A parametric approach to the estimation of cointegration vectors in panel data

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(εitε⁰_it). 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” α_i 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

r_r

T

2 log I ∑i (β) I .                        (2)

i=1

where c₀ is some constant and

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 L_c (β) is equivalent to maximizing IΣ^b I, which leads to
a simple eigenvalue problem. For N > 1, however, we have to maximize
the expression ^Q_i^N₌₁ IΣ^bi(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

<<   1   2   3   4   5   6   7   >>

More intriguing information