698 Panel Methods to Test for Unit Roots and Cointegration
13.3.1.6 Single equation estimation of the cointegrating vector
In this sub-section we consider estimation of the cointegrating vectorβfor data
given by (13.18′) and (13.22′), that is, when abstracting from the presence of cross-
sectional dependence via factors and without structural breaks in the deterministic
component. This is a topic of further research, as is the study of systems estimators
of cointegrating vectors in a similar set-up (see section 13.3.2).
Note that in this case (that is, without cross-sectional dependence or breaks)
we have already imposed the assumption that the cointegrating vectors are cross-
sectionally identical, that is,βi=βfor alli=1,...,N. This restriction appears
reasonable (although it may be generalized) since in order to gain by resorting to
panel methods some of the coefficients should be considered identical for all cross-
section members. Given that cointegration is the prime focus, it appears natural to
assume identical cointegrating relationships and to allow for heterogeneity in the
other characteristics of the DGP.
It has to be noted, however, that, as distinct from the pure time series case, the
pooled OLS estimator ofβin (13.18′)also converges to a well-defined limit when
the cointegrating vectors are not cross-sectionally identical and, more interestingly,
converges to a well-defined limit even in the spurious regression case. This limit is
given by the so calledaverage long-run regression coefficient(for a detailed discussion
and the precise assumptions, see Theorems 4 and 5 of Phillips and Moon, 1999). As
in the time series case, the limiting distribution of the OLS estimator depends upon
so called second-order bias terms, which necessitates the use of modified estima-
tion methods to result in mean zero mixed normal limiting distributions, which are
required, for example, to perform valid inference. The literature has, similar to the
time series case, proposed two modifications of the OLS estimator to take account
of second-order bias. These are given by fully modified OLS (FM-OLS) estimation,
as proposed by Phillips and Hansen (1990), and D-OLS estimation, introduced in
Saikkonen (1991) (to which we have already referred in section 13.3.1.4). Further-
more, similar to the cointegration tests, estimation can be performed in a pooled
or group mean fashion.
For simplicity, we focus in the description of the estimation procedures on the
casem=2, that is, the case including only fixed effects and note that the other
two cases form– no deterministic components or both fixed effects and individual
specific linear trends – can be handled analogously. Ifm=1 the original observa-
tions are used as inputs in the procedure and ifm=3 the variables are individually
demeaned and detrended first. Lety ̄i=N^1
∑T
t= 1 yi,t,x ̄i=1
N∑T
t= 1 xi,tand denote
the cross-sectionally demeaned variables byy ̃i,t=yi,t−y ̄iandx ̃i,t=xi,t− ̄xi.^28
Fully Modified OLS
Estimation of the cointegrating vector in the panel context by using FM-OLS esti-
mation is discussed in Phillips and Moon (1999), Kao and Chiang (2000) and
Pedroni (2000). In a first step obtain estimatorsˆuv,i,ˆv,iand%ˆv,ifrom the
residuals(uˆi,t,vi′,t)′. The residuals can be obtained by either individual specific
OLS estimation or by using the least squares dummy variable (LSDV) estimator