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

13 Appendices

A Proof of Lemma 1

It is convenient to introduce ξ = a'₁ζp₁, so that a₁₁ = a₁ξ and a₁₂ = a₁ξ₁. Because
the estimates of p,τ,β,δ are superconsistent we can find the asymptotic variance of
a₁₁ and a₁₂ by finding derivatives of a₁₁ and a ₂ with respect to a and ζ, and then
transform the asymptotic variance of (a, ¢) by an application of the δ- method.

It is convenient to express the matrix expansion

(a + u)1 = aɪ — aɪ ua ' + 0(∣u∣²)

in terms of differentials

(dal) = — aɪ (da)a '.                              (25)

As an example of the δ-method we can find the asymptotic variance of a₁ from (16)
and (25) as

asVar(a₁) = a(I_r; 0)Φ(I_r; 0)'a' 0 a'₁Ωa₁.

Applying (25) we find derivatives of ξ = a'₁ζp₁,ξ₁, and a₁ :

dξ = —a1(da)a 'ζ p₁ + a^l₁(dζ )p₁,
dξ± = lp'ι(ζ 'ɑ(da)' — (dζ )')a12,
da ₁ = —a(da)'a ₁.

Hence from a₁₂ = a₁ξ ₁, we get

da12 = (da±)ξ ₁ + a1(dξ₁) = — (I_p — a1ξp'₁ ζ')a; a1ξp'₁)
where we find

0 = aιξp'ι = a1a^,1 ζpι(p1ζ 'aɪ a1CpJ~¹p1,
which gives the expression (17). We next find

da₁₁ = d(a ₁)a1ζp₁ + a ₁d(a1)ζp₁ + a ₁a1 (dζ )p₁,             (26)

= —a(da)' a₁₁ — a ₁a1(da)a 'ζp₁ + a ₁a'₁(dζ )p₁.

In order to find the elements of the asymptotic variance of a ₁₁ it is enough to have
an expression for the asymptotic variance of v'a ₁₁u for any vectors v,u. We find from
(26)

v^lda₁₁ u = — u'a(da)'a₁₁v — u'a ₁a1(da)a 'ζ p₁v + u'a ₁a^l₁(dζ )p₁u

= — v'a'₁₁(da)a 'u + u'a ₁a1[(dζ ) — (da)a 'ζ ]p₁u

= N + N2

Then Var(v'da₁₁u) = Var(N₁) + Var(N₂) + 2Cov(N₁, N₂), which is the expression in
(18).

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