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

degrees of freedom. ɪɪɪ(p — ^r ~ ^mi)∙ From (32), (29), and (33) if follows that at the
true value we have

(β °l₁H₁b₁₁(dφ₁₁B ), ∙ ∙ .,β ↑₁H_rb_rl(dφ_lrB )),

(αι(dφ₂ι),... ,ar(dφ_r)),

We want to determine the rank of these matrices to determine the number of parameters
in the restricted model. We now need the result, see Johansen (2007).

Lemma 3 If β_i = hi + H_i^,φ_i is identified, then the (p — r) × m_i matrix β^,₁H_i has rank
m_i.

-Γ. e -ɪ-ɪ         , 1 , τsθ∕ 7-7-1             1                1 , 1       r          IjI           , .

It follows that β ₁H_i has rank m_i, and therefor so has the matrix

β↑H_i(b_i.M = (β⁰₁₁,βl₂)'Hi(5i,b,,ɪ) = ( ^β°χβ^{bl βb}'ⁱ ɔ .

m        i r _             Ji        i r /ɔθ TTi ∙   __          m r

The rank of a_i is g_i so the rank of β _llH_ib_i± is m_i — g_i. Therefore

This shows that the number of parameters is s_ls₂+ ∑r=_l ^_i, and hence that the test
for the identifying restrictions is asymptotically distributed as χ² with J2r=_l(P — r — ^m_i)
degrees of freedom.

32

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