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



4 Inference

In practice, the number of cointegration relationships is often unknown. It is
therefore interesting to test hypotheses on the cointegration rank. Larsson,
Lyhagen and Lothgren (2001) suggest a “LR-bar” statistic that is based on
the standardized mean of the cross section LR statistics for the hypothesis
H0 : r = r0 against the alternative HA : r > r0 . This test statistic assumes
that the cointegration vectors are different across
i (heterogenous cointegra-
tion), whereas our framework assumes that the cointegration vectors are the
same for all cross section units (homogenous cointegration). To improve the
power of the test in the case
βi = β for all i, the homogeneity assumption
can be imposed.

Following Saikkonen (1999) a simple test procedure is constructed, where
the restriction of a homogeneous cointegration relationship can easily be
imposed. To nest the null and the alternative hypotheses we write

δyit = αiβ0yi,t-1 + γiβyi,t-1 + εit ,                   (7)

where γi is a k × (k - r) matrix with full column rank. Under the null hy-
pothesis it is assumed that
γi = 0 yielding (1), whereas under the alternative
γi is unrestricted so that the matrix

Πi = [αiβ0 , γiβ0 ]

has full rank for at least one i {1, . . . , N}. Pre-multiplying (7) with the
orthogonal complement
α0i, yields

uit = δiwi,t-1 + eit ,                                (8)

where uit = α'i,∆yü, δi = α,i,γi, wit = βyit, and ea = α,i,1εit. To test the
hypothesis
r = r0 the equation (8) is estimated by ordinary least-squares and
a LR, Wald or LM statistic can be constructed to test the hypothesis
δi = 0
for all
i.

In practice the matrices αi, and β are unknown and must be replaced
by consistent estimators. This can be done by computing orthogonal com-
plements of the estimates of
αi from the first step and the estimate of β from
the second step of the estimation procedure proposed in section 3. The fol-
lowing theorem states, that the limiting null distribution of the test statistic
is similar to the one derived by Lyhagen et al. (2001).



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