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DISCUSSION PAPERS IN ECONOMICS DISCUSSION PAPERS IN ECONOMICS
Working Paper No. 99-09
Series-specific Tests for a Unit Root in a Panel Setting
with an Application to Real Exchange Rates
Janice Boucher Breuer
Department of Economics, The Darla Moore School of Business
University of South Carolina, Columbia, SC
Robert McNown
Department of Economics, University of Colorado at Boulder
Boulder, Colorado
Myles Wallace
Department of Economics, Sirrine Hall, Clemson University
Clemson, SC
June 1999
Center for Economic Analysis
Department of Economics
University of Colorado at Boulder
Boulder, Colorado 80309
© 1999 Janice Boucher Breuer, Robert McNown, Myles Wallace Abstract
A unit root testing procedure is presented that exploits the well-established power advantages of panel
estimation while rectifying a deficiency in other panel unit root tests. This procedure, which takes into
account contemporaneous cross-correlation and heterogeneous serial correlation of the regression residuals,
allows determination of which members of the panel reject the null hypothesis of a unit root and which do
not. Applying the procedure to real exchange rates yields results that are in broad agreement with those
obtained from single-equation unit root tests. There is little evidence that a unit root can be rejected in
dollar-based real exchange rates for the floating rate period. 1
I.
Introduction
Prior to 1996, single-equation augmented Dickey-Fuller tests for a unit root in real exchange rates
failed to produce much support for purchasing power parity among industrialized countries. The common
finding across such studies had been that real exchange rates contain a unit root. Post-1996 evidence offers
a different conclusion. A revival of empirical support for purchasing power parity has occurred over the
last three years with the application of panel unit root estimation procedures to testing for a unit root in real
exchange rates. Studies by Sarno and Taylor (1998), Taylor and Sarno (1998), Wu and Wu (1998),
Papell (1997), Oh (1996), Wu (1996), and others, using a cross-section of real exchange rates, present
evidence for the floating rate period that favors stationarity. Real exchange rates are found to be mean
reverting in these studies, and so purchasing power parity is corroborated. OConnell (1998), Papell
1
(1997) and Papell and Theodoridis (1998) provide dissenting evidence in their panel unit root tests..
The evidence of mean reversion is often bittersweet for proponents of purchasing power parity; the
half-life estimates for shocks to the real exchange rate range between two to four years, suggesting
considerable persistence in the departures from purchasing power parity. The conclusions from panel
studies should also be regarded as only weakly supportive of purchasing power parity since the extant
panel studies (including tests presented by Im, Pesaran, and Shin (1997), Maddala and Wu (1997), and
Taylor and Sarnos (1998) MADF test) are tests of a joint hypothesis that all real exchange rates in the
panel contain a unit root. When rejection of the joint null hypothesis materializes, it is possible that only
one member of the panel contributes to the finding thus making any proclamations favorable to purchasing
power parity less strong.
This paper presents a procedure that can be used to help identify the time series properties of the
individual series in a panel and thereby avoids the joint null hypothesis problem endemic to extant panel
unit root tests. The methodology used is panel estimation which takes account of contemporaneous cross- 2
correlations of the regression residuals. Also treated is heterogeneous serial correlation across panel
members. To set the stage for this procedure, the next section contains a brief review of some current panel
tests.
2

II.
A Brief Review of Some Panel Unit Root Tests
Panel unit root studies of real exchange rates come on the heels of the development of statistical
approaches to conducting unit root tests with cross-sectional and time series data (panel data) formally
introduced by Quah (1990), Breuting and Meyer (1991), and Levin and Lin (1992, 1993). However, the
proper genesis of panel unit root tests predates the works of Levin and Lin and Quah, and actually begins
with Abuaf and Jorion (1990) who introduced a panel unit root estimation procedure to test purchasing
power parity. Their motivation, as with subsequent panel unit root studies, has been a desire to increase
the power properties of single-equation unit root tests that are based on a limited time series dimension.
For a panel of N countries with T time series observations, Abuaf and Jorion used the panel estimation
procedure of Seemingly Unrelated Regressions (SUR) to estimate
q
i,t
=
"
i
+
Dq
q
i,t-1
+ u
i,t
t = 1, ... ,T
(1)
.
.
q
N,t
=
"
N
+
Dq
q
N,t-1
+u
N,t
t = 1, ... ,T
where q
i,t
is the real exchange rate for currency i at time t. The null hypothesis is
D
= 1 implying a unit root
for each and every member of the panel. An inability to reject the null hypothesis of a unit root in a panel
of real exchange rates is evidence against purchasing power parity since it implies no reversion of the real
exchange rates back to a (possibly non-zero, country-specific) mean. On the other hand, rejection of the
null hypothesis is taken as evidence favorable to purchasing power. Using this test, Abuaf and Jorion
(1990) report evidence favorable to purchasing power parity for the G-10 countries over the period 1973- 3
1987.
Levin and Lin (1992, 1993) offer several different specifications for testing the unit root hypothesis
with panel data. One of the more parsimonious of the Levin and Lin (1992) panel unit root specifications
is a renormalized version of equation (1) and is presented below.
)
q
1,t
=
"
1
+
$

q
q
1,t-1
+ u
1,t
t = 1,...,T
(2)
.
.
.
)
q
N,t
=
"
N
+
$

q
q
N,t-1
+ u
N,t
t = 1,...,T
The panel unit root test is now a test of
$
= 0 (i.e.
D
=1) where critical values in Levin and Lin
(1992) must be consulted. The critical values depend on the magnitudes of N and T and whether fixed
effects (different intercepts) are introduced. While lagged augmentation terms can be added to their
specification to correct for serial correlation of the error terms, no allowance is made for contemporaneous
cross-correlation of the errors.
Studies by Wu (1996) and Oh (1996) use variants of the Levin and Lin procedure to test for a unit
root in real exchange rates. While both studies find evidence of PPP for the recent float, these studies
impose a homogeneous lag structure across panel members and they do not account for contemporaneous
cross correlation of the error terms. In the case that the error terms are serially correlated or
contemporaneously cross-correlated, the Levin and Lin critical values used in these studies will not be
appropriate.
Papell (1997) adapts the Levin and Lin specification above to allow for heterogenous serial
correlation of the errors in testing for purchasing power parity.
3
Lags of
)
q
it
are introduced into the LL
specification of equation (2) as in an augmented Dickey-Fuller test and the lag structure is permitted to
differ (be heterogeneous) across panel members. Equation (2) thus becomes: 4
)
q
1,t
=
"
1
+
$

q
q
1,t-1
+
G
j=1
k1

*
1,j

q

)
q
1,t-j
+ u
1,t
t = 1,...,T; j = 1,...,k
1
.
(3)
.
)
q
N,t
=
"
N
+
$

q
q
N,t-1
+
G

j=1
kN

*
N,j

q

)
q
N,t-j
+ u
N,t
t = 1,...,T; j = 1,...,k
N
Using this modification to the panel unit root test, Papell finds evidence favorable to purchasing power
parity during the floating rate era for a panel of twenty countries with monthly and annual (but not
quarterly) data regardless of the choice of the price index used to construct real exchange rates.
OConnell (1998) adapts the LL specification, and simulates critical values based on equation (2) that
allow for contemporaneous cross-sectional dependence in the data. OConnell thus uses the panel
estimation of seemingly unrelated regressions (SUR) to estimate
$
. OConnell also incorporates lags of
)
q
i,t
to account for potential serial correlation but forces the lag structure across panel members to be
homogeneous so that the
*
s (and ks) are identical. He finds that once cross-sectional dependence is
accounted for, a unit root in real exchange rates cannot be rejected. He conducts tests for panels of up to
64 countries.
A serious problem remains with the adaptations of the Levin and Lin test discussed above. The
problem is that all panel members are forced to have identical orders of integration. Either all panel
members are integrated as specified by the null hypothesis, or all are stationary. This all-or-nothing
characteristic of the tests follows from the restriction that the autoregressive parameter is identical for all
members of the panel, which is inappropriate in applications to panels that may contain a mix of integrated
and stationary series. Breuer, McNown, and Wallace (1998) demonstrate with Monte Carlo simulations
how the Levin and Lin procedure performs in panels with mixed orders of integration. With as few as one
stationary member of the panel, the rejection rate rises above the nominal size of the test, and continues to
increase with the number of I(0) series in the panel. Although the null hypothesis that all series have a unit
root is correctly rejected, the alternative of all stationary is also false in these mixed p