1. INTRODUCTION
The use of cointegration techniques to test for the presence of
long-run relationships among integrated variables has enjoyed growing
popularity in the empirical literature. Unfortunately, a common dilemma
for practitioners has been the inherently low power of many of these
tests when applied to time series available for the length of the
postwar period. Research by Shiller and Perron (1985), Perron (1989,
1991), and recently Pierce and Snell (1995) has generally confirmed that it is the span of
the data, rather than the frequency, that matters for the power of these
tests. On the other hand, expanding the time horizon to include prewar data
can risk introducing unwanted changes in regime for the data relationships.
In light of these data limitations, it is natural to question whether a
practical alternative might not be to bring additional data to bear
upon a particular cointegration hypothesis by drawing upon data from
among similar cross-sectional data in lieu of additional time
periods.
For many important hypotheses to which cointegration methods have
been applied, data are in fact commonly available on a time series
basis for multiple countries, for example, and practitioners could
stand to benefit significantly if there existed a straightforward
manner in which to perform cointegration tests for pooled time series
panels. Many areas of research come to mind, such as the growth and
convergence literature and the purchasing power parity (PPP)
literature, for which it is natural to think about long-run properties
of data that are expected to hold for groups of countries.
Alternatively, time series panels are also increasingly available for
industry level data and stock market data. For applications where the
cross-sectional dimension grows reasonably large, existing systems
methods such as the Johansen (1988, 1991) procedure are likely to become infeasible,
and panel methods may be more appropriate.
On the other hand, pooling time series has traditionally involved a
substantial degree of sacrifice in terms of the permissible
heterogeneity of the individual time series.1
See, for example, Holz-Eakon, Newey, and Rosen (1988) on the dynamic homogeneity restrictions
required typically for the implementation of panel vector
autoregression (VAR) techniques.
To ensure broad applicability
of any panel cointegration test, it will be important to allow for as
much heterogeneity as possible among the individual members of the
panel. Therefore, one objective of this paper will be to construct
panel cointegration test statistics that allow one to vary the degree
of permissible heterogeneity among the members of the panel, and in the
most general case pool only the multivariate unit root information,
leaving the form of the time series dynamics and the potential
cointegrating vectors entirely heterogeneous across individual
members.
Initial work on nonstationary panels was done in the context of
testing for unit roots. For example Quah (1994) derives asymptotically normal distributions
for standard unit root tests in panels for which the time series and
cross-sectional dimensions grow large at the same rate. Levin, Lin, and
Chu (2002) extend this work for the case in
which both dimensions grow large independently and derive asymptotic
distributions for panel unit root tests that allow for heterogeneous
intercepts and trends across individual members. Im, Pesaran, and Shin
(2003) develop a panel unit root estimator
based on a group mean approach. Since the original versions of this
paper, many other works have extended the literature on nonstationary
panels, and we refer readers to recent surveys by Banerjee (1999), Phillips and Moon (2000), and Baltagi and Kao (2000).
In earlier working paper versions of this work, Pedroni (1995, 1997a, 2001a), we examined the properties of spurious
regressions and residual-based tests for the null of no cointegration
for both homogeneous and heterogeneous panels and studied special
conditions under which tests for the null of no cointegration with
homogeneous slope coefficients are asymptotically equivalent to raw
panel unit root tests. In the interest of space, in this version we
focus on the most general of these results, namely, the tests for the
null of no cointegration for panels with heterogeneous dynamics and
heterogeneous slope coefficients. In particular, we study both between
dimension and within dimension residual-based test statistics. Each of
these tests is able to accommodate individual specific short-run
dynamics, individual specific fixed effects and deterministic trends,
as well as individual specific slope coefficients. We derive limiting
distributions for these under the null and show that each is standard
normal and free of nuisance parameters. We also provide Monte Carlo
evidence to document and compare their small sample performance and
illustrate their use in testing weak PPP for a panel of
post–Bretton Woods exchange rate data.
The remainder of the paper is organized as follows. Section 2
presents the underlying theory and asymptotic results for each of the
test statistics. Section 3 studies the small sample properties of these
tests under a variety of different scenarios for the error processes,
and Section 4 demonstrates a brief empirical application of the tests
to the hypothesis of PPP for a panel of post–Bretton Woods
exchange rate data. Section 5 ends with a few concluding remarks. The
derivations for each of the results in Section 2 are collected in the
Appendix.
2. ASYMPTOTIC PROPERTIES
In its most general form, we will consider the following type of
regression:
for a time series panel of observables
yit and Xit
for members i = 1,…,N over time periods
t = 1,…,T, where
Xit is an m-dimensional column
vector for each member i and βi is an
m-dimensional row vector for each member i. The
variables yit and
Xit are assumed to be integrated of order
one, denoted I(1), for each member i of the panel,
and under the null of no cointegration the residual
eit will also be I(1). The
parameters αi and δi
allow for the possibility of member specific fixed effects and
deterministic trends, respectively. The slope coefficients
βi are also permitted to vary by individual, so
that in general the cointegrating vectors may be heterogeneous across
members of the panel.
For regressions of the form given in (1), we will be interested in
studying the properties of tests for the null hypothesis
H0: “all of the individuals of the panel are
not cointegrated.” For the alternative hypothesis, it is worth
noting that if the underlying data generating process (DGP) is assumed
to require that all individuals of the panel be either uniformly
cointegrated or uniformly not cointegrated, then the natural
interpretation for the alternative hypothesis is simply
H1: “all of the individuals are
cointegrated.” On the other hand, if the underlying DGP is
assumed to permit individual members of the panel to differ in whether
or not they are cointegrated, then the natural interpretation for the
alternative hypothesis should be H1: “a
significant portion of the individuals are cointegrated.”
In earlier versions of this work we also studied the properties of
tests for the null of no cointegration in panels for which the slope
coefficients βi are constrained to be
homogeneous across all individuals. For such panels in which the
estimated slope coefficients are constrained to be homogeneous, we
showed that under the special case of strict exogeneity for the
regressors, the distribution for residual-based tests of the null of no
cointegration is asymptotically equivalent to the distribution for raw
panel unit root tests despite the fact that the residuals are
estimated.2
For the case in which the regressors are
endogenous, the asymptotic equivalence result no longer holds for
panels with homogeneous slope coefficients, and it is necessary to
adjust for the asymptotic bias induced by the estimated regressor
effect. In fact, the work in Kao (
1999) has
since examined the properties of such a test for the null of no
cointegration that adjusts for the bias term due to the estimated
regressors effect under endogeneity for the special case in which both
the slope estimates and the short-run dynamics are constrained to be
homogeneous across members of the panel. The difficulty with this
approach arises when we attempt to interpret the resulting tests for
the null of no cointegration in the case when the true DGP is such that
the slope coefficients are not common across individual members of the
panel. Specifically, in this case if we impose a common slope
coefficient despite the fact that the true slopes are heterogeneous,
then the estimated residuals for any member of the panel whose slope
differs from the average long-run regression correlation will be
nonstationary, even if in truth they are cointegrated. In many
situations the true slope coefficients are likely to vary across
individuals of the panel, and the implications for constraining the
coefficients to be common are unlikely to be acceptable for tests of
the null of no cointegration.
For this reason, we present here a set of residual-based test
statistics for the null of no cointegration that do not pool the slope
coefficients of the regression and thus do not constrain the estimated
slope coefficients to be the same across members of the panel. These
statistics will be applicable as tests for the null of no cointegration
in the general case in which the regressors are fully endogenous and
the slope coefficients are permitted to vary across individual members
of the panel. Since both the dynamics and the cointegrating vector
itself are permitted to vary across individual members of the panel,
one can think of the test as effectively pooling only the information
regarding the possible existence of the cointegrating relationship as
indicated by the stationarity properties of the estimated residuals.
To study the distributional properties of such tests, we will
describe the DGP in terms of the partitioned vector
zit′ ≡
(yit,Xit′)
such that the true process zit is
generated as zit =
zit−1 + ξit,
for ξit′ ≡
(ξity,ξitX′).
We then assume that for each member i the following condition
holds with regard to the time series dimension.
Assumption 1.1 (Invariance Principle). The process
ξit′ ≡
(ξity,ξitx′)
satisfies
,
for each
member i as T → ∞, where ⇒ signifies
weak convergence and
Bi(Ωi) is vector
Brownian motion with asymptotic covariance Ωi
such that the m × m lower diagonal block
Ω22i > 0 and where the
Bi(Ωi) are taken
to be defined on the same probability space for all i.
Assumption 1.1 states that the standard functional central limit
theorem is assumed to hold individually for each member series as
T grows large. The conditions on the error process required
for this convergence are relatively weak and include the class of all
stationary autoregressive moving average (ARMA) processes.
3 See standard references, for example,
Phillips (1986, 1987), Phillips and Durlauf (1986), and Phillips and Solo (1992), for further discussion of the conditions
under which Assumption 1.1 holds more generally.
The
(
m + 1) × (
m + 1) asymptotic covariance matrix
is given by
and can be decomposed as
Ωi ≡
Ωio +
Γi + Γi′, where
Ωio and
Γi represent the contemporaneous and dynamic
covariances, respectively, of ξit for a given
member i. The matrix is partitioned to conform with the
dimensions of the vector ξit′ ≡
(ξity,ξitx′)
so that the Ω22i element is an m
× m-dimensional matrix. The off-diagonal terms
Ω2li capture the feedback between the
regressors and the dependent variable, and in keeping with the
cointegration literature we do not require that the regressors
xit be exogenous. The fact that
Ωi is permitted to vary across individual
sections of the panel reflects the fact that we will permit all
dynamics that are absorbed in the asymptotic covariance matrix to be
heterogeneous. Finally, by requiring that Ω22i
> 0, we are ruling out cases where the regressors are themselves
cointegrated with one another in the event that we have multiple
regressors.
In addition to the conditions for the invariance principle with
regard to the time series dimension, we will also assume the following
condition in keeping with a basic panel data approach.
Assumption 1.2 (Cross-Sectional Independence). The individual
processes are assumed to be independent and identically distributed (i.i.d).
cross-sectionally, so that E
[ξitξjs′]
= 0 for all s,t,i ≠ j. More
generally, the asymptotic long-run variance matrix for a panel of size
N × T is block diagonal positive definite with
the ith diagonal block given by the asymptotic covariances for
member i, such that
diag(Ω1,…,ΩN) . The
process ξit is taken to be generated by a
linear process ξit =
Ci(L)ηit,
where Ωi =
Ci(1)Ci(1)′
and where the white noise innovations ηit and
the random coefficients Ci(L) are
independent of one another and each i.i.d. over both the i and
t dimensions.
This condition will allow us to apply standard central limit theorems
in the cross-sectional dimension in the presence of heterogeneous
errors in a relatively straightforward fashion. As in Phillips and Moon
(1999), we take the coefficients
Ci(L) as being drawn from a
distribution that is i.i.d. over the i dimension and
independent from the ηit innovations. In the
empirical illustration of Section 4 we also discuss some possibilities
for dealing with the case in which such independence is violated in
practice. Finally, note that the condition that
Ωi > 0 ensures that there is no
cointegrating relationship between yit and
the Xit, as will be the case under the
null hypothesis that we consider throughout this study.
Together, Assumptions 1.1 and 1.2 will provide us with the basic
conditions for investigating the asymptotic properties of the various
statistics as the dimensions T and N grow large. The
first assumption will allow us to make use of standard asymptotic
convergence results over the time series dimension for each of the
individual members. In particular, we will make use of the fact that
the following convergencies, developed in Phillips and Durlauf (1986) and Park and Phillips (1988), must also hold for each of the individual
members i = 1,…,N as T grows large,
so that
where Zi(r) ≡
(Vi(r),Wi(r)′)′
is vector Brownian motion, such that
Vi(r) and
Wi(r) are independent standard
Wiener processes for all i, where
Wi(r) is itself an m
× 1 dimensioned vector, Γi is as
previously defined, and Li is a lower
triangular decomposition of Ωi such that
The convergence results in equations (2) and (3) hold under standard
assumptions regarding the initialization of
zio, and for convenience we will take
these to be common across the panel such that
zio = 0 for all i.
The second key assumption, Assumption 1.2, will then allow us to
apply simple averaging arguments over the cross-sectional sums of the
corresponding Brownian motion functionals that are used to construct
the panel statistics. In particular, we will make use here of
sequential limit arguments to investigate the properties of the panel
statistics as the time series and cross-sectional dimensions grow
large. Specifically, this means that in computing the limiting
properties for the panel statistics, we will first take the limit as
T grows large, followed sequentially by the limit as
N grows large. Thus, for the typical double sum statistic
involved in constructing the panel
statistics, we write
where
.
Let Ri
be the limit of the standardized sum of
SiT as T → ∞. For the
sequential limit, we first compute Ri as
T → ∞ and then compute the limit of the
standardized sum of
as N → ∞. The
sequential limit is a convenient method for computing the limit
distribution for a double index process. However, it is not the most
general method. Phillips and Moon (1999)
formalize the notion of sequential limits for nonstationary double
index processes and also compare it to the more general method of joint
limits. In contrast to the sequential limit, the joint limit allows
both indexes to pass to infinity concurrently, rather than in sequence.
As Phillips and Moon (1999) point out, this
implies that the joint limit distribution characterizes the limit
distribution for any monotonic expansion rate of T relative to
N. Phillips and Moon (1999) also
provide a specific set of conditions that are required for sequential
convergence to imply joint convergence. Following Phillips and Moon, we
will denote sequential limits as (T,N →
∞)seq.
In our context, the sequential limit substantially simplifies the
derivation of the limit distribution for two important reasons. One
reason is that it allows us to control the effect of nuisance
parameters associated with the serial correlation properties of the
data in the first step as T → ∞ by virtue of the
standard multivariate invariance principle. This substantially
simplifies the computation of the limit as N → ∞
for the panel statistics in the second step because it implies that we
can typically characterize the heterogeneity of the standardized sum of
random variables
in terms of a single nuisance parameter
associated with the conditional long-run variance of the differenced
data, L11i2. Another reason the
sequential limit simplifies the derivation is that applying the limit
as T → ∞ in the first step allows one to focus only
on the first-order terms of the limit in the time series dimension,
since the higher order terms are eliminated prior to averaging over the
N dimension. This second feature is particularly convenient
for the purposes of computing the limit for the panel. However, this
feature can also be deceptive in its simplicity because it hides the
need to control the relative expansion rate of the two dimensions as is
often the case for the more general joint limit. In practical terms,
the relative expansion rate can also be an important indicator for the
small sample properties of the statistics for different dimensions of
N and T. In Section 3, we illustrate this in terms of
a series of Monte Carlo experiments that examine the size properties of
the statistics as N and T grow large along various
diagonal paths characterized by different monotonic rates of expansion
of T relative to N.
In particular, we consider two classes of statistics. The first class
of statistics is based on pooling the residuals of the regression along
the within dimension of the panel, whereas the second class of
statistics is based on pooling the residuals of the regression along
the between dimension of the panel. The basic approach in both cases is
to first estimate the hypothesized cointegrating relationship
separately for each member of the panel and then pool the resulting
residuals when constructing the panel tests for the null of no
cointegration. Specifically, in the first step, one can estimate the
proposed cointegrating regression for each individual member of the
panel in the form of (1), including idiosyncratic intercepts or trends
as the particular model warrants, to obtain the corresponding residuals
.
In the second step, the
way in which the estimated residuals are pooled will differ among the
various statistics, which are defined as follows.
DEFINITION 1 (Panel and Group Mean Cointegration Statistics for
Heterogeneous Panels). Let
where
is estimated from a model
based on the regression in (1). Then we can define the following test
statistics for the null of no cointegration in heterogeneous panels:
where
for some choice of lag window
where
such that
is a
consistent estimator of Ωi.
The first three statistics are based on pooling the data across the
within dimension of the panel, which implies that the test statistics
are constructed by summing the numerator and denominator terms
separately for the analogous conventional time series statistics.
4 In earlier versions of this study, we
presented these panel statistics in a form in which each of the
component statistics of the numerator and denominator was weighted by
the member specific long-run conditional variances
L11i2, whereas here we present
versions of the statistics that are not weighted by
L11i2. The distinction between
weighted and unweighted statistics is a fairly common occurrence in
panels, and the limit distribution is the same for both types. However,
in Monte Carlo simulations, we found that the unweighted statistics
consistently outperformed the weighted statistics in terms of the small
sample size properties. Consequently, we present here only the
unweighted statistics. We are thankful to an anonymous referee for
suggesting the unweighted form of the statistics.
Thus, for
example, the
“panel-rho” statistic is analogous to the semiparametric
“rho” statistic studied in Phillips and Perron (1988) and Phillips and Ouliaris (1990) for the conventional time series case, and
the panel statistics can be constructed by taking the ratio of the sum
of the numerators and the sum of the denominators of the analogous
conventional time series “rho” statistic across the
individual members of the panel. Likewise, the
“panel-t” statistic, and the
“panel variance ratio” statistics are analogous to the
semiparametric t-statistic and the long-run variance ratio
statistic, each of which was also studied in Phillips and Ouliaris
(1990) for the conventional time series
case.
The next two statistics are constructed by pooling the data along the
between dimension of the panel. In practice this implies that the
statistics can be constructed by first computing the ratio
corresponding to the conventional time series statistic and then
computing the standardized sum of the entire ratio over the N
dimension of the panel. Consequently, these statistics in effect
compute the group mean of the individual conventional time series
statistics. We have presented here two group mean statistics,
,
that are analogous to the rho-statistic and t-statistic
studied in Phillips and Ouliaris (1990) for
the conventional time series case. In principle it is also possible to
construct a group mean variance ratio statistic analogous to the one
presented for the pooled panel cointegration statistics in Definition
1. We also experimented with such a statistic and found it to be
dominated by the other two in terms of the small sample size
properties. Consequently we present here only the semiparametric
group-rho and group-t statistics.
In the interest of space and simplicity, we have only presented here
the forms of the statistics that correspond to the nonparametric
treatment of the nuisance parameters. However, it should be apparent
that the nuisance parameters can also be treated parametrically for
both the panel and group mean statistics, in which case the same limit
distributions presented in the following proposition still apply. We
refer readers to earlier versions of this work for a discussion of the
parametric treatment of these in the form of panel and group mean
augmented Dickey–Fuller (ADF) statistics.
5 For example, the parametric analogue to the
ZtNT-statistic would
take the form of the standard ADF correction. For more details on the
parametric ADF version of the panel and group mean statistics, we refer
readers to an earlier version of the paper, Pedroni (1997a), which examines the small sample properties
of the parametric ADF version of the statistics, and to Pedroni (1999), which discusses in more detail the
construction of the ADF.
We discuss in more detail the
estimation of the various nuisance parameters
in Section 3 in conjunction with the small sample properties of the
statistics.
In the proposition that follows, we present the limiting
distributions for these test statistics under the null. In particular,
for the following proposition, we posit
,
respectively, to be finite means and
covariances of the appropriate vector Brownian motion functional. As
the following proposition indicates, when the statistics are
standardized by the appropriate values for N and T,
then the asymptotic distributions will depend only on known parameters
given by
.
PROPOSITION 1 (Asymptotic Distributions of Residual-Based Tests for
the Null of No Cointegration in Heterogeneous Panels). Let
Θ,Ψ signify the mean and covariance for the vector Brownian
motion functional
,
where
,
j = 1,2,3 refers to the
j × j upper submatrix of ψ. Similarly,
let
signify the mean and variance for
the vector Brownian motion functional
.
Then under the null of no cointegration the asymptotic distributions of
the statistics presented in Definition 1 are given by
as (T,N → ∞)seq,
where the values for φ(j) are given
as φ(1)′ =
−Θ1−2,
φ(2)′ =
(−Θ2Θ1−2,Θ1−1),
and φ(3)′ =
(−½Θ2Θ1−3/2(1
+
Θ3)−1/2,Θ1−1/2(1
+ Θ3)−1/2,
−½Θ2Θ1−1/2(1
+ Θ3)−3/2).
These results are fairly general and give the nuisance parameter free
asymptotic distributions simply in terms of the corresponding moments
of the underlying Brownian motion functionals, which can be computed by
Monte Carlo simulation, much as is done for the conventional single
equation tests for the null of no cointegration. Note that we require
only the assumption of finite second moments here provided that we
apply sequential limit arguments such that first T →
∞ so that this produces sums of i.i.d. random variables
characterized as Brownian motion functionals to which standard
Lindeberg–Levy central limit arguments can be applied for large
N.
The result applies in general for any of the models associated with
regression (1) and for any number of regressors when the slope
coefficients are estimated separately for each member of the panel. On
the other hand, the specific values for the moments
depend on the
particular form of the model, such as whether heterogeneous intercepts
or trends have been included in the estimation, and on the number of
integrated regressors, m. Accordingly, Table 1 gives large finite sample moments for the
leading bivariate cases of interest based on the simulated Brownian
motion functionals of Proposition 1 so that we can evaluate the
corresponding formulas under these conditions.
Large finite sample moments
Let
signify the
means and covariances for the vector Brownian motion functionals
defined in Proposition 1. Then the approximations shown in Table 1 are obtained on the basis of Monte Carlo
simulations for 100,000 draws from pairs of independent random walks
with T = 1,000, N = 1, where cases 1, 2, and 3 refer,
respectively, to the same functionals constructed from standard Wiener
processes, demeaned Wiener processes, and demeaned and detrended Wiener
processes, respectively. We then use these simulations to approximate
the asymptotic distributions for the panel cointegration statistics as
N grows large on the basis of Proposition 1. The results are
summarized in the following corollary.
COROLLARY 1 (Empirical Distributions). Let
so that based on Proposition 1,
for each
of the k = 1,…,5 statistics of χ. Based on the
empirical moments given in Table 1
for large T, the following approximations obtain as N → ∞
under the null of no cointegration:
where cases 1, 2, and 3 refer, respectively, to statistics
constructed from estimated residuals
, from the standard case, the
case with estimated fixed effects,
, and the case with estimated
fixed effects and estimated trends
.
The usage for these statistics is the same as for the single series
case. For the panel-v statistics, large positive values indicate
rejections, whereas for the panel-rho statistics and panel-t
statistics, large negative values indicate rejection of the null. The computed
moments in Table 1 and the corresponding
distributions in Corollary 1 are for the leading case in which a single
regressor is included. For the moments and empirical distributions
corresponding to cases with various numbers of additional regressors, we refer
readers to Pedroni (1999), which reports these for
cases ranging from m = 2 through m = 7 regressors.
The bias correction terms given by μ in the corollary are required
to ensure that the distribution does not diverge as the N
dimension grows large. The need for these stems from the fact that
functionals for the underlying Weiner processes have nonzero means,
which must be accommodated when averaging over the N dimension
to ensure convergence. In comparing the distributions for the panel-rho
and panel-t statistics to the ones applicable for raw panel unit
root tests reported in Levin et al. (2002), we
see that the consequence of using estimated residuals is to affect not only
the asymptotic variance but also the rate at which the mean of the unadjusted
pooled statistics diverge asymptotically. In these cases, ignoring the
consequences of the estimated regressors problem for the asymptotic bias in
panels would lead the raw panel unit root statistic to become divergent when
applied to estimated residuals.
3. MONTE CARLO EXPERIMENTS
In this section we study some of the small sample properties of the
statistics for variously dimensioned panels. We also study the
empirical properties of the statistics as the sample dimensions grow
large at different relative expansion rates. In particular, to study
the small sample size properties we will employ the following DGP under
the null hypothesis.
Data Generating Process 1.1. Let zit =
(yit,xit)′,
t = 1,…,T, i = 1,…,N
be generated by
where xit is a scalar series so that
m = 1.
This DGP is particularly convenient because it allows us to easily
model and observe the consequences of heterogeneity in the dynamics for
Δzit = ξit
under the null hypothesis in terms of the long-run covariance matrix
Ωi for ξit.
Specifically, by drawing ξit from a vector
moving average process and setting θ12i =
θ22i = 0 while allowing
θ11i and θ21i to vary
across the individual members of the panel we obtain a fairly simple
mapping between these coefficients and the long-run covariance matrix
Ωi, while at the same time permitting
substantial heterogeneity in the key features of the dynamics. In
particular, since Ωi = (I +
θi)E
[ηitηit′](I
+ θi)′, this implies that when
θ12i = θ22i = 0, we
can characterize the conditional long-run variance of the spurious
regression in terms of the simple ratio
L11i2 = (1 +
θ21i2)−1(1 +
θ11i)2. Thus, by varying the values
for θ11i and θ21i we
control the values for Ωi and
L11i2. For example, for the
special case in which the DGP is i.i.d., so that
θ11i and θ21i are
both zero, then Ωi = I and
L11i2 = 1. The two extremes
then occur as either one of the parameters
θ11i or θ21i
approaches its upper bound. For example, when
θ21i is at its minimum value of zero and
θ11i is at its maximum value of 0.5, then
L11i2 = 2.25. At the other
extreme, when θ11i is at its minimum value of
zero and θ21i is at its maximum value of 0.5,
then L11i2 = 0.80.
To implement each of the test statistics described in Section 2, we
must obtain the various nuisance parameter estimates
as presented in Definition 1. Thus, to obtain these, we first estimate
by ordinary least squares (OLS)
separately for each member of the panel, and then we estimate
using the
Newey–West kernel estimator, which allows us to construct
.
6 Note that it is also possible to construct these estimates
by imposing the null value ρi = 1 and
estimating the nuisance parameters directly from
.
However, Phillips
and Ouliaris (1990) recommend against this,
and we follow the same recommendation here by first estimating
in order to estimate the
nuisance parameters.
In setting the lag length for the band
width of the kernel estimator, we followed the recommendation from
Newey and West (
1994) and set the lag
truncation as a function of the sample length to the nearest integer
given by
K = 4(
T/100)
2/9. The
parameter
L11i2 is based on the
triangularization of the long-run covariance matrix for the vector
error process ξ
it =
zit −
zit−1. Analogous to the way in which
we obtain estimates for the first three nuisance parameters, we first
estimate ξ
it by OLS separately for each member
of the panel based on the autoregression of
zit,
7 The
same issue discussed in the previous note applies here in that we could
also impose a unit root on the zit data
and estimate the nuisance parameters directly from
ξit = Δzit.
Instead, however, we have followed the recommendation in Phillips and
Ouliaris (1990) here by first estimating
.
and then we estimate
the long-run variance for these estimated residuals using the
Newey–West kernel estimator using the same rule for determining
the lag truncation for the band width.
In our analysis of the small sample properties of the statistics, we
are interested in particular in comparing the behavior of the
statistics for different dimensions of the panel. We are also
interested in observing the empirical consequences for the statistics
as the cross-section and time series dimensions grow large at different
relative rates. To illustrate these features, we focus here on a few
key empirical properties, which we illustrate graphically. For more
extensive Monte Carlo results presented in tabular form for the various
other statistics, including the ADF versions of the
t-statistics, we refer the reader to an earlier working paper
version of this study, Pedroni (1997a).8
The DGP used in Pedroni (1997a) was based on various parameterizations of
the one proposed by Haug (1996) for the
conventional time series case. We have simplified the DGP in this
version, which enables us to more easily characterize the long-run
variance of ξit in terms of the
DGP.
In the first two figures, we study the nominal sizes of the three
different constructions of the rho-statistics as the dimensions of the
panel vary. In Figure 1, we depict the
empirical sizes for the nominal 5% tests as the T dimension
grows large for a fixed value of N. Specifically, in this case
we set N = 20 and allowed T to vary in increments of
10, with 10,000 independent draws from the DGP 1.1 described
earlier.9
In all figures, the curves
representing the behavior of each of the statistics have been smoothed
slightly by means of a moving average of neighboring points.
The figure shows that for this DGP, the two
t-statistics
converge from above, whereas the other statistics converge from below.
In other words, at very small values for
T, the two
t-statistics are somewhat oversized in the range of 10% for
T = 40, whereas the other statistics are somewhat undersized.
By the time the
T dimension reaches around
T = 150,
all of the statistics have converged to a range of around 4% and 7.5%
when the
N dimension is fixed at
N = 20.
Figure 2 examines the reverse case when
T
is fixed and
N increases, which we varied by increments of 1
in our simulation. In this case, the figure shows that when
T
is fixed at
T = 250, the panel-
v and panel-rho
statistics converge from above, whereas the others are already close to
nominal size when
N is small. Specifically, at
N = 10
the nominal sizes range from around 4.5% to 8.5%. At
N = 50
they range from a little over 3% to 6%. Notice that since one index is
held fixed in both of the first two figures, there is no anticipation
that the series will fully converge to the nominal size even as the
other index becomes very large. For example, if
N is fixed at
N = 20, then no matter how large
T becomes, the
statistics are likely to retain some minimal size distortion.
Empirical size as T varies for 5% tests, N =
20.
Empirical size as N varies for 5% tests, T =
250.
In the next set of figures, we study the empirical size properties as
both the N and T dimensions grow large at various
relative rates of expansion. These experiments are particularly
interesting, because they tell us something about the behavior of the
statistics for different rates of expansion, which the sequential limit
analysis that we used to obtain the limit distributions of the previous
section does not address. In the following diagrams we depict results
for the empirical sizes of the nominal 5% test using only a single
statistic in each diagram, but for different relative rates of
expansion. In particular, we illustrate this for two different
statistics, the panel-rho statistic in Figure
3 and the group-rho statistic in Figure
4. Thus, the figures illustrate what happens for the
empirical size of the particular statistic as the two dimensions grow
large at different relative rates ranging from N =
T1/2 to N =
T5/6. Specifically, in Figures 3 and 4 the horizontal
axis reports the value for T, and the various curves then
report the empirical sizes for the statistic when N is given
by N = T1/2, N =
T2/3, N =
T3/4, and N =
T5/6, respectively. In each case, we allowed
T to increase by increments of 10 and then assigned N
a value indicated by the corresponding expansion rate rounded to the
nearest integer and constructed the statistics based on 10,000
independent draws from DGP 1.1.
Empirical size along various expansion paths for 5% panel-rho
test.
Empirical size along various expansion paths for 5% group-rho
test.
From Figure 3 we can see that among these
expansion rates, convergence appears to occur most quickly for the
panel-rho statistic when N = T3/4 and
appears to occur most slowly when N =
T1/2. In each case, convergence is from above.
Figure 4 depicts the same experiments done
for the group-rho statistic. Interestingly, the group-rho statistic
appears to exhibit a hump-shaped feature for each of these rates of
expansion, in that the size first rises before falling and appears to
peak at around T = 100. Again, among these expansion rates,
convergence appears to be slowest for the case when N =
T1/2. However, for the group-rho statistic, the
case when N = T5/6 appears to do best
and also exhibits the least of the hump-shaped feature. In comparison
to the panel-rho statistic, the empirical sizes for the group-rho
statistic appear to be much closer to nominal size for very short
panels along any of the expansion paths. For the case when N =
T5/6, the empirical size begins at its lowest
point at around 4.8% when T = 60 and peaks at its highest
point of around 5.8% when T = 100.
We also experimented with rates of expansion with powers of
T equal to and in excess of 1.0. In these cases the T
dimension grows faster than the N dimension, and, as
anticipated, the empirical sizes do not converge to nominal size along
these expansion paths. This result is consistent with the fact that for
more general joint convergence results, it is often necessary to impose
the condition that the ratio N/T → 0 to
eliminate bias terms that otherwise explode when
T/N → 0. However, it is also interesting to
note that in these cases we found that both of the statistics remain
undersized relative to nominal size and eventually go to zero as the
sample dimensions go to infinity, with the speed increasing as the
exponent, a, for N = Ta
increases. The fact that the statistics become undersized in these
cases is reassuring in that it tells us that in practice the tests
simply tend to become overly conservative in finite samples in which
the N dimension exceeds the T dimension. Finally, as
a separate issue, it is also worth noting that for more extreme cases
in which large negative moving average components are present,
modifications in the form of those studied by Ng and Perron (1997) for the conventional single cointegration
equation context may also be helpful in further reducing small sample
size distortions in the panel context.
Next, we study the power properties of the statistics against various
alternative hypotheses. To simulate data under the alternative, we use
the following DGP.
Data Generating Process 1.2. Let
yit,xit,t
= 1,…,T,i = 1,…,N be generated
by
where xit is a scalar series so that
m = 1 and where we vary the value for φ across
experiments.
Note that we have imposed the alternative hypothesis in DGP 1.2 by
ensuring that the residuals eit are
stationary. Furthermore, in this case, rather than using a moving
average process for the errors, instead we use an autoregressive
process. The reason for this is because the power of the tests is
primarily sensitive to the autoregressive coefficient φ of the
residuals eit. In conventional time series
tests, the small sample power tends to be weak against alternatives
that imply near unit root behavior for the residuals, and we are
interested in knowing the extent to which the small sample power
improves against such near unit root alternatives in the case of the
panel tests. Consequently, we examine the empirical power of the 5%
tests against near unit root alternatives for the residuals, ranging
from φ = 0.9 to φ = 0.99.
Again, we are interested in studying this for different combinations
of the panel dimensions, N and T. As a general rule,
we find that the power rises most rapidly as the N dimension
increases. Accordingly, in the first set of figures, we depict the
power of each of the various tests as the T dimension
increases for a given value of N when the autoregressive
parameter for the regression residuals is 0.9 and 0.95. When we later
consider the extreme case such that the autoregressive parameter for
the regression residuals is 0.99, we depict both the case when
T increases for fixed N and the case when N
increases for fixed T. Thus, in Figure
5 we depict the raw power of the 5% nominal tests for the null
of no cointegration against the alternative hypothesis that the members
of the panel are cointegrated when the AR(1) coefficient for the
regression residuals is φ = 0.9. Specifically, for Figure 5 we set N = 20 and allowed
T to vary in increments of 5, with 10,000 independent draws
from the DGP 1.2 described previously for the case when φ = 0.9.
The results show that in this case the empirical power for all of the
tests rises rapidly as T increases, with the group-rho test
reaching 100% power at the slowest rate, by the time T = 70,
and all of the other tests achieving near 100% power by the time
T reaches around T = 50. In Figure
6, we examine the empirical power properties for the case in
which the alternative is closer to the null, such that φ = 0.95.
In this case, we allowed T to increase by increments of 10,
with N fixed at N = 20. Each of the statistics shows
the same relative patterns as in Figure 5,
except that they now require larger values for T to achieve a
given level of power. The panel-v test achieves 100%
power the quickest, at around T = 90. The group-rho test is
again the slowest to obtain power as the T dimension
increases, but it achieves nearly 100% power by the time T =
130. When comparing the raw power of the statistics for very small
values of T, we should keep in mind however that according to
Figures 1 and 2,
the group-rho statistic is also the most conservative in terms of
empirical size so that the difference in power is likely to be less
extreme for size adjusted power. Of course as the dimensionality of the
panel increases and the size distortion decreases, this plays less of a
role.
Empirical power as T varies for 5% tests, rho = 0.9,
N = 20.
Empirical power as T varies for 5% tests, rho = 0.95,
N = 20.
Finally, we are interested in considering the power properties of the
statistics for the extreme case in which the alternative is extremely
close to the null and the regression residuals exhibit near unit root
properties with φ = 0.99. In such cases, conventional time series
tests for the null of no cointegration have very little power even in
relatively sizable samples. In Figure 7, we
see that even in panels, when N = 20 we still require a fairly
long time dimension before the tests achieve high power. The panel-v
test reaches 100% power most quickly, at around T = 350, whereas the
panel-rho is the next quickest to reach 100% power at around T = 500.
However, Figure 8 illustrates how it is
possible to achieve near 100% empirical power even in the extreme case
when φ = 0.99 by considering increases in the N dimension
in lieu of the T dimension. Specifically, for Figure 8 we set T = 250 and varied
N by increments of 1. In this case, we can see that the
panel-v statistic reaches nearly 100% power already by the time
N = 45, and the two other panel statistics exceed 90% power by the
time N = 100, and the two group statistics exceed 90% power by the
time N = 120. These results are potentially very promising for
empirical research. In terms of monthly data, they imply that with little
more than 20 years of data it may be possible to distinguish even the most
extreme cases from the null of no cointegration when the data are
pooled across members of panels with these dimensions.
Empirical power as T varies for 5% tests, rho = 0.99,
N = 20.
Empirical power as N varies for 5% tests, rho = 0.99,
N = 250.
Taken together, the Monte Carlo results from this section can also be
helpful in deciding among the best uses for the various statistics
presented in the previous section. For example, in very small panels,
if the group-rho statistic rejects the null of no cointegration, one
can be relatively confident of the conclusion because it is slightly
undersized and empirically the most conservative of the tests. On the
other hand, if the panel is fairly large so that size distortion is
less of an issue, then the panel-v statistic tends to
have the best power relative to the other statistics and can be most
useful when the alternative is potentially very close to the null. The
other statistics tend to lie somewhere in between these two extremes,
and they tend to have minor comparative advantages over different
ranges of the sample size. Finally, it is worth noting that the
simulations here have been conducted for the case in which
heterogeneous intercepts are estimated. An important avenue of further
research will be to consider the small sample properties of the test
statistics in the presence of member specific heterogeneous trends,
which are likely to affect the power and size.
4. AN EMPIRICAL APPLICATION TO THE
PURCHASING POWER PARITY HYPOTHESIS
The PPP hypothesis has long been popular as an initial area of
investigation for new nonstationary time series techniques, and in
keeping with this tradition we illustrate here a fairly simple example
of the application of the statistics proposed in this paper to a
version of the hypothesis known as weak long-run PPP. This version of
the PPP hypothesis posits that although nominal exchange rates and
aggregate price ratios may move together over long periods, there are
reasons to think that in practice the movements may not be directly
proportional, leading to cointegrating slopes different from 1.0. For
example, the presence of such factors as international transportation
costs, measurement errors, differences in price indices, and
differential productivity shocks has been used to explain why under the
weak version of PPP the cointegrating slope may differ from unity.10
See earlier versions of this study, Pedroni
(1995, 1997a), for
a more detailed discussion of the PPP application of this section. In
separate work, Pedroni (1996, 2000), a panel fully modified ordinary least
squares (FMOLS) method for testing hypotheses regarding cointegrating
vectors in such panels is developed and subsequently applied in Pedroni
(2001b) to test the strong version of PPP for
a similar data set, which is strongly rejected.
Because these
factors do not generally indicate a specific value for the
cointegrating slope, under this version of the theory, the
cointegrating slopes must be estimated, and a test of the weak form of
PPP is interpreted as a cointegration test among the nominal variables.
Furthermore, because factors leading to a nonunit value for the
cointegrating slope coefficient can be expected to differ in magnitude
for different countries, it is important to allow the slope
coefficients to vary β
i to vary by individual
country. Thus, the empirical specification becomes
where sit is the log nominal bilateral
U.S. dollar exchange rate at time t for country i and
pit is the log price level differential
between country i and the United States at time t,
and a rejection of the null of no cointegration in this equation is
taken as evidence in favor of the weak PPP hypothesis.
Table 2 reports both the conventional
individual country results for a test of the null of no cointegration
and the results of the panel and group mean statistics for the null of
no cointegration. We employ both monthly and annual IFS data on nominal
exchange rates and CPI deflators for the post–Bretton Woods
period from June 1973 to December 1994 for between 20 and 25 countries
depending on availability and reliability of the data. Results for both
annual data, T = 20, and monthly data, T = 246, are
reported side by side for each statistic, with the results for the
monthly data reported in italics. For the semiparametric tests we have
used the Newey and West (1994) recommendation
for truncating the lag length for the kernel bandwidth. We also report
the individual, panel, and group mean parametric ADF for comparison,
where we have used a standard step down procedure, starting from
K = 12 for the monthly data and K = 2 for the annual
data. Because this results in a different truncation for each country,
we report these in the last column. For the panel and group mean
statistics we report results both for the raw data and for data that
have been demeaned with respect to common time effects to accommodate
some forms of cross-sectional dependency, so that in place of
sit,pit we
use
where
.
Individual, panel, and group tests for weak purchasing power
parity
A few results are worth noting in particular. First, the point
estimates for the slopes and intercepts appear to vary greatly among
different countries, and second, as expected, the number of rejections
based on the individual country tests is relatively low, so that on
this basis alone the evidence does not appear to favor even weak PPP.
By comparison, we see that for the annual data the panel-rho statistic
and the two ADF statistics reject the null for the standard case,
whereas all but the group rho reject the null for the case when the
time means are subtracted. For the monthly data, each of the panel
statistics rejects the null for the standard case, and the group
statistics also reject the null for the case when the time means are
subtracted. Thus, in contrast to the individual time series tests, both
the panel and group statistics appear to provide fairly strong support
in favor of the likelihood that weak PPP holds for at least a
significant portion of countries in the post–Bretton Woods
period.
An important caveat worth noting is that not all forms of
cross-sectional dependency are necessarily accommodated by simple
common time effects. This approach assumes that the disturbances for
each member of the panel can be decomposed into common disturbances
that are shared among all members of the panel and independent
idiosyncratic disturbances that are specific to each member.11
We should note that the estimation of
common time effects potentially further complicates the analysis of
limiting distributions because the number of parameters to be estimated
for the time effects grows with the time dimension, T. We
report these estimates for our empirical illustration here simply for
the case of comparison.
For many cases this may be
appropriate, as, for example, when common business cycle shocks impact
the data for all countries of the panel together. In other cases,
additional cross-sectional dependencies may exist in the form of
relatively persistent dynamic feedback effects that run from one
country to another and that are not common across countries, in which
case common time effects will not account for all of the dependency. A
general solution to the issue of cross-sectional dependency is beyond
the scope of this paper. However, if the time series dimension is long
enough relative to the cross-sectional dimension, then one practical
solution in such cases may be to employ a generalized least squares
(GLS) approach based on the estimation of the panel-wide asymptotic
covariance for the weighting matrix. This approach is examined in
Pedroni (
1997b), which finds that empirically
cross-sectional dependency does not appear to play a large role in
panel-based exchange rate tests. Another promising approach based on a
bootstrap for cases in which the time series dimension is not as long
is examined in Chang (
2000) for panel unit
root tests. Most recently, papers by Bai and Ng (
2002), Moon and Perron (
2003), and Phillips and Sul (
2003) have studied the possibility of using various
factor model approaches to modeling the cross-sectional dependence for
the purpose of panel unit root tests and could presumably also be
adapted for the case of residual-based tests for the null of no
cointegration in this context. These promising approaches are also able
to accommodate more general forms of long-run dependency that may exist
among members.
5. CONCLUDING REMARKS
We have studied in this paper properties of residual-based tests for
the null of no cointegration for panels in which the estimated slope
coefficients are permitted to vary across individual members of the
panel. These statistics allow for heterogeneous fixed effects and
deterministic trends and also for heterogeneous short-run dynamics. The
sequential limiting distributions under the null are shown to be normal
and free of nuisance parameters. We have also studied the small sample
behavior of the proposed statistics under a variety of different
scenarios in a series of Monte Carlo experiments, and we have showed
how these statistics could be applied in an empirical application to
the PPP hypothesis. Finally, we note that the study is intended as an
initial investigation into the properties of such statistics and that
in so doing it raises many important additional issues of both a
practical and a technical nature that we hope will be of interest for
future research on the theory and application of nonstationary panel
data techniques.
APPENDIX
Proof of Proposition 1. Let
where now
.
By
expanding
R1,iT,R2,iT
in terms of the convergencies given in (2) and (3), it can be shown
that as T → ∞
where Q is defined in terms of V,W as in
the statement of the proposition. (See the appendix in Pedroni (1997a) for more details regarding this
calculation.)
Similarly, we can evaluate
as follows. First,
note that
.
Under the null,
is
Op(T−1) so that
the second term will be eliminated asymptotically as T →
∞ and will not impact the sequential limit distribution. For
convenience in notation, we drop these terms and write
as
where
Because ξit is strictly stationary we know
that
Using the convergencies from (2) and (3) and
Ωi =
Li′Li,
following some algebra, we see that as T → ∞,
Thus, letting
gives
as T →
∞.
Now, let RiT =
(R1iT,R2iT,R3iT)′
and note that the first three statistics of the proposition can be
written as different combinations of the standardized sums of these
elements over the N dimension. Specifically,
where the individual elements of
RiT,i = 1,…,N are
i.i.d. over the i dimension. Next, define the mean of the
values for L11i2 averaged over
the i dimension to be E
[L11i2] = π. It
should be apparent that
as (T,N →
∞)seq because
as T →
∞ and
as N
→ ∞ in the second stage limit. Next, to determine the
limiting distribution of each of the panel statistics as
(T,N → ∞)seq, we use
the delta method, which provides the limiting distribution for
continuously differential transformations of i.i.d. vector sequences.
Toward this end, we first expand each of the statistics as follows:
where the elements of Θ correspond to the means of the vector
functional
as defined in the proposition. Thus, because we take the coefficients
generating ξit =
Ci(L)ηit
to be i.i.d. over the i dimension and independent of the
innovations, we have E
[RiT] =
π(Θ1,Θ2,1 +
Θ3)′ as T → ∞ for any
i.
Now, for the next stage of the sequential limit, as N →
∞, the summations in parentheses converge to the means of the
respective random variables by virtue of a law of large numbers. This
leaves the expressions involving each of the standardized square
bracketed terms as a continuously differentiable transformation of a
sum of i.i.d. random variables. In general, for a continuously
differential transformation ZN of an
i.i.d. vector sequence Xi, with vector
mean
and covariance Σ, the delta method tells us
that
as N → ∞, where the jth element of the
vector α is given by the partial derivative
.
Thus, in terms of our notation for the moments of
RiT, we set
for each of the
statistics, which produces the limiting distributions stated in the
proposition as (T,N →
∞)seq.
The cases with demeaned or demeaned and detrended data can be
obtained in similar fashion by defining
and the elements of RiT conformably in
terms of the demeaned or demeaned and detrended data. In this case, the
elements of the vector
become defined analogously in terms of demeaned Brownian motion,
V*,W*, or demeaned and detrended Brownian motion,
V**,W**, and the derivation in terms of the
corresponding moments proceeds accordingly.
To establish the limiting distribution for the two group mean
statistics, let
Using similar notation it should be apparent then that
as T → ∞. Next, expand each of the statistics as
which converge to
,
respectively, as
N → ∞ by the same type of arguments. █
Proof of Corollary 1. Expanding the terms for the variances in
Proposition 1 gives
where ζ = Θ1−1(1 +
Θ3)−1ψ22 +
¼Θ22Θ1−3(1
+ Θ3)−1ψ11 +
¼Θ22Θ1−1(1
+ Θ3)−3ψ33 −
Θ2Θ1−2(1 +
Θ3)−1ψ12 −
Θ2Θ1−1(1 +
Θ3)−2ψ23 +
½Θ22Θ1−2(1
+ Θ3)−2ψ13.
Substituting the empirical moments for large T,N = 1
into these expressions gives the reported approximations for the
asymptotic distributions as N → ∞. The results for
the group mean statistics follow immediately upon substituting the
empirical moments for large T,N = 1 into the
expressions of Proposition 1. █
