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 + γiβ⊥yi,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 , γiβ⊥0 ]
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 = β⊥yit, 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).