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                                                                         , ,

-ɪ B2∣_ψ=_ψo = 0.                            (31)

9Φ_ib

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

The derivative of B₁ = β ι₁β (0) with respect to ф_1В is

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

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_1В which locally determines B₁ and
φ_1c which locally determines C. Thus at the true value it should holds that

^d-Bι = 0 ∂2B2 = 0
∂Φ1     ^, ∂φ[

We find from (30) and (31) that (36) is satisfied. Because B₁ does not depend on
φ_1c, we have &B = 0. On the other hand C does depend on ф_1В, 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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