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



so that from (28) we find

2                                                                         , ,

B2ψ=ψo = 0.                            (31)

ib

m 1         j∙ r τ^^>       Z>0/ /3∕.∕∖ ∙ji             j j √

The derivative of B1 = β ι1β (0) with respect to ф is

∂φB 1ψ=ψ0


= β⅛M‰ι),.. .,3tiHrbrl{dφ1Br)] = 0,


(32)


which will be discussed below, and finally we find the derivative of C

ɪe∣, .o n n
∂η C T=Ψ0=η0>p=p0

= βdηMp1 = 1c,

(33)

-d-C ∣, .0 n n
∂^ψ C p=p0=η0,p=p0

= β 12 ∂iβ ^∣^0 P0 P1 = 0,

(34)

—C ∣, .0    n    n

∂pc P=P0=η0,p=p0

= β 1⅛0.     p + η0 .      p ]P1 = 0,

∂p          ∂p

(35)

because p0'p01 = 0, β12β0 = 0, and β12η0 = 0.

B.4 Conditions for asymptotic χ2 distribution

The conditions are expressed in terms of the derivative at the true value, see Johansen
(2006, Theorem 5). There are two conditions, the first states that although
B2 may
depend on
φ1, the dependence is very small close to the true value, and the second
implies that we can split the parameter
φ1 into ф which locally determines B1 and
φ1c which locally determines C. Thus at the true value it should holds that

d-Bι = 02B2 = 0
∂Φ1     , ∂φ[

(36)


∂B1

1C


= 0,


∂C
1B


(37)


We find from (30) and (31) that (36) is satisfied. Because B1 does not depend on
φ1c, we have &B = 0. On the other hand C does depend on ф, but from (34) we
have
Φ^C~ = 0 at the true value so that (37) holds. The consequence of this is that
asymptotic inference is
χ2 and we only have to find the degrees of freedom, that is, the
difference between the number of identified parameters with and without restrictions.

B.5 Number of parameters

The number of parameters in the unrestricted parameters B1, B2, and C are s1r, s2r
and s1s2 respectively. We next want to show that when β is identified by the linear
restrictions, the number of parameters in the model is
s1s2 + ɪɪɪ m^ which gives the

31



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