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where e_it is a mean-zero regression error term and a_it represents the
deterministic part of the model. i=1, ..., N indexes panels, and t=1,
..., T indexes time. a_it may include panel-specific intercepts (fixed
effects), a panel-specific time trend, or nothing, in which case y_it is
predicated to have mean zero for all panels.
All the tests except for the Hadri LM test investigate null hypotheses of
the general form Ho: rho_i = 1 versus Ha: rho_i < 1, though they differ
in precisely how Ha is specified. The Hadri LM test, rather than
assuming a unit root under the null hypothesis, assumes that the data are
stationary (rho_i < 1) versus the alternative that the data contain a
unit root.
Here we provide a brief overview of the salient features of each test;
see [XT] xtunitroot for additional information.
Remarks are presented under the following headings:
Levin-Lin-Chu test
Harris-Tzavalis test
Breitung test
Im-Pesaran-Shin test
Fisher-type tests
Hadri LM stationarity test
Levin-Lin-Chu test
The Levin-Lin-Chu (LLC) (2002) test assumes that all panels have the same
autoregressive parameter, i.e., that rho_i = rho for all i. Then the
alternative hypothesis is simply that rho < 1.
The LLC test requires that the panels be strongly balanced.
The LLC test is based on a regression t statistic, but because the data
are nonstationary under the null hypothesis, the asymptotic mean and
standard deviation of the t statistic depend on the specification of the
deterministic part of the model.
Levin, Lin, and Chu recommend using their procedure for moderate-sized
panels, with perhaps between 10 and 250 individuals and 25 to 250
observations per individual. If the time-series dimension of the panel
is very large, then standard unit-root tests can be applied to each
panel, because the gains from aggregation are likely to be small.
Formally, if there is no deterministic term in the model (a_it = 0), then
the test allows the number of time periods, T, to tend to infinity at a
slower rate than the number of cross-sectional units, N, though T must go
to infinity sufficiently fast that sqrt(N)/T tends to 0. If fixed
effects or time trends are included in the deterministic part of the
model, then T must tend to infinity at a rate faster than N so that N/T
tends to 0.
Harris-Tzavalis test
The Harris-Tzavalis (HT) (1999) test is similar to the LLC test in that
it assumes that all panels have the same autoregressive parameter so that
the alternative hypothesis is simply rho < 1. Differing from the LLC
test, the HT test assumes that the number of time periods, T, is fixed.
Like the LLC test, the HT test requires that the panels be strongly
balanced.
Baltagi (2008, 278) mentions that T being fixed is the typical case in
micropanel studies. Here you may have a panel dataset of firms, and it
may be more natural to think that if you could increase the sample size
of your dataset, you would do so by collecting data on more firms, though
the number of time periods available for each firm is fixed.
Breitung test
The LLC and HT tests are based on regression t statistics that are
subsequently adjusted to reflect the fact that under the null hypothesis,
the t statistics have a nonzero mean because of the inclusion of
panel-specific means or trends. The Breitung (2000) test takes a
different approach, transforming the data before computing the
regressions so that the standard t statistics can be used.
The Breitung test requires that the panels be strongly balanced.
When the robust option is specified, a version of the t statistic that is
robust to cross-sectional correlation of the error terms is reported.
This statistic has an asymptotically normal distribution when first T
tends to infinity followed by N tending to infinity.
The Breitung test assumes that all panels have a common autoregressive
parameter. The null hypothesis is that all series contain a unit root.
The alternative hypothesis is that rho < 1 so that the series are
stationary. Breitung and Das (2005) remark that the test also has power
in the heterogeneous case, where each panel is allowed to have its own
autoregressive parameter, though the test is optimal in the case where
all panels have the same autoregressive parameter.
The Breitung (2000) Monte Carlo simulations suggest that his test is
substantially more powerful than other panel unit-root tests for the
modest-size dataset he considered (N=20, T=30).
Im-Pesaran-Shin test
A major limitation of the LLC, HT, and Breitung tests is the assumption
that all panels have the same value of rho. The Im-Pesaran-Shin (IPS)
(2003) test relaxes the assumption of a common rho and instead allows
each panel to have its own rho_i. The null hypothesis is that all panels
have a unit root (Ho: rho_i = 0 for all i). The alternative hypothesis
is that the fraction of panels that are stationary is nonzero.
Specifically, if we let N_1 denote the number of stationary panels, then
the fraction N_1/N tends to a nonzero fraction as N tends to infinity.
This allows some (but not all) of the panels to possess unit roots under
the alternative hypothesis.
The IPS test does not require strongly balanced data, but there can be no
gaps in each individual time series.
When the errors are assumed to be serially uncorrelated, imposed by
either specifying the lags(0) option or not specifying the lag() option
at all, xtunitroot ips reports IPS's t-bar, t-tilde-bar, and
Z_t-tilde-bar statistics. These statistics assume that the number of
time periods, T, is fixed. When there are no gaps in the data,
xtunitroot ips reports exact critical values for the t-bar statistic that
are predicated on the number of panels, N, also being fixed. The other
two statistics assume N tends to infinity.
For the asymptotic normal distribution of Z_t-tilde-bar to hold, T must
be at least 5 if the dataset is strongly balanced and the deterministic
part of the model includes only panel-specific means, or at least 6 if
time trends are also included. If the data are not strongly balanced,
then T must be at least 9 for the asymptotic distribution to hold. If
these limits on T are not met, the p-value for Z_t-tilde-bar is not
reported.
When serial correlation in the error terms is accommodated by using the
lags() option with xtunitroot ips, then IPS's W_t-bar statistic is
reported. This statistic is asymptotically normally distributed when
first T tends to infinity followed by N tending to infinity.
Fisher-type tests
Fisher-type tests approach testing for panel-data unit roots from a
meta-analysis perspective. That is, these tests conduct unit-root tests
for each panel individually, and then combine the p-values from these
tests to produce an overall test. xtunitroot fisher supports ADF tests
with the dfuller option and Phillips-Perron tests with the pperron
option. Any options allowed by dfuller or pperron can be specified
(except the regress option has no effect).
xtunitroot fisher does not require strongly balanced data, and the
individual series can have gaps.
These tests assume that T tends to infinity. If the number of panels, N,
is fixed, then these tests are consistent against the alternative that at
least one panel is stationary. If we allow N to tend to infinity, then
the number of panels that do not have a unit root must grow at the same
rate as N for the tests to be consistent.
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