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Using panel data, powerful tests for a unit root in the autoregressive representation of the series can be constructed. Following Dickey and Fuller (1979), the unit root hypothesis can be tested by performing the regression:
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(12.25) |
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(12.26) |
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
heteroscedasticity 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 modified 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
The quantlet computing all these unit root statistics is called
output = panunit(z,m,p,d {,T})The parameters necessary for computing the statistics are as follows. The parameter m indicates the number of variable in the data set which is 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. For example, assume that the second variable in a balanced data set with T=32 is to be tested including a constant and a lagged difference. Then the respective command is
output = panunit(z,2,1,1,32)The string output first gives an output table of a pooled Dickey-Fuller regression. In a second table, the four unit root statistics are presented.