an intersection test for panel unit roots · this paper proposes a new panel unit root test based...

$: An Intersection Test for Panel Unit Roots · This paper proposes a new panel unit root test based on Simes’ [Biometrika 1986, \An Improved Bonferroni Procedure for Multiple Tests$
An Intersection Test for Panel Unit Roots∗

Christoph Hanck†

First Version: May 2008

This Version: October 2010

Abstract

This paper proposes a new panel unit root test based on Simes’ [Biometrika 1986,“An Improved Bonferroni Procedure for Multiple Tests of Significance”] classical inter-section test. The test is robust to general patterns of cross-sectional dependence andyet is straightforward to implement, only requiring p-values of time series unit roottests of the series in the panel, and no resampling. Monte Carlo experiments showgood size and power properties relative to existing panel unit root tests. Unlike previ-ously suggested tests, the new test allows to identify the units in the panel for whichthe alternative of stationarity can be said to hold. We provide an empirical applicationto real exchange rate data.

Keywords: Multiple Testing, Panel Unit Root Test, Cross-Sectional Dependence

JEL classification: C12, C23

∗I am grateful to two anonymous referees, conference and seminar participants in Cologne, Graz andMaastricht as well as to Matei Demetrescu and Franz Palm for constructive and helpful comments.†Rijksuniversiteit Groningen, Nettelbosje 2, 9747 AE Groningen, The Netherlands. Tel. (+31) 50-3633836,

[email protected].

1 Introduction

Testing for unit roots in heterogeneous panels has attracted much attention in recent years.

‘First generation’ tests [Maddala and Wu, 1999; Im et al., 2003; Levin et al., 2002] assume

the individual time series in the panel to be cross-sectionally independent [see Banerjee,

1999, for an early review]. It is, however, now widely recognized that this assumption is

not met in typical macroeconometric panel data sets. For instance, common global shocks

induce cross-sectional dependence among the test statistics [O’Connell, 1998].

The aim of ‘second generation’ panel unit root tests therefore is to provide reliable inference

under cross-sectional dependence. Phillips and Sul [2003], Moon and Perron [2004] and Bai

and Ng [2004] assume the dependence to be driven by (multiple) common factors. Suitable

‘de-factoring’, e.g. via principal components, asymptotically removes the factors, then allow-

ing for the application of standard panel tests. Breitung and Das [2005] propose a feasible

generalized least-squares approach that can be applied when T > n, where T denotes the

number of time series observations on each of the n series. Pesaran [2007] adds the cross-

section averages of lagged levels and of first differences of the individual series to Augmented

Dickey-Fuller [1979] (ADF) regressions. Panel tests can then be based on averages of the

individual cross-sectionally augmented ADF statistics. Chang [2004] uses a bootstrap to

condition on the dependence structure in the data. The meta-analytic p-value combination

test by Demetrescu et al. [2006] is most closely related to the one to be proposed here.

In view of the complexities induced by the cross-sectional panel dimension, many of these

tests require non-trivial decisions by the user, which complicate implementation. These

concern, e.g., the choice of the number of factors, of a joint lag length in pooled regressions,

or of a resampling scheme in bootstrap tests. In addition, available tests do not provide

guidance as to the identity of the cross-section units that are stationary [Breitung and

Pesaran, 2008]. Hence, it is difficult to properly interpret a rejection of a panel test.

Our goal is to provide a new panel unit root test that alleviates these potential shortcomings.

The test is based on Simes’ [1986] classical intersection test of the ‘global’ null hypothesis H0

that all individual null hypotheses Hi,0, i = 1, . . . , n, are true. (Here, that all n time series

have a unit root.) Simes’ [1986] test is widely applied in, among many other areas, genetical

micro-array experiments [e.g., Dudoit et al., 2003]. The test is robust to general patterns of

1

cross-sectional dependence and yet is straightforward to implement, only requiring p-values

of unit root tests on the n series in the panel. It compares the ordered p-values to suitably

increasing critical points and rejects the panel unit root null if at least one p-value is smaller

than the corresponding critical point.

Importantly, the new test allows to identify the units in the panel for which the alternative of

stationarity appears to hold. Doing so, it still controls the ‘Familywise Error Rate’ (FWER),

i.e. the probability to falsely reject at least one true individual time series null hypothesis,

at some chosen level α. This would not be achieved by the widely applied strategy to reject

for all those time series unit root test statistics that exceed some fixed level-α critical value,

as this approach ignores the multiple testing nature of the problem.1

Section 2 develops the new test. Section 3 reports Monte Carlo results on the test as well as

on other popular panel unit root tests. Section 4 presents an application to a panel of real

exchange rates. The last section concludes.

2 The Panel Unit Root Test

Consider the first-order autoregressive panel model

yi,t = µi(1− φi) + φiyi,t−1 + εi,t (i ∈ Nn, t ∈ NT ), (1)

where j ∈ Na is shorthand for j = 1, . . . , a, φi ∈ (−1, 1], i ∈ Nn, and n denotes the number

of series. The panel unit root null H0 states that all n series have a unit root, H0 : φi = 1

(i ∈ Nn), or, equivalently, that all single time series hypotheses Hi,0 : φi = 1 are true:

H0 =⋂i∈Nn

Hi,0, (2)

where⋂i∈Nn

denotes the intersection over the n individual hypotheses. Simes [1986] provides

a simple test of the ‘intersection’ null hypothesis (2). Suppose p-values pi, i ∈ Nn, of suitable

test statistics Ti (i ∈ Nn) for the Hi,0 are available. Let p(1) 6 . . . 6 p(n) be the ordered

p-values. Then, Simes’ test (S test, for short) rejects H0 at level α if and only if

p(j) 6 j · α/n for some j ∈ Nn. (3)

1Recently, procedures taking multiplicity into account have begun to find their way into the econometricsliterature. Romano et al. [2008] provide a survey of available methods and Hanck [2009] an application.

2

That is, one sorts the p-values from most to least significant, compares these to gradually

less challenging critical points jα/n and rejects if there exists at least one sufficiently small

p-value. The S test is level α when the Ti are independent and pi ∼ U [0, 1], i ∈ Nn (with

U the uniform distribution) [Simes, 1986]. As argued above, the independence assumption

will not be met in typical applications. Fortunately, Sarkar [1998] shows that it can be

weakened substantially. Concretely, he shows that the S test is level α if the test statistics are

multivariate totally positive of order 2 (MTP2). A vector of test statistics T = (T1, . . . , Tn)′

is MTP2 if its joint density f satisfies

f(min(T1, U1), . . . ,min(Tn, Un)

)· f(max(T1, U1), . . . ,max(Tn, Un)

)>

f(T1, . . . , Tn) · f(U1, . . . , Un),(4)

for any two points (T1, . . . , Tn) and (U1, . . . , Un). The MTP2 class is rather large, including

the multivariate normal with nonnegative correlations, specific absolute-valued multivari-

ate normal, multivariate gamma, and F distributions. Whether dependent unit root test

statistics Ti satisfy (4) is not known. In view of their nonstandard distributions, and the cor-

responding complicated structure of even the marginal densities, condition (4) is difficult to

verify analytically. We do not provide a proof here. However, Sarkar [1998] shows that even

(4) is not necessary for validity of the S test. Also, Section 3.3 sheds light on whether the

S test controls size for patterns of cross-sectional dependence typically assumed in dynamic

panel models.

It is important to properly interpret rejections of H0. Some tests have the alternative of a

completely stationary panel [e.g., Levin et al., 2002], i.e. maxi |φi| < 1, suggesting that a

rejection implies that the entire panel is stationary. However, these tests also have power

against ‘mixed’ panels, where only a subset n1 < n of the yi,t is stationary [see Taylor and

Sarno, 1998; Boucher Breuer et al., 2001]. To avoid unwarranted conclusions, it is thus

preferable to employ the more conservative alternative of a partially stationary panel:

HA : |φ(`)| < 1, ` ∈ Nn1 , 0 < n1 6 n,

with φ(`) the ordered coefficients. The S test belongs to this latter category, as

HA = Hc0 =

(⋂i∈Nn

Hi,0

)c=⋃i∈Nn

Hci,0 =

⋃i∈Nn

Hi,A,

where c denotes the complement and Hi,A the individual alternative |φi| < 1.

3

Remark 1. The S test is consistent, as T → ∞, for any n < ∞. To see this, let IA the

index set of the wrong hypotheses. Then, IA 6= ?, as |φi| < 1 for at least one i ∈ Nn. Using

consistent Ti for the p-values in S, there exists some pj, j ∈ IA, such that pj ≡ p(1) →p 0.

Hence, limT→∞ Pr(p(1) < α/n) = 1. Note that this result also holds for other p-value

combination procedures relying on some transformed sum of p-values, such as Fisher’s [1932],

the inverse normal test of Stouffer et al. [1949] or its second-generation variant studied by

Hartung [1999] and Demetrescu et al. [2006] in the panel unit root test context. To see this,

consider the Fisher test (the argument is analogous for the other tests) Pχ2 = −2∑n

i=1 ln(pi).

Then, p(1) →p 0 implies that Pχ2 →p +∞ even if pi = Op(1) for i 6= j.

Remark 2. Unlike other panel unit root tests [e.g., Im et al., 2003; Pesaran, 2007], the S test

does not require n1/n → θ1 > 0 as n → ∞ for test consistency (see Remark 1). It does,

however, require p(`) = op(`/n) as n, T → ∞, for some ` ∈ Nn1 . That is, the fraction of

stationary series may tend to zero, as long as there are sufficiently small p-values. Again, a

similar result could be established for other p-value combination tests. Provided the smallest

p-values tend to zero sufficiently fast, e.g. Pχ2 would diverge fast enough so as to still exceed

the critical values which grow in n due to the χ22n null distribution (under independence).

Due to the nonlinear transformations of the p-values employed in Hartung [1999] and Fisher

[1932] it however seems unclear how to precisely relate the necessary divergence rates to

those required for Simes’ test. Simulation evidence to be reported in Section 3 suggest that

while these tests also have power against nearly nonstationary panels, S performs better for

panels with few stationary units.

Existing tests are uninformative about which units are stationary under HA.2 Once H0 is

rejected, the S test does identify the units for which Hi,A can be said to hold. Hommel [1988]

proves the following procedure to control the FWER at α whenever S has level α.

Hommel’s Procedure

A. Compute

j = max{i ∈ Nn : p(n−i+k) > kα/i for k ∈ Ni}. (5)

B. If p(n) 6 α, reject all Hi,0. Else, reject all Hi,0 with pi 6 α/j.

For concreteness, consider an illustrative example where n = 3. We find j as the largest i

2Most tests are also silent about the number of stationary units n1. Ng [2008] is a recent exception,discussed in Section 3.2.

4

such that all adjacent conditions in Table I hold. If such a j cannot be found, then even

p(n) 6 α, so that we can ‘safely’ reject all hypotheses. If the p-values are given by, say,

3α/5, 2α, α/5, then j = 2 such that we only reject H3,0.

Table I.—Hommel’s [1988] procedure for n = 3

i = 1: k = 1 p(3−1+1) = p(n) > α

i = 2: k = 1 p(3−2+1) = p(n−1) > α/2k = 2 p(3−2+2) = p(n) > α

i = 3: k = 1 p(3−3+1) = p(1) > α/3k = 2 p(3−3+2) = p(n−1) > 2α/3k = 3 p(3−3+3) = p(n) > α

3 Monte Carlo Evidence

This section provides experimental evidence on the procedures discussed here. Section 3.1

investigates the S test under scenarios where validity of the S test is known to hold, such

as independent and defactored panels. To gauge the effectiveness of S, we compare it to

Maddala and Wu’s [1999] Fisher-type test, defined by Pχ2 = −2∑n

i=1 ln(pi)H0∼ χ2(2n), as

well as the Hartung [1999]-type test of Demetrescu et al. [2006]. Section 3.2 studies Hommel’s

procedure. Section 3.3 provides evidence regarding the performance of the test under various

cross-sectionally dependent DGPs. Since we have not formally established the validity of the

S test under such scenarios, the results shed some experimental light on whether the test

also works reliably in these settings. Future work could attempt a more formal discussion of

the validity of the S test under the scenarios discussed in Section 3.3.

3.1 Size and Power of the S test

We use the following simple data generating process (DGP):

yi,t = µi + xi,t

xi,t = φixi,t−1 + εi,t(i ∈ Nn, t ∈ NT )

We discard 30 initial observations to mitigate the effect of initial conditions under HA. For a

benchmark in which the MTP2 condition is naturally satisfied, DGP A creates independent

panels.

5

A. Independence: Let εt = (ε1,t, . . . , εn,t)′, In the identity matrix and εt ∼ N (0, In).

We also study the tests’ performance under Bai and Ng [2004]-type factor models, where yi,t

is driven by a single I(1) common factor Ft and independent idiosyncratic components ei,t.

DGP B :

yi,t = λiFt + ei,t, Ft = Ft−1 + ut, ei,t = φiei,t−1 + εi,t, εi,t, ut ∼ N (0, 1)

We apply the tests to the estimated idiosyncratic components ei,t, computed using the prin-

cipal components procedure of Bai and Ng [2004].3 Again, this setting is not only of indepen-

dent interest but also a useful benchmark where S test is known to be valid asymptotically,

as the ei,t are asymptotically independent when the Ft are estimated, and hence extracted,

precisely.4 (Whether Ft is I(1) can be tested as shown by Bai and Ng [2004].)

When φn ≡ (φ1, . . . , φn)′ = ın, H0 =⋂i∈Nn

Hi,0 is true. We then study size. If mini |φn| <1, we analyze power. Specifically, we investigate the heterogeneous alternatives φn =

(ı′n/2, φ′n/2)

′ and φn =(ı ′3n/4, φ

′n/4

)′. The components of φ are distributed as (φ)i ∼

U(3/4, 1). Finally, we consider the ‘weakly stationary’ panel alternative φ+n = (ı ′n−m, φ

′m)′,

where m is a small number. For µi = 0 (i ∈ Nn), we calculate n ADF τ -statistics from

regressions of yi,t on yi,t−1.5 We use MacKinnon [1996]-type response surfaces to obtain the

p-values.

In a further set of experiments, we modify DGPs A and B by introducing serial correlation

into the error terms. Concretely, we redefine εi,t to be the MA(1) process εi,t = ξi,t + γξi,t−1,

where γ ∈ {−1/2, 1/2}. Of course, we then need to take this serial correlation into account

when constructing the underlying time series unit root tests. We construct S and Pχ2

from ADF and Phillips and Perron [1988] (PP)-type tests, choosing the number of lagged

differences (ADF) and autocorrelations when computing the spectrum (PP, using a standard

3We assumed the number of common factors to be known. In practice one could and should employinformation criteria (IC) such as those of Bai and Ng [2002]. Bai and Ng [2004, p. 1145] show that thisdoes not materially affect the accuracy of their procedure. In view of the relatively large T and the orderof magnitude of calculations O(T 3) involved in defactoring the yi,t a full search over a number of possiblefactors using IC would have been rather demanding in terms of computation time.

4If Ft ∼ I(1), the tests can be seen as panel non-cointegration tests, as λjyi,t−λiyj,t ∼ I(0) if ei,t ∼ I(0)[Bai and Ng, 2004].

5To investigate non-zero intercepts, we simulate µi ∼ U [0, 0.2] and calculate τ -statistics from regressionsof yi,t on 1, yi,t−1. The results, which we do not report for brevity but which are available upon request,are similar to those to be reported. As expected, fitting a constant results in loss of power throughout.Furthermore, we analyzed φn = 0.95 · ın for a homogenous and φn = φn for a fully stationary alternative.

6

Fejer, or triangular, kernel) with the Schwert [1989]-type criterion b4 · (T/100)1/4c. tρ∗,κ is

built using ADF statistics throughout.

Table II reports the size of S, Pχ2 and tρ∗,κ. Panel I confirms that all tests control size

very well under both DGP A and B. This is true for any panel size, including small panel

dimensions that are known to produce unreliable sizes for many other panel tests. Panel II

gives results for the case of serial correlation. Subpanel (a) discusses the challenging case of

negative moving average roots (γ = −1/2). Clearly, the panel tests based on PP unit root

tests do not perform adequately. This is not surprising as Phillips and Perron [1988, p. 344]

show the unit root tests to suffer from a severe upward size distortion under such a DGP

(this result is robust to other bandwidth choices, see the working paper version of the paper).

Such distortions then inevitably carry over to the panel tests via erroneously small p-values.

The ADF-based versions perform better. The S test exhibits stable size for sufficiently large

T and any n. On the other hand, the size of Pχ2 tends to deteriorate with n. For instance,

the size distortion of Pχ2 at T = 30 and DGP B goes from 7% for n = 8 to 14% for n = 24,

while that of S remains more stable at around 3% and 4%, respectively. While this result

may seem surprising at first, it actually is perfectly in line with the theoretical results of

Hanck [2008]. He shows that meta panel tests such as Pχ2 that are constructed from a sum

of p-values will accumulate seemingly minor size distortions of the underlying time series

unit root tests so as to produce a panel test with pronounced size distortion when n is large.

Interestingly, the S test does not appear to suffer from this drawback to such an extent.

As such, it may be more robust in empirically relevant scenarios such as the one discussed

here.6 Subpanel (b) shows that positive MA roots are well handled by the ADF-based tests.

In particular, in view of much smaller time series size distortions, Pχ2 also accumulates these

much more slowly. The PP-based tests are often quite undersized, especially under DGP

B. Again, this is not surprising in view of the conservativeness of the PP tests under this

scenario [Phillips and Perron, 1988, p. 344]. As expected, size is relatively less well-controlled

under DGP B particularly for small T . This is because the common factors are then not

extracted very precisely. Nevertheless, the ADF-based S test still is quite accurate at least

6This result is robust to other methods to choose the lag lengths. Unreported results show that if theseare chosen so high that (seemingly slightly) undersized time series unit root tests result, Pχ2 will accumulatethese distortions to become increasingly conservative with n, thus wasting power (Also see subpanel (b) forDGP B, which shows that Pχ2 is undersized by 2.7% for n = 8 and T = 30 under DGP B, but already by3.9% for n = 48.)

7

for moderate and large T (and any n). tρ∗,κ seems to be somewhat less accurate than S in

particular for small T . To summarize, these results strongly recommend basing the panel

tests on ADF time series unit root tests.

Table III gives power results for S and Pχ2 under various specifications of the vector of

autoregressive coefficients φn.7 Panel I reports results for a half-stationary panel. Here,

Pχ2 and tρ∗,κ outperform S for small T . This is not entirely surprising as Pχ2 is known

to enjoy certain optimality properties when applied to independent data [Littell and Folks,

1971]. In turn, tρ∗,κ, the variant of the closely related inverse normal of Demetrescu et al.

[2006] performs similarly. However, when increasing the fraction of nonstationary series to

3/4, the power difference is much less pronounced (cf. Panel II). Of course, all tests then

also become absolutely less powerful. Indeed, when moving to the ‘weakly stationary’ panel

alternative φ+n for m = 1, the power of S exceeds that of Pχ2 and tρ∗,κ. Again, this result

has an intuitive interpretation. When n grows, so does the χ2(2n) critical value of Pχ2 . On

the other hand, the test statistic remains comparatively small, as it consists of n−1 large p-

values. This results in low power. Conversely, the S test only requires one single sufficiently

small p-value to reject H0, and its performance is less affected by the other large p-values.

Hence, although other p-value combination tests also have power against ‘near-nonstationary

panels’ (cf. Remark 2) it appears that S is most effective in such a scenario. (Unreported

simulations show that the advantage of S increases with the strength of the stationarity of

the stationary yi,t. For instance, S outperforms Pχ2 for any n and T when φm = 0.5. The

entry for T = 30 and n = 8 results in a power difference of 13 percentage points in favor of

S, as opposed to the difference of -6% reported here for φm ∼ U(3/4, 1).)

3.2 Detecting Stationary Units

We now gauge the effectiveness of Hommel’s procedure to estimate the fraction of stationary

units θ1. Most panel unit root tests are uninformative about θ1. Recently, however, Ng

[2008] put forward such a strategy relying on properties of the time average of cross-sectional

variances of yi,t. (See her paper for a detailed discussion.) We compare Hommel’s estimates

of θ1 to those of Ng’s approach. Results are based on DGPs A and DGP B. In the latter

7Here, we focus on the case of no serial correlation. Unreported additional simulations, which are availableupon request, draw a qualitatively similar picture under serial correlation and size-adjusted critical values.

8

case, we defactor the yi,t and apply the estimators to the ei,t. Tables IV reports the average

estimated θ1, θ1,H and θ1,Ng, and the corresponding mean squared errors (MSE).

For small T , Hommel’s θ1,H underestimates θ1. This is not surprising as the p-values are not

yet close to their T -probability limits of 0. As such, (5) will yield a relatively large j, hence

fewer rejections. However, Hommel’s θ1,H is consistent, tending to the correct classification

of the yi,t with increasing T .8 Ng’s θ1,Ng is approximately unbiased provided T is sufficiently

large. However, this comes at the expense of a large MSE. θ1,H clearly outperforms θ1,Ng in

terms of MSE for small θ1, and is mostly superior with increasing T for any scenario.9 Indeed,

the asymptotic normality of θ1,Ng implies nothing prevents it from exceeding one, or being

less than zero, helping explain the larger MSE. For instance, we find that θ1,Ng is outside

the admissible interval [0, 1] in a bit more than a quarter of the replications for T = 100 and

n = 8 when θ1 = 1/4 under DGP A. (See the working paper version for detailed results.)

This effect decreases with T and n, which is of course in line with Ng’s asymptotics. For

instance, the figure corresponding to the above case is roughly .08 for T = 2000 and n = 48.

It also turns out to decrease with θ1, explaining why Hommel’s procedure outperforms Ng’s

in particular for small T , small n and small θ1. Interestingly, Ng’s procedure produces more

extreme estimates under independence (DGP A) than under common factors (DGP B),

again helping explain why Hommel’s procedure outperforms Ng’s more clearly under DGP

A than B. Overall, given the larger bias of θ1,H for moderate to sizeable T and the larger

MSE of θ1,Ng, the choice between the two procedures will therefore depend to some extent on

the analyst’s loss function: a small bias is not helpful if one is far away from the true value

with the single sample one has in practice—nor is of course a small MSE if one obtains an

underestimate. In view of the additional advantage of Hommel’s procedure that it directly

identifies the series for which the alternative Hi,A holds, it appears to be a useful tool for

the applied macroeconometrician.

8Since the singular value decompositions involved in defactoring the yi,t for DGP B are of order O(T 3),they become computationally very burdensome for T > 1000. We therefore waive to analyze DGP B forvery large T .

9Further research could investigate whether recent powerful resampling-based multiple tests by, e.g.,Romano and Wolf [2005] can mitigate the bias of θ1,H.

9

3.3 Some evidence for cross-sectionally dependent DGPs

We now provide evidence regarding the performance of the tests under various DGPs for

cross-sectionally dependent panels. We compare the tests to popular second generation panel

unit root tests.10 Concretely, we consider tests by Demetrescu et al. [2006] (tρ∗,κ), Moon and

Perron [2004] (t∗a and t∗b), Pesaran [2007] (the truncated CIPS ∗ test, denoted C∗ here) and

Breitung and Das [2005] (trob). DGPs C-E generate cross-sectional correlation among the

εi,t, where we apply the tests to the observed data yi,t. Modify DGP A as follows.

C. Equicorrelation: Let εt ∼ N (0,Σ), Σ = δını′n+(1−δ)In and ın = (1, . . . , 1)′, (n×1).

D. ‘Spatial’ Correlation: Let εt ∼ N (0,Σ) and Σij = ψ|i−j|.

E. Factor Structure: εi,t = λi · νt + ξi,t, where ξi,t, νtiid∼ N (0, 1) and λi ∼ U(−1, 3).

Table V reveals that the first generation Pχ2 test has a pronounced size distortion. This is

as expected, as the χ2(2n) null distribution hinges on independence of the units. t∗a and t∗b

also do not perform entirely satisfactorily, especially under equi- and spatial correlation (for

which they are not designed, of course). S is level-α throughout, being slightly conservative

under equicorrelation (cf. C). Its size is very accurate in the other experiments (cf. A,D-E).

C∗ is sometimes somewhat oversized, while tρ∗,κ and trob perform very well.

Table VI reports power results for φn = (ı′n/2, φ′n/2)

′. In view of its size distortion under

C-E, we do not report power results for Pχ2 here. The power of t∗a and t∗b is disappointing

here. The tρ∗,κ test is powerful in the setups A, C,D and E. The performance of C∗ is

convincing, too. For small T , S is somewhat less powerful. That is unsurprising since the

p-values cannot yet be close to their probability limit 0. Hence, condition (3) is less likely

to be satisfied. This may well be a price worth paying given the S test’s additional ability

to identify individual false hypotheses. Moreover, S does very well for large T , as maybe

expected from Remarks 1 and 2. S’s power then is the highest of all tests, also outperforming

the alternative p-value combination test tρ∗,κ.

Table VII increases the fraction of nonstationary series to 3/4 and corroborates the excellent

performance of the S test. Of course, all tests are now less powerful. However, it is note-

worthy that the loss of power relative to Table VI is smallest for the S test; its performance

10See, e.g., Breitung and Pesaran [2008], Gengenbach et al. [2010] or the working paper version (availablefrom the author’s website) for surveys and comparisons of the tests used here.

10

is now the best for many relevant configurations.11

Table VIII(a) supplies additional results for DGP B. It shows that with the exception of

t∗a and t∗b , all tests control size quite well (for T sufficiently large for C∗). As regards power

(panel (b)), we again find S, Pχ2 (cf. Table III) and tρ∗,κ to outperform the other tests.

In many applications, the relevant question is to test the observed data yi,t for unit roots

rather than the ei,t. As however pointed out in for instance Gengenbach et al. [2010, p. 127],

the other panel unit root tests considered here in some way account for the common factor

and hence amount to a test of a panel unit root in the idiosyncratic component. Table

IX provides results for DGP B when directly testing the observed data yi,t without prior

defactoring. The table shows that S exhibits analogous behavior to these tests in that it has

power against panels with ei,t ∼ I(0) and controls size in panels with ei,t ∼ I(1). Relative to

the other tests, size control and power of S seem convincing in particular when we expect it

to be so, i.e. for panels with few stationary units (panel (c)) and relatively large T .

Banerjee et al. [2004] alternatively parameterize nonstationary dependent panels via cross-

cointegration. We employ DGP F to investigate the robustness of the tests under this setup.

DGP F : Stack Yt = (y1,t, . . . , yn/2,t)′ and Xt = (yn/2+1,t, . . . , ynt)

′. Let(∆Yt∆Xt

)= αβ′

(Yt−1

Xt−1

)+ εt,

α = −α ·(In/2 0n/20n/2 In/2

)and β′ =

(In/2 −In/20n/2 Bn/2

).

This yields n/2 ‘within-country’ cointegration relationships yi,t − yn/2+i,t (i ∈ Nn/2). ‘Cross-

cointegration’ obtains when rk(Bn/2) > 0. If, say, the first row of Bn/2 is (1, −1, 0′n/2−2),

then yn/2+1,t and yn/2+2,t cointegrate with (1,−1)′. We put α = 0.1 and rk(Bn/2) = 2. Table

X shows that, quite remarkably, only S seems to control size under cross-cointegration.

(Since DGP F ensures I(1)-ness for all yi,t, no power study applies here.) Several tests are

quite severely distorted, but also alternative p-value combination tests such as tρ∗,κ loose

some precision under this setup, unlike S.

11Results for the remaining power experiments mentioned in footnote 5, which were similar, are availableupon request. Overall, the C∗ performs marginally best there, and the Moon and Perron [2004] tests aresubstantially better. For near-integrated homogenous alternatives, the power of S test is relatively lower.This is unsurprising because we then have p(1) ≈ p(n), and it is unlikely that p(n) 6 α for φn close to ın.

11

In a final set of (unreported) simulations, we also checked that Hommel’s procedure is ef-

fective also under the DGPs considered in this subsection. Summarizing, the S test offers a

convincing performance across a wide range of relevant scenarios.

4 Application

We now apply the tests to the relative Purchasing Power Parity (PPP) hypothesis. As

discussed above, the S test can be constructed with any unit root test for which p-values

are available. To maximize the test’s ability to reject false nulls, preference should be given

to the most powerful time series unit root tests. The test regressions in the present PPP

application require an intercept. We therefore use Elliott et al.’s [1996] efficient GLS unit

root tests.

Let pi,t be the (log) price index in country i and period t, p∗t the ‘foreign’ (log) price index

of the reference country in the panel and si,t the (log) nominal exchange rate between the

currencies of country i and the reference country (the U.S.A). The real exchange rate is then

given by

ri,t = pi,t − p∗t − si,t (i ∈ Nn, t ∈ NT )

The PPP hypothesis is naturally tested with unit root tests on the real exchange rate [Rogoff,

1996]. We revisit the dataset used by Taylor [2002], which includes annual data for the

nominal exchange rate, CPI and the GDP deflator. (See Taylor [2002] for further details.)

Table XI lists the countries contained in our panel. We report results using CPI series.

The left portion of Table XI reports results for the S test. Obviously, condition (3) is satisfied,

so that the panel unit root null can be rejected. The left portion of Table XII reports similar

findings for the other tests. Here, we also calculate Pesaran’s [2007] untruncated CIPS test,

but find only minor differences (as in some unreported Monte Carlo) to his truncated C∗

statistic. Hommel’s [1988] method allows us to identify those countries with a stationary

real exchange rate, while controlling for multiplicity. Using the results from Table XI and (5)

we find j = 10. Thus, we reject Hi,0 for Argentina, Sweden, Norway, Mexico, Italy, Finland,

France, Germany, Belgium and the U.K.

Of course, if the panel contains a nonstationary common factor, direct application of the

tests to ri,t is not valid. We therefore redo the analysis after extracting a factor Ft. Bai and

12

Ng’s [2004] ADF cF

test statistic applied to Ft is -2.95. Hence, Ft is found to be stationary

and working with ri,t directly is valid. Applying the tests to the idiosyncratic components

ei,t confirms the previous finding of panel stationarity, as expected (cf. the right portions

of Tables XI and XII). Of course, depending on the factor loadings and the size of the

different components, individual country findings now differ—although the previous ones

are remarkably robust. Hommel’s j is now 12; thus, we reject for Finland, Norway, Mexico,

Argentina, Italy and Sweden.

Let us compare these conclusions to those from the traditional strategy of rejecting all Hi,0

whose p-value satisfies pi 6 α, which ignores the multiple nature of the testing problem. This

method rejects all Hi,0 except for Denmark, Switzerland and Japan (cf. the left part of Table

XI). The results of Hommel’s procedure suggest that several of the additional rejections may

be spurious, i.e. solely due to the traditional strategy not controlling the FWER.

5 Conclusion

We propose a new test for a panel unit root against the alternative of a partially stationary

panel, making use of Simes’ [1986] classical intersection test. The test is intuitive, straight-

forward to implement and yet is robust to general patterns of cross-sectional dependence.

Simulations show it to compare well in terms of finite sample size and power with recent

panel unit root tests. Further work could investigate whether typical unit root test statistics

satisfy the MTP2 condition sufficient for validity of the test.

Unlike other tests, Simes’ [1986] allows to shed light on the important question for which

units in the panel the alternative can be said to hold if the intersection null is rejected.

Hence, the it permits researchers to decide, for each unit individually, whether to model the

respective time series in levels or first differences. An application to the questions of real

exchange rate stationarity demonstrates the practical usefulness of the test.

Since the panel test only requires p-values of individual time series test statistics, the frame-

work is very flexible. For instance, it can be extended rather easily to test for panel coin-

tegration, the analysis of which is still at an earlier stage of its development [Breitung and

Pesaran, 2008]. As such, it could be applied to issues such as savings-investment correlation

or spot and forward exchange rates, that have so far been approached with other techniques.

13

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15

Table II.—Size results for S and Pχ2 , DGPs A and B.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

I. No Serial CorrelationA. Independence

S .056 .053 .051 .049 .057 .052 .051 .048 .059 .052 .055 .054Pχ2 .053 .049 .048 .051 .053 .050 .050 .050 .051 .054 .050 .051tρ∗,κ .054 .049 .050 .050 .048 .047 .047 .047 .036 .038 .036 .034

B. Common Factors

Pχ2 .057 .055 .054 .057 .057 .055 .053 .054 .057 .051 .049 .054tρ∗,κ .050 .051 .050 .051 .050 .049 .045 .045 .037 .035 .036 .035

II. Serial Correlation(a) γ = −1/2

A. Independence

SADF .067 .061 .053 .060 .067 .059 .056 .059 .081 .064 .063 .069Pχ2,ADF .070 .064 .055 .064 .092 .071 .066 .070 .111 .077 .067 .083SPP .514 .605 .645 .620 .605 .714 .758 .740 .759 .856 .902 .873Pχ2,PP .595 .676 .707 .688 .737 .808 .836 .823 .927 .958 .971 .958tρ∗,κ .069 .066 .057 .065 .081 .061 .058 .062 .077 .053 .046 .059

B. Common Factors


(b) γ = 1/2A. Independence


B. Common Factors

SADF .027 .050 .042 .038 .025 .053 .033 .039 .028 .057 .033 .038SADF .028 .051 .041 .041 .027 .051 .044 .037 .027 .059 .034 .037Pχ2,ADF .021 .059 .039 .043 .020 .063 .037 .036 .011 .070 .028 .034SPP .005 .005 .016 .023 .003 .005 .013 .019 .002 .002 .006 .015Pχ2,PP .005 .012 .021 .025 .005 .011 .013 .017 .001 .003 .007 .011tρ∗,κ .022 .055 .063 .040 .018 .056 .065 .030 .006 .043 .053 .023

Nominal level α = 0.05, 10000 draws. S is by Simes [1986], Pχ2 by Maddala and Wu[1999] and tρ∗,κ by Demetrescu et al. [2006].

16

Table III.—Power of S and Pχ2 for partially stationaryheterogeneous panels, DGPs A and B.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

I. φn =(ı ′n/2, φ

′n/2

)′ and (φn/2)i ∼ U(.75, 1)A. Independence

S .208 .440 .904 .995 .220 .475 .952 .999 .246 .544 .991 1.000Pχ2 .455 .747 .968 .998 .594 .877 .996 1.000 .864 .991 1.000 1.000tρ∗,κ .446 .735 .961 .997 .561 .857 .991 1.000 .781 .977 1.000 1.000

B. Common Factors

S .096 .150 .389 .714 .096 .176 .484 .840 .105 .203 .645 .970S .101 .155 .391 .711 .101 .179 .486 .972 .106 .209 .648 .968Pχ2 .166 .288 .566 .814 .222 .413 .754 .973 .362 .680 .967 .999tρ∗,κ .144 .268 .557 .808 .185 .373 .725 .919 .268 .575 .944 .998

II. φn =(ı ′3n/4, φ

′n/4

)′ and (φn/4)i ∼ U(.75, 1)A. Independence

S .129 .268 .688 .924 .142 .287 .761 .973 .151 .327 .887 .998Pχ2 .193 .339 .680 .910 .257 .441 .817 .972 .412 .699 .971 .999tρ∗,κ .202 .345 .623 .837 .246 .417 .744 .924 .346 .600 .896 .990

B. Common Factors

S .068 .097 .215 .425 .080 .111 .279 .565 .082 .128 .387 .801S .072 .105 .213 .437 .077 .109 .271 .573 .077 .123 .402 .802Pχ2 .101 .140 .248 .448 .117 .179 .353 .616 .174 .289 .621 .894tρ∗,κ .090 .135 .249 .422 .106 .168 .335 .569 .129 .238 .545 .819

III. φ+n , m = 1

A. Independence

S .094 .170 .441 .729 .090 .134 .400 .698 .076 .097 .311 .650Pχ2 .109 .163 .324 .617 .094 .128 .263 .531 .081 .101 .186 .382tρ∗,κ .114 .163 .294 .462 .087 .122 .214 .357 .054 .068 .121 .199

B. Common Factors

S .060 .072 .136 .250 .063 .078 .123 .249 .054 .064 .115 .255S .060 .073 .133 .256 .064 .072 .125 .261 .061 .060 .111 .256Pχ2 .075 .088 .132 .222 .071 .082 .116 .207 .070 .072 .109 .171tρ∗,κ .068 .079 .129 .203 .061 .073 .105 .177 .046 .051 .075 .117

See notes to Table II. No serial correlation.

17

Table IV.—Estimating the Stationary Fraction.

n 8 12 24 48

T θM1,H θMSE1,H θM1,Ng θMSE

1,Ng θM1,H θMSE1,H θM1,Ng θMSE



1,Ng

θ1 = 1/4DGP A

30 .130 .016 .180 .810 .087 .028 .126 .613 .044 .043 .101 .333 .022 .052 .078 .19250 .137 .015 .250 .477 .093 .026 .208 .343 .047 .042 .172 .189 .024 .051 .154 .103100 .169 .011 .295 .296 .134 .017 .255 .208 .092 .027 .224 .114 .066 .035 .213 .059200 .204 .007 .319 .224 .186 .008 .292 .156 .168 .009 .258 .081 .154 .010 .240 .043500 .232 .004 .334 .188 .226 .003 .302 .134 .218 .002 .271 .069 .212 .002 .257 .0361000 .243 .002 .340 .182 .240 .002 .305 .123 .235 .001 .276 .065 .231 .001 .261 .0332000 .250 .002 .346 .173 .246 .001 .309 .120 .244 .000 .277 .063 .241 .000 .263 .032

DGP B

30 .130 .015 .051 .419 .086 .028 .011 .304 .042 .044 -.015 .194 .021 .053 -.030 .14350 .129 .016 .155 .289 .086 .028 .145 .179 .043 .043 .111 .111 .022 .052 .092 .071100 .136 .015 .236 .218 .094 .026 .228 .143 .052 .040 .194 .075 .029 .049 .179 .042200 .149 .013 .278 .211 .115 .021 .259 .139 .082 .030 .234 .067 .068 .034 .220 .036500 .166 .011 .308 .191 .146 .015 .279 .129 .136 .016 .259 .065 .139 .014 .244 .033

θ1 = 1/2DGP A

30 .135 .136 .437 .581 .090 .170 .396 .417 .044 .208 .378 .235 .022 .229 .371 .12850 .158 .123 .494 .312 .109 .157 .460 .233 .056 .198 .436 .128 .028 .223 .421 .069100 .273 .066 .522 .195 .232 .083 .503 .142 .179 .109 .473 .074 .134 .136 .464 .041200 .385 .026 .536 .149 .368 .027 .521 .102 .341 .030 .491 .057 .314 .037 .487 .029500 .457 .008 .556 .119 .450 .007 .521 .088 .438 .006 .509 .044 .427 .007 .499 .0241000 .481 .004 .560 .116 .477 .003 .539 .079 .470 .002 .512 .041 .464 .002 .501 .0222000 .493 .002 .559 .111 .491 .002 .542 .076 .486 .001 .521 .040 .483 .001 .507 .021

DGP B

30 .129 .139 .240 .330 .083 .175 .219 .250 .041 .211 .184 .190 .021 .229 .168 .15650 .132 .137 .362 .210 .091 .169 .332 .156 .046 .207 .315 .099 .022 .228 .299 .073100 .157 .123 .449 .150 .114 .153 .428 .107 .068 .188 .404 .060 .042 .211 .402 .035200 .211 .094 .503 .122 .179 .112 .483 .086 .155 .124 .458 .047 .141 .132 .449 .026500 .288 .061 .522 .126 .285 .059 .514 .081 .294 .050 .490 .044 .299 .044 .484 .021

θ1 = 3/4DGP A

30 .142 .374 .701 .317 .093 .434 .680 .241 .046 .496 .667 .130 .023 .529 .663 .07050 .189 .327 .721 .176 .128 .393 .710 .128 .067 .469 .703 .069 .034 .513 .693 .039100 .408 .145 .748 .097 .358 .174 .734 .074 .281 .229 .726 .039 .216 .289 .719 .020200 .590 .045 .762 .073 .566 .048 .747 .054 .528 .057 .736 .029 .490 .072 .734 .015500 .691 .012 .769 .058 .684 .011 .764 .041 .666 .011 .749 .024 .652 .012 .745 .0121000 .722 .006 .774 .054 .718 .004 .765 .039 .710 .003 .755 .021 .702 .003 .749 .0112000 .739 .003 .775 .052 .736 .002 .772 .036 .731 .001 .755 .020 .727 .001 .752 .011

DGP B

30 .130 .386 .409 .291 .089 .439 .395 .241 .043 .500 .381 .194 .021 .531 .367 .17750 .143 .373 .551 .150 .094 .432 .539 .118 .046 .496 .521 .090 .023 .529 .513 .076100 .195 .320 .657 .079 .146 .372 .641 .064 .092 .436 .629 .043 .061 .476 .620 .031200 .321 .211 .714 .062 .290 .230 .700 .046 .260 .249 .690 .027 .231 .274 .681 .017500 .491 .100 .755 .055 .497 .086 .747 .037 .499 .074 .726 .022 .490 .073 .729 .011

10000 replications. θM1,H is the mean of Hommel’s estimator and θMSE1,H its MSE. θM1,Ng and θMSE

1,Ng are Ng’s.18

Table V.—Size results, DGPs C-E.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

C. Equicorrelation

S .037 .036 .032 .035 .037 .036 .031 .033 .032 .032 .028 .028Pχ2 .195 .202 .193 .197 .227 .227 .225 .222 .261 .261 .265 .265trob .065 .064 .061 .066 .068 .068 .061 .063 .071 .073 .072 .065C∗ .087 .072 .062 .063 .057 .051 .041 .040 .062 .052 .048 .043tρ∗,κ .056 .058 .050 .055 .058 .061 .055 .055 .060 .060 .057 .049t∗a .077 .157 .175 .167 .051 .155 .172 .186 .040 .061 .153 .162t∗b .066 .119 .125 .111 .051 .119 .131 .139 .043 .056 .125 .121

D. Spatial Correlation


E. Factor Structure


Nominal level α = 0.05, 10000 draws. In C, δ = 0.98. In D, ψ = 0.75. S is by Simes[1986], Pχ2 by Maddala and Wu [1999], C∗ by Pesaran [2007], trob by Breitung andDas [2005], tρ∗,κ by Demetrescu et al. [2006] and t∗a and t∗b by Moon and Perron [2004].

19

Table VI.—Power results for partially stationary heterogeneouspanels, φn =

(ı ′n/2, φ

′n/2

)′and (φn/2)i ∼ U(.75, 1).

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

C. Equicorrelation

S .145 .295 .784 .987 .140 .288 .804 .996 .123 .265 .808 .999trob .142 .165 .226 .273 .137 .169 .218 .281 .149 .182 .233 .296C∗ .794 .945 .994 .999 .825 .961 .997 1.00 .894 .986 1.00 1.00tρ∗,κ .258 .449 .791 .950 .267 .480 .823 .966 .282 .497 .856 .975t∗a .184 .289 .302 .274 .177 .317 .332 .371 .219 .251 .352 .367t∗b .136 .236 .253 .220 .148 .284 .297 .345 .197 .227 .337 .346


S .194 .393 .852 .992 .207 .430 .907 .999 .228 .483 .966 1.00trob .177 .231 .309 .377 .194 .256 .334 .399 .243 .315 .398 .479C∗ .242 .350 .636 .899 .207 .307 .602 .901 .250 .366 .686 .951tρ∗,κ .365 .612 .894 .983 .428 .711 .950 .995 .600 .872 .993 1.00t∗a .204 .294 .374 .422 .239 .329 .418 .464 .336 .458 .536 .584t∗b .223 .275 .318 .350 .286 .335 .390 .416 .416 .499 .537 .569

E. Factor Structure

S .211 .435 .893 .994 .207 .454 .937 .999 .224 .506 .975 1.00trob .192 .247 .316 .387 .196 .258 .344 .408 .227 .292 .386 .463C∗ .209 .311 .579 .841 .171 .288 .594 .889 .219 .379 .760 .969tρ∗,κ .415 .691 .934 .992 .502 .777 .971 .997 .632 .895 .989 1.00t∗a .166 .237 .309 .363 .142 .237 .332 .363 .160 .265 .377 .449t∗b .184 .223 .261 .293 .179 .239 .299 .309 .239 .303 .367 .416

See notes to Table V.

20

Table VII.—Power results for partially stationary heterogeneouspanels, φn =

(ı ′3n/4, φ

′n/4

)′and (φn/4)i ∼ U(.75, 1).

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

C. Equicorrelation

S .108 .212 .606 .906 .095 .205 .654 .953 .090 .188 .672 .991trob .098 .105 .118 .130 .093 .112 .117 .141 .105 .114 .128 .147C∗ .609 .778 .906 .964 .609 .799 .935 .979 .693 .868 .972 .996tρ∗,κ .122 .207 .425 .656 .123 .208 .463 .698 .130 .212 .497 .736t∗a .119 .201 .197 .170 .105 .232 .234 .266 .138 .156 .261 .277t∗b .094 .147 .132 .098 .091 .190 .180 .207 .128 .133 .228 .226


S .126 .247 .647 .919 .138 .256 .714 .960 .141 .300 .825 .997trob .097 .114 .137 .160 .098 .113 .145 .170 .099 .131 .159 .194C∗ .176 .216 .344 .534 .144 .180 .285 .478 .170 .202 .316 .537tρ∗,κ .189 .315 .560 .774 .221 .361 .656 .843 .291 .500 .797 .938t∗a .109 .173 .232 .266 .122 .188 .262 .308 .154 .257 .323 .376t∗b .140 .171 .193 .199 .180 .210 .243 .259 .253 .311 .336 .361

E. Factor Structure

S .134 .260 .671 .920 .141 .283 .749 .973 .141 .313 .860 .997trob .094 .112 .140 .172 .096 .114 .146 .171 .108 .137 .170 .202C∗ .141 .176 .279 .468 .108 .140 .267 .474 .129 .179 .346 .611tρ∗,κ .205 .334 .600 .805 .255 .413 .691 .874 .332 .533 .803 .924t∗a .069 .117 .162 .202 .053 .096 .153 .183 .044 .083 .152 .200t∗b .094 .116 .130 .145 .089 .113 .134 .147 .104 .122 .154 .174


Table VIII.—Rejection Rates under Nonstationary Common Factors.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

(a) Size

trob .033 .039 .045 .051 .025 .031 .043 .044 .012 .020 .031 .041C∗ .112 .091 .075 .070 .081 .062 .053 .048 .087 .068 .058 .052tρ∗,κ .051 .044 .050 .049 .043 .046 .043 .050 .033 .034 .039 .037t∗a .153 .163 .174 .176 .124 .131 .141 .138 .100 .109 .108 .108t∗b .086 .098 .106 .107 .073 .079 .088 .093 .064 .072 .073 .075

(b) Power : φn =(ı ′3n/4, φ

′n/4

)′ and (φn/4)i ∼ U(.75, 1)

trob .072 .099 .130 .146 .062 .091 .141 .175 .052 .102 .182 .229C∗ .130 .132 .162 .229 .098 .100 .135 .231 .123 .127 .197 .334tρ∗,κ .090 .131 .250 .413 .100 .163 .342 .568 .123 .238 .548 .825t∗a .219 .247 .280 .278 .201 .218 .257 .268 .217 .261 .287 .300t∗b .134 .158 .182 .184 .129 .143 .182 .194 .148 .190 .225 .233


21

Table IX.—Rejection Rates under Nonstationary Common Factors,tests applied to observed yi,t.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200

(a) Size

S .055 .056 .052 .052 .056 .056 .051 .047 .058 .054 .053 .050trob .047 .053 .052 .054 .047 .045 .049 .055 .044 .049 .053 .056C∗ .104 .088 .078 .067 .072 .060 .046 .046 .070 .062 .058 .051tρ∗,κ .064 .068 .067 .062 .075 .071 .066 .066 .076 .078 .076 .075t∗a .035 .070 .100 .119 .020 .043 .076 .094 .011 .026 .053 .075t∗b .060 .076 .080 .083 .048 .059 .071 .069 .043 .052 .060 .066

(b) Power : φn =(ı ′n/2, φ

′n/2

)′ and (φn/2)i ∼ U(.75, 1)

S .090 .130 .266 .425 .097 .132 .297 .494 .101 .143 .367 .629trob .102 .126 .152 .170 .111 .130 .160 .173 .131 .157 .180 .204C∗ .196 .255 .374 .519 .186 .239 .399 .585 .252 .351 .572 .763tρ∗,κ .156 .227 .331 .450 .180 .245 .379 .495 .208 .282 .424 .567t∗a .166 .268 .378 .464 .170 .261 .398 .446 .216 .387 .512 .585t∗b .186 .250 .317 .382 .211 .271 .364 .377 .310 .437 .512 .560

(c) Power : φn =(ı ′3n/4, φ

′n/4

)′ and (φn/4)i ∼ U(.75, 1)

S .076 .091 .167 .275 .073 .098 .183 .323 .078 .104 .237 .432trob .070 .079 .088 .094 .074 .083 .096 .104 .080 .087 .104 .117C∗ .139 .150 .185 .245 .113 .121 .159 .245 .131 .162 .238 .357tρ∗,κ .117 .135 .203 .290 .126 .161 .246 .334 .161 .206 .307 .403t∗a .071 .123 .190 .239 .062 .111 .182 .228 .056 .113 .200 .242t∗b .100 .122 .155 .176 .106 .127 .164 .180 .119 .162 .208 .220


Table X.—Size under Cross-Cointegration.

n 8 12 24T 30 50 100 200 30 50 100 200 30 50 100 200


See notes to Table V. Two cross-unit cointegrating relationships, α = 0.1.

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Table XI.—DF-GLSµ-p-values for the real exchange rate series

ri,t ei,tcountry p-values Simes crit. p-values countryArgentina 0.0001 0.0026 0.0001 FinlandSweden 0.0001 0.0053 0.0001 NorwayNorway 0.0001 0.0079 0.0001 MexicoMexico 0.0001 0.0105 0.0001 ArgentinaItaly 0.0001 0.0132 0.0025 ItalyFinland 0.0001 0.0158 0.0025 SwedenFrance 0.0050 0.0184 0.0050 FranceGermany 0.0050 0.0211 0.0050 BelgiumBelgium 0.0050 0.0237 0.0050 U.K.U.K. 0.0050 0.0263 0.0150 BrazilBrazil 0.0175 0.0289 0.0200 NetherlandsAustralia 0.0175 0.0316 0.0200 AustraliaNetherlands 0.0200 0.0342 0.0375 PortugalPortugal 0.0250 0.0368 0.0400 CanadaCanada 0.0400 0.0395 0.0500 GermanySpain 0.0500 0.0421 0.0550 SpainDenmark 0.0575 0.0447 0.0550 DenmarkSwitzerland 0.2375 0.0474 0.2825 SwitzerlandJapan 0.2475 0.0500 0.2825 Japan

‘p-values’ are for the DF-GLSµ time series unit root tests [Elliott et al., 1996]. ‘Simes crit.’ isthe value that needs to exceed the adjacent p-value in at least once for a rejection of H0, cf. (3).

Table XII.—Panel Unit Root Tests for the real exchange rate panel

ri,t critical points ei,ttrob −3.703 −1.64 −6.780tρ∗,κ −7.578 −1.64 −7.367CIPS −3.084 −2.16 −2.990C∗ −3.077 −2.16 −2.990t∗a −0.032 −1.64 −14.79t∗b −0.290 −1.64 −5.189

See notes to Table V. Nominal level is α = 0.05. The column ‘criticalpoints’ gives the α level critical value. ri,t gives the test statisticsapplied to the real exchange rates and ei,t those applied to extractedidiosyncratic components. CIPS is by Pesaran [2007].

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an intersection test for panel unit roots · this paper proposes a new panel unit root test based...

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