Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 647

non-asymptotic values ofT)is therefore restricted to balanced panels and equal lag
lengths in all units of the panel. For large values ofT, the use of simulated critical
values (specific to the nuisance parameters) can be avoided and the asymptotic
critical values can be used instead.
Similar principles apply to the use of theIPSLM test under the additional
restriction that limN/T=c>0, as firstTand thenN→∞.

13.2.1.3 Fisher tests – Maddala and Wu (1999) and Choi (2001)

Continuing to focus on cross-sectional independence, Maddala and Wu (1999) and
Choi (2001) developed tests based on combiningp-values, an idea due to Fisher
(1932). Fisher started his considerations by observing that thep-values of a con-
tinuous test statistic are uniformly distributed over the unit interval. Combining
this with the fact that minus two times the log of a uniform distribution over the

unit interval is distributed as aχ(^22 )random variable, panel unit root tests can easily
be constructed for cross-sectionally independent panels. If theNunits are indepen-
dent, the sum of thep-values,− 2
∑N
i= 1 logpi∼χ

2
( 2 N), under the null hypothesis of
a unit root in each unit of the panel. As long asp-values are computable for a test
for a unit root on the individual units, either asymptotic, or via means of simula-
tions or response surfaces (see, for example, MacKinnon, 1994, and MacKinnon,
Haug and Michaelis, 1995, for augmented Dickey–Fuller tests), the Fisher test can
be used to test for unit roots in panels. The key assumption remains that of cross-
sectional independence, although its relaxation is possible in some cases (to allow
for certain special forms of short-run dependence as described in section 13.2.2.1)
by means of bootstrap techniques.

13.2.1.4 Tests with stationarity as the null hypothesis

Tests have also been developed which take as the null that of (heterogeneous)
stationary roots against the alternative of a unit root in all cross-section members.
Among this class of tests are those due to Hadri (2000) and Hadri and Larsson
(2005), which apply the idea developed in Kwiatkowskiet al.(1992) to the panel
framework.
Looking at the specification where a linear trend is present under the null hypoth-
esis, the LM-tests are based on looking at the partial sums of the residuals of the
regressions (estimated for each unit):

yi,t=μi+γit+ui,t, (13.11)

where, under the null hypothesis, and only for the sake of illustration,ui,tmay be
taken to be a serially uncorrelated stationary process.
Denoting byuˆi,tthe estimated residuals in (13.11), and their partial sums by

Si,t=

∑t

j= 1 uˆi,j, the Hadri statistic, denoted byHLM, is given by:

HLM=

1

NT^2

∑N

i= 1

∑T

t= 1

S^2 i,t

σˆu^2 ,i

,