short-run dynamics and deterministic terms:
p-1
∆yit = Ψidt + αiβ0yi,t-1 + Γi,j ∆yi,t-j + εit ,
j=1
where dt is a vector of deterministic variables (such as a constant, trend or
dummy variables) and Ψi is a k × k matrix of coefficients.
If Ψi, Γi,1, . . . , Γi,p-1 are unrestricted matrices, they can be “partialled
out” from the likelihood function (cf. Johansen 1988). Let ∆yeit (yei,t-1)
denote the residual vectors from a least squares regression of ∆yit (yi,t-1)
on ∆yi,t-1 , . . . , ∆yi,t-p+1 and dt. The two-step estimator of the long-run
parameters is obtained from the regression
zei+t = Byei(,2t)-1 + veit i= 1,...,N; t= 1,...,T, (13)
where zei+t = yei(,1t)-1 - (αbi0Σbi-1αbi)-1αb0iΣbi-1∆yeit and vit is defined analogously. The
asymptotic distribution of the two-step estimator Bb2S resulting from (13) is
the same as in Theorem 1.
As in the usual time series case with N = 1 the asymptotic distri-
bution of the cointegration rank statistic are affected by the determinis-
tic terms. For example, if dt is a constant so that (8) includes a con-
stant, then the Brownian motions Wk-r(a) in Theorem 2 are replaced by
Wk-r(a) = Wk-r(a) — R01 Wk-r(a)da. If dt represents a polynomial in time,
then the asymptotic expressions can be derived by using the results of Ou-
liaris et al. (1989). Appendix B contains the respective values of μr and σ2
for a model with a constant and a linear trend.
An important problem of multi-country panel data sets is the apparent
contemporaneous correlation among the errors (e.g. O’Connell 1998, Wu
and Wu 2001). For panel unit root tests simulation techniques are applied
to control for such correlation among the errors. For the FM-OLS approach,
however, cross-section correlation imply more fundamental problems that
have not been resolved yet.2 Since the second step of the parametric ap-
proach is based on an ordinary least-square regression, it is straightforward
to account for possible contemporaneous correlation. First, one may use a
feasible GLS procedure to estimate the set of seemingly unrelated regression
2Phillips and Moon (1999, p. 1092) state that “... when there are strong correlations
in a cross section (as there will be in the face of global shocks) we can expect failures in
the strong laws and central limit theory arising from the nonergodicity.”
10