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

∂B1 (^ ʌ
ɑ,
lB)
lB

∂B2 ʌ

∙,.ιd'φb
∂φ2

∂C мж ʌ
(,   (dφιc )

lc


(β °l1H1b11(dφ11B ), ∙ ∙ .,β ↑1Hrbrl(dφlrB )),

(αι(2ι),... ,ar(dφr)),

lc


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 βi = hi + Hi,φi is identified, then the (p r) × mi matrix β,1Hi has rank
mi.

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

It follows that β 1Hi has rank mi, and therefor so has the matrix

β↑Hi(bi.M = 011l2)'Hi(5i,b,,ɪ) = ( β°χβbl βb'i ɔ .

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

The rank of ai is gi so the rank of β llHibi± is mi gi. Therefore

rank(


∂vec(Bl)
d(^
B)


rank(


∂vec(B2)
∂vec(φ
2)


rank(


∂vec(C )
d(Ac)


r                                r

rank(βɪɪHφi ) = y∑(mi gi),
i=l                        i=l

r                     r

ɪʌank(aə =   gi,

i=l               i=l

SlS2.


This shows that the number of parameters is sls2+ ∑r=l ^i, and hence that the test
for the identifying restrictions is asymptotically distributed as
χ2 with J2r=l(P r mi)
degrees of freedom.

32



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