin, and Chu (2002) we proof asymptotic validity of the wild bootstrap implementation. It turns out that the bootstrap approach to homogeneous PURTs does not rely on the fully asymptotic case with N → ∞ but just requires the time dimension to approach infinity. We study the empirical features of homogenous PURT variants under alternative settings with regard to the underlying error term distributions. To preview the simulation results, it turns out that the proposed modifications yield (substantial) reductions of finite sample size biases under cross sectional correlation. The application of the wild bootstrap to obtain critical values for a particular test statistic adds further improvements of the empirical size features. In particular, implementing the resampling design with residuals implied by the null hypothesis performs best in terms of achieving the nominal significance level empirically. Size adjusted power results indicate that the proposed modifications do not involve any power loss under 3

the (nearly integrated) alternative hypothesis. The remainder of this paper is organized as follows: Section 2 reviews different forms of cross sectional dependence. Section 3 introduces the analyzed test statistics and our proposed modifications. Section 4 describes the Monte Carlo study and presents first results. In Section 5, we provide the wild bootstrap design along with simulation results. Section 6 concludes. Formal proofs of the asymptotic validity of the wild bootstrap approximation are collected in the Appendix as well as tabulated Monte Carlo results.

2

Cross sectional dependence in panel data

Starting from a microeconometric perspective, traditional methods for estimation and inference in panel data rely on the assumption of cross sectionally independent error terms. In developing the GLS-SUR approach, Zellner (1962) pointed out that this assumption might be too restrictive even for many microeconometric applications. When turning to macroeconometric analysis, where the cross sections are usually countries or states, prevalence of cross sectional dependence is more the rule than the exception. Consequently, a number of different concepts to specify cross sectional dependence have been forwarded. In this section, we review particular concepts of cross sectional dependence, namely common factors and spatial correlation. We start with introducing the general testing problem.

2.1

The general model

We assume a first order autoregressive (AR(1)) panel data model yit = ρi yit−1 + uit ,

t = 1, . . . , T,

i = 1, . . . , N,

(1)

where yit denotes the observation on the ith cross section member at time t and the initial value yi0 is assumed to be known. The autoregressive dynamics of the panel are described by the coefficient ρi which is allowed to vary across panel members. Finally, uit is a serially uncorrelated, mean zero error term which, however, is allowed 4

to be contemporaneously correlated along the cross sectional dimension,   ω2 , t = s ij E[uit ] = 0 ∀ i, t, E[uit ujs ] = , Ω = [ωij ].  0 else

(2)

The latter model formalizes a set of restrictive simplifying assumptions. First of all, we only consider the case of a pure AR(1) excluding any additional serial correlation in the error terms. Obviously, short term dynamics are an important feature of many real life macro econometric examples. Yet, despite the undoubted importance of residual serial correlation, introducing short term dynamics complicates the implementation of PURTs. In this paper, we focus on the effects of contemporaneous correlation and leave the additional complications of introducing short run dynamics aside. Similar arguments apply for deterministic terms and exogenous regressors in the data: Concentrating on the effects of cross sectional dependence on the performance of various PURTs, we keep the model of interest as simple as possible. The unit root null hypothesis of ρi = ρ = 1 is usually tested by means of the regression ∆yit = φi yit−1 + uit ,

(3)

with ∆yit = yit − yit−1 and, under H0 , φi = φ = 0. Alternative PURTs allow for a varying degree of heterogeneity of φi under the alternative hypothesis. Three different possibilities might be encountered: H1a : φi = φ < 0

(4)

H1b : φi < 0 ∀ i

(5)

H1c : φi < 0, i = 1, . . . , N − R,

φi = 0, i = N − R + 1, . . . , N.

(6)

Hypothesis H1a is the homogeneous alternative considered by Levin et al. (2002) (LLC henceforth). It restricts the process for all cross section members not only to be stationary but also to have the same autoregressive parameter. Obviously, this is a very restrictive assumption, especially if cross section members are countries or regions of different size and with different customs and institutions. A less restrictive 5

version is given by alternative H1b . This hypothesis still restricts the process to be stationary for all cross sectional units, albeit it allows heterogeneity of the specific (stationary) AR parameters. The least restrictive alternative hypothesis is given by H1c . It allows for an even higher degree of heterogeneity, permitting random walks for a fraction of the cross sectional members. This framework, however, comes at the cost of a losing interpretational clarity under the alternative. As J¨onsson (2005) points out, this is especially the case if economic hypotheses are tested by means of a PURT. For the latter reason, we focus on the class of homogeneous tests in the following.

2.2

Common factors

One approach of modeling cross sectional dependence is by means of observed or unobserved common factors. Examples of second generation PURTs fitting into a common factor approach are Pesaran (2005), Bai and Ng (2004), Moon and Perron (2004), Phillips and Sul (2003) and Choi (2002). It is worthwhile pointing out that the scope of common factor approaches is quite general as it allows for correlation of the common effect with the observable regressors, heterogeneous impacts of the common effect and it can be extended to cases of multivariate and possibly serially correlated common effects. As in Pesaran (2005) we focus on the case of a single unobserved common effect in the following for simplicity of exposition. A more general treatment including observable common factors is given in Pesaran (2003). To impose cross sectional correlation through a factor structure, we define uit in (1) to be generated according to uit = γi ft + ²it ,

E[²it ] = 0, ∀ i, t,

  σ 2 , i = j, t = s ² , E[²it ²js ] =  0 else

(7)

where ft denotes the unobserved common effect with E[ft ] = 0. The individual factor loadings γi measure the impact of the common effect on cross section i and ²it is a white noise error term. The resulting contemporaneous covariance between two observations on cross sections i and j, then, depends on the factor loadings γi 6

and γj and is given by E[uit ujt ] = γi γj E[ft2 ].

(8)

Let ut = [u1t , . . . , uN t ]0 denote the N × 1 vector of error terms and Γ = [γ1 , . . . , γN ]0 the N × 1 vector of factor loadings stacked over the cross section. Without loss of generality we may set the variance of the unobserved factor to unity and obtain the full N × N contemporaneous covariance matrix as ΩCF = E[ut u0t ] = ΓΓ0 + σ²2 IN ,

(9)

where IN is an N × N identity matrix. Following Breitung and Pesaran (2006), this form of contemporaneous correlation is called strong cross sectional dependence since the largest eigenvalue of ΩCF is O(N ) and, thus, unbounded as N → ∞.

2.3

Spatial dependence

Another way of modeling cross sectional dependence is the concept of spatial dependence. The approach is widely used in regional and urban economics (see e.g. Anselin et al., 2004). Its application can be motivated whenever the structure of cross sectional dependence is ascribable to some observable measure of location or distance. Location and distance in this context can be interpreted quite generally, including measures of economic distance. In the following, we concentrate on the most common specification of the latter category, namely the spatial autoregressive (SAR) error model as formalized for panel data by Elhorst (2003). Consider panel model (1) or (3) and let ut be defined as above. Spatially autocorrelated error terms are then given by ut = (IN − θW )−1 ²t ,

²t ∼ iid (0, σ²2 IN ) ∀ t,

(10)

where θ, |θ| < 1, is the spatial AR coefficient, measuring the strength of the spatial dependence and W denotes the so-called N × N contiguity matrix containing the spatial weights. The structure of W is principally unrestricted except for the main diagonal that contains zero elements by convention. It is common practice, however, 7

to normalize column or row sums of W to unity. One particular form of W is the ”j ahead and j behind” structure, where shocks in one entity spill over onto the next j neighbors. For the case of a circular world, where the first and last of N = 20 entities share a common border Figure 1 visualizes the structure of W for j = 1 and j = 5. Defining B = IN − θW , the contemporaneous covariance matrix becomes Figure 1: Contiguity matrices 0

0

5

5

10

10

15

15

20

20 0

5

10

15

20

0

Spatial AR(1)

5

10

15

20

Spatial AR(5)

ΩSAR = E[ut u0t ] = σ²2 (B 0 B)−1 .

(11)

It is noteworthy that, even if W is a sparse matrix, all cross sections will be contemporaneously correlated with each other but the strength of the correlation decays with the distance between two cross section units. Moreover, and in contrast to the common factor model, since |θ| < 1 the eigenvalues of ΩSAR are bounded. In the terminology of Breitung and Pesaran (2006), the SAR type cross sectional correlation formalizes weak dependence.

3

Reviewing some panel unit root tests

We analyze robustified versions of the homogenous PURT introduced by LLC. All variants are based on the same pooled OLS regression and differ with respect to the chosen covariance estimator. J¨onsson (2005) and Breitung and Das (2005) propose a 8

test based on panel corrected standard errors (Beck and Katz, 1995) which is shown to be asymptotically normally distributed under weak cross sectional dependence. We advocate a generalized version of this test that relies on a panel generalization of the White (1980) heteroscedasticity consistent covariance estimator. Moreover, an additional modification of the latter two variants is motivated that employs bias corrections similar to those suggested in MacKinnon and White (1985).

3.1

The Levin-Lin-Chu (2002) test

The LLC procedure assumes a homogenous panel, i.e. φi = φ ∀ i, which allows ˆ LLC originally to construct a test statistic for the pooled regression estimator, φ. developed their test for a more involved model where only the first order autoregressive dynamics are restricted to be the same for all cross sectional members but all other model features such as deterministic components, higher order autoregressive dynamics and the error term variances are allowed to vary along the cross sectional dimension. Their test is then obtained from a three step testing procedure. Focussing on the pure AR(1) case with cross sectional homoscedasticity, the ˆ Formally, under test statistic reduces to the pooled regression OLS t-ratio of φ. homogeneity we obtain the pooled regression from model (3) as ∆yt = yt−1 φ + ut ,

(12)

where ∆yt = [∆y1t , . . . , ∆yN t ]0 , yt = [y1t , . . . , yN t ]0 and ut = [u1t , . . . , uN t ]0 are N × 1 vectors of observations and error terms stacked over the cross section. Thus, the t-ratio of φˆ is computed as tOLS

where σ ˆ 2 = (N T )−1

PT

t=1 (∆yt

PT 0 t=1 yt−1 ∆yt , = qP T 0 σ ˆ t=1 yt−1 yt−1

(13)

ˆ Under the assumptions of ˆ 0 (∆yt − yt−1 φ). − yt−1 φ)

cross sectional independence and E[u4it ] = c¯ < ∞, LLC show that the statistic d

tOLS −→ N (0, 1),

as T → ∞,

9

followed by N → ∞.

(14)

Moreover, if both T and N tend to infinity jointly, asymptotic normality holds as √ long as N /T → 0. The asymptotic normality formalized in (14) relies crucially on the presumption of underlying cross sectionally uncorrelated disturbances. In the more realistic case of contemporaneous error correlation tOLS will loose asymptotic pivotalness. The results in Breitung and Das (2005) imply that under weak forms of cross sectional correlation the asymptotic distribution of tOLS could be given as a weighted average of N specific functionals of Brownian motions. In this case the latter weights depend on the underlying covariance structure. Moreover, the stated asymptotic normality relies crucially on the fully asymptotic case with N → ∞. In this scenario tOLS is actually a sum of N suitably scaled random variables with mean zero and unit variance such that standard central limit theory applies to obtain the result in (14). Since single components entering the tOLS statistic fail (asymptotic) normality, the limit distribution under a finite cross sectional dimension can hardly be evaluated analytically.

3.2

The panel corrected standard errors test

Even before O’Connell (1998) highlighted the caveats of first generation PURTs, Beck and Katz (1995) assessed the problem of contemporaneously correlated and potentially heteroscedastic error terms in standard panel data models. They show that in these instances, common OLS covariance estimators tend to underestimate the true variance. While, in principle, feasible GLS (FGLS) would be an efficient way of estimation in such cases, its usefulness is rather limited: Under cross sectional correlation FGLS is only applicable if T > N . Moreover, Beck and Katz (1995) show that FGLS provides reasonable variance estimates only when T is roughly three times as large as N . Hence, Beck and Katz (1995) propose the Panel Corrected Standard Errors (PCSE) estimates to account for both contemporaneous correlation and group wise heteroscedasticity. To robustify the OLS statistic against contemporaneous cross sectional correlation J¨onsson (2005) and Breitung and Das (2005) propose a modification of the LLC procedure, employing the PCSE. As McGarvey and Walker (2003) point out, PCSE can be seen as special cases of the robust covariance esti10

mator proposed by Driscoll and Kraay (1998), that additionally accounts for serial correlation in the spirit of Newey and West (1987). Following their notation, the resulting test statistic is given by tSC2

PT y 0 ∆yt φˆ = p = qPt=1 t−1 T νˆφ ˆ y 0 Ωy t=1

where

t−1

,

(15)

t−1

T T 1X 1X 0 ˆ ˆ0 ˆ ˆt = ˆ tu Ω = u (∆yt − yt−1 φ)(∆y t − yt−1 φ) . T t=1 T t=1

(16)

Breitung and Das (2005) formally proof that under weak cross sectional dependence d

tSC2 −→ N (0, 1),

as T → ∞,

followed by N → ∞.

(17)

For the asymptotic result in (17) the assumption of weak dependence is essential. If some eigenvalues of Ω are of order O(N ), the denominator of tSC2 fails to converge as N → ∞.

3.3 3.3.1

Finite sample modifications A ’White’ correction

ˆ Regarding the construction of the covariance estimator in (16) it is evident that Ω might be a poor approximation of the true error covariance in cases where N À T . In such instances, as N (N + 1)/2 nontrivial (co)variances are estimated using only N T distinct pieces of information. In the spirit of White (1980) we accordingly propose a test statistic that avoids explicit estimation of Ω. The employed covariance estimator is also a special case of Driscoll and Kraay (1998). Hence, according to the notation of McGarvey and Walker (2003) the resulting tests statistic is then given by tSC1

PT 0 φˆ t=1 yt−1 ∆yt = p = qP T ν˜φ ˆ0 y ˆu y0 u t=1

t−1

t

t t−1

PT ,

ν˜φ =

0 ˆ 0t yt−1 ˆ tu t=1 yt−1 u . P 0 yt−1 )2 ( Tt=1 yt−1

(18)

Proposition 1 states the limiting distribution of tSC1 under the assumptions in Breitung and Das (2005).

11

Proposition 1 Let the DGP be given by (12) with E[ut u0t ] = Ω. If the largest eigenvalue of Ω, λ1 = c¯ < ∞ and T → ∞ followed by N → ∞, in addition with E[u8it ] < ∞ the statistic tSC1 as defined in (18) has a standard normal limiting distribution. The proof is deferred to the Appendix. It is noteworthy that, from its construction, the covariance estimator employed in (18) also allows moderate forms of time variation of second order moments as it originated from robustifying common OLS significance tests against heteroskedasticity of unknown form. 3.3.2

Improving finite sample residuals

MacKinnon and White (1985) discuss three transformations of regression residuals that reduce the finite sample bias of heteroscedasticity consistent covariance estimators in classical (i.e. N = 1) regression models. In a number of studies (MacKinnon and White, 1985, Chesser and Jewitt, 1987 and Davidson and Flachaire, 2001) one particular refinement tends to yield most accurate bias reductions. We generalize the latter to the panel case and apply it to the two robust PURT statistics in (15) and (18). Following Davidson and Flachaire (2001) the preferable residual transformation (HC3 in their notation) could be adapted to the panel autoregression as ˜ t = [˜ u u1t , . . . , u˜N t ]0 ,

u˜it = uˆit /(1 − hit ),

0 0 hit = yi,t−1 (yi,− yi,− )−1 yi,t−1 ,

(19)

ˆ t in (16) and (18) by u ˜ t obtains the where yi,− = [yi,0 , . . . , yi,T −1 ]0 . Substituting u modified refined test statistics denoted t˜SC2 and t˜SC1 .

4 4.1

Monte Carlo study The simulation design

We analyze the finite sample performance of the proposed PURT statistics under a variety of scenarios of cross sectional dependence. To estimate the empirical size, 12

we generate data according to model (1) with ρi = ρ = 1. For the error processes uit , the following DGPs are considered: DGP1: uit = ²it ,

²it ∼ iid N (0, 1) ∀ i and t,

DGP2: uit = (IN − θW )−1 ²it , DGP3: uit = γi ft + ²it , DGP4: uit = γi ft + (IN − θW )−1 ²it . DGP1 is the benchmark case with spherical disturbances. It is expected that in this case all tests have empirical size estimates close to the nominal levels. DGP2 introduces spatially autocorrelated disturbances. We run the simulation for the two particular versions of the spatial weights matrix W visualized in Section 2.3 with choosing j = 1(W1) and j = 5(W5). We use row normalized versions of these matrices in our simulations. In the first set of simulations, the spatial correlation coefficient θ is set to 0.8. DGP3 addresses cross sectional dependence through one common factor. In generating γi and ft we follow Pesaran (2005), drawing γi ∼ iid U (0.5, 1) ∀ i and ft ∼ iid N (0, 1) ∀ t. Here, γi is drawn only once to keep the structure of cross sectional correlation constant across tests and replications. DGP4 finally combines the factor model with spatially dependent error terms formalized via W1. Potential size distortions might depend on the strength of cross sectional dependence. We therefore complement the first set of results by a sensitivity analysis where we reduce the strength of the cross sectional correlation setting θ = 0.4 or sampling γi ∼ iid U (0.1, 0.5) ∀ i. To reduce effects of initial conditions, we generate T + 51 observations for every experiment and set the initial values yi,−51 = 0. We run the simulation for every combination of N ∈ [10, 20, 50, 100], T ∈ [10, 20, 30, 50, 100] and generate 5000 replications throughout.

4.2

Empirical size

Empirical size estimates for alternative test procedures are reported in Tables 1 through 5. For all experiments the nominal level is γ = 5%. As all tests are one13

sided we determine the empirical size by comparing the test statistics against the critical value given by the 5% quantile of the Gaussian distribution. To facilitate interpretation of the results, bold entries in the Tables indicate that nominal and empirical size differ with 5% significance. Significant size distortions are diagnosed in case the rejection frequencies under H0 are not covered by a confidence interval p [4.4, 5.6] constructed as γ ± 1.96 γ(1 − γ)/5000. Table 1 reports results for the iid case. The results show some incidence of small sample bias for all tests in case N = 10. The largest distortions are visible for tSC1 , while test decisions based on the refined statistic t˜SC2 deviate least from the nominal level. As the cross sectional dimension increases, the overall performance of the tests improves. For N = 20, inference based on the statistics tSC2 and t˜SC2 yields empirical rejection probabilities for almost all values of T that cannot be distinguished from the nominal level by statistical criteria. It is noteworthy, that for panels with N ∈ [50, 100] the performance of the PCSE based test deteriorates showing severe undersizing. As noted above, if N À T , the applied covariance estimator might be a poor approximation of the true covariance matrix. In fact, our results indicate that in these instances, the PCSE tests overestimate the true covariance. Decisions based on the OLS and t˜SC1 statistics show convergence to the nominal level as N becomes large without being adversely affected by small values of T . Employing the tSC1 statistic small sample biases vanish only if N and T increase jointly. Summarizing the simulation results for the benchmark scenario (DGP1) it is worthwhile to recall that all results provided for robust tests are indicative for the risk of ’falsely’ applying robust procedures in iid settings. We infer from our results as t˜SC1 does not perform worse than tOLS it is not associated with the risk of being ’falsely’ applied in the iid case. The risk of ’falsely’ applying tSC1 is moderate with a less than 2% higher rejection probability compared to tOLS in most instances while the risk of ’falsely’ applying both variants of tSC2 clearly depends on the relative size of N and T . Results for the spatially correlated DGP2 (W1) are reported in Table 2. The results on the LLC test in the first column resemble the results of Baltagi et al. 14

(2005) and clearly document that spatial dependence restricts the applicability of first generation PURTs. In contrast, all other considered tests yield empirical size estimates that are much closer to the nominal level. Some larger biases are only notable for small N , and especially for the tSC1 statistic showing up to 9.7% rejections under H0 . For medium to large cross sectional dimensions, finite sample biases decrease for all four robust tests. From almost all instances, it is evident that the refined statistics t˜SC2 and t˜SC1 outperform their raw counterparts tSC2 and tSC1 as their empirical size comes closer to the nominal benchmark. Results obtained under spatial correlated disturbance sampled with W5 (Table 3) are qualitatively similar to the corresponding W1 case. Size distortions are slightly more pronounced for all tests and tSC2 and t˜SC2 yield most accurate empirical size features. Modeling cross sectional dependence by means of the common factor approach as formalized by DGP3 and DGP4 render the robust tests asymptotically invalid. However, J¨onsson (2005) notes that even though not converging to a nuisance free limiting distribution the PCSE test substantially reduces the bias of the LLC test if contemporaneous correlation is introduced through a common factor. Our results in Table 4 support this finding. While the size of the LLC test is obviously diverging, the robust tests are only moderately oversized, conditional on the chosen factor loadings. Again, reliance on the refined statistics t˜SC2 and t˜SC1 leads to a further bias reduction as compared to tSC2 and tSC1 . However, as the statistics do not converge asymptotically to a Gaussian distribution, this bias does not vanish asymptotically, and, as noted by J¨onsson (2005), remains largely unaffected by the degree of cross sectional correlation. Finally, Table 5 displays the results for DGP4, a common factor model with spatially correlated error terms. While empirical rejection frequencies for the LLC test range between 21% and 26%, all robust tests have empirical type 1 errors of mostly less than 9%. As before, t˜SC2 tends to be the preferable test statistic with empirical size estimates falling in the 95% confidence band constructed around the nominal level in about one third of considered cases.

15

4.3

Size adjusted power

Under the stationary alternative hypothesis, we consider H1b that restricts ρi to be less than unity for all cross sections, but does not impose cross sectional homogeneity. Moreover, we follow Baltagi et al. (2005) who allow for cross section specific intercepts under the alternative hypothesis. Specifically, yit is given as yit = (1 − ρi )µi + ρi yit−1 + uit ,

i = 1, . . . N, t = 1, . . . T,

(20)

where µi ∼ iid U (0, 0.02) and ρi ∼ iid U (0.95, 0.99). The values of µi and ρi are drawn only once and kept constant across replications, tests and panel sizes. The specific choice of ρi ∼ iid U (0.95, 0.99) is made to guard against trivial power estimates. The alternative distributions of the error terms uit are the same as discussed for simulations under H0 . To enhance comparability of alternative test procedures, we only report size-adjusted power. Size adjusted power is calculated by comparing the test statistics under H1 against the critical values corresponding to the empirical 5% quantile of the test statistic’s distribution under H0 . Whenever the empirical size of a test differs largely from the nominal level, one should be careful in interpreting size adjusted power.

In the iid case, the LLC test displays the highest power. For the smallest sample, the estimated power is 26.1%, quite rapidly increasing for larger values of T and N . Depending on N , the relative advantage over the second most powerful test, the tSC1 statistic, ranges from 3% to 8% for T = 10 and almost vanishes for values of T ≥ 30. The least powerful test is the robust test of J¨onsson and Breitung and Das with an up to 10% smaller rejection probability for small values of T . Power differences are most pronounced for the case N = 50 and T = 10, while for larger values of T , rejection probabilities for all tests converge to unity. For the cases of cross sectional correlation, we refrain from interpreting (size adjusted) power estimates for the LLC test in the light of overly large size estimates. Compared to the case of independence, the power of the remaining four 16

tests decreases to values around 12% for the smallest sample size, while the relative performance remains basically unchanged: the tSC1 and the refined t˜SC1 statistic are usually slightly more powerful than both versions of tSC2 . The relative advantage of both versions of tSC1 over tSC2 is largest as N À T , where up to 10% higher rejection frequencies are reported for tSC1 in comparison with tSC2 variants.

4.4

Sensitivity analysis

To check the sensitivity of the results with respect to the parameters governing the strength of cross sectional correlation, θ and γi , we also evaluate empirical test features for scenarios of weakened cross sectional dependence (θ = 0.4 and γi ∼ iid U (0.1, 0.5)). As expected, the size bias of all tests shrinks when the strength of spatial correlation decreases. Yet, the empirical size of the LLC test still differs significantly from the nominal level in all experiments. For the spatially correlated DGPs, the bias is roughly constant over all panel dimensions and varies between 3% to 4% and 2% to 3% in the W1 and W5 case, respectively. In the case of a common factor model, the bias shows an increasing tendency, with around twice the nominal level for the largest panel dimension. This result indicates that even low degrees of contemporaneous correlation call for robust test procedures. Detailed results for weaker patterns of cross sectional dependence are not tabulated but available from the authors upon request.

5 5.1

Robust bootstrap inference Setup of bootstrap algorithm

The preceding analysis demonstrated the asymptotic invalidity of the LLC test in cross sectionally dependent panels. Robust tests as obtained by using PCSE and the proposed modifications are asymptotically valid in case of weak cross sectional dependence and, as demonstrated, reduce finite sample biases in the case of strong dependence. Moreover, noting the widespread use of factor models it is of immediate

17

interest to have a test procedure at hand that works in the semiasymptotic case of a finite cross section. Generally, the robust tests are subject to small sample biases with empirical size estimates exceeding significantly the nominal benchmarks especially if data dimensions N or T are small. Resampling methods have proven to be a valuable tool to obtain asymptotically valid significance levels in cases where the true limiting distribution of some test statistic is unknown or hard to derive analytically. Moreover, applying bootstrap based critical values in test decisions might improve finite sample properties (Horowitz, 2001) in the sense that actual significance levels converge faster to their nominal counterparts as it is the case for first order asymptotic approximations of pivotal test statistics. Considering the issue of testing for individual effects in stationary panel models Herwartz (2006) shows that the wild or external bootstrap (Wu, 1986) is suitably immunized against the failure of pivotalness under cross sectional dependence. Liu (1988) and Mammen (1993) established asymptotic validity of the wild bootstrap for significance tests in the framework of the linear regression model with heteroscedastic error terms. Regarding unit root testing, Ferretti and Romo (1996) prove an asymptotically valid bootstrap algorithm with replacement. For the panel case, Chang (2004) proposes bootstrap tests to overcome the difficulties associated with cross sectional dependence and formally proves their asymptotic validity. The latter are also based on resampling with replacement. Other examples of resampling PURTs with replacement can be found in Breitung and Das (2005) and Smith et al. (2004). In the following, we suggest the wild bootstrap as an asymptotically valid alternative for unit root testing in dependent panels. From the proof of its asymptotic validity it will become evident that the wild bootstrap approximation will be valid in case of a finite cross sectional dimension. The wild bootstrap implementation of homogeneous PURTs proceeds along the following steps: 1. Calculate the chosen PURT statistic, ψ, for a given data set and obtain the ˆ ˆ t = ∆yt − yt−1 φ. estimated residuals from the pooled OLS regression, u

18

2. For s = 1, . . . , S with S sufficiently large, ˆ t as • draw bootstrap residuals u∗t preserving the second order features of u u∗t = (u∗1t , u∗2t , . . . , u∗N t )0 = ηt (ˆ u1t , uˆ2t , . . . , uˆN t )0 ,

ηt ∼ (0, 1),

(21)

where ηt , t = 1, . . . , T , is a sequence of independent scalar random variables; • From u∗t construct the bootstrap sample yit∗ from the DGP presumed under H0 as ∗ yt∗ = yt−1 + u∗t ,

y0∗ = y0 ;

• Calculate the bootstrap version ψ ∗ of the original statistic ψ. 3. Decision: Reject H0 with significance γ if ψ < c∗γ , the γ quantile of ψ ∗ . The central feature of the wild bootstrap design is that through resampling without replacement, potential time heteroscedasticity of underlying model disturbances is imitated by the bootstrap sample. Numerous choices for ηt are available from the literature (Liu, 1988 and Mammen, 1993). Davidson and Flachaire (2001) powerfully illustrate the particular merits of the Rademacher distribution to implement the wild bootstrap (Liu, 1988),   1 with Pr 0.5 ηt = ,  −1 with Pr 0.5

E[ηt ] = 0, E[ηt2 ] = 1,

t = 1, . . . , T.

(22)

Vector resampling by means of the scalar random variables ηt , moreover, ensures that the bootstrap quantities u∗t mimic the first two moments of the original error terms ut . Specifically one can show that, T T T T 1X 1X 1X 1X 0 ∗ 0 ˆt = ˆ tu Cov(ut ) = u ut ut + oP (1) = Cov(ut ) + oP (1), (23) T t=1 T t=1 T t=1 T t=1

hence, on average, the wild bootstrap reflects the true underlying covariances. As described above wild bootstrap samples are generated from unrestricted residˆ To improve the finite ˆ t implied by the pooled parameter estimate φ. ual estimates u 19

sample properties of the resampling scheme it appears promising to generate bootstrap innovations from the restricted residuals obtained from the data under H0 , i.e. ˜ t = ∆yt . Note that in the framework of the pure AR(1) model, building bootstrap u ˜ t by means of the Rademacher ηt amounts to drawing bootstrap samples from u series yt∗ from the ’true’ residuals ut . Davidson and Flachaire (2001) point out this particular merit of the Rademacher distribution for significance testing in regression models comprising only one explanatory variable. While it promises further size improvements on the one hand the use of residual estimates implied by H0 may go at the cost of power loss. Evaluating the latter will be an issue of the Monte Carlo exercises.

5.2

Asymptotic validity

The following Proposition states the asymptotic validity of the wild bootstrap for the case of the tOLS statistic. Proposition 2 Let yt = [y1t , . . . , yN t ]0 denote an N × 1 vector of AR(1) processes, yit = ρi yit−1 + uit , generated from possibly cross sectionally correlated, E[ut u0t ] = Ω, but serially uncorrelated mean zero vector innovations. By assumption the eigenvalues of Ω are at most of order O(N ). Then, under H0 : ρi = 1, ∀i, as T → ∞ d

L(t∗OLS |y0 , y1 . . . , yT ) → L(tOLS ),

(24)

where tOLS is given in (13) and t∗OLS is the corresponding wild bootstrap counterpart.

The proof of Proposition 2 is given in the Appendix. The asymptotic Gaussian distribution of tOLS in the case considered by LLC has already been discussed in Section 3.1. Proposition 2 goes beyond the pivotal case in two particular directions: First, for the bootstrap approximation the cross sectional dimension N is allowed to be finite. Moreover, the underlying model disturbances are allowed to be characterized by cross sectional dependence. For both generalizations nuisance parameters

20

will characterize the actual asymptotic distribution L(tOLS ). The bootstrap approximation, however, will be shown to cope with the nuisance appropriately. Within the fully asymptotic case (N → ∞) the approach followed by Breitung and Das (2005) can be seen as a trial to immunize L(tOLS ) against contemporaneous error correlation by (implicitly) estimating the nuisance parameters. The derived pivotal test statistic tSC2 may also be subjected to resampling in order to improve its performance in case N is finite. Moreover, in the light of the results reviewed in Horowitz (2001) resampling pivotal test statistics might be preferable as it could offer faster convergence to nominal significance levels as first order asymptotic approximations.

5.3

Empirical features

In the simulation study, we set the number of bootstrap replications S to 299. Tables 6 to 9 and 10 to 13 document rejection frequencies results for wild bootstrap implementations under spatially correlated disturbances and common factor models, respectively. The results for the spatially correlated DGPs (Tables 6 to 9) document the potential of the wild bootstrap to improve the finite sample size properties of PURTs under cross sectional error correlation. Throughout the empirical size features of all test procedures are markedly improved. More specifically, using the estimated ˆ t , t∗OLS , still shows some moderate positive residuals for the bootstrap, u∗t = ηt u size bias. Especially if the time dimension is small, rejection frequencies under H0 are up to 8%. Even better size features are obtained for t˜∗SC1 where for each choice of W only two scenarios obtain significant deviations of the empirical from the nominal level. Generating the bootstrap sample under φ = 0, u∗t = ηt ∆yt offers further improvements of empirical rejection frequencies under H0 for all test statistics except tSC2 . Similar results also hold for the case of common factor models as documented in Tables 10 to 13. If estimated residuals are used to generate the bootstrap sample, t∗OLS based inference shows some slight upward bias for small values of T while for restricted residuals, t∗SC2 shows some bias, especially for large N . Tests based on 21

t∗SC1 and both modified variants t˜∗SC2 and t˜∗SC1 show hardly any bias in either cases. With regards to rejection probabilities under H1 , we find that application of the bootstrap does not affect the power function adversely. Size adjusted power estimates are very close to those obtained via first order asymptotic approximations. In particular, it is noteworthy that size improvements offered by resampling from H0 implied residuals does not come at the cost of (adjusted) power loss.

6

Conclusions

We analyze the performance of some first and second generation homogeneous PURTs under cross sectionally dependent model disturbances. We focus on scenarios of spatially autocorrelated errors, common factors and combinations thereof. Our simulation study confirms that the PURT of Levin, Lin, and Chu (2002) (LLC) loses control over actual significance levels under cross sectional dependence. A second generation PURT suggested by J¨onsson (2005) and Breitung and Das (2005) is characterized by substantial size violations in particular scenarios where the cross sectional dimension exceeds the time dimension of the panel. We suggest finite sample modifications of robust tests building upon former work of MacKinnon and White (1985) and Davidson and Flachaire (2001) that turn out to improve finite sample features which are particularly apparent for scenarios with N large compared with T . Noting from Herwartz and Neumann (2005) that wild bootstrap inference could immunize test statistics against nuisance invoked by cross sectional correlation we formally proof the validity of the wild bootstrap approximation of the LLC test. Considering the semiasymptotic case our asymptotic results merely rely on T → ∞ and hold under cross sectional dependence in its weak or strong form. Monte Carlo analyses underscore the virtue of bootstrap inference even for the cases of rather small panel dimensions. Simulation results for the bootstrapped tests support the proposed small sample modifications as the corresponding empirical size estimates are accurately close to their nominal benchmarks. Bootstrapping the robust tests proposed by J¨onsson (2005) and Breitung and Das (2005) remains oversizes for

22

scenarios where the cross section dimension is large in comparison with the time dimension. Admittedly in this paper we address the restrictive case of pure AR(1) models. A natural aspect of future research is to investigate if the robustness of the proposed modifications carries over when testing for unit roots in more complex models including deterministic terms and higher order autoregressive dynamics. Moreover, the proposed modified covariance estimator and the wild bootstrap implicitly allow for some variation of second order moments along the time dimension. As time variation of cross sectional dependence is likely relevant in empirical practice, it is tempting to enhance robust inference in this particular direction. We regard the latter as a further issue of future research.

23

References Anselin, L. (1988). Spatial Econometrics: Methods and Models. Kluwer Academic Publishers, Dordrecht. Anselin, L., R. J. G. M. Florax and S. J. Rey (2004). Advances in Spatial Econometrics. Springer, Berlin. Bai, J. and S. Ng (2004). A panic attack on unit roots and cointegration. Econometrica 72 , 1127–1178. Baltagi, B. H., G. Bresson and A. Pirotte (2005). Panel unit root tests and spatial dependence. Manuscript, Syracuse University, New York. Beck, N. and J. N. Katz (1995). What to do (and not to do) with time-series cross-section data. American Political Science Review 89 , 634–647. Breitung, J. and S. Das (2005). Panel unit root tests under cross sectional dependence. Statistica Neerlandica 59, 414–433. Breitung, J. and H. M. Pesaran (2006). Unit roots and cointegration in panels. In L. Matyas and P. Sevestre (Ed.), Forthcoming in: The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice. Dordrecht: Kluwer Academic Publishers. Chang, Y. (2002). Nonlinear IV unit root test in panels with cross sectional dependency. Journal of Econometrics 110, 261–292. Chang, Y. (2004). Bootstrap unit root test in panels with cross sectional dependency. Journal of Econometrics 120, 263–294. Chesser, A. and I. Jewitt (1987). The bias of a heteroskedasticity consistent covariance matrix estimator. Econometrica 55, 1217–1222. Choi, I. (2002). Combination unit root tests for cross-sectionally correlated panels. Mimeo, Hong Kong University of Science and Technology. 24

Davidson, R. and E. Flachaire (2001). The wild bootstrap, tamed at last. GREQAM Document de Travail 99A32. Driscoll, J. C. and A. C. Kraay (1998). Consistent covariance matrix estimation with spatially dependent data. The Review of Economics and Statistics 80, 549–560. Elhorst, J. P. (2003). Specification and estimation of spatial panel data models. International Regional Science Review 26, 244–268. Ferretti, N. and J. Romo (1996). Unit root bootstrap tests for AR1 models. Biometrika 83 , 849–860. Fisher, I. (1930). The Theory of Interest. Macmillan, New York. Herwartz, H. (2006). Testing for random effects in panel data under cross sectional error correlation - A bootstrap approach to the Breusch Pagan test. Computational Statistics and Data Analysis 50, 3567–3591. Herwartz, H. and M. H. Neumann (2005). Bootstrap inference in single equation error correction models. Journal of Econometrics 128, 165–193. Herwartz, H. and H. E. Reimers (2006). Modelling the Fisher hypothesis: World wide evidence. Economic Working Paper 2006-04, University of Kiel. Horowitz, J. L. (2001). The bootstrap. In J. J.. Heckman and E.. Leamer (Eds.), Handbook of Econometrics. Amsterdam: Elsevier. Hurlin, C. and V. Mignon (2004). Second generation panel unit root tests. Manuscript, THEMA-CNRS, University of Paris X. Im, K. S., H. M. Pesaran and Y. Shin (2003). Testing for unit roots in heterogenous panels. Jounral of Econometrics 115, 53–74. J¨onsson, K. (2005). Cross-sectional dependency and size distortion in a small-sample homogeneous pane data unit root test. Oxford Bulletin of Economics and Statistics 67 , 369–392. 25

Levin, A., C. F. Lin and C. J. Chu (2002). Unit root tests in panel data: asymptotic and finite-sample prperties. Journal of Econometrics 108, 1–24. Liu, R. Y. (1988). Bootstrap procedures under some non-i.i.d. models. Annals of Statistics 16, 1696–1708. MacKinnon, J. G. and H. White (1985). Some heteroscedasticity consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics 29, 305–329. Mammen, E. (1993). Bootstrap and wild bootstrap for high dimensional linear models. Annals of Statistics 21, 255–285. McGarvey, M. G. and M. B. Walker (2003). An alternative estimator for fixed effects spatial models. Georgia State University, Working Paper 03-03. Moon, H. R. and B. Perron (2004). Testing for a unit root in panels with dynamic factors. Journal of Econometrics 122, 81–126. Newey, W. K. and K. D. West (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703– 708. O’Connell, P. G. J. (1998). The overvaluation of purchasing power parity. Journal of International Economics 44, 1–19. Pesaran, M. H. (2003). Estimation and inference in large heterogenous panels with cross section dependence. CESifo working paper series 869. University of Munich. Pesaran, M. H. (2005). A simple panel unit root test in the presence of cross section dependence. Cambridge Working Papers in Economics, University of Cambridge 0346. Phillips, P. C. B. and D. Sul (2003). Dynamic panel estimation and homogeneity testing under cross section dependence. Econometrics Journal 6 (1).

26

Smith, L. V., S. Leybourne, T.-H. Kim and P. Newbold (2004). More powerful panel data unit root tests with an application to mean reversion in real exchange rates. Journal of applied econometrics 19, 147–170. Taylor, A. M. and M. P. Taylor (2004). The purchasing power parity debate. Journal of Economic Perspective 18, 135–158. White, H. (1980). A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica 48 , 817–838. Wu, C. F. J. (1986). Jacknife bootstrap and other resampling methods in regression analysis. Annals of Statistics 14, 1261–1295. Zellner, A. (1962). An efficient method for estimating seemingly unrelated regressions and tests of aggregation bias. Journal of the American Statistical Association 58, 977–992.

27

A A.1

Appendix Asymptotic distribution of tSC1

PROOF of Proposition 1. We first note that the numerator of tSC1 is the same as in tSC2 . Hence, we only have to show that the denominator of tSC1 has the same probability limit as tSC2 . Both terms differ only with respect to the choice of the estimator of the covariance matrix. Define ξ := T

−2

T X

†

0 Ωyt−1 yt−1

and ξ := T

−2

t=1

T X

0 yt−1 ut u0t yt−1 .

t=1

Under the assumption of E[u4it ] < ∞, Breitung and Das (2005) show that for fixed N and T → ∞ N

−1

d

ξ −→ N

−1

N X

Z λ2i

i=1

1

Wi (r)2 dr,

0

¯ 2 = limN →∞ N −1 with λi denoting the eigenvalues of Ω. Hence as λ

PN i=1

R1 λ2i , E[ 0 Wi (r)2 dr] =

0.5 and N → ∞, p ¯ 2 /2. N −1 ξ −→ λ

If E(ξ) = E(ξ † ) it directly follows that p ¯ 2 /2. N −1 ξ † −→ λ

(25)

From the law of iterated expectation one can show T T X X 0 0 E( yt−1 ut ut yt−1 ) = E( yt−1 Ωyt−1 ).

(26)

t=1

t=1

It remains to be shown that for T → ∞ T

−2

T X

0 ˆ 0t yt−1 ˆ tu u yt−1

d

−→ T

−2

T X

0 ut u0t yt−1 . yt−1

t=1

t=1

From properties of the OLS estimator we have !−1 Ã X X 0 0 ˆ t = ut − yt−1 ut yt−1 yt−1 u yt−1 t

t

= ut − yt−1 q,

where q =

à X t

28

!−1 0 yt−1 yt−1

X t

0 ut = Op (T −1 ).(27) yt−1

Inserting yields T

−2

T X

0 0 0 0 0 0 (yt−1 ut u0t yt−1 − yt−1 ut qyt−1 yt−1 − yt−1 yt−1 qu0t yt−1 + yt−1 yt−1 q 2 yt−1 yt−1 ).

t=1

(28)

Consider the second or third term in (28) and define ξ=

T X

0 0 yt−1 . ut yt−1 yt−1

(29)

t=1

Independence of ut and yt−1 directly establishes E[ξ] = 0 and some algebra yields Var[ξ] = O(T 4 ). Hence, T

1 X 0 0 q y ut yt−1 yt−1 ∼ op (1). T 2 t=1 t−1

(30)

With respect to the last term in (28) we define ς=

T X

0 0 yt−1 yt−1 yt−1 yt−1 .

(31)

t=1

It can be shown that conditional expectations E[ς|q] = O(T 3 ) and E[ς 2 |q] = O(T 6 ). P Hence, T −2 q 2 E[ Tt=1 ς] = op (1), concluding the proof that T

−2

T X

0 ˆ tu ˆ 0t yt−1 yt−1 u

d

−→ T

−2

t=1

T X

0 yt−1 ut u0t yt−1 + op (1).

(32)

t=1

¤

A.2

Asymptotic validity of the wild bootstrap approximation

PROOF of Propositions 2. To proof the asymptotic validity of the wild bootstrap approximation applied to the LLC test we proceed in two steps. First, we consider a wild bootstrap version of the common Dickey Fuller statistic and then turn to the case addressed by LLC. With regard to the latter it will become evident that the bootstrap approximation of the LLC statistic is valid irrespective of the prevalence of contemporaneous error term correlation. Moreover, for the bootstrap approximation it is not necessary to regard the fully asymptotic case with N → ∞. We first consider the DF test: 29

Proposition 3 Let yt be a random walk process, generated as yt = yt−1 +ut , E[ut ] = 0, E[u2t ] = σ 2 , E[u4t ] = σ 4 . yt∗ is the wild bootstrap counterpart of yt . As T → ∞ it holds under H0 that d

L(t∗DF |y0 , y1 , . . . , yt ) → L(tDF ), where tDF

PT ∗ ∗ PT t=1 yt−1 ut ∗ t=1 yt−1 ut = qP , tDF = qP T T ∗ 2 2 σ ˆ σ ˆ t=1 (yt−1 ) t=1 (yt−1 )

and σ ˆ is the usual OLS based residual variance estimator. Under the null hypothesis the DGP is a random walk without any deterministic component yt = yt−1 + ut . The DF statistic obeys the following representation and asymptotic distribution: tDF

PT yt−1 ut d (1/2){[W (1)]2 − 1} aT = √ = qt=1 o1/2 . PT 2 −→ nR 1 bT 2 σ ˆ y [W (r)] dr t=1 t−1 0

Accordingly, the wild bootstrap version of tDF is PT ∗ ∗ a∗T t=1 yt−1 ut ∗ tDF = p ∗ = qP , T bT ∗ 2 σ ˆ (y ) t=1 t−1

(33)

(34)

where ∗ yt∗ = yt−1 + u∗t ,

u∗t = ηt uˆt ,

  ηt =

1

with probability 0.5

 −1

with probability 0.5

and uˆt is the residual from the OLS regression of ∆yt on yt−1 . Owing to the Rademacher distribution of ηt E[(u∗t )4 ] = uˆ4t . From properties of the OLS estimator we have uˆt = ut − yt−1

à X t

= ut − yt−1 q,

!−1 2 yt−1

X t

where q =

yt−1 ut

à X t

!−1 2 yt−1

X

yt−1 ut = Op (T −1 ).

(35)

t

Note that the random variable q is a population quantity. Thus, when discussing the distributional features of the bootstrap approximation we condition throughout 30

on q. Now consider the numerator in (34) ( t−1 ) T X X a∗T = (ηs us − ηs ys−1 q) (ηt ut − ηt yt−1 q) =

t=1 s=1 (Ã t−1 T X X t=1

! η s us

ηt ut − q

s=1

+q 2

à t−1 X

à t−1 X

! ηs us ηt yt−1 − q

s=1

!

à t−1 X

! ηs ys−1 ηt ut

s=1

)

ηs ys−1 ηt yt−1

s=1 ∗ ∗ ∗ ∗ = S1T − S2T − S3T + S4T .

(36)

To prove the asymptotic validity of the bootstrap approximation we first show that, d

as T → ∞, a∗T → aT . For this purpose we consider the asymptotic properties of d

∗ ∗ → SkT , k = 1, . . . , 4 in turn. To preview the results it will turn out that T −1 S1T ∗ T −1 aT while T −1 SkT = op (1) for k = 2, 3, 4. We obtain ∗ S1T : Apparently,

∗ 1/T S1T = T −1

à t−1 T X X t=1

! d

ηs us ηt ut −→ (1/2)σ 2 {[W (1)]2 − 1},

(37)

s=1

since {ηt ut }Tt=1 is an uncorrelated mean zero process sharing the variance (and fourth order moments) of the innovation sequence ut driving yt under H0 . ∗ ∗ , S2T : Consider the single components qψ2,t entering S2T Ã t−1 ! T T X X X ∗ S2T = q ψ2t = q ηs us ηt yt−1 . t=1

t=1

(38)

s=1

By construction, {ψ2t }Tt=1 are serially uncorrelated and have mean zero. Therefore ∗ |q] E[S2T

= 0,

∗ |q] Var[S2T

2

= q E[

T X

2 ]. ψ2,t

t=1

Regarding the latter term some algebra reveals that 2 E[ψ2t ] = (t − 1)σ 4 + (t − 1)(t − 2)(σ 2 )2 .

(39)

∗ ∗ Thus, the variance of S2T increases at rate O(T ) such that T −1 S2T is op (1).

31

∗ ∗ S3T : Analogously, we have for components entering S3T , Ã ! T T t−1 X X X ∗ S3T =q ψ3,t = q ηs ys−1 ηt ut , t=1 ∗ E[S3T ]

= 0,

∗ Var[S3T |q]

= E[q

t=1 2

T X

(40)

s=1

2 2 ψ3,t ], with E[ψ3,t ] = (σ 2 )2 (t − 1)(t − 2)/2.

t=1

Thus,

∗ ] Var[S3T

∗ = op (1). = O(T ) and, consequently, T −1 S3T

∗ : Finally, S4T ∗ S4T = q2

T X

ψ4,t =

t=1

T X

q2

t=1

à t−1 X

! ηs ys−1 ηt yt−1 ,

s=1

yielding ∗ E[S4T |q]

= 0,

∗ |q] Var[S4T

2 2

= (q ) E

" T X

# 2 ψ4,t

.

t=1

£ 2 ¤ Since E ψ4,t = σ 4 (t − 2)2 (t − 1)/2 + (σ 2 )2 (t − 1)(t − 2)/2, we obtain T X 2 ∗ ∗ E[ ψ4,t ] = O(T 4 ) and, thus, Var[S4T ] = O(1) and T −1 S4T = op (1). t=1 ∗ Summarizing the results for SkT , k = 1, . . . , 4 we have

T −1 a∗T = T −1 aT + op (1).

(41)

Now consider the denominator in (34) that obeys the following decomposition: ( t−1 )2 T X X b∗T = (ηs us − ηs ys−1 q) t=1

=

s=1 Ã T  X t−1 X t=1



!2 ηs us

− 2q

s=1

t−1 X

à ηs2 us ys−1 +

q

t−1 X

s=1

s=1

!2   ηs ys−1



∗ ∗ ∗ = R1T − R2T + R3T

(42) d

∗ ∗ As before, we proceed by item RkT , k = 1, 2, 3, and prove that T −2 R1T → T −2 bT , ∗ while T −2 RkT , k = 2, 3, are op (1). ∗ R1T : Properties of {ηt ut }Tt=1 and functional CLTs for random walks imply à t−1 !2 Z 1 T X X d −2 ∗ −2 2 T R1T = T η s us −→ σ [W (r)]2 dr. t=1

s=1

32

0

(43)

∗ ∗ R2T : For components entering R2T we get

∗ R2T

= 2q

T X

ϕ2,t =

t=1

T X

2q

t=1

t−1 X

ηs2 us ys−1 .

(44)

s=1

By construction, E[ϕ2,t ] = 0,

∗ E[R2T |q]

= 0,

∗ Var[R2T |q]

= 4q

2

T X

E[ϕ22,t ].

t=1

After some algebra and using E[ηt4 ] = 1 we obtain E[ϕ22,t ] = (σ 2 )2 (t − 2)(t − ∗ ∗ 1)/2 = O(T 2 ). Thus, Var[R2T ] = O(T ) and T −2 R2T = op (1). ∗ : With respect to the last component entering b∗T we have R3T

∗ R3T = q2

T X

ϕ3,t =

t=1

T X

q2

t=1

à t−1 X

!2 ηs ys−1

.

(45)

s=1

Observing that ζs−1

t−1 T X √ 1 X 2 =√ ηs ys−1 ∼ Op (1), it follows that q ( t − 2ζs−1 )2 ∼ Op (1). t − 2 s=1 t=1 (46)

∗ We can thus deduce that T −2 R3T = op (1) concluding the proof that

T −2 b∗T = T −2 bT + op (1).

(47)

Now observe that the numerator and denominator of tDF and t∗DF can be augmented √ with T −1 and T −2 . Moreover, as it is the case for the raw DF statistic, the innovations driving the numerator and denominator of the bootstrap approximation are identical ({ηt ut }Tt=1 ) such that ’joint’ convergence of the latter quantities to the asymptotic Dickey Fuller distribution is ensured. ¤ Having proved the asymptotic validity of the wild bootstrap in the case of a single DF regression (N = 1), its generalization to the panel test statistic it straightforward in case of a finite cross section dimension N . Since the bootstrap random variable ηt is scalar the derivation of the leading terms in the numerator and denominator 33

of the bootstrap approximation of the LLC statistic is completely analogous to the univariate case. The decomposition of the numerator and denominator of PT ∗ 0 ∗ ∗ a t=1 yt−1 ∆yt t∗OLS = ∗T = qP , bT T ∗ 0 ∗ σ ˆ y y t=1 t−1 t−1 is only notationally more demanding. Following the proof for the DF statistic by item we obtain in particular for the leading terms ! Ã t−1 ! Ã t−1 T T X X X X d ∗ u0s ut = T −1 S1T T −1 S1T = T −1 ηs u0s ηt ut → T −1 t=1

s=1

t=1

(48)

s=1

and ∗ T −2 R1T = T −2

à t−1 T X X t=1

s=1

ηs us

!0 Ã t−1 X

! η s us

s=1

= T −2 R1T .

d

−→ T −2

à t−1 T X X t=1

s=1

us

!0 Ã t−1 X

! us

s=1

(49)

Owing to joint convergence of the numerator and denominator of the bootstrap LLC statistic to the corresponding population quantities the conditional distribution L(t∗OLS |y0 , y1 , . . . , yT ) converges to the unconditional distribution L(tOLS ). For the latter convergence it is irrelevant if tOLS is pivotal as originally considered by LLC or not. ¤

34

A.3

Tabulated simulation results Table 1: Empirical properties under independence N 10

T tOLS 10 6.1 20 6.5 30 6.4 50 6.8 100 6.5 20 10 5.8 20 6.1 30 6.1 50 6.4 100 5.7 50 10 5.4 20 6.0 30 5.8 50 6.0 100 5.0 100 10 5.1 20 5.0 30 5.8 50 5.8 100 5.1

Empirical tSC2 tSC1 7.1 8.6 6.8 8.1 6.3 7.3 6.3 7.3 6.2 6.8 6.1 8.0 5.5 7.6 5.2 7.4 5.5 7.0 4.8 5.7 3.9 8.2 3.8 7.8 3.9 6.7 3.8 6.5 3.3 5.4 1.9 7.6 1.4 6.3 1.8 7.0 2.0 6.3 2.4 5.6

size t˜SC2 5.2 5.8 5.7 5.9 6.0 4.7 4.7 4.6 5.2 4.7 3.0 3.1 3.6 3.6 3.1 1.2 1.2 1.6 1.9 2.3

t˜SC1 6.1 6.6 6.4 6.7 6.5 5.8 6.1 6.3 6.5 5.4 5.9 6.3 6.0 6.0 5.1 5.2 5.2 5.7 5.7 5.2

tOLS 26.1 41.1 57.3 81.6 99.0 39.9 63.6 82.7 96.5 100.0 80.2 97.8 99.9 100.0 100.0 97.1 100.0 100.0 100.0 100.0

Size adjusted power tSC2 tSC1 t˜SC2 t˜SC1 20.7 24.4 21.0 22.9 35.9 39.4 36.0 38.8 53.4 55.7 53.5 54.8 79.1 80.7 79.2 80.4 98.9 98.9 98.9 98.9 30.5 36.4 30.6 33.4 55.7 60.4 56.0 59.2 77.3 80.5 77.4 80.0 95.4 95.9 95.4 95.7 100.0 100.0 100.0 100.0 62.2 72.6 62.9 70.4 94.0 96.5 94.4 96.3 99.7 99.8 99.7 99.8 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 88.6 94.6 89.2 94.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

Notes: All results are based on 5,000 replications. Empirical size is estimated at the nominal 5% level, bold numbers indicate significant deviations from the nominal level.

35

Table 2: Empirical properties under SAR1 N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

tOLS 18.8 20.8 21.8 22.0 24.0 18.8 20.9 20.4 22.6 21.8 19.9 20.1 19.7 21.3 21.3 19.5 19.8 19.7 21.1 20.3

Empirical tSC2 tSC1 8.2 9.7 8.5 9.7 7.6 9.0 7.7 8.7 8.3 8.6 7.3 8.7 6.9 8.0 6.8 7.9 7.7 8.3 7.0 7.7 6.5 8.6 5.7 7.6 5.7 7.3 5.6 7.6 6.5 7.3 4.8 8.0 4.3 7.3 4.4 6.9 4.9 7.3 5.1 6.4

size t˜SC2 6.2 7.3 6.7 7.2 8.1 5.5 5.9 6.2 7.1 6.8 5.0 5.0 5.0 5.2 6.2 3.9 3.8 3.8 4.5 5.0

t˜SC1 6.6 7.7 7.4 7.9 8.1 5.9 6.5 6.9 7.5 7.1 6.0 6.3 6.5 6.8 7.0 5.6 6.1 6.0 6.7 6.2

Size adjusted power tOLS tSC2 tSC1 t˜SC2 t˜SC1 12.9 11.5 13.7 11.6 12.7 20.0 17.5 17.9 17.6 17.1 27.6 24.9 24.9 24.8 24.5 42.3 35.0 35.5 34.9 35.2 75.6 61.6 61.7 61.6 61.6 18.6 16.5 19.0 16.8 18.1 30.7 26.9 28.8 26.9 27.8 42.5 36.1 38.0 36.2 37.2 63.0 56.1 56.5 56.1 55.9 93.5 88.2 87.9 88.3 87.7 40.8 29.3 36.2 29.9 35.1 66.3 56.9 62.5 57.1 61.3 84.4 78.2 79.9 78.4 79.4 97.4 95.4 95.5 95.4 95.4 100.0 100.0 100.0 100.0 100.0 65.2 47.9 58.3 49.3 56.7 90.0 82.7 86.6 83.1 86.0 98.1 96.0 96.9 96.2 96.8 100.0 99.9 99.9 99.9 99.9 100.0 100.0 100.0 100.0 100.0

Table 3: Empirical properties under SAR5 N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

tOLS 24.6 27.0 28.5 30.7 34.0 23.5 25.6 26.5 28.5 31.0 23.6 24.4 24.7 25.9 27.1 23.7 25.1 25.4 25.9 26.4

Empirical tSC2 tSC1 8.5 10.7 8.8 10.4 7.4 9.2 8.8 9.7 8.6 9.0 7.5 9.4 7.9 9.5 7.6 9.1 7.7 8.8 7.4 8.3 6.8 9.3 5.7 7.8 6.2 7.6 6.3 7.8 6.7 7.5 5.5 8.6 5.1 7.8 4.9 7.2 5.5 7.7 6.1 7.6

Size t˜SC2 6.2 7.4 6.6 8.5 8.5 5.6 6.5 6.7 7.2 7.1 5.0 4.8 5.5 5.9 6.5 4.3 4.3 4.4 5.3 5.8

t˜SC1 tOLS 7.0 11.0 7.6 14.8 7.2 21.0 8.6 34.8 8.3 64.4 6.0 14.6 7.3 23.4 7.8 35.6 7.8 59.3 7.9 87.3 6.4 33.5 6.2 58.5 6.6 75.9 7.1 92.7 7.3 99.8 6.0 52.6 6.5 79.5 6.3 92.4 7.1 99.2 7.1 100.0

36

Size adjusted tSC2 tSC1 10.8 11.7 14.9 15.8 20.1 20.2 22.2 22.7 40.4 39.1 15.0 17.7 23.4 24.1 29.1 29.0 42.2 41.0 65.1 64.9 26.6 31.3 48.7 51.0 64.4 65.8 83.8 83.5 98.6 98.5 38.4 45.9 67.8 71.0 86.1 86.9 97.8 97.5 100.0 100.0

Power t˜SC2 t˜SC1 10.8 10.5 14.7 15.5 20.1 19.7 22.2 22.7 40.4 39.0 15.2 16.3 23.6 23.6 29.4 28.7 42.3 40.8 65.1 64.8 27.0 29.4 49.0 50.2 64.3 65.2 83.9 83.6 98.6 98.5 39.1 43.2 68.1 70.0 86.1 86.7 97.8 97.5 100.0 100.0

Table 4: Empirical properties, static factor model N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

tOLS 11.7 13.0 13.0 13.6 12.5 16.7 17.9 18.9 19.9 20.1 23.1 25.3 27.4 28.6 28.9 29.8 34.4 35.1 39.1 41.6

Empirical tSC2 tSC1 7.3 9.4 7.5 8.9 7.3 8.5 7.1 8.4 6.6 7.2 7.7 9.5 6.5 8.2 6.3 8.0 7.2 8.4 6.8 7.5 6.5 9.1 5.2 8.2 6.4 9.1 5.7 7.7 6.3 7.8 5.6 9.0 4.7 8.0 5.1 8.1 5.9 8.5 5.5 7.5

Size t˜SC2 5.3 6.7 6.6 6.7 6.4 5.8 5.5 5.4 6.7 6.5 4.6 4.4 5.7 5.3 6.1 4.2 3.9 4.4 5.5 5.4

t˜SC1 6.2 7.3 7.5 7.6 6.8 6.2 6.4 6.7 7.5 7.0 6.1 5.8 7.5 6.9 7.3 5.6 5.9 6.7 7.6 6.9

Size adjusted Power tOLS tSC2 tSC1 t˜SC2 t˜SC1 16.0 15.2 17.2 15.4 16.0 24.5 22.3 23.9 22.5 23.4 35.2 31.7 31.7 31.7 31.4 58.2 49.4 49.7 49.2 49.5 88.8 78.4 77.5 78.4 77.4 25.2 21.3 26.6 21.7 24.3 45.6 39.9 42.8 40.4 41.7 68.8 55.5 54.2 55.3 53.6 85.0 68.5 67.4 68.5 67.5 98.5 91.9 91.1 91.9 91.2 27.5 27.8 32.2 28.3 30.4 54.9 49.2 49.6 49.5 48.9 73.7 58.5 55.3 58.3 55.1 90.1 74.7 71.9 74.7 72.0 99.3 92.6 91.9 92.6 91.9 29.1 34.5 39.4 35.2 37.5 68.1 56.4 54.2 56.3 54.5 81.8 65.8 62.1 65.7 62.2 93.4 78.3 75.6 78.4 75.5 99.7 95.7 94.8 95.7 94.8

Table 5: Empirical properties, static spatial factor model N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

tOLS 21.3 22.4 23.6 24.5 24.9 21.4 22.5 23.1 24.8 25.1 22.4 22.8 23.0 24.5 23.8 22.8 24.1 24.5 25.8 25.5

Empirical tSC2 tSC1 8.9 10.6 8.2 9.6 7.6 8.9 7.7 8.5 7.6 8.3 8.0 9.4 7.5 8.6 7.0 8.4 7.2 8.0 7.1 7.6 6.4 8.9 5.7 7.3 5.8 7.5 6.0 7.2 6.8 7.6 5.5 8.8 4.8 7.6 5.1 7.3 5.2 7.6 5.5 7.0

Size t˜SC2 6.5 7.3 6.8 7.1 7.3 6.2 6.4 6.4 6.6 6.8 4.6 4.9 5.3 5.5 6.5 4.1 3.9 4.5 4.8 5.3

t˜SC1 tOLS 6.8 10.9 7.6 14.8 7.7 23.1 7.5 37.0 7.8 63.9 6.8 21.3 7.1 34.8 7.2 53.9 7.2 78.5 7.3 97.8 5.7 36.1 6.0 59.8 6.7 77.8 6.5 94.0 7.3 99.9 5.7 55.2 6.2 84.1 6.1 96.0 6.8 99.5 6.7 100.0

37

Size adjusted tSC2 tSC1 10.9 12.5 14.4 16.0 20.9 21.1 29.8 30.3 50.7 50.2 17.0 21.2 30.6 32.7 45.3 46.9 66.5 65.6 93.0 92.8 27.4 34.5 50.3 54.8 68.1 69.4 89.1 89.5 99.3 99.2 42.5 52.9 75.4 78.7 90.6 92.1 98.3 98.3 100.0 100.0

Power t˜SC2 t˜SC1 11.1 11.4 14.7 15.1 21.0 20.2 29.9 30.0 50.7 50.0 16.8 19.0 30.4 31.5 45.2 46.6 66.5 65.1 93.0 92.7 28.0 32.0 50.3 53.8 68.0 68.9 89.2 89.1 99.3 99.2 44.0 50.2 75.3 78.1 90.7 91.8 98.3 98.3 100.0 100.0

ˆ SAR1 Table 6: Empirical properties of bootstrap tests, φ, T t∗OLS 10 7.3 20 5.9 30 5.5 50 5.5 100 5.2 20 10 8.0 20 5.5 30 5.7 50 5.6 100 5.5 50 10 7.2 20 5.5 30 5.9 50 5.7 100 5.4 100 10 7.9 20 5.6 30 5.9 50 6.0 100 4.8 N 10

Empirical t∗SC2 t∗SC1 5.5 5.2 5.3 5.2 5.3 5.1 5.2 5.1 5.0 5.3 6.0 5.7 4.4 4.6 5.3 5.4 5.3 5.4 5.1 5.3 5.6 5.1 5.0 4.7 5.2 5.2 5.6 5.4 5.3 5.3 6.4 5.4 4.8 4.7 5.3 5.5 5.7 5.8 5.0 4.7

Size t˜∗SC2 5.4 5.3 5.4 5.2 5.0 6.0 4.4 5.3 5.3 5.1 5.7 5.0 5.3 5.5 5.3 6.2 4.7 5.3 5.7 5.0

t˜∗SC1 t∗OLS 5.2 13.9 5.2 22.6 5.1 30.2 5.1 43.5 5.2 76.0 5.7 17.6 4.5 31.4 5.4 43.4 5.3 62.8 5.3 93.4 5.1 37.7 4.7 66.8 5.2 83.8 5.5 97.5 5.3 100.0 5.5 56.1 4.7 91.3 5.5 97.9 5.8 99.9 4.7 100.0

Size adjusted t∗SC2 t∗SC1 13.4 13.6 19.8 19.3 26.1 25.3 36.8 35.5 62.6 61.5 17.3 17.3 29.8 28.0 39.3 38.3 54.3 55.3 88.8 88.0 37.6 35.5 63.8 63.2 81.5 78.6 96.1 95.9 100.0 100.0 56.7 55.4 90.6 87.8 97.7 96.7 99.9 99.9 100.0 100.0

Power t˜∗SC2 t˜∗SC1 14.3 13.3 19.9 19.0 26.2 25.3 36.9 35.6 62.5 61.6 18.6 16.9 29.8 28.1 39.3 38.2 54.2 55.1 88.8 87.9 37.0 35.2 63.6 63.0 81.5 78.7 96.1 95.3 100.0 100.0 58.3 53.1 90.5 87.6 97.7 96.7 99.9 99.9 100.0 100.0

Notes: All results are based on 5,000 replications and 299 Bootstrap replications.

ˆ SAR5 Table 7: Empirical properties of bootstrap tests, φ, T t∗OLS 10 6.0 20 5.9 30 5.3 50 5.0 100 5.2 20 10 7.8 20 5.2 30 5.6 50 5.5 100 6.2 50 10 7.2 20 5.6 30 5.5 50 5.9 100 5.3 100 10 7.8 20 5.2 30 5.8 50 6.2 100 5.6 N 10

Empirical t∗SC2 t∗SC1 5.6 5.2 5.7 5.7 5.4 5.2 5.2 5.3 5.4 5.6 6.1 5.7 4.8 4.7 5.0 4.8 5.4 5.3 5.8 5.5 5.7 5.2 5.0 4.6 5.0 4.9 5.6 5.7 5.4 5.4 6.2 5.5 4.8 4.8 5.4 5.3 5.9 5.7 5.3 5.5

Size t˜∗SC2 5.3 5.7 5.4 5.2 5.4 6.0 4.7 5.0 5.5 5.8 5.7 5.0 5.0 5.7 5.4 6.1 4.8 5.4 5.9 5.4

t˜∗SC1 t∗OLS 5.1 15.2 5.6 24.0 5.1 32.6 5.4 43.6 5.6 67.4 5.6 19.3 4.6 37.3 4.8 50.1 5.3 65.7 5.5 88.0 5.3 36.4 4.6 62.3 4.9 79.0 5.7 93.5 5.5 99.8 5.4 48.9 4.7 82.6 5.3 92.8 5.8 99.1 5.4 100.0

38

Size adjusted t∗SC2 t∗SC1 13.5 12.2 17.2 15.4 21.4 19.7 26.2 24.8 40.0 40.5 17.9 16.4 28.2 25.4 35.3 34.1 43.8 41.5 63.3 61.6 33.0 30.8 53.6 51.4 69.6 65.4 85.2 82.3 98.8 98.5 46.9 44.5 77.5 73.3 89.0 85.6 98.0 97.0 100.0 100.0

Power t˜∗SC2 t˜∗SC1 13.8 12.0 17.1 15.4 21.3 19.4 26.2 24.7 41.9 40.5 18.9 16.5 28.0 25.4 37.0 34.2 43.7 41.7 63.2 63.5 34.3 31.2 55.2 53.3 67.6 65.7 85.1 82.2 98.8 98.5 46.1 43.3 77.3 73.4 88.9 85.8 98.0 97.1 100.0 100.0

Table 8: Empirical properties of bootstrap tests, φ = 0, SAR1 N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

t∗OLS 5.1 5.1 4.7 5.4 5.1 5.5 4.4 5.0 5.3 5.3 5.2 4.5 5.2 5.5 5.2 5.8 4.6 5.2 5.7 4.7

Empirical t∗SC2 t∗SC1 3.1 5.1 4.5 5.2 5.0 5.0 5.2 5.1 5.1 5.2 4.4 5.5 4.4 4.6 5.2 5.2 5.6 5.2 5.1 5.1 5.1 5.0 5.2 4.5 5.8 5.4 5.8 5.5 5.5 5.3 6.5 5.4 5.9 4.7 6.2 5.1 6.5 5.8 5.4 4.7

Size t˜∗SC2 4.8 5.0 5.1 5.2 5.1 5.6 4.4 5.1 5.4 5.0 5.1 4.8 5.1 5.5 5.3 5.5 4.6 5.3 5.6 4.9

t˜∗SC1 t∗OLS 5.2 14.3 5.2 22.4 5.0 32.4 5.1 45.0 5.2 75.9 5.5 17.9 4.6 33.8 5.2 44.0 5.2 64.1 5.2 93.4 4.8 38.2 4.5 68.3 5.4 84.5 5.5 97.6 5.3 100.0 5.5 58.7 4.6 91.1 5.2 97.9 5.8 99.9 4.7 100.0

Size adjusted t∗SC2 t∗SC1 15.1 13.2 20.1 18.8 26.0 27.0 36.4 35.3 62.4 61.4 20.3 17.9 32.0 27.9 40.1 37.7 57.6 55.1 89.2 87.8 40.5 35.1 66.8 63.1 81.7 79.0 96.6 95.3 100.0 100.0 60.2 55.0 90.5 88.1 97.4 96.8 99.9 99.9 100.0 100.0

Power t˜∗SC2 t˜∗SC1 14.2 13.1 19.2 18.7 26.3 27.0 36.5 35.4 62.6 61.4 18.4 18.0 30.7 28.2 39.0 37.8 56.5 55.0 89.9 87.8 36.3 37.1 65.3 63.0 81.0 79.0 96.5 95.3 100.0 100.0 57.1 54.8 89.9 88.0 97.5 96.9 99.9 99.9 100.0 100.0

Table 9: Empirical properties of bootstrap tests,φ = 0, SAR5 T t∗OLS 10 5.2 20 5.5 30 4.8 50 4.9 100 5.3 20 10 5.6 20 4.5 30 5.2 50 5.2 100 6.0 50 10 5.1 20 4.8 30 5.1 50 5.5 100 5.3 100 10 5.4 20 4.5 30 5.3 50 6.0 100 5.4 N 10

Empirical t∗SC2 t∗SC1 3.4 5.4 5.0 5.5 5.0 5.2 4.9 5.0 5.4 5.5 4.5 5.5 4.6 4.5 5.1 4.8 5.6 5.2 5.9 5.5 4.9 4.9 5.1 4.5 5.3 4.7 6.0 5.6 5.6 5.5 6.2 5.4 5.3 4.7 6.3 5.3 6.4 5.7 6.0 5.5

Size t˜∗SC2 4.9 5.5 5.1 4.8 5.5 5.6 4.3 4.9 5.4 5.7 5.0 4.7 4.7 5.6 5.3 5.4 4.7 5.3 5.8 5.5

t˜∗SC1 t∗OLS 5.5 15.6 5.5 23.1 5.2 33.0 5.0 44.9 5.5 67.1 5.4 21.2 4.5 38.2 4.9 49.2 5.3 65.4 5.5 88.0 5.0 37.4 4.5 62.7 4.7 79.1 5.6 93.8 5.5 99.8 5.5 50.9 4.6 82.8 5.2 92.7 5.7 99.2 5.5 100.0

39

Size adjusted t∗SC2 t∗SC1 13.3 12.2 16.4 15.3 21.0 19.4 26.7 24.6 40.9 40.9 20.5 16.0 28.7 26.6 35.3 33.7 41.7 41.4 63.1 61.2 36.9 32.2 55.7 51.3 69.5 65.5 85.0 82.1 98.7 98.2 48.4 44.4 78.5 73.3 88.4 85.8 97.7 97.2 100.0 100.0

Power t˜∗SC2 t˜∗SC1 14.1 12.4 17.8 15.4 20.8 19.4 27.7 24.5 41.7 40.9 18.2 16.1 28.2 26.8 36.5 34.0 43.5 41.6 63.1 61.4 32.8 31.0 54.7 53.3 69.3 65.8 85.1 82.2 98.8 98.2 47.3 44.9 78.3 73.6 88.7 85.9 97.8 97.2 100.0 100.0

ˆ static factor model Table 10: Empirical properties of bootstrap tests, φ, T t∗OLS 10 7.6 20 5.6 30 5.5 50 5.3 100 5.0 20 10 6.8 20 5.4 30 6.3 50 5.4 100 4.9 50 10 7.3 20 5.8 30 5.5 50 5.3 100 5.4 100 10 6.6 20 6.3 30 5.2 50 5.2 100 5.1 N 10

Empirical t∗SC2 t∗SC1 5.7 5.4 5.2 4.9 5.4 5.2 5.5 5.4 5.4 5.3 5.2 4.7 5.0 4.6 5.7 5.6 5.3 5.2 4.7 4.7 5.9 5.4 5.5 5.3 5.3 4.8 5.8 5.6 6.0 5.7 5.6 5.1 6.0 6.1 4.7 4.6 5.4 5.3 5.0 4.9

Size t˜∗SC2 5.6 5.1 5.4 5.4 5.3 5.2 5.0 5.7 5.3 4.7 5.9 5.4 5.2 5.8 6.0 5.5 6.0 4.7 5.4 5.0

t˜∗SC1 5.5 5.0 5.1 5.4 5.3 4.6 4.6 5.7 5.2 4.7 5.1 5.1 4.8 5.5 5.7 5.0 6.0 4.6 5.4 4.9

Size adjusted Power t∗OLS t∗SC2 t∗SC1 t˜∗SC2 t˜∗SC1 16.9 17.0 16.6 16.8 16.2 30.6 27.2 27.6 27.0 27.7 41.5 34.5 32.9 34.4 32.9 60.6 49.2 47.6 49.2 47.6 89.4 76.4 74.6 76.3 74.6 33.9 30.3 27.6 29.8 27.0 56.7 45.7 41.7 44.1 41.8 70.5 55.7 52.0 55.5 52.2 87.8 71.0 67.6 70.9 67.8 98.5 91.6 90.4 91.5 90.4 44.0 36.3 32.2 37.1 33.1 68.0 54.6 47.6 54.3 48.6 81.1 64.5 59.5 64.3 59.9 91.9 74.7 71.4 74.6 71.8 99.2 93.0 91.2 93.0 91.2 57.1 46.5 38.9 46.9 40.9 76.4 59.7 51.4 59.3 52.3 86.7 71.6 64.3 71.3 65.0 95.1 80.7 76.1 80.5 76.5 99.7 96.1 94.7 96.1 94.8

ˆ static spatial factor model Table 11: Empirical properties of bootstrap tests, φ, T t∗OLS 10 6.8 20 6.1 30 5.4 50 5.5 100 5.2 20 10 7.3 20 5.5 30 5.9 50 6.0 100 4.7 50 10 7.4 20 6.3 30 5.4 50 5.5 100 5.6 100 10 7.7 20 6.6 30 5.7 50 5.7 100 5.6 N 10

Empirical t∗SC2 t∗SC1 5.7 5.6 5.5 5.3 5.3 5.1 5.4 5.2 5.2 5.1 5.7 5.5 5.2 5.3 5.3 5.4 5.9 5.9 4.9 5.0 6.0 5.5 5.3 5.2 4.7 5.1 5.3 5.2 5.6 5.7 5.8 5.6 5.5 5.6 5.2 5.2 5.6 5.3 5.2 5.3

Size t˜∗SC2 5.6 5.4 5.3 5.4 5.2 5.7 5.3 5.4 5.9 4.9 5.9 5.3 4.7 5.3 5.6 5.7 5.5 5.2 5.6 5.5

t˜∗SC1 t∗OLS 5.4 12.0 5.2 18.6 5.1 26.3 5.2 37.4 5.0 66.0 5.4 22.3 5.2 39.2 5.4 54.4 5.9 74.9 5.1 97.8 5.5 31.6 5.2 59.4 5.1 78.8 5.1 94.4 5.7 99.9 5.5 55.4 5.5 84.8 5.2 96.0 5.3 99.6 5.1 100.0

40

Size adjusted t∗SC2 t∗SC1 11.3 10.6 18.0 17.0 20.9 19.6 29.5 28.4 50.4 49.2 21.0 20.5 34.1 31.6 47.1 44.5 65.3 64.1 93.4 92.1 32.5 29.5 55.8 53.6 73.4 68.9 89.7 88.4 99.6 99.5 54.8 48.0 80.9 75.9 93.2 91.3 98.7 98.4 100.0 100.0

Power t˜∗SC2 t˜∗SC1 11.3 11.3 18.0 16.8 20.8 19.6 29.6 28.5 50.4 49.2 20.8 20.2 33.9 31.4 47.1 44.8 65.4 64.1 93.5 92.0 32.4 29.9 55.7 53.2 73.2 69.2 89.7 88.4 99.6 99.4 54.5 48.4 81.9 77.9 93.2 91.3 98.9 98.4 100.0 100.0

Table 12: Empirical properties of bootstrap tests, φ = 0, static factor model N 10

T 10 20 30 50 100 20 10 20 30 50 100 50 10 20 30 50 100 100 10 20 30 50 100

t∗OLS 5.7 4.9 5.0 5.0 4.8 4.6 4.6 5.7 5.2 4.7 5.6 4.9 4.9 5.1 5.3 5.4 5.7 5.3 5.2 5.0

Empirical t∗SC2 t∗SC1 4.1 5.4 4.8 4.9 5.1 5.2 5.5 5.4 5.3 5.1 4.1 4.5 5.0 4.7 5.9 5.5 5.6 5.2 4.8 4.7 5.6 5.5 6.1 5.0 5.6 4.8 6.2 5.5 6.3 5.6 6.2 5.1 6.8 5.9 5.7 4.6 6.2 5.3 5.5 4.9

Size t˜∗SC2 5.1 4.8 5.1 5.3 5.1 4.5 4.8 5.6 5.2 4.5 5.4 5.1 4.8 5.6 5.9 5.2 5.7 4.6 5.3 4.9

t˜∗SC1 5.4 5.0 5.2 5.5 5.1 4.6 4.7 5.6 5.2 4.7 5.5 4.9 4.9 5.6 5.6 5.1 5.9 4.6 5.3 4.9

Size adjusted Power t∗SC2 t∗SC1 t˜∗SC2 t˜∗SC1 16.6 18.5 16.3 17.1 16.5 31.6 28.9 27.2 28.0 27.4 40.3 35.0 32.8 34.1 32.6 60.2 47.5 47.9 49.0 47.8 89.1 76.7 74.6 76.4 74.7 35.0 32.3 27.1 30.2 27.5 56.4 45.2 41.3 45.4 41.6 70.7 56.4 51.8 55.6 53.9 87.7 69.8 67.5 71.0 67.5 98.5 91.6 90.4 91.6 90.4 44.4 37.6 32.4 36.7 33.2 67.9 53.6 47.7 53.5 49.6 81.0 63.6 59.5 65.1 59.9 91.8 73.7 70.5 74.1 70.6 99.4 92.4 91.9 92.9 91.9 54.9 44.5 39.4 45.6 40.6 75.7 58.9 51.8 58.8 51.1 86.2 69.7 65.9 71.3 66.4 95.0 80.1 76.4 81.5 76.7 99.7 95.8 94.7 96.2 94.7

t∗OLS

Table 13: Empirical properties of bootstrap tests, φ = 0, static spatial factor model T t∗OLS 10 5.2 20 5.3 30 4.9 50 5.3 100 5.2 20 10 5.4 20 4.7 30 5.5 50 5.7 100 4.7 50 10 5.4 20 5.2 30 4.9 50 5.1 100 5.5 100 10 5.1 20 5.6 30 5.2 50 5.4 100 5.5 N 10

Empirical t∗SC2 t∗SC1 3.8 5.4 4.8 5.3 4.9 5.0 5.2 5.3 5.2 4.9 4.3 5.2 4.9 5.0 5.4 5.5 6.0 5.9 5.0 5.0 5.5 5.4 5.5 5.2 5.3 5.0 5.7 5.1 6.0 5.6 5.8 5.7 6.2 5.4 6.1 5.1 6.2 5.3 5.7 5.4

Size t˜∗SC2 5.3 5.1 5.0 5.3 5.1 5.2 5.0 5.4 5.8 4.9 5.7 5.1 4.8 5.3 5.7 5.2 5.3 5.1 5.5 5.3

t˜∗SC1 t∗OLS 5.3 12.5 5.3 19.7 5.0 27.1 5.3 37.0 5.0 65.7 5.3 23.2 5.0 40.7 5.5 53.0 5.9 76.0 5.0 97.8 5.6 34.5 5.2 59.7 5.0 78.9 5.1 94.2 5.6 99.9 5.6 56.0 5.4 84.3 5.2 96.0 5.3 99.6 5.2 100.0

41

Size adjusted t∗SC2 t∗SC1 11.5 11.0 17.9 16.7 21.8 21.2 29.2 28.5 49.8 51.7 23.0 20.0 36.0 31.7 47.6 42.4 65.8 64.1 92.6 92.8 35.0 30.9 58.1 52.6 73.6 70.7 89.5 88.4 99.6 99.4 57.5 50.8 81.3 77.8 93.5 91.2 98.8 98.3 100.0 100.0

Power t˜∗SC2 t˜∗SC1 11.1 11.3 17.4 16.6 21.2 19.7 29.5 28.4 50.0 51.6 21.0 20.0 33.5 32.0 47.1 42.4 65.3 64.0 93.3 92.0 30.3 29.3 54.9 52.8 73.2 69.1 89.6 88.3 99.6 99.4 52.9 48.6 81.3 78.0 93.1 91.2 98.9 98.3 100.0 100.0