12.5 Unit Root Tests for Panel Data
- output =
panunit(z, m, p, d{, T})
- computes various panel unit root statistics for the m-th variable in
the data set z with p lagged and different deterministic term
indicated by d.
|
In the previous sections we implicitely assumed that real exchange rates
are difference stationary variables and the real interest rates
are stationary in levels.
This assumption is also made by MacDonald and Nagayasu (1999),
for example. In the recent literature of dynamic panel data tests have
been suggested to test such hypotheses. Following
Dickey and Fuller (1979), the unit root hypothesis can be
tested by performing the regression:
 |
(12.9) |
and testing
for all
. The test procedure of
Breitung and Meyer (1994) assumes that
and
for
. Thus, as in traditional panel data models,
heterogeneity is represented solely by an individual specific intercept.
Under this simplifying assumptions a pooled autoregression can be
estimated and the lag order can be chosen with respect to the highest
significant lag. Therefore, we propose to select the lag order of the
model by testing the last lag in the autoregressive specification. This
procedure is equivalent to select the lag order by using the highest
significant partial autocorrelation. Similarly, the deterministic
terms can be selected.
A simple test for the unit root hypothesis is obtained by running the
regression
 |
(12.10) |
The
-statistic for
is asymptotically
standard normal as
. Since this procedure is also
valid for small
(e.g.
), this test called BM in
the panunit output is recommended if only a small number of
time periods is available.
Levin and Lin (1993) extend the test procedure to individual
specific time trends and short run dynamics. At the first stage
the individual specific parameters are ``partialled out'' by
forming the residuals
and
from a regression
of
and
on the deterministics and the
lagged differences. To account for heteroskedasticity the
residuals are adjusted for their standard deviations yielding
and
. The final regression is of
the form
If there are no deterministics in the first-stage regressions, the
resulting
-statistic for the hypothesis
is
asymptotically standard normally distributed as
and
. However, if there is a constant or a time trend in
the model, then second-stage
-statistic tends to infinity as
, even if the null hypothesis is true.
Levin and Lin (1993) suggest a correction of the
-statistic to remove the bias and to obtain an asymptotic
standard normal distribution for the test statistic. Monte Carlo
simulations indicate that the test may perform poorly for
and, thus,
the test should only be used for
or
(see
Im, Pesaran, and Shin (1997) and Breitung (1999)).
Another way to deal with the bias problem of the
-statistic is
to adopt a different adjustment for the constant and the time
trend. The resulting test statistics are called the unbiased
Levin-Lin statistic. In the model with a constant term only, the
constant can be removed by using
instead of
. The first stage regression only uses the lagged
differences as regressors. At the second stage, the regression is
and the resulting
-statistic for
is asymptotically
standard normal as
and
. If there is a
linear trend in the model the nuisance parameters are removed by
estimating the current trend value by using past values of the
process only. Accordingly, the series are adjusted according to
Note that
is an estimate of the drift parameter.
Again, the resulting modification yields a
-statistic with a
standard normal limiting distribution Breitung (1999).
Im, Pesaran, and Shin (1997) further extended the test procedure
by allowing for different values of
under the
alternative. Accordingly, all parameters were estimated separately
for the cross-section units. Let
denote the
individual
-statistic for the hypothesis
. As
and
we have
where
and
is the expectation and standard
deviation of the statistic
. Im, Pesaran, and Shin (1997)
present the mean and variances for a wide range of
and
.
The quantlet
panunit
uses estimated values for
and
that are obtained from regressions on
,
,
and
and
.
Generally, the quantlet computing all these unit root statistics
is called as follows:
output = panunit(z, m, p, d{, T})
The parameters necessary for computing the statistics are as
follows. The parameter m indicates the column number of the
variable to be tested for a unit root. The parameter p
indicates the number of lagged differences in the model. The
parameter d indicates the kind of deterministics used in the
regressions. A value of d=0 implies that there is no
deterministic term in the model. If d=1, a constant term is
included and for d=2 a linear time trend is included.
Finally, if a balanced panel data set is used, the common time
period T is given.
In our application, we test for unit roots in the interest rate
differential. The unit root tests for the long-term interest
spread (second variable) including a constant and a single lagged
difference are obtained using the command
panunit(z, 2, 1, 1)
The results can be found in the output table:
[ 1,] "====================================================="
[ 2,] "Pooled Dickey-Fuller Regression: 2'th variable "
[ 3,] "====================================================="
[ 4,] "PARAMETERS Estimate robust SE t-value"
[ 5,] "====================================================="
[ 6,] "Lag[1]= -0.2696 0.0296 -9.117"
[ 7,] "Delta[ 1]= -0.0863 0.0551 -1.566"
[ 8,] "const= 0.1109 0.1137 0.976"
[ 9,] "====================================================="
[10,] "N*T= 378 N= 16 With constant "
[11,] "Unit root statistics: "
[12,] "STATISTIC Value crit. Value (5%) mean variance"
[13,] "====================================================="
[14,] "B/M (1994) -2.453 -1.65 0.000 1.000"
[15,] "L/L (1993) -3.563 -1.65 -0.560 0.856"
[16,] "mod. L/L -3.711 -1.65 0.000 1.000"
[17,] "I/P/S (1997) -5.313 -1.65 -1.493 0.756"
[18,] "====================================================="
All four unit root tests clearly reject the hypotheses of a unit
root in the long-term interest spread. Similar result are obtained for the
short-term interest differential (not reported). These results are in line
with macroeconomic theory on the international term structure.