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



B Proof of Lemma 2

We apply the maximum likelihood parameterization (12), especially the parameters
p, τ and β = τp. In order to apply the results in Johansen (2006) we have to normalize
τ and β on T0 and β respectively, that is, so that β'β = Ir and τ,T0 = Ir+s. We split
the parameters
τ,p into the variation free parameters β = τp, η = τp1 and p, and let
β = β() be given by (19)

β(φ) = (h1 + H1 φ1,... , hr + Hrφr).

B.l Normalization of parameters

We define the normalized versions of β, τ and the corresponding p, as functions of the
free parameters
φ, η, p, using the decomposition τ = τpp' + τp1p'1 = β(φ')p' + ηp'1 :

β = ~βW = β (Ψ)(β 0'β Wr1

τ = τφφ,η^ = φβ (f)p'+ ηp1Xp0'β (W)p'+ p0''∕p, l,
p = p(φ)= τoβ (φ)(β 0'β (φ))-1.

which satisfies β°'β = Ir, τ0'τ = Ir+sι, and β = τp.

The asymptotic theory for hypotheses on the parameters in the I(2) model, Jo-
hansen (2006), is developed in terms of the parameters
— ∩/ ~ ,

Bi = β "         Bi (Φib2),

B2 = β 1'2β(φ)= B2 1B 2),

C = β "l    = Φ1c,

where φ1 = (Φib1c) and φ2, are defined below, in such a way that the estimators
for
B2 and φ2 are T2 consistent and the estimators for B1, C, and φ1 = (Φ1b , Φ2b) are
T consistent. Moreover the asymptotic distribution of (TB1,T2B2) is mixed Gaussian
and so is the asymptotic distribution of
TC. The asymptotic distribution of T2φ2 and
1 are only mixed Gaussian under some further conditions, which we discuss below.

B.2 The parameters φ2, φ, and φιc

We start defining the parameters φ2 which determine the variation of B2 through

T 1,2 β (φ) = β ⅛ + H1Φ1, ...,hr + Hr φr ) = β 12H1(φ1 - φ01),. . .,β l2Hr (φr - Φ0)

0           0                    0                             τ0∕

In order to find the effective number of parameters we write β 12Hi = aib'i, where
0                              0 7∖0∕

ai, s1 × gi, and b'i, gi × mi, are of rank gi = rank(β12Hi) mrn(mi, s2). We define the
∑r=1 gi parameters

φ2 = 2i,.. .2r) = 11 - φlφ.. .,b'r (φr - φ,

29



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