2 ML estimimation
For the ease of exposition we first consider a cointegrated VAR(1) model
with the VECM presentation
∆yit = αiβ0yi,t-1 + εit , t = 0, 1, . . . , T; i = 1, . . . , N , (1)
where εit is an k-dimensional white noise error vector with E(εit) = 0 and
positive definite covariance matrix Σi = E(εitε0it). Furthermore, it is assumed
that the number of time periods is the same for all cross section units (bal-
anced panel). Various extensions of this model will be considered in Section
5.
In this specification the cointegration vectors β are the same for all cross
section units, whereas the “loading matrix” αi is allowed to vary across i. A
similar setup is considered by Pesaran et al. (1999), Pedroni (1995, 2000)
and Phillips and Moon (1999). Assuming normally distributed errors, we can
concentrate the log-likelihood function with respect to the individual specific
parameters α1, . . . , αN and Σ1, . . . , ΣN yielding
rr
T
2 log I ∑i (β) I . (2)
i=1
where c0 is some constant and
Σbi(β)
εeit(β)
T
2-1 εeit (β)εeit (β)0
t=1
∆yit -
X
t=1
∆yityi0,t-1
β Xt=1
β0yi,t-1yi0,t-1β
-1
β0yi,t-1
The problem with this criterion function is that it cannot be maximized by
solving a simple eigenvalue problem. In the pure time series case with N = 1,
the maximization of Lc (β) is equivalent to maximizing IΣb I, which leads to
a simple eigenvalue problem. For N > 1, however, we have to maximize
the expression QiN=1 IΣbi(β)I, which cannot be solved by a simple eigenvalue
problem.
Nevertheless, it is possible to maximize Lc(β) in (2) by using numerical
techniques. It is well known that r2 restrictions are required to identify the
cointegration vectors. Following Johansen (1995, p. 72) the cointegration
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