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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:
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(12.9) |
A simple test for the unit root hypothesis is obtained by running the regression
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(12.10) |
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
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
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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
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.