Testing Hypotheses in an I(2) Model with Applications to the Persistent Long Swings in the Dmk/$ Rate

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

al₂Ωa_x2 0 [(Ip - θζ')a; θ]⅛[(I_p - 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(I_r; 0)Φ(I_r; 0)'a'u)(n'a±₁Ωa±₁n)                               (18)

+(v'ρ'±(ζ'a, -I_p)⅛(ζ'a, -I_p)'ρ±v)(u'a±a±Ωa±a±«)
+2(v^,p^,±(ζ'a, -I_p)⅛(I_r; 0)'a^,u)(v^,a^,±₁Ωa±a±u).

Note that by choosing the unit vectors и = e_i and v = Cj we find the asymptotic
variance of the element (a±₁)_ij∙ = e(α±₁ε_j∙, and for и = e_i + e_k ,v = Cj we can then
also find the asymptotic covariance between (a±₁)_ij∙ and (a±₁)⅛_j∙. 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 h_i is p × 1 and H_i is p × m_i both known, and ψ_i is an unknown parameter of
dimension m_i × 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.

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