Palgrave Handbook of Econometrics: Applied Econometrics

648 Panel Methods to Test for Unit Roots and Cointegration

where:

σˆu^2 ,i=

1

T

∑T

t= 1

uˆ^2 i,t.

Under the null, using sequential convergence (that is, as before,T→∞followed
byN→∞), the re-centered and re-scaled Hadri statistic is given by:

ZLM=

√

N(HLM−ξ)

ζ

⇒N(0, 1).

The correction terms depend on the specification of the deterministic process and
are given by Hadri (2000). The extension to the case of serially correlated (but
stationary under the null) errors is easily dealt with by using an estimator of the
long-run variance ofui,t.
Hadri and Larsson (2005) allow for fixedTby deriving the finite sample mean

and variance ofκi,T=T^12

∑T

t= 1

Si^2 ,t

σˆu^2 ,i

, so that, by a simple application of central limit

theory, it follows that:

HT=

1

√

N

∑N

i= 1

(

κi,T−Eκi,T

var(κi,T)

)

⇒N(0, 1)asN→∞.

As before, the correction terms depend on the specification of the deterministic
component and are given by Hadri and Larsson (2005). Note also that the closed
form expressions for the correction factors derived in Hadri and Larsson are only
correct for the case of serially uncorrelated errors, which limits the usefulness of
the test to some extent.

13.2.1.5 A summary of simulation evidence (Hlouskova and Wagner, 2006)

We describe briefly here the results of a large-scale simulation study undertaken by
Hlouskova and Wagner (2006) on many of the tests described above. For further
details, we refer the reader to their paper, which provides a meticulous account
of the behavior of the tests for unit roots in panels that are cross-sectionally inde-
pendent as a function of numerous features of the DGP and estimation methods,
including the dimensions ofTandN, lag selection algorithms in the augmentation
of the Dickey–Fuller tests, the presence of moving average terms in the error pro-
cesses, and certain forms of cross-sectional correlation, given by either constant
correlation or geometrically declining correlations by assuming the correlation
matrix to be of Toeplitz form.
Hlouskova and Wagner conclude that the best performance in terms of power,
where the evidence is based on simulations, for the case where the model has
an intercept, is displayed by the LLC test or its modification proposed by Bre-
itung (2000). For short panels, the Harris and Tzavalis (1999) modification of
LLC offers some gains in panels where the errors are not serially correlated or the
amount of serial correlation is small. They also observe that, whenTis small