This paper illustrates analytically the effects of cross-unit cointegration using as an example the conventional pooled least squares estimate in the spurious panel regression case. The results suggest that the usual result of asymptotic normality depends critically on the absence of cross-unit cointegration.econometrics;

Urbain, Jean-Pierre

Westerlund, Joakim

English

Research Papers in Economics

Jean-Pierre Urbain, Joakim Westerlund 
 
Spurious Regression in Nonstationary Panels 
with Cross-Unit Cointegration 
 
RM/06/057 
 
 
JEL code : C13, C33 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Maastricht research school of Economics 
of TEchnology and ORganizations 
 
Universiteit Maastricht 
Faculty of Economics and Business Administration 
P.O. Box 616 
NL - 6200 MD Maastricht 
 
phone  : ++31 43 388 3830 
fax  : ++31 43 388 4873  
 
 
 
 Spurious Regression in Nonstationary Panels with
Cross-Unit Cointegration∗
Jean-Pierre Urbain† and Joakim Westerlund‡
December 14, 2006
Abstract
This paper illustrates analytically the eﬀects of cross-unit cointegration
using as an example the conventional pooled least squares estimate in the
spurious panel regression case. The results suggest that the usual result of
asymptotic normality depends critically on the absence of cross-unit cointe-
gration.
JEL Classiﬁcation: C13; C33.
Keywords: Panel Cointegration; Spurious Regression; Cross-Unit Cointegra-
tion; Cross-Section Dependence.
1 Introduction
The use of nonstationary panel data techniques to infer both spurious and coin-
tegrated regressions has basically exploded during the last couple of years. The
main reason for this being the more powerful and straightforward asymptotic re-
sults based on pooling across a large cross-section of independent units. In fact, as
shown by Phillips and Moon (1999), since this independence eﬀectively works by
smoothing out the usual unit root dependency for each unit, normal inference is
usually possible, even in the usually so diﬃcult spurious regression case.
∗A preliminary version of the paper was presented at the Conference on Common Feature in
Maastricht, at Ente ”Luigi Einaudi” in Rome, ESEM2004 in Madrid and at the Econometric
Seminar in Rotterdam. The author wants to thank Christian Gengenbach, St´ ephane Gr´ egoir,
Richard Paap, Franco Peracchi and Fabio Busetti for helpful comments and suggestions. The
second author would also like to thank the Maastricht Research School of Economics of Technology
and Organizations for its hospitality during a visit at the Department of Quantitative Economics
at the University of Maastricht, where a part of this paper was written. Thank you also to the
Jan Wallander and Tom Hedelius Foundation for ﬁnancial support under research grant number
W2006-0068:1. The usual disclaimer applies.
†Corresponding author: Department of Quantitative Economics, Universiteit Maastricht, P.O.
Box 616, 6200 MD Maastricht, The Netherlands. Telephone: +31 43 388 3660; Fax: +31 43 388
48 74; E-mail address: j.urbain@ke.unimaas.nl
‡Department of Economics, Lund University, P.O.Box 7082, S-220 07 Lund, Sweden. Tele-
phone: +46 46 222 8670; Fax: +46 46 222 4118; E-mail address: joakim.westerlund@nek.lu.se.
1However, as argued by Banerjee at al. (2006), assuming independence in this
way is usually not warranted, especially when considering macroeconomic or ﬁnan-
cial data with strong intra-economic linkages. In particular, by using simulation
methods, they show that the presence of cross-unit cointegration, a reasonable
assumption in most applications, can have quite drastic eﬀects on inference.
In this paper, we complement analytically the simulation results of Banerjee at
al. (2006) taking as an example the pooled least squares slope estimator in the
simple bivariate case.1 We consider two possibilities, one is when the cointegration
occurs across the whole panel, while the other is when the cointegration occurs
within groups that are otherwise independent of each other, which seems like a
reasonable scenario when studying for example convergence clubs. In both cases,
it is shown that the presence of cross-unit cointegration in both the dependent and
independent variables leads to statistics that diverge with the size of the cross-
section. On the other hand, if at least one of the variables does not cointegrate
across units, then normality is again possible, with the center of the distribution
located at the long-run average regression coeﬃcient. For the case with cointe-
gration within groups, this naturally leads to the deﬁnition of a long-run average
group coeﬃcient.
The next two sections present our main ﬁndings, while Section 4 concludes. All
proofs are regelated to the appendix.
2 Panel common unit roots
Suppose that yit and xit are two a scalar variables such that
xit = λxifxt + δxiexit + uxit and yit = λyifyt + δyieyit + uyit, (1)
where t = 1,...,T and i = 1,...,N indexes the time series and cross-sectional
units, respectively. Note that while the ﬁrst term on the right-hand side of these
equations are common across i, the second is not. The parameters λxi, δxi, λyi
and δyi are assumed to be mutually independent, and randomly distributed across
i with expected values λx, δx, λy and δy, respectively. We further assume that
the disturbances uxit and uyit are stationary processes, while the remaining right-
hand side variables in (1) are pure unit root processes that satisfy the following
invariance principles as T → ∞ for each i
1
√
T

fxt
fyt

⇒

Bx
By

and
1
√
T

exit
eyit

⇒

Bxi
Byi

,
where ⇒ signiﬁes weak convergence. The vectors of common and idiosyncratic
Brownian motions, B and Bi say, are assume to be independent with covariance
matrices Ωf and Ωe
i, respectively, where Ωe
i is a random matrix with expectation
1This work is related to that of Gengenbach et al. (2006). However, while they focus speciﬁcally
on the case when testing for cointegration, our focus is more conceptual, and our interest in the
least squares slope estimate is just for illustration.
2Ωe. All variables and random parameters indexed by i are independent along the
cross-sectional dimension.
Although very simple, as we shall see, this data generating process yields signif-
icant insight. In particular, note how (1) nests a variety of spurious regression cases
both with and without cross-unit cointegration, depending on the values taken by
λxi, δxi, λyi and δyi.2 In fact, it is not diﬃcult to see that (1) is actually a system
of two Bai and Ng (2004) factor models, one for xit and one for yit. Deﬁne
b β =
 
N X
i=1
T X
t=1
xityit
!
.
 
N X
i=1
T X
t=1
x2
it
!
.
The objective of this paper is to show how b β, the conventional pooled least squares
slope estimator in a regression of yit on xit, behaves in this context.
Let us deﬁne the following two quantities
Qxx = λ2
x
Z
B2
x

+
1
2
δ2
xΩe
xx,
Uxy =
1
6

λ2
xδ2
yΩf
xxΩe
yy + λ2
yδ2
xΩf
yyΩe
xx + δ2
yδ2
xΩe
yyΩe
xx

+
1
2
λyλxδyδxΩf
xyΩe
xy.
Given Qxx and Uxy, if we let →p signify convergence in probability, then the
asymptotic distribution of b β can be summarized in the following way.
Theorem 1. Under the assumptions laid out above, as N, T → ∞
(a) b β →p
1
Qxx

λyλx
Z
ByBx +
1
2
δxδyΩe
xy

(b)
√
N b β −
√
N
1
Qxx

λyλx
Z
ByBx +
1
2
δxδyΩe
xy

⇒ N

0,
1
Q2
xx
Uxy

Note how Theorem 1 expresses the asymptotic distribution of b β in terms of the
parameters of (1). This is very instructive.
Firstly, note that the mean of the distribution depends critically on the com-
mon components of xit and yit. In particular, note that if λx is zero, so that
only yit has a common component, then
√
N b β converges to a normal variate with
mean δyΩe
xy/δxΩe
xx, which is the long-run average regression coeﬃcient derived by
Phillips and Moon (1999) in the cross-sectionally independent case. On the other
hand, if λx and λy are both nonzero, then b β is no longer consistent for δyΩe
xy/δxΩe
xx,
and in fact diverges as N grows. This suggests that much of the empirical work
based on the pooled least squares estimator in nonstationary panel data with a
possible common factor structure needs to be reevaluated, see for example Coakley
et al. (2005).
2The case with cointegration between yit and xit at is also covered, and amounts for example to
setting λxi and λyi to zero for all i while imposing equality or proportionality of the idiosyncratic
unit roots exit and exit.
3This clearly illustrates the importance of the common components, both for
inferential and cointegration testing purposes. For example, with both yit and xit
aﬀected, the asymptotic distribution of
√
N b β is no longer normal, but dependent
on Brownian motion, which makes it look more like something that one would
expect to see in the conventional time series case. Similarly, it is not diﬃcult to
see how tests for the absence of cointegration based on the consistency of b β, such
as those of Kao (1999), become invalid in this case.
Secondly, the variance of the asymptotic distribution of
√
N b β depends on both
the idiosyncratic and common components. However, note that if there are no
common components, then the variance reduces to 2δ2
yΩe
yy/3, which is likely to be
smaller than for the case with common components, and it is deﬁnitely smaller
than 2(λ2
yΩf
yy +δ2
yΩe
yy)/3, the variance when λx is zero. Hence, the presence of the
common components aﬀects the asymptotic distribution of
√
N b β by increasing its
variance.
3 Group speciﬁc unit roots
Next, we examine the case with group speciﬁc common unit roots. Speciﬁcally,
suppose that there are M groups with K units in each, and let fxjt be the common
unit root component of xit for group j = 1,...,K containing the units i = 1 +
(j − 1)K,...,jK. The group speciﬁc common component of yit is constructed in
the same way, and is denoted fyjt. Let Bxj and Byj be the Brownian motions
associated with fxjt and fyjt, and assume that these have covariance Ω
f
j and that
they are independent across j.
In view of Theorem 1, it is not diﬃcult to see that as N, T → ∞ with M ﬁxed
b β →p

λyλx
1
M
M X
j=1
Z
ByjBxj +
1
2
δxδyΩe
xy
.
λ2
x
1
M
M X
j=1
Z
B2
xj +
1
2
δ2
xΩe
xx

.
By contrast, if M is permitted to grow with N, then we get
b β →p

λyλxΩf
xy + δxδyΩe
xy
.
λ2
xΩf
xx + δ2
xΩe
xx

,
where Ωf is the expected value of Ω
f
j. This second result is very interesting.
Speciﬁcally, suppose that δx is zero so that xit has no idiosyncratic unit root, then
the probability limit of b β reduces to λyΩf
xy/λxΩf
xx, which might be thought of as a
long-run average group coeﬃcient. Thus, if the number of groups are permitted to
grow with N, then we end up with something that is very similar to what we had
earlier in the case with panel common unit roots. This seems very reasonable since
in this case the group essentially takes the role of the individual unit. Moreover,
as in Section 2, it is possible to show that if M is permitted to grow with N and
T, then
√
N b β is asymptotically normal.
44 Concluding remarks
It is now well-known that the conventional assumption of independent cross-sections
is very hard to maintain when conducting inference in nonstationary panels.
In this paper, we study the eﬀects of cross-unit cointegration taking as an exam-
ple the pooled least squares estimator in the spurious regression case. We consider
both the case when the cointegration takes place across the panel as a whole, and
when it takes place within groups. Our ﬁndings suggest that the presence of cross-
unit cointegration can have dramatic eﬀects for the usual asymptotic results based
on cross-section independence, and that the consistency and asymptotic normal-
ity of the pooled least squares estimator may even be lost. These results have
important implications, not only for theoretical, but also for applied work. Take
for example the study of Coakley et al. (2005), in which the authors seek to in-
fer purchasing power parity using pooled least squares methods in the presence of
common factors. Our results indicate that this type of analysis can be potentially
very deceptive.
As a ﬁnal note, although the simplicity of the data generating process used
in this paper naturally raises some concerns, we would like to stress that most
assumptions are only for ease of exposure, and can in principle be relaxed with-
out aﬀecting the main conclusions. For example, it can be easily generalized to
accommodate deterministic intercept, trend terms and additional dynamics.
References
Bai, J., and S. Ng (2004). A Panic Attack on Unit Roots and Cointegration.
Econometrica 72, 1127-1177.
Banerjee, A., M. Marcellino and C. Osbat (2004). Some Cautions on the use of
Panel Methods for Integrated Series of Macroeconomic Data. Econometrics
Journal 7, 322-340.
Coakley, J., R. P. Flood, A. M. Fuertes and M. P. Taylor (2005). Purchasing
Power Parity and the Theory of General Relativity: The First Tests. Journal
of International Money and Finance 24, 293-316.
Gengenbach, C., F. C. Palm and J-P. Urbain (2006). Panel Cointegration Test-
ing in the Presence of Common Factors. Oxford Bulletin of Economics and
Statistics 68, 683-719.
Kao, C. (1999). Spurious Regression and Residual-Based Tests for Cointegration
in Panel Data. Journal of Econometrics 90, 1-44.
Phillips, P. C. B., and H. R. Moon (1999). Linear Regression Limit Theory of
Nonstationary Panel Data. Econometrica 67, 1057-1111.
5Mathematical appendix
In this appendix, Theorem 1 is proven. For brevity, only necessary details are
provided. Unreported results can be obtained from the corresponding author upon
request.
Proof of Theorem 1.
Consider (a). The least squares estimator of β can be written as
√
N b β =
 
1
NT2
N X
i=1
T X
t=1
x2
it
!−1
1
√
NT2
N X
i=1
T X
t=1
xityit. (A1)
Consider the denominator, whose limit as N, T → ∞ is given by
1
NT2
N X
i=1
T X
t=1
x2
it = λ2
xi
1
T2
T X
t=1
f2
xt + δ2
xi
1
NT2
N X
i=1
T X
t=1
e2
xit
+ 2λxiδxi
1
NT2
N X
i=1
T X
t=1
fxtexit + Op(T−1)
→p λ2
x
Z
B2
x

+ δ2
xE
Z
B2
ix

+ 2λxδxE
Z
BxBix

= λ2
x
Z
B2
x

+
1
2
δ2
xΩe
xx, (A2)
where the last equality follows from the independence of Bx and Bix, and the
moments of Brownian motion.
Similarly, for the numerator, we have
1
√
NT2
N X
i=1
T X
t=1
xityit =
√
Nλxiλyi
1
T2
T X
t=1
fxtfyt + λxiδyi
1
√
NT2
N X
i=1
T X
t=1
fxteyit
+ λyiδxi
1
√
NT2
N X
i=1
T X
t=1
fytexit
+ δyiδxi
1
√
NT2
N X
i=1
T X
t=1
eyitexit + Op(T−1), (A3)
which has the following limit as T → ∞
1
√
NT2
N X
i=1
T X
t=1
xityit ⇒
√
N

λxiλyi
Z
BxBy

+
1
√
N
N X
i=1

λxiδyi
Z
BxByi + λyiδxi
Z
ByBxi + δyiδxi
Z
ByiBxi

=
√
N

λxiλyi
Z
BxBy

+
1
√
N
N X
i=1
Ui, say. (A4)
6Thus, passing N → ∞, it is clear that
1
NT2
N X
i=1
T X
t=1
xityit −

λxλy
Z
BxBy

=
1
N
N X
i=1
Ui →p
1
2
δxδyΩe
xy.
This establishes (a).
Next, consider (b). By using (a) and the Lindberg-L´ evy theorem, as N, T → ∞
1
√
NT2
N X
i=1
T X
t=1
xityit −
√
N

λxλy
Z
BxBy +
1
2
δxδyΩe
xy

⇒ N
 
0, lim
N→∞
1
N
N X
i=1
E(U2
i )
!
.
For the variance, we have
E(U2
i ) = E
 
λxδy
Z
BxByi + λyδx
Z
ByBxi + δyδx
Z
ByiBxi
2!
= λ2
xδ2
yE
Z
BxByi
2
+ λ2
yδ2
xE
Z
ByBxi
2
+ δ2
yδ2
xE
Z
ByiBxi
2
+ 2λyλxδyδxE
Z
BxByi
Z
BxiBy

=
1
6

λ2
xδ2
yΩf
xxΩe
yy + λ2
yδ2
xΩf
yyΩe
xx + δ2
yδ2
xΩe
yyΩe
xx

+
1
2
λyλxδyδxΩf
xyΩe
xy. (A5)
where the second equality follows from the fact that all cross-products except the
squared ones have expectation zero, while the second follows from the moments of
Brownian motion.
Putting everything together, we obtain
√
N b β −
√
N
1
Qxx

λxλy
Z
BxBy +
1
2
δxδyΩe
xy

⇒ N

0,
1
Q2
xx
Uxy

.
This completes the proof of (b). 
7

A Panic Attack on Unit Roots and Cointegration.

Panel Cointegration Testing in the Presence of Common Factors.

Purchasing Power Parity and the Theory of General Relativity: The First Tests.

Some Cautions on the use of Panel Methods for Integrated Series of Macroeconomic Data.

Spurious Regression and Residual-Based Tests for Cointegration in Panel Data.

Spurious Regression in Nonstationary Panels with Cross-Unit Cointegration

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Spurious Regression in Nonstationary Panels with Cross-Unit Cointegration

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