where
= E ( β>Xt^i
+ ≠½ τ 'δ.
τ '∆xt^ι
ʌ / β'χt-1 + ,ψ'τ±τ±δ¾-1
T ∖ T '∆xt-1
From the expression α11 = a±a'±ζp± and a±2 = a±(a±ζp±)± we can therefore find
the asymptotic variances of the estimators for these from those of a and ζ using the
δ- method, see Paruolo (2002).
Lemma 1 Let the asymptotic variance of a,ζ be Φ 0 Ω. The asymptotic variance of
a±2 is given by
al2Ωax2 0 [(Ip - θζ')a; θ]⅛[(Ip - θζ')a; θ]', (17)
where θ = a±a,±ζp±(p,±ζ'a a' fp ) 1 p' .
For any vectors u,v the asymptotic variance of u'a±1v is given by
(u'a(Ir; 0)Φ(Ir; 0)'a'u)(n'a±1Ωa±1n) (18)
+(v'ρ'±(ζ'a, -Ip)⅛(ζ'a, -Ip)'ρ±v)(u'a±a±Ωa±a±«)
+2(v,p,±(ζ'a, -Ip)⅛(Ir; 0)'a,u)(v,a,±1Ωa±a±u).
Note that by choosing the unit vectors и = ei and v = Cj we find the asymptotic
variance of the element (a±1)ij∙ = e(α±1εj∙, and for и = ei + ek ,v = Cj we can then
also find the asymptotic covariance between (a±1)ij∙ and (a±1)⅛j∙. The proof is found in
Appendix A.
4.3 Tests on β
We consider in Section 9 test for linear restrictions on each β vector
β = (h1 + H1β1,...,hr + Яг ' ) (19)
where hi is p × 1 and Hi is p × mi both known, and ψi is an unknown parameter of
dimension mi × 1.
Lemma 2 Under the assumption that the restrictions (19) are identifying the asymp-
totic distribution of the likelihood ratio test is χ2(∑r=1(p - r - mi))∙
The proof is given in Appendix B. The hypothesis will be applied to simplify the
estimated polynomial cointegration relations. The hypothesis does not involve the
coefficient δ, because the asymptotic theory for such hypotheses has not been worked
out.