degrees of freedom. ɪɪɪ(p — r ~ mi)∙ From (32), (29), and (33) if follows that at the
true value we have
∂B1 (^ ʌ
ɑ, ∖dφlB)
dφlB
∂B2 ʌ
∙,.ι ⅛d'φb
∂φ2
∂C мж ʌ
(, (dφιc )
dφlc
(β °l1H1b11(dφ11B ), ∙ ∙ .,β ↑1Hrbrl(dφlrB )),
(αι(dφ2ι),... ,ar(dφr)),
dφlc
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 = (β011,βl2)'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)
dvœ(^B)
rank(
∂vec(B2)
∂vec(φ2)
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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