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 = I_r and τ^,T⁰ = I_r+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 + Hrφ_r).

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'⁺ ηp1X^p0'^{β (W)}p'^{+ p0}''∕p^, ’ ^l,
p = p(φ)= τ^oβ (φ)(β ⁰'β (φ))-¹.

which satisfies β°'β = I_r, τ⁰'τ = I_r+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 (Φi_b ,φ₂),

B₂ = β 1'₂β(φ)= B₂ (Φ1B ,φ₂),

C = β "_l    = Φ1c,

where φ₁ = (Φ_ib ,φ_1c) 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 = (Φ_1b , Φ_2b) 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
Tφ₁ are only mixed Gaussian under some further conditions, which we discuss below.

B.2 The parameters φ₂, φ_lβ, and φ_ιc

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 ₁₂H_i = a_ib'_i, where
0                              0 7∖0∕

a_i, s₁ × g_i, and b'_i, g_i × m_i, are of rank g_i = rank(β₁₂H_i) < mrn(m_i, s₂). We define the
∑r=₁ g_i parameters

^φ2 = ^(φ2i^,.. .^,φ2r⁾ = ^(Ь1^(ф1 ^{- φ}l^φ.. .^,b'r (^φr ^{- φ0φ,}

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