Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 705

set-up without deterministic components for illustration, these parameters are con-
tained in the matricesαiandi. Based on the new estimates ofαiandi, equation
(13.34) can be re-estimated. This can be repeated until convergence, according to
some numerical criterion, occurs.
Such an iterative procedure corresponds by and large to the iterative estima-
tor proposed in Larsson and Lyhagen (1999), with the only difference being that
Larsson and Lyhagen propose usingβˆ 2 = N^1
∑N
i= 1 βˆi,2as the initial estimator.
βˆi,2denotes as before the (not-normalized coordinates of the) individual specific
Johansen (1995) estimates. Let us note as a side remark that the set-up of Larsson
and Lyhagen (1999) is more general, since these authors consider, in the VAR(1)
case without deterministic components, the specification:

⎡

⎢⎢

⎣

Y1,t

..

.

YN,t

⎤

⎥⎥

⎦=

⎡

⎢⎢

⎣

α 11 ··· α 1 N

..

.

..

.

..

.

αN 1 ··· αNN

⎤

⎥⎥

⎦

⎡

⎢⎢

⎣

β

′

..

.

β

′

⎤

⎥⎥

⎦

⎡

⎢⎢

⎣

Y1,t− 1

..

.

YN,t− 1

⎤

⎥⎥

⎦+

⎡

⎢⎢

⎣

w1,t

..

.

wN.t

⎤

⎥⎥

⎦,

with a full covariance matrix of the stacked noise process and all necessary assump-
tions such that the joint process is anI(1) process. Thus, Larsson and Lyhagen
(1999) consider a VAR model for the stacked vector of variables, where only the
cointegrating space is restricted to be cross-sectionally identical and where no cross-
unit cointegration occurs (meaning, in the notation of Appendix B, that the B
matrix that collects all cointegrating relationships is block-diagonal).
It is not clear whether such an almost unrestricted VAR model for the stacked
process should really be interpreted as a panel model or as a time series model for
a high-dimensional process with some cross-equation restrictions. Pragmatically,
the loose parameterization implies that the time dimension of the panel has to be
very large compared to the cross-sectional dimension. Similar comments apply to
the work of Groen and Kleibergen (2003), who also consider a very general (panel)
VAR model and consequently present simulation evidence for a bivariate example
forN=1, 3 and 5 andT=1,000.
Breitung (2005) shows that his two-step estimator,β ̃ 2 say, is asymptotically
normally distributed in the sequential limit with firstT→∞followed byN→∞:

N^1 /^2 Tvec(β ̃ 2 −β 2 )⇒N(0,− 21 ⊗α),

where⊗denotes the Kronecker product,

 2 = lim

N→∞

lim

T→∞

E[

1

NT^2

∑N

i= 1

∑T

t= 1

Yi^2 ,t− 1 (Yi^2 ,t− 1 )

′

]

andα= lim
N→∞

1

N

∑N

i= 1

(

α

′

i

− 1

i αi

)− 1

, with these limits, in case of 2 non-singular,

implicitly assumed to exist by Breitung (2005).
Let us now turn to the test for cointegration that Breitung considers, which is
based on Saikkonen (1999). The main difference to the test of LLL discussed above is