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



and the J2⅛(mi - 9i) parameters

φ1B = (φ1B1, ∙∙∙, ΦiBr ) = (b1±Cφ1 - Φι ), ∙∙∙, b',Φr -θ)),

and finally the s1s2 parameters

φ1C = β ±2∏∙

Thus the number of parameters is ɪɪɪ m+ s1s2Note that

φi - r'' = bibi(ψi - ψ0^ + 6ii- ψ0^ = biφ2i + ⅛±φiBz

The parameters ф1В, φ1c, 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 ()]tβ 0'β (r 1 + β "⅛w.)[ / (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) = Ir,
which means that because ɪɪβ(L) = 0, we find

σ d1 .
/Lb21ψ=ψ°

σ2 D

2 2


= β1'2[H(d≠1), ∙ ∙ ∙, Hr(dφr)] = (αιb'1(d≠1), ∙ ∙ ∙, <⅛b'r(dφr)),

(27)

(28)


(29)

(30)


= 2β01'2[    (L)w>][ɪ(β            . ]

/l        /L

This implies, using i = b(2i) + bjɪ(1Bi), that

• I             n

2 B2|.=.°

I           n

1b B2|...°


(θ∙1 (21) , ∙ ∙ ∙ , ®r (2rD>

(α1b'1b1±(1B1), ∙ ∙ ∙, ^rξφr(wr)) = 0

Similarly

β ^2 /j.   β Cφ=y°

σφ1B


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

= (u1b1b1χ(1B1), ∙ ∙ ∙ , ^rbrbr(1Br)) = 0

30



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