Palgrave Handbook of Econometrics: Applied Econometrics

Anindya Banerjee and Martin Wagner 713

structure of the second moments, that is, the covariance, respectively correla-
tion, structure. Obviously, this focus stems from the fact that when using theI(1)
framework we are dealing with cumulated weakly stationary processes which are
characterized by their second moment structure. Focusing on the second moment
structure is enough for studying the structure of dependencies although, in general,
it will not be enough for the statistical analysis, since, for example, the cross-
section members might be uncorrelated but not independent. However, for our
discussion here we use dependence synonymously with correlation.
Considering the non-stationary panel (for any given cross-sectional dimension
N)as a high-dimensional time series, we will first define cross-unit cointegration
(following Wagner and Hlouskova, 2007), which will then be used to define short-
run and long-run dependence.
Let us start our discussion with panels of univariate time series, noting here that
the concept of cross-unit cointegration becomes more interesting for panels of
multivariate time series. Take without loss of generality in the panel of time series
the firstN 1 series to be (jointly) stationary and the remainingN−N 1 series to be
I(1). With the firstN 1 series stationary, a set oftrivialcointegrating relationships
emerges foryt, with a basis given by[IN 1 ,0]′. Cross-unit cointegration is present
if, over and above these, cointegrating relationships that are not contained in the
above trivial cointegrating space are present.
This concept can be formalized by denoting asB(a basis of) the cointegrating
space of the vectoryt. Then, we define thecross-unit cointegrating spaceas the pro-
jection ofBon the orthogonal-complement of the trivial cointegrating space (that

is,[IN 1 ,0]′,) given byBCU=

[

00

0 IN−N 1

]

B, which in our currently simple frame-

work of panels of univariate series amounts to nothing but the cointegrating space
of theN−N 1 integrated series contained in the panel, assumed to be ordered last
in the discussion. The dimension of this space is called the cross-unit cointegrating
rank.
Let us now turn to panels of multivariate time series and consider a panel com-
prised ofNm-dimensional vectors of time series that are assumed to be jointlyI(1),

respectively stationary, that is,Yt=(Y

′

1,t,...,Y

′

N,t)

′∈RNm. (^32) Next denote with
βi∈Rm×kitheki-dimensional cointegrating space ofYi,t. As before we denote the
cointegrating space of the stacked process asB∈R
Nm×
∑
i
ki

. We stack the individual
specific cointegrating spaces in:

β-=

⎡

⎢⎢

⎢

⎢⎢

⎣

β 1 0 ··· 0

0 β 2

..

.

..

.

..

.

..

.

..

. 0
0 ··· 0 βN

⎤

⎥⎥

⎥

⎥⎥

⎦

∈R

Nm×

∑

i

ki

.

By definition it holds thatsp{B}⊇sp{β-}. The cross-unit cointegrating space is
defined as the projection ofBon the orthogonal-complement ofβ-, that is, it is