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

and the J2⅛=ι(^mi - 9i) parameters

^φ1B = (^φ1B1^, ∙∙∙, ΦiB_r ) = (^b1±C^φ1 - Φι ), ∙∙∙^{, b}'rΛ^,Φr - ≠θ)),

and finally the s₁s₂ parameters

^φ1C = ^β ±2∏∙

Thus the number of parameters is ɪɪɪ m⅞ + s₁s₂∙ Note that

φi ^{- r}'' = b_ibi(ψ_i ^- ψ⁰^ ⁺ 6_ii⅛ ^- ψ⁰^ = b_i^φ_2i ⁺ ⅛±^φiBz∙

The parameters ф_1В, φ_1c, and φ₂ are varying freely.

B.3 Derivatives of parameter functions

We first investigate the derivatives of B₂ with respect to the parameters l, and find

JLb₂ = β °₁'₂[ /Lβ (⅛)]_tβ ⁰'β (⅛_r ¹ + β "⅛_w.)[ / (3 ⁰'β _w._r ^ι]

/ B₂ = β⁰₁'₂[ /,β(≠)](β⁰'β(L))^-1 + β⁰l₂β(L)[ ∕∙(β'
σψ          σψ                      σψ

+2β 2 ^φ-β(≠)][ɪ(β⁰'β(L))^-1]∙
σψ σψ

For L = L⁰ we have β(^φ0) = β⁰, so that βθ'₂β(^,L₀) = βθ'₂β⁰ = 0, and β⁰'β(L⁰) = I_r,
which means that because ɪɪβ(L) = 0, we find

= β1'₂[H(d≠₁), ∙ ∙ ∙, Hr(d^φ_r)] = (αιb'₁(d≠₁), ∙ ∙ ∙, <⅛b'_r(d^φ_r)),

= 2β⁰₁'₂[    (L)∣_w>][ɪ(β            . ]∙

/l        /L

This implies, using dψ_i = b⅛(dφ_2i) + bjɪ(dφ_1Bi), that

(θ∙1 (^dφ21^{) ,} ∙ ∙ ∙ ^, ®r (^dφ2rD>

(α1b'1b1±(dφ1B1), ∙ ∙ ∙, ^rξ^φr(dφ_wr)) = 0∙

Similarly

/   7/7   ∕7/    ∖          7∕7∕7/ W ΓX

= (u1b1b1χ(dφ1B1), ∙ ∙ ∙ , ^rbrbr(dφ1Br)) = 0

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