Convergence in TFP among Italian Regions - Panel Unit Roots with Heterogeneity and Cross Sectional Dependence

where V (θ) is the estimated variance of θ. In relationship to panel unit root
tests, the test developed by Levin, Lin and Chu (2002) imposes homogeneity
of no unit root under the alternative. This may be potentially restrictive and
subject to the heterogeneity bias mentioned above.⁷

The test developed by Im, Pesaran, and Shin (IPS, 2003) tries to overcome
this problem. IPS propose estimating individual-specific ADF tests and then
pooling the t-statistic of each test, into a t — statistic. They then compute
the exact critical values of this statistic and show that after transformation
by factors provided in the paper, the W[ɪ] statistic is distributed standard
normal under the null hypothesis of non-stationarity.

As in the LLC test, the test developed by IPS assumes that all series are
non-stationary under the null hypothesis, after allowing for individual effects,
time trends, and common time effects. As in LLC, lags of the dependent
variable may be introduced to allow for serial correlation in the errors. Unlike
the LLC test, however, the IPS test does not assume that all series are
stationary under the alternative, but is consistent under the alternative that
only a fraction of the series are stationary.

These two tests are the most common in the literature, but other equiva-
lent are available. Maddala and Wu (MW, 1999), for example, have proposed
a test known as Fisher’s test, which is similar to that of IPS, in that it com-
bines the p-values from N independent unit root tests.⁸ Hadri (2000), on the
other hand develops a Lagrange Multiplier (LM) test where the test statistic
is distributed as standard normal under the null of stationarity.

Sarno and Taylor (ST, 1998) provide a test particularly useful for the
purposes of our analysis. The Authors develop a multivariate augmented
Dickey-Fuller test that can be considered similar to that of LLC, because

⁷The Hausman test will provide information on the similarity of cross sectional esti-
mates. Therefore it will provide information on whether there is a similar speed of conver-
gence across Italian regions or whether we should pool regions on the basis of convergence
clubs.

⁸The major advantage of this test is that it does not require the panel to be balanced.
This property, however, is not required in our case.

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