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Cross-sectional correlation robust testsfor panel cointegrationChristoph Hanck aa Universiteit Maastricht , Maastricht, The NetherlandsPublished online: 18 Jun 2009.

To cite this article: Christoph Hanck (2009) Cross-sectional correlation robust tests for panelcointegration, Journal of Applied Statistics, 36:7, 817-833, DOI: 10.1080/02664760802510042

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Journal of Applied StatisticsVol. 36, No. 7, July 2009, 817–833

Cross-sectional correlation robust testsfor panel cointegration

Christoph Hanck∗

Universiteit Maastricht, Maastricht, The Netherlands

(Received 18 December 2007; final version received 28 September 2008)

We use meta-analytic procedures to develop new tests for panel cointegration, combining p-values fromtime-series cointegration tests on the units of the panel. The tests are robust to heterogeneity and cross-sectional dependence between the panel units. To achieve the latter, we employ a sieve bootstrap procedurewith joint resampling of the units’ residuals. A simulation study shows that the tests can have substantiallysmaller size distortion than tests ignoring the presence of cross-sectional dependence while preserving highpower. We apply the tests to a panel of post-Bretton Woods data to test for weak purchasing power parity.

Keywords: panel cointegration tests; cross-sectional dependence; sieve bootstrap

1. Introduction

The application of unit root and cointegration tests in macroeconometric practice is often hamperedby the lack of large sample sizes. The (first-order) asymptotic approximation used for derivingthe distribution of the test statistics may then be rather inaccurate. One solution to improve thepower and reduce the error-in-rejection probability (ERP, or size distortion) of these tests is topool several time series into a panel data set and develop panel unit root and cointegration tests.

Pedroni [33] and Kao [23] generalize the residual-based tests of Engle and Granger [14] andPhillips and Ouliaris [34], Larsson et al. [25] extend Johansen’s [22] tests to panel data whileMcCoskey and Kao [29] propose a test for the null of panel cointegration in the spirit of Shin [37].All these tests, however, rely on the assumption that the different cross-sectional units of thepanel are independent or, at most, exhibit dependence of a rather simple form. This assumption,characterizing the so-called first generation tests, greatly simplifies the derivation of limitingdistributions of the panel test statistics, but may not hold in practice. Phillips and Sul [35], Moonand Perron [30], Bai and Ng [1] and Gengenbach, Palm and Urbain [16] put forward factorapproaches to deal with cross-sectional correlation in panel unit root and cointegration testing.Their approach may, however, require the validity of the factor structure assumption modeling thecorrelation structure of the panel units (see also [41] for a recent criticism of the PANIC approach).

∗Email: c.hanck@ke.unimaas.nl

ISSN 0266-4763 print/ISSN 1360-0532 online© 2009 Taylor & FrancisDOI: 10.1080/02664760802510042http://www.informaworld.com

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As argued by Breitung and Das [6], size distortions may result if this assumption, which is hardto verify, is not met.

The main contribution of the present paper, therefore, is to suggest new tests for panel cointegra-tion that are robust to cross-sectional dependence (or, synonymously, cross-sectional correlation)of a general form. The main idea of the testing principle has been used in meta-analytic studiesfor a long time [15,18] and was introduced into the panel literature by Maddala and Wu [27] andChoi [11], who propose meta-analytic panel unit root tests. Consider the testing problem on thepanel as consisting of N testing problems, one for each unit of the panel. Conduct N separatetime-series tests and obtain the corresponding p-values of the test statistics. Then, combine thep-values of the N -tests (in a sense to be made precise below) into a single panel test statistic. Weextend their framework to the panel cointegration setting.

To robustify the panel cointegration tests against cross-sectional correlation of a general form,we use a bootstrap scheme that jointly resamples entire cross-sections of residuals to preservethe cross-sectional correlation structure in the panel. We provide some simulation evidence todemonstrate the effectiveness of the suggested procedure. We also compare it with recently pro-posed tests by Westerlund [42], which follow a similar idea to the one put forward here. We findthat the bootstrap tests can have dramatically smaller ERPs than tests ignoring the presence ofcross-sectional dependence. At the same time, the bootstrap tests preserve high power.

We use the tests to investigate the weak purchasing power parity (PPP) hypothesis for a panel ofpost-Bretton Woods exchange rate data. Our main result is that using cross-sectional correlation-corrected critical values may make an important difference in econometric practice.

The remainder of the paper is organized as follows. The next section establishes notation andreviews the meta-analytic p-value combination tests for panel cointegration under the assumptionof cross-sectional independence. Section 3 discusses the bootstrap algorithm used to robustify thetests against general forms of cross-sectional dependence. Section 4 summarizes the simulationevidence on the effectiveness of the bootstrap tests under cross-sectional dependence. Section 5illustrates the use of the tests. The final section concludes.

2. p-value combination tests for panel cointegration

Consider the multivariate time-series regression

yi,t = αi + κit + ϑit2 + βixi,t + ui,t (i ∈ NN, t ∈ NTi

) (1)

for each of the N units of a possibly unbalanced panel. (a ∈ Nbc is shorthand for a = b, . . . , c,

omitting b if b = 1.) The (K × 1) column vector xi,t = (xi1t , . . . , xiKt )� (i ∈ NN, t ∈ NTi

)

collects the observations on the K regressors for given i and t . yi,t and xi,t are integrated oforder 1, I (1) (i ∈ NN). The row vector βi , αi, κi and ϑi may vary across i, allowing for het-erogeneous cointegrating vectors and time polynomials of order up to 2, i.e. constants, trend andsquared trend terms. We make the following Functional Central Limit Assumption on the datagenerating process (DGP) of the variables.

Assumption 1 (Invariance principle) Let zi,t := (yi,t , x�i,t )

� and ξi,t := (ξiyt , ξi1t , . . . , ξiKt )�.

The true process zi,t is generated as zi,t = zi,t−1 + ξi,t , (i ∈ NN, t ∈ NTi). ξi,t satisfies

T −0.5i

∑[Tir]t=1 ξi,t ⇒ Bi,�i

(r) (i ∈ NN) as Ti → ∞, where r ∈ [0, 1] and ⇒ denotes weak con-vergence. [x] is the integer part of x and Bi,�i

(r) is vector Brownian motion with asymptoticcovariance matrix �i . Also, the K × K lower right submatrix of �i , �xx,i , has full rank.

This assumption ensures, among other things, that while all variables are individually integrated,there are no cointegrating relationships among the regressors in Equation (1).1 pi denotes the

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Journal of Applied Statistics 819

p-value of a time-series cointegration test applied to the ith unit of the panel. Let θi,Tibe a

time-series cointegration test statistic on unit i for a sample size of Ti . Let FTidenote the null

cumulative distribution function (cdf) of θi,Ti. Since the tests considered here are one-sided,

pi = FTi(θi,Ti

) if the test rejects for small values of θi,Tiand pi = 1 − FTi

(θi,Ti) if the test rejects

for large values of θi,Ti. We only consider time-series tests with the null of no cointegration.

We test the following null hypothesis:

H0: There is no cointegration for any i (i ∈ NN),

against the alternative

H1: There is cointegration for at least one i.

Under H0, {ui,t }t in Equation (1) is an I (1) stochastic process (i ∈ NN). The alternative H1 statesthat there are 1 to N cointegrated units in the panel. That is, a rejection neither allows us toconclude that the entire panel is cointegrated nor does it provide information about the numberof units of the panel that exhibits cointegrating relationships.

We make the following assumptions:

Assumption 2 (Continuity) Under H0, θi,Tihas a continuous distribution function (i ∈ NN).

Assumption 3 (Cross-sectional uncorrelatedness) E[ξi,t ξ�l,s] = 0 (s, t ∈ NTi

, i �= l). The errorprocess ξi,t is generated as a linear vector process ξi,t = Ci(L)ηi,t , where L is the lag operatorand ηi,t is vector white noise.

Remarks

• Assumption 2 is a regularity condition which ensures a uniform distribution of the p-valuesof the time-series test statistics under H0: pi ∼ U[0, 1] (i ∈ NN) (see, e.g. [4, Sec. 4.1]). It issatisfied by the time-series tests considered in this paper [33].

• Assumption 3 is strong (see, e.g. [2]). It implies that the different units of a panel must not belinked to each other beyond relatively simple forms of correlation such as common time effects.These can be eliminated by demeaning across the cross-sectional dimension. This assumptionis likely to be violated in many typical macroeconomic panel data sets. For instance, consider apanel data set consisting of exchange rates vis-à-vis the US dollar. The exchange rates of, say,the Euro and the Mexican Peso generally do not react identically to a macroeconomic shock inthe USA, given the very different structure of financial and trade links with the USA.

• We emphasize that Assumption is sufficient, but not necessary. Section 3 presents an approachthat allows to dispense with this assumption.

We now present the test statistics for panel cointegration put forward in this paper. Combinethe N p-values of the individual time-series cointegration tests, pi (i ∈ NN), as follows:

Pχ2 = −2N∑

i=1

ln(pi), (2a)

P−1 = N−0.5N∑

i=1

−1(pi), (2b)

where −1 denotes the inverse of the cdf of the standard normal distribution.2 When considered asa group, we refer to Equations (2a) and (2b) as P -tests. Furthermore, we refer to P -tests relying

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820 C. Hanck

on Assumption 3 as ‘simple’ P -tests. The P -tests, via pooling p-values, provide convenienttests for panel cointegration by imposing minimal homogeneity restrictions on the panel. Forinstance, the panel can be unbalanced. Furthermore, the evidence for (non-)cointegration is firstinvestigated for each unit of the panel and then compactly expressed with the p-value of the time-series cointegration test. Hence, the coefficients describing the relationship between the differentvariables for each unit of the panel can be heterogeneous across i. Thus, the availability of large T

time series allows for pooling the data into a panel without having to impose strong homogeneityrestrictions on β as in traditional panel data analysis.3

We now turn to the asymptotic distributions of the test statistics, which can be proved by anapplication of the transformation theorem for absolutely continuous random variables (see, e.g.[5, Theorem 4.2]).

Theorem 1 Under the null of no panel cointegration and Assumptions 1–3, as Ti → ∞ (i ∈NN), the P -tests are asymptotically distributed as

(i)Pχ2 →d χ22N ;

(ii)P−1 →d N (0, 1).

Using consistent (as Ti → ∞) time-series cointegration tests, pi →p 0 under the alternativeof cointegration. Hence, quite intuitively, the smaller pi , the more it contributes towards rejectingthe null of no panel cointegration. The decision rule therefore is to reject the null of no panelcointegration when the realized test statistic Pχ2 exceeds the critical value from a χ2

2N distributionat the desired significance level. For Equation (2b) one would reject for large negative valuesof the panel test statistic P−1 . We also see from the proof that the tests have a well-definedasymptotic distribution for any finite N . This feature is attractive because in many applicationsof panel cointegration analysis like the above example, the assumption of N , the number of unitsin the panel, going to infinity may not be a natural one. To rationalize the alternative hypothesisH1, note that a small fraction of the units of the panel exhibiting strong evidence of time-seriescointegration, thus yielding low p-values, might lead to a rejection of the null hypothesis. Thus, asstated above, rejection of H0 should not be taken as evidence of the entire panel being cointegrated.

We now discuss how to obtain the p-values required for computation of the P -test statistics.The null distributions of residual-based cointegration tests generally converge to functionals of theBrownian motion. Hence, analytic expressions of the distribution functions are not available andp-values of the tests cannot simply be obtained by evaluating the corresponding cdf. In the time-series case, it is now a fairly standard practice to obtain p-values of unit root and cointegrationtests using response surface regressions. We use p-values of the augmented Dickey–Fuller (ADF)cointegration tests [14] as provided by MacKinnon [26].4 Suppressing deterministic trend termsfor brevity, the p-values are derived from the t-statistic of �i − 1 in the OLS regression

�ui,t = (�i − 1)ui,t−1 +Ji∑

j=1

νj�ui,t−j + εJi ,i,t , (3)

where � := 1 − L and Ji is the number of lagged differences required to render εJi ,i,t white noise.Here, ui,t is the usual OLS residual from the first-stage Engle and Granger [14] regression (1).However, as should be clear from the above discussion, the P -tests are general enough to accom-modate any time-series cointegration test for which p-values are available. Alternatively, onecould, for instance, capture serial correlation by the semiparametric approach of Phillips andOuliaris [34].5 We focus on the Engle and Granger [14] ADF cointegration test because of itspopularity and widespread availability.

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Journal of Applied Statistics 821

3. Allowing for cross-sectional error dependence

We now relax Assumption 3. As can be seen from the proof of Theorem 1, this assumptionguarantees the correct null distributions of the test statistics. Theorem 1 no longer holds undergeneral forms of cross-sectional dependence. We suggest a bootstrap approach to capture thedependence structure in the panel with the aim to construct panel cointegration tests robust tothe presence of cross-sectional correlation.6 We employ the sieve bootstrap.7 The sieve bootstrapapproximates ui,t with a finite-order autoregressive process, where the order increases with samplesize, and resamples from the residuals. Under the following assumption, the sieve bootstrap yieldsan accurate approximation [9].

Assumption 4 (Linearity) The first differences of the equilibrium errors are generated as(possibly heterogeneous) linear processes, �ui,t = φi(L)εi,t , where φi(z) := ∑∞

�=0 φi,�z�.

More precisely, the bootstrap algorithm proceeds as follows.

(1) Compute the P -test statistic(s) according to Equations (2a) and (2b). Denote the realizationsby Pχ2 and P−1 .

(2) Suppressing deterministic trend terms and denoting estimates by a , estimate Equation (1)by OLS:

yi,t = αi + βixi,t + ui,t (i ∈ NN, t ∈ NTi).

(3) Fit an autoregressive process to �ui,t (i ∈ NN, t ∈ N2Ti). It is natural to use the Yule–Walker

procedure because it always yields an invertible representation [7, Sections 8.1–2]. Letting�ui := (Ti − 1)−1 ∑Ti

t=2 �ui,t , compute

γi (j) := 1

Ti − 1 − j

Ti−j∑t=2

(�ui,t − �ui)(�ui,t+j − �ui) (i ∈ NN, j ∈ Nq)

the empirical autocovariances of �ui,t up to order q. Defining

�i,q :=⎛⎜⎝

γi (0) · · · γi (q − 1)...

. . ....

γi (q − 1) · · · γi (0)

⎞⎟⎠

and γi := (γi(1), . . . , γi(q))�, obtain the AR coefficient vector as

(φq,i,1, . . . , φq,i,q)� := �−1

i,q γi (i ∈ NN).

(4) The residuals are, as usual, given by

εq,i,t := �ui,t −q∑

�=1

φq,i,��ui,t−�, (i ∈ NN, t ∈ Nq+2Ti

).

Center εq,i,t to obtain

εq,i,t := εq,i,t − 1

Ti − q − 1

Ti∑g=q+2

εq,i,g, (i ∈ NN, t ∈ Nq+2Ti

).

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822 C. Hanck

(5) Resample non-parametrically from εq,i,t to get ε∗q,i,t . To preserve the empirical cross-sectional

dependence structure, jointly resample residual vectors

εq,�,t := (εq,1,t , . . . , εq,N,t ), (t ∈ Nq+2Ti

).

(6) Recursively construct the bootstrap samples as8

�u∗q,i,t =

q∑�=1

φq,i,��u∗q,i,t−� + ε∗

q,i,t , (i ∈ NN, t ∈ Nq+2Ti

).

(7) It is necessary to impose the null of a unit root when generating the artificial data in bootstrapunit root tests to achieve consistency [3]. Accordingly, impose the null of non-cointegrationby integrating �u∗

i,t to obtain u∗i,t and form

y∗q,i,t = αi + βixi,t + u∗

q,i,t , (i ∈ NN, t ∈ NTi).

(8) Perform the P -tests using the artificial data set (y∗q,i,t , x�

i,t )�. Denote the realizations of the

test statistics by, e.g. P b∗χ2 .

(9) Repeat Steps 4–8 many, say B, times.(10) Denote the indicator function by 1{ } and choose a significance level α. Then, reject H0 of

the Pχ2 or the P−1 -test if

1

B

B∑b=1

1{P b∗χ2 > Pχ2} < α or

1

B

B∑b=1

1{P b∗−1 < P−1} < α, (4)

respectively.

Remarks

• We provide no formal proof of the consistency of this bootstrap procedure. It might be con-jectured from the proofs of bootstrap consistency for unit root tests [9,38] and for inferencein cointegrating regressions [10]. The latter authors argue that their results may hold moregenerally, e.g. for panel cointegration models.

• Steps 4 and 6, respectively, ‘prewhiten’ (or ‘sieve’) and ‘recolor’ the residuals using the sievebootstrap. Thus, we attempt to generate a valid bootstrap distribution of the data across thetime-series dimension using a parametricAR approximation to the true DGP. There is, however,no plausible parametric approximation of the dependence structure across the cross-sectionaldimension. We therefore resample entire cross-sections of residuals to preserve the cross-sectional dependence structure of the data. The resampling scheme across the cross-sectionaldimension is thus similar to block bootstrap procedures [24].

• The selection of the lag order q in Step 3 can be based on any of the well-known selectioncriteria such as the Akaike information criterion or a top–down procedure. The goal of the sievebootstrap is to prewhiten the residuals across t to obtain random resamples from εq,i,t . Hence,a selection scheme based on the whiteness of εq,i,t is also an appealing choice.

• It is possible to let q vary over i, qi �= q, to capture heterogeneity in the error processes.We do not make this possibility explicit in the notation.

• It is also possible to compute the bootstrap P -tests based on resamples from prewhitenedresiduals of a vector autoregressive (VAR) regression of (�ui,t , �x�

i,t )�. For a related purpose,

Chang et al. [10] advocate a similar scheme in order to capture endogeneity of the regressorsxi,t . Our simulation results are however similar to the ones to be presented for the sieve bootstrapin the next section. The VAR approach is computationally considerably more expensive and istherefore not discussed in detail. Results are available upon request.

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Journal of Applied Statistics 823

4. Monte-Carlo study

We perform a Monte-Carlo study of the tests proposed in the previous sections. The main resultsare as follows. The simple P -tests can have high ERPs when the units of the panel exhibit cross-sectional correlation of a general form.9 Compared with the simple P -tests, bootstrapping theP -tests as outlined in the previous section can strongly reduce the ERP.

4.1 Factor dependence

The DGP used here is an extension of a design which has been used in many Monte-Carlo studiesof (panel) cointegration tests; see Engle and Granger [14] and, for the extension to the panel datasetting, Kao [23]. For simplicity, consider the bivariate case, i.e. K = 1.

yi,t − αi − βixi,t = vi,t , a1yi,t − a2xi,t = wi,t (i ∈ NN), (5)

where

vi,t = ρivi,t−1 + fzi,t , �wi,t = ewi,t + πiewi,t−1,

fzi,t = ezi,t + λiεt

and ⎛⎝ezi,t

ewi,t

εt

⎞⎠ iid∼ N

⎛⎝

⎡⎣0

00

⎤⎦ ,

⎡⎣ 1 ψσ 0

ψσ σ 2 00 0 1

⎤⎦

⎞⎠ .

Remarks

• When |ρi | < 1, the equilibrium error vi,t in Equation (5) is stationary such that yi,t and xi,t arecointegrated with cointegration vector (1 αi βi)

�. Further, ρi need not be constant across i.We do, however, choose a common ρ in the simulations to limit the number of experimentsand to facilitate the interpretation of the results.

• Solving the system of Equations (5) for xi,t , we can write

xi,t = a1αi + a1vi,t − wi,t

a2 − a1βi

.

xi,t is weakly exogenous when a1 = 0.• The panel is cross-sectionally dependent because of the common factor εt and the idiosyncratic

factor loadings λi ∼ U[ζ1, ζ2], where U[ζ1, ζ2] denotes the uniform distribution with lowerbound ζ1 and upper bound ζ2.10 Similar factor structures have been employed by Bai and Ng[1] and Phillips and Sul [35]. Even though we generate cross-sectional dependence via a singlefactor, the validity of the suggested bootstrap tests does not depend on knowledge of the latentfactor structure.

• Since λi �= λ = cst, it is not possible to remove the cross-sectional dependence by subtractingtime-specific means N−1 ∑

i zi,t , as would be possible under the stronger assumptions of, e.g.Westerlund [40].

• If πi �= 0, there is a moving-average component in the errors. In particular, values −1 < πi < 0are well-known to have a potentially severe size-distorting effect on unit root and cointegrationtests [36].

The dimensions of the panel are N ∈ {10, 30} and, after having discarded 75 initial observations,T ∈ {50, 100}. These are representative for the dimensions of data sets often encountered in

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824 C. Hanck

macroeconometric applications. For a given cross-sectional dimension, we draw the unit-specificintercepts as αi ∼ U[0, 5] and keep them fixed for different T . We choose βi = U[1, 2], a2 = −1and σ = 1 and investigate all combinations of the following values for the parameters of the model:a1 ∈ {0, 1}, ψ ∈ {0, 0.5}, π ∈ {−0.5, 0} and ρ ∈ {0.8, 0.9, 1}. Further, there are three differentdegrees of cross-sectional dependence: (ζ1, ζ2) = (0, 0), (ζ1, ζ2) = (0, 1) and (ζ1, ζ2) = (1, 4),corresponding to no, ‘weak’and ‘strong’cross-sectional dependence [28]. That is, 72 experimentsare conducted for each of the four panel dimensions.

To limit the computational burden, we perform the simulation experiments using the recentlyproposed Warp-speed bootstrap by Giacomini et al. [17]. This method only requires B = 1 boot-strap replications for each of M Monte-Carlo replications (see [13] for a similar idea related to thedouble bootstrap). The B = M resamples are then used to construct the bootstrap distribution usedfor the test decision in Equation (4). It thus offers a tremendous reduction in the computational costof performing the simulations, as compared with the conventional approach of constructing onebootstrap distribution per Monte-Carlo replication, which requires M · B draws. Giacomini et al.[17] show that, under rather general conditions, the Warp-speed bootstrap accurately estimatesthe finite-sample rejection probability of bootstrap tests. Intuitively, the method works becausethe marginal distribution of each Monte-Carlo replicate is the same for a given set of parameters.Hence, employing one bootstrap replication from each Monte-Carlo replication asymptoticallyleads to the same bootstrap distribution as in the conventional approach, and thus to the samerejection probabilities for the tests.

Of course, in practice, only one sample is available and thus some B > 1 need to be chosen.We perform part of the experiments in the conventional way, with M = 5000 and B ∈ {200,

500, 1000}. The results for all B were very close to each other and to the ones to be reported here,suggesting that (i) B = 200 will be sufficient in practice and (ii) the Warp-speed bootstrap yieldsthe same conclusions as the conventional approach. Detailed results are available upon request.

The p-values for the P -tests are obtained from the Engle and Granger [14] ADF test, selectingthe number of lagged differences Ji in the second-stage ADF regressions (3) according to theautomatic procedure suggested by Ng and Perron [31]. We record a rejection if (i) the Pχ2 -teststatistic exceeds the 5% critical value of the χ2

2N distribution, the P−1 -test statistic falls belowthe 5% quantile of the cdf of the standard normal distribution, −1.645, or (ii) for the bootstrapversion of the tests, if Equation (4) applies.

To summarize, the DGP used here simultaneously addresses several issues that have provedboth relevant for empirical work and challenging for unit root and cointegration tests. Hence,while cautioning that interpretation of Monte-Carlo results should strictly speaking be confinedto the DGP at hand [20], we are optimistic that tests which perform well under this experimentaldesign may be useful for applied studies.

To put our results into perspective, we compare the rejection rates of the Pχ2 and the P−1 -testwith that of related tests recently proposed by Westerlund [42]. These use a quite similar bootstrapapproach to robustify the test statistics against cross-sectional dependence. They differ from theones proposed here in that they employ an error-correction representation

�yit = δi + αiyi,t−1 + λ�i xi,t−1 +

pi∑j=1

αij�yi,t−j +pi∑

j=1

γ�ij�xi,t−j + eit , (6)

instead of the residual-based representation considered here. The null of no panel cointegrationis then equivalent to αi = 0 (i ∈ NN). H0 can then be tested with, e.g.

Gα = 1

N

N∑i=1

T · αi

1 − ∑pi

j=1 αij

,

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Journal of Applied Statistics 825

where αi and 1 − ∑pi

j=1 αij are OLS and spectral based estimates of the parameters in (6), respec-

tively. Reject H0 if 1/B∑B

b=1 1{Gb∗α < Gα} < α, where Gb∗

α denotes the bootstrap estimates.See Westerlund [42] for a precise description of his bootstrap procedure and the full set of theproposed test statistics.

Table 1 reports the rejection rates from the Monte-Carlo study of the bootstrap and the simpleP -tests. For brevity, we only give the (representative) results for the serially correlated case(π = −0.5) and ψ = 0.5. In order to illustrate the importance of a suitable lag order q whenapplying the sieve bootstrap, we choose q = 2 (columns BS2) and the data-dependent rule q =[4 · (T /100)0.25] (columns BSs). The columns ‘Sim’ refer to the simple P -tests.

The main findings may be summarized as follows. The simple P -tests overreject in the pres-ence of cross-sectional correlation. The strength of the dependence – see the rows ρ = 1 inpanels (a) and (b) – does matter. The ERPs are particularly severe in the presence of endoge-nous regressors (see panels (ii) vs. (i)). Under ‘strong’ correlation (panel (a)) and endogeneity(a1 = 1), the simple P -tests even seem to be biased as they reject more frequently for samplesgenerated under H1 than for samples generated under H0. On the other hand, the difference inthe performance of the tests between the uncorrelated case (c) and the weakly correlated case(b) is small, suggesting that correlation robust tests may be most expedient when one suspectsstrong forms of dependence in the data. Generally, the sieve bootstrap is capable of removingor at least substantially reducing the ERP. Size control usually is more effective in columns BSs

than in columns BS2. Thus, as expected, it is necessary to suitably prewhiten the residuals witha data-dependent lag order selection scheme for q. This is in line with many other studies in thenon-stationary panel literature. For instance, Hlouskova and Wagner [19] find that selection ofthe lag length in panel unit root ADF regressions plays an important role in the behavior of manypopular tests.

Concerning the power of the tests, consider rows ρ = 0.9, 0.8. Power increases with T , asexpected. The increase in power with growing N is more pronounced when the dependence inthe data is smaller. This is intuitive because under strong dependence, the amount of independentinformation in the panel is smaller for a given N .

The corresponding results for Westerlund’s [42] tests may be found in Table 2. We generallyfind the tests by Westerlund to control size quite well. When regressors are exogenous, theirpower is comparable to that of the P -tests. It however also appears that Westerlund’s [12] testshave little power in the challenging case of cross-sectional dependence (panels (a) and (b)),endogeneity (a1 = 1) and negative serial correlation (π < 0). This effect is stronger the strongerthe cross-sectional dependence and the stronger the negative serial correlation.11

4.2 Cross-unit cointegration

Of course, the DGP considered in and below Equation (5) is only one way in which cross-sectional correlation can arise. To check the robustness of our results to variations in the trueDGP, we run an extra set of simulations where the dependence among the different units is drivenby cross-unit cointegration.12 Cross-unit cointegration arises for instance when there exists acointegration relation among regressors xit and xjt , i �= j . Banerjee et al. [2] have shown thatcross-unit cointegration has a strong size-distorting effect on first-generation panel unit root tests.More specifically, stack Yt = (y1t , . . . , yNt )

� and Xt = (x1t , . . . , xNt )�. We then consider the

following error-correction model

(�Yt

�Xt

)= αβ�

(Yt−1

Xt−1

)+ �

(�Yt−1

�Xt−1

)+ εt .

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826 C. Hanck

Table 1. Rejection rates of the P -tests.

Pχ2 P−1

N = 10 N = 30 N = 10 N = 30

T BS2 BSs Sim BS2 BSs Sim BS2 BSs Sim BS2 BSs Sim

(a) λi ∼ U[1, 4](i) a1 = 0ρ = 1 50 0.079 0.057 0.086 0.100 0.079 0.142 0.117 0.104 0.122 0.149 0.131 0.200

100 0.112 0.077 0.107 0.140 0.095 0.168 0.141 0.108 0.149 0.178 0.139 0.233ρ = 0.9 50 0.162 0.123 0.225 0.176 0.148 0.324 0.228 0.194 0.319 0.248 0.215 0.444

100 0.436 0.257 0.554 0.464 0.291 0.686 0.521 0.334 0.688 0.545 0.384 0.812ρ = 0.8 50 0.346 0.242 0.499 0.386 0.281 0.625 0.437 0.341 0.617 0.478 0.392 0.745

100 0.845 0.625 0.924 0.869 0.662 0.966 0.882 0.714 0.965 0.901 0.748 0.986

(ii) a1 = 1ρ = 1 50 0.186 0.061 0.283 0.256 0.113 0.445 0.189 0.099 0.231 0.234 0.155 0.359

100 0.218 0.082 0.163 0.333 0.081 0.303 0.225 0.117 0.177 0.307 0.134 0.335ρ = 0.9 50 0.184 0.103 0.183 0.248 0.127 0.251 0.191 0.131 0.117 0.193 0.150 0.162

100 0.261 0.093 0.091 0.356 0.107 0.125 0.265 0.130 0.091 0.362 0.170 0.127ρ = 0.8 50 0.205 0.114 0.150 0.318 0.159 0.195 0.248 0.170 0.095 0.307 0.227 0.101

100 0.387 0.139 0.136 0.631 0.230 0.253 0.465 0.243 0.154 0.725 0.427 0.298

(b) λi ∼ U[0, 1](i) a1 = 0ρ = 1 50 0.036 0.018 0.017 0.040 0.019 0.012 0.075 0.041 0.018 0.085 0.057 0.011

100 0.090 0.036 0.031 0.113 0.039 0.027 0.119 0.064 0.031 0.163 0.091 0.028ρ = 0.9 50 0.175 0.103 0.105 0.304 0.206 0.170 0.336 0.252 0.144 0.586 0.498 0.281

100 0.740 0.443 0.610 0.960 0.810 0.910 0.884 0.719 0.772 0.994 0.960 0.980ρ = 0.8 50 0.554 0.348 0.493 0.885 0.722 0.836 0.788 0.660 0.641 0.978 0.948 0.939

100 0.998 0.978 0.997 1.000 1.000 1.000 1.000 0.998 0.999 1.000 1.000 1.000

(ii) a1 = 1ρ = 1 50 0.156 0.054 0.198 0.308 0.095 0.356 0.184 0.113 0.123 0.365 0.222 0.212

100 0.218 0.066 0.113 0.434 0.097 0.194 0.241 0.105 0.114 0.478 0.189 0.194ρ = 0.9 50 0.159 0.126 0.147 0.395 0.185 0.266 0.235 0.208 0.088 0.571 0.402 0.169

100 0.518 0.213 0.314 0.876 0.506 0.589 0.641 0.389 0.380 0.966 0.808 0.736ρ = 0.8 50 0.424 0.242 0.364 0.777 0.521 0.636 0.617 0.428 0.346 0.927 0.848 0.643

100 0.948 0.700 0.895 1.000 0.989 0.999 0.985 0.891 0.955 1.000 1.000 1.000

(c) λi ≡ 0(i) a1 = 0ρ = 1 50 0.041 0.012 0.027 0.031 0.008 0.005 0.057 0.043 0.027 0.065 0.046 0.002

100 0.096 0.044 0.018 0.136 0.023 0.012 0.081 0.074 0.026 0.178 0.069 0.009ρ = 0.9 50 0.113 0.047 0.085 0.339 0.198 0.101 0.315 0.181 0.128 0.658 0.612 0.208

100 0.721 0.514 0.543 0.996 0.902 0.957 0.883 0.810 0.711 1.000 0.998 0.995ρ = 0.8 50 0.551 0.352 0.443 0.928 0.828 0.852 0.826 0.706 0.544 1.000 0.985 0.969

100 1.000 0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

(ii) a1 = 1ρ = 1 50 0.182 0.070 0.190 0.275 0.092 0.292 0.204 0.094 0.108 0.375 0.233 0.185

100 0.225 0.067 0.108 0.451 0.070 0.196 0.303 0.106 0.115 0.518 0.184 0.157ρ = 0.9 50 0.215 0.119 0.200 0.452 0.188 0.329 0.271 0.211 0.157 0.619 0.442 0.250

100 0.592 0.309 0.326 0.926 0.611 0.733 0.734 0.453 0.434 0.987 0.887 0.863ρ = 0.8 50 0.552 0.243 0.480 0.839 0.628 0.765 0.699 0.476 0.524 0.964 0.927 0.815

100 0.997 0.871 0.979 1.000 0.999 1.000 1.000 0.976 0.997 1.000 1.000 1.000

Note: ψ = 0.5, πi = −0.5. M = 10,000 replications. Panels (a), (b) and (c) correspond to ‘strong’, ‘weak’ and nocorrelation. Panels (i) and (ii) correspond to the exogenous and endogenous case. BS2, BSs and Sim are the sievebootstrap with q = 2, q data dependent and simple, resp.

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Journal of Applied Statistics 827

Table 2. Rejection rates of Westerlund’s tests.

N = 10 N = 30

T Gα Gτ Pα Pτ Gα Gτ Pα Pτ

(a)λi ∼ U[1, 4](i) a1 = 0ρ = 1 50 0.075 0.034 0.091 0.063 0.085 0.046 0.108 0.086

100 0.073 0.045 0.078 0.067 0.085 0.049 0.101 0.078ρ = 0.9 50 0.186 0.122 0.210 0.197 0.218 0.151 0.262 0.252

100 0.376 0.312 0.407 0.420 0.434 0.377 0.472 0.501ρ = 0.8 50 0.336 0.261 0.378 0.396 0.411 0.324 0.472 0.477

100 0.745 0.729 0.759 0.822 0.793 0.767 0.819 0.860

(ii) a1 = 1ρ = 1 50 0.078 0.052 0.069 0.077 0.108 0.106 0.100 0.118

100 0.075 0.074 0.069 0.079 0.077 0.084 0.071 0.090ρ = 0.9 50 0.042 0.028 0.042 0.035 0.027 0.020 0.022 0.024

100 0.015 0.019 0.012 0.015 0.005 0.006 0.002 0.002ρ = 0.8 50 0.027 0.014 0.019 0.015 0.008 0.005 0.007 0.005

100 0.009 0.009 0.010 0.011 0.002 0.002 0.001 0.002

(b) λi ∼ U[0, 1](i) a1 = 0ρ = 1 50 0.032 0.002 0.033 0.013 0.033 0.002 0.042 0.012

100 0.049 0.017 0.054 0.040 0.041 0.009 0.047 0.030ρ = 0.9 50 0.206 0.041 0.251 0.181 0.413 0.096 0.548 0.496

100 0.810 0.559 0.868 0.897 0.983 0.927 0.992 0.995ρ = 0.8 50 0.573 0.235 0.670 0.665 0.906 0.617 0.954 0.966

100 0.998 0.995 0.999 0.999 1.000 0.999 1.000 1.000

(ii) a1 = 1ρ = 1 50 0.083 0.063 0.072 0.086 0.095 0.093 0.086 0.115

100 0.066 0.065 0.050 0.058 0.067 0.065 0.052 0.070ρ = 0.9 50 0.037 0.044 0.031 0.053 0.035 0.045 0.043 0.080

100 0.020 0.046 0.025 0.072 0.010 0.042 0.022 0.167ρ = 0.8 50 0.031 0.055 0.045 0.129 0.052 0.107 0.087 0.294

100 0.085 0.228 0.131 0.499 0.156 0.561 0.353 0.933

(c) λi ≡ 0(i) a1 = 0ρ = 1 50 0.042 0.000 0.029 0.012 0.022 0.000 0.027 0.006

100 0.042 0.014 0.039 0.032 0.033 0.005 0.045 0.029ρ = 0.9 50 0.182 0.023 0.233 0.164 0.444 0.059 0.669 0.601

100 0.877 0.636 0.952 0.969 1.000 0.980 1.000 1.000ρ = 0.8 50 0.681 0.267 0.759 0.730 0.974 0.606 0.998 0.998

100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000(ii) a1 = 1ρ = 1 50 0.052 0.040 0.046 0.046 0.055 0.071 0.055 0.072

100 0.056 0.050 0.046 0.055 0.069 0.057 0.057 0.075ρ = 0.9 50 0.032 0.042 0.040 0.074 0.033 0.048 0.033 0.094

100 0.020 0.067 0.033 0.138 0.015 0.090 0.037 0.335ρ = 0.8 50 0.029 0.089 0.059 0.173 0.064 0.151 0.121 0.523

100 0.180 0.450 0.265 0.762 0.356 0.858 0.634 0.999

Note: ψ = 0.5, πi = −0.5. M = 10000 replications. Panels (a), (b) and (c) correspond to ‘strong’, ‘weak’and no correlation. Panels (i) and (ii) correspond to the exogenous and endogenous case. pi is chosen as[4 · (T /100)0.25].

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828 C. Hanck

Further, we let

α = −α ·(

IN 0N

0N IN

)and β� =

(IN −IN

0N BN

).

For α > 0, this formulation ensures cointegration relationships yi,t − xi,t (i ∈ NN), such thatwe analyze the power of the tests. By increasing α, the degree of reversion to the cointegrationrelationship increases, such that the tests should become more powerful. Cross-unit cointegrationobtains when there are non-zero rows in BN , e.g.

BN =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 −1 0 · · · 00 1 −1 0 · · · 0...

. . .. . . · · · 0

0 0 · · · 0 · · · 0...

.... . .

...

0 0 · · · 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

such that xi,t and xi+1,t are cointegrated with (1, −1)� for the non-zero rows of BN . To analyzethe size of the tests, we correspondingly let

β� =(

0N 0N

0N BN

),

Table 3. Rejection rates of the P -tests under cross-unit cointegration.

Pχ2 P−1

N = 10 N = 30 N = 10 N = 30

T BS2 BSs Sim BS2 BSs Sim BS2 BSs Sim BS2 BSs Sim

(a) Size50 0.024 0.018 0.002 0.015 0.009 0.000 0.032 0.025 0.002 0.027 0.017 0.000

α = 0 100 0.051 0.027 0.004 0.048 0.021 0.000 0.070 0.042 0.004 0.079 0.044 0.00150 0.029 0.021 0.002 0.014 0.008 0.000 0.037 0.028 0.002 0.023 0.018 0.000

α = 0.05 100 0.054 0.025 0.004 0.049 0.023 0.001 0.071 0.040 0.003 0.087 0.041 0.00150 0.022 0.018 0.003 0.017 0.012 0.000 0.029 0.028 0.003 0.023 0.020 0.000

α = 0.1 100 0.045 0.028 0.006 0.051 0.018 0.001 0.061 0.039 0.006 0.083 0.043 0.00050 0.023 0.023 0.003 0.014 0.009 0.000 0.027 0.028 0.003 0.024 0.015 0.000

α = 0.15 100 0.044 0.028 0.004 0.050 0.022 0.001 0.058 0.033 0.006 0.090 0.043 0.00050 0.027 0.021 0.004 0.014 0.012 0.000 0.031 0.029 0.004 0.025 0.019 0.000

α = 0.25 100 0.050 0.031 0.004 0.054 0.020 0.000 0.065 0.040 0.006 0.084 0.035 0.00150 0.027 0.017 0.003 0.013 0.008 0.000 0.028 0.023 0.003 0.025 0.013 0.000

α = 0.3 100 0.051 0.028 0.007 0.042 0.023 0.000 0.069 0.043 0.010 0.073 0.047 0.001

(b) Power50 0.023 0.018 0.003 0.020 0.008 0.000 0.033 0.033 0.002 0.042 0.024 0.000

α = 0.05 100 0.100 0.041 0.008 0.163 0.065 0.003 0.192 0.102 0.021 0.448 0.255 0.01950 0.036 0.030 0.007 0.061 0.027 0.002 0.088 0.071 0.010 0.206 0.131 0.007

α = 0.1 100 0.462 0.221 0.174 0.938 0.679 0.478 0.708 0.497 0.389 0.998 0.968 0.89750 0.105 0.061 0.035 0.228 0.144 0.034 0.237 0.162 0.069 0.647 0.508 0.133

α = 0.15 100 0.925 0.726 0.791 1.000 0.998 1.000 0.991 0.929 0.948 1.000 1.000 1.00050 0.574 0.368 0.396 0.963 0.864 0.886 0.813 0.703 0.644 1.000 0.995 0.992

α = 0.25 100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00050 0.803 0.604 0.742 0.999 0.990 0.999 0.948 0.879 0.896 1.000 1.000 1.000

α = 0.3 100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Note: � = 0.15IN . M = 10000 replications. There are six cross-unit cointegrating relationships. BS2, BSs and Sim arethe sieve bootstrap with q = 2, q data dependent and simple, resp.

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Journal of Applied Statistics 829

Table 4. Rejection rates of Westerlund’s tests under cross-unit cointegration.

N = 10 N = 30

T Gα Gτ Pα Pτ Gα Gτ Pα Pτ

(a) Size50 0.043 0.017 0.033 0.024 0.040 0.010 0.028 0.023

α = 0 100 0.051 0.034 0.046 0.040 0.038 0.019 0.038 0.03250 0.050 0.021 0.041 0.029 0.035 0.010 0.025 0.021

α = 0.05 100 0.044 0.027 0.033 0.031 0.042 0.024 0.031 0.02950 0.046 0.021 0.038 0.031 0.039 0.010 0.029 0.025

α = 0.1 100 0.052 0.036 0.049 0.049 0.043 0.019 0.037 0.03050 0.049 0.025 0.040 0.032 0.047 0.016 0.037 0.025

α = 0.15 100 0.048 0.027 0.043 0.040 0.042 0.019 0.035 0.03150 0.053 0.025 0.041 0.033 0.039 0.009 0.038 0.026

α = 0.25 100 0.042 0.028 0.042 0.040 0.042 0.016 0.033 0.03050 0.047 0.027 0.037 0.037 0.035 0.008 0.033 0.023

α = 0.3 100 0.047 0.032 0.044 0.043 0.046 0.018 0.042 0.039

(b) Power50 0.117 0.047 0.111 0.079 0.194 0.042 0.210 0.167

α = 0.05 100 0.553 0.315 0.655 0.617 0.951 0.703 0.981 0.98650 0.403 0.184 0.423 0.381 0.794 0.429 0.871 0.904

α = 0.1 100 0.990 0.972 0.996 1.000 1.000 1.000 1.000 1.00050 0.750 0.542 0.810 0.841 0.990 0.950 0.999 1.000

α = 0.15 100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00050 0.980 0.978 0.991 0.999 1.000 1.000 1.000 1.000

α = 0.25 100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.00050 0.995 0.998 0.998 1.000 1.000 1.000 1.000 1.000

α = 0.3 100 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Note: � = 0.15IN . M = 10000 replications. pi is chosen as [4 · (T /100)0.25]. There are six cross-unit cointegratingrelationships.

such that there are no within-unit, but still potentially size-distorting cross-unit cointegration rela-tionships. We again consider the cross-sectional panel dimensions N ∈ {10, 30}. For the degree ofmean reversion, we choose α ∈ {0, 0.05, 0.1, 0.15, 0.25, 0.3}. Results are summarized in Tables 3and 4. We see from panels (a) and (b) that under cross-unit cointegration with six non-zero rowsin BN , all robust tests control size. The non-robust simple tests, on the other hand, are undersized,confirming the need for cross-sectional correlation robust tests also under cross-unit cointegration.As regards power, the robust tests appear to be consistent, where Westerlund’s tests are somewhatmore powerful for smaller T . This may not be so surprising in view of the fact that this DGPvery closely mirrors the error-correction representation (6) for which Westerlund’s tests havebeen derived.

To summarize the Monte-Carlo results, we find that both P -tests and Westerlund’s tests performwell in a variety of challenging cases, with neither set of tests consistently outperforming the other.These results suggest that the P -tests can be seen as a useful addition to the literature.

5. An empirical application to the PPP hypothesis

In this section, we reconsider the PPP hypothesis, which has attracted wide attention in theliterature [see 39, for a recent survey]. Assuming the law of one price to hold at least in the longrun, its absolute version implies that the ratio of domestic price level Pi,t and foreign price levelP ∗

i,t should be close to the exchange rate Si,t . Equivalently, the real exchange rate Ri,t should benear one. Empirically, denoting natural logs by lowercase letters, the PPP hypothesis therefore

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830 C. Hanck

postulates that

ri,t = pi,t − p∗i,t − si,t (i ∈ NN)

is a stationary process. Due to factors such as transportation costs, one often allows for non-unitarycoefficients to obtain an equation of the form

p∗i,t = ai + βi1pi,t + βi2si,t + ei,t (i ∈ NN) (7)

referred to as the weak PPP hypothesis. Since the variables typically are I (1), the weak PPPhypothesis implies that ei,t is stationary, or, equivalently, that p∗

i,t , pi,t and si,t are cointegrated.We use quarterly post-Bretton Woods data for a panel of OECD countries, observing the spot

exchange rate and the consumer price index ranging from 1973:1 until 1998:2.13 The numerairecountry (with respect to which the spot exchange rate is sampled) is the USA. As has been pointedby, inter alia, O’Connell [32], the choice of a numeraire country naturally induces a commoncomponent in the set of regressions (7). It is therefore interesting to investigate the effect of usingcross-sectional correlation robust tests of the PPP hypothesis. Table 5 reports the estimation results.The first three columns give the estimates of ai , βi1 and βi2. The results differ across countries,making it attractive to use a panel procedure that allows for heterogeneity. The number of laggeddifferences Ji chosen by the procedure of Ng and Perron [31] is reported in column 4. The fifthcolumn gives the results for the t-statistics of the individual ADF cointegration regressions (3). All

Table 5. Tests for weak PPP.

Countries ai β1i β2i Ji t�i−1

Australia 16.848 0.833 3.982 2 −1.912Austria −23.109 1.216 −0.115 2 −2.412Belgium −3.907 1.114 −0.205 1 −1.626Canada −5.292 0.943 10.491 1 −0.809Denmark 12.461 0.974 −1.005 0 −0.751Finland 10.224 0.915 0.455 3 −1.841France 18.154 0.959 −2.014 1 −0.446Germany −40.631 1.425 −3.421 3 −2.778Greece 38.940 −0.036 0.352 0 −2.273Ireland 30.309 0.875 −23.705 0 −1.082Italy 28.505 0.659 0.003 3 −2.815Japan 6.847 1.098 −0.119 0 −1.222Netherlands −21.331 1.290 −6.165 0 −3.010New Zealand 26.845 0.734 3.613 2 −1.727Norway 11.707 0.888 0.489 1 −1.500Portugal 38.443 0.383 0.155 3 −2.821Spain 27.919 0.657 0.046 1 −2.340Sweden 14.692 0.766 1.426 3 −2.423Switzerland −30.012 1.313 −2.622 0 −1.203UK 16.231 0.819 5.693 1 −2.002US Numeraire country

Bootstrap critical values for q =Panel results Simple cr. val 2 4 6 8 10

Pχ2 16.751 55.758 46.018 33.160 38.908 37.408 41.151P−1 3.483 −1.645 −0.522 1.190 0.383 0.432 −0.052

Note: First three columns are estimates of (7), Ji are number of lagged differences chosen for unit root tests on ei,t . t�i−1

is t-statistic from (3). Bottom part gives realized test statistics (left), simple (middle) and bootstrap (right) 5% criticalvalues for different AR orders q in the sieve bootstrap procedure.

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Journal of Applied Statistics 831

statistics fail to produce evidence in favor of PPP, clearly exceeding the critical value of −3.837.The Netherlands have the smallest statistic t�Neth−1 = −3.010, implying a p-value of 0.251 [26].The panel test statistics are reported in the bottom of Table 5. In view of the individual outcomes,it is not surprising to see that the simple and bootstrap p-value combination tests also do not rejectthe null hypothesis of no PPP.

Even though neither tests rejects the null, the critical values differ substantially. In particular,the bootstrap critical values are almost all closer to the realized test statistic, indicating that usingthe critical values of the simple tests would be overly conservative in the present application. Weconclude that using cross-sectional correlation robust tests may well make an important differencein econometric practice.

6. Conclusion

We suggest new meta-analytic p-value combination tests for panel cointegration. The tests arerobust to the cross-sectional dependence of a general form. To achieve robustness, a bootstrapprocedure is used. Further advantages compared with other widely used panel cointegration testsare that they are flexible, intuitively appealing and comparatively easy to compute.

The simulation study reveals that the approach is generally effective in improving the reliabilityof the ‘simple’ P -tests. However, care is needed in properly selecting the lag order in the sievebootstrap scheme. This finding is analogous to the well-known result that the size and powerof time-series unit root and cointegration tests are fairly sensitive to the selection of the correctnumber of lagged differences in the ADF regressions (or, selection of lag truncations in estimatinglong-run variances).

A further interesting finding is that power gains from using panel tests are smaller the stronger thecross-sectional correlation in the data.We interpret this as reflecting that combining correlated timeseries leads to less additional information than combining independent series.This could imply thatthe impressive power gains reported in simulations of first-generation panel unit root tests vs. time-series unit root tests are partially an artifact of the experimental design relying on independence ofthe units. Our empirical study of the weak PPP hypothesis reveals that cross-sectional correlationrobust critical values for the P -tests can differ substantially from their simple counterparts. Itwould be interesting to investigate how other commonly used panel cointegration tests such asthose developed by Pedroni [33] or Kao [23] perform under general forms of cross-sectionalcorrelation.

Another task for future research is to establish whether the bootstrap procedure suggested inthis paper is consistent in the sense of leading to the same limiting distributions for the bootstrapstatistics as for the original ones. It may be possible to establish bootstrap consistency along thelines of Chang and Park [9], Chang [8], Chang et al. [10] and Swensen [38].

Acknowledgements

I would like to thank conference participants at the FEMES 2006, Beijing, the Statistische Woche 2006, Dresden, PavelStoimenov and two anonymous referees for comments that helped improve the paper, as well as Joakim Westerlund forproviding GAUSS code. Partial support by DFG under Sonderforschungsbereich 475 is gratefully acknowledged.

Notes

1. See Pedroni [33] for further discussion.2. See also Maddala and Wu [27] and Choi [11].3. For an overview of panel data models relying on N → ∞ asymptotics, see [21].

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832 C. Hanck

4. MacKinnon improves upon his prior work by using a heteroscedasticity and serial correlation robust technique toapproximate between the estimated quantiles of the response surface regressions.

5. It is also possible to employ the p-values for Johansen’s [22] λtrace and λmax tests. The corresponding P -tests performpoorly, however, as Johansen’s [22] time-series cointegration tests severely overreject for the time-series lengthsusually available in a macroeconometric practice. Accordingly, the corresponding p-values are not distributed asU[0, 1] under H0 and hence not suitable for the P -tests.

6. Note that our goal is not to provide bootstrap tests that yield asymptotic refinements, as there is no valid first-orderasymptotic theory available in our setup.

7. For important early work, see Maddala and Wu [27]. See also Swensen [38].8. We run the recursion for 30 initial observations before using the �u∗

q,i,t to mitigate the effect of initial conditions.9. Suppose the DGP is in the null hypothesis set, the tests are performed at a nominal level α and the rejection

frequency, or type I error rate, of the test is R(α). Then, ERP := |R(α) − α|. The term size distortion is often usedsynonymously [12].

10. Uniform random numbers are generated using the KM algorithm from which Normal variates are created withthe fast acceptance–rejection algorithm, both implemented in GAUSS. Part of the calculations are performed withCOINT 2.0 by Peter Phillips and Sam Ouliaris.

11. We also performed the simulations with π = −0.8. Results were, as expected, somewhat inferior. The main findingsare that both P - and Westerlund’s tests need to be constructed with a higher q (resp. pi in his notation, seeEquation (6)) to control size, e.g. q = [8 · (T /100)0.25]. This is not surprising in view of the more persistent errors.Detailed results are available upon request.

12. I am grateful to an anonymous referee for suggesting this extension.13. The data is from the IMF Financial Statistics CD-ROM. The list of included countries can be found in Table 5.

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