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 H_A : 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 = ^αi^β0^yi,t-1 + ^γi^β⊥^yi,t-1 ^{+ ε}it ^,                   (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
γ_i 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 α⁰_i,⊥ yields

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

where uit = α'_i,⊥∆y_ü, δi = α^,_i,⊥γ_i, wit = β⊥yit, and e_a = 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 δ_i = 0
for all i.

In practice the matrices α_i,⊥ 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).

<<   5   6   7   8   9   10   11   >>

More intriguing information