Palgrave Handbook of Econometrics: Applied Econometrics

644 Panel Methods to Test for Unit Roots and Cointegration

ensure consistency of estimation by choosing the lag lengths in order to eliminate
serial correlation in the error terms; that is, to have, at least asymptotically, white-
noise processesνi,t. In practice this typically means that information criteria are
used to choose the lag lengths to ensure that the estimated residualsνˆi,tshow no
evidence of serial correlation.^11
In order to construct the LLC test, for chosen lag lengthspi, two auxiliary
regressions are initially estimated – the first consists of regressingyi,ton its lags
yi,t−k,k=1, 2,...,pi, and the deterministic terms; denote the residuals from this
regression by ̃ei,t. The second consists of regressingyi,t− 1 on the same set of regres-

sors, to yield residuals denoted byf ̃i,t− 1. Finally, ̃ei,tis regressed onf ̃i,t− 1 and the
regression standard error from this equation, denotedσˆν,i, is used to construct the

standardized residualseˆi,t=e ̃i,t/σˆv,iandfˆi,t− 1 =f ̃i,t− 1 /σˆv,i. This standardization is
needed to remove the effects of cross-sectional heterogeneity of the processesvi,t
in (13.10) on the limiting distributions.
The next step is to estimate the long-run variance ofyi,t. Recall that under the
null hypothesis,

yi,t=μi+δit+

∑pi

k= 1

φi,kyi,t−k+νi,t,i=1, 2,...,N;t=pi+2,...,T,

so that a direct estimate of the long-run variance of yi,t is given by:

σˆv^2 ,i( 1 −
∑pi
k= 1

φˆi^2 ,k)−^2.

In practice the following estimate is preferred in order to improve the size and
the power of the test in finite samples. Denoting byuˆi,t=yi,t−δˆmidmt, where
dmtdenotes the specification of the deterministic terms, the long-run variance

is estimated byσˆLR^2 =T−^1

∑T

t= 1 uˆ

2

i,t+

2

T

∑L

j= 1 w(j,L)

∑T
t=j+ 1 uˆi,tuˆi,t−j, with the
lag truncation parameter chosen by criteria given in Andrews (1991) or Newey
and West (1994). The weightsw(j,L)are, in most applications, given byw(j,L)=

1 −L+j 1 , referred to as the Bartlett kernel. For future reference, definesˆ^2 i=ˆσLR^2 /σˆv^2 ,i

andSˆN,T=N−^1

∑N
i= 1 sˆi.
The essential component of the LLC test statistic (under all three deterministic
specifications) is given by:

ρˆ=

∑N

i= 1

∑T

t=pi+ 2 ˆei,tfˆi,t− 1

∑N

i= 1

∑T

t=pi+ 2

fˆ^2

i,t− 1

,

computed from the pooled regression ofˆei,tonˆfi,t− 1. Thet-form of this statistic,
to test the null hypothesisH 0 :ρ=0, is given by standardizingρˆby the standard
deviation ofρˆ, denotedSTD(ρ)ˆ, from the pooled regression.
For the case with no deterministic terms,tρ= 0 ⇒N(0, 1)asT→∞followed
byN→σ.^12 However, when either a constant or trend (or both) is present in the
model, thet-statistic diverges (due to the presence of the so-called Nickell bias; see
Nickell, 1978) to minus infinity (even under the null) and consequently needs to