The Integration Order of Vector Autoregressive Processes

In the next proposition we consider the I(2) case:

Proposition 4.2 (I(2) case). A necessary and sufficient condition for
α⊥∏(1)β⊥ = ξη' and ∖α'2θβ2∖ = 0, where ξ and η arep—r ×s matrices of
full rank s < p — r, θ = ^π2^l +Π(1)βα'∏(1), α2 = α⊥ξ⊥ and β2 = β⊥η⊥
is that

a = m — 2

in Π_a(z) = (z — 1)^aH(z) and H(1) = 0_p. The explicit expression for
H (1) is

H (1) = g (1) β2( α2 θβ2)^-1 α2.

PROOF. Assume a = m — 2 so that d = 2; the first derivative
of Π(z)H(z) = H(z)Π(z) = (z — 1)2g(z)I_p implies α⊥∏(1)β⊥ζ2 =
ζ2α⊥∏(1)β⊥ = 0_p-r and thus ∖ζ2∖∖α⊥∏(1)β⊥∖ = 0; if ∖α⊥∏(1)β⊥∖ = 0
then ζ2 = 0_p-r contradicts H(1) = 0_p; thus ∖α⊥∏(1)β⊥∖ = 0, α⊥∏(1)β⊥ =
ξι∣' where ξ and η are p — r × s matrices of full rank s < p — r and
ζ 2 = η⊥ψξ⊥ for some p—r—s×p—r—s matrix ψ of rank 0 < t ≤ p—r—s ;
then H (1) becomes

H (1) = β⊥η⊥ψξ⊥ α⊥ = β 2 ψα'2.

Note that the first derivative of (4.1) provides the equality

•                                        a

β'H(1) = α ' ∏(1) H (1).

The the second derivative of (4.1) implies

a a                                                                   a                   a

(4.3)         α⊥ ∏(1) H (1) + 2 α⊥ ∏(1) H(1) = 2 g (1) α⊥.

Using I_p = β⊥β'⊥ + ββ' we see that α⊥∏(1)TH^(1) = ξη'β⊥H(1) +
a                 ___ a                                             ___ a                                            a                 ___ a

α⊥∏(1)/3β'HD = ξη'β'⊥H(1) + α⊥∏(1)/3α'∏(1)H(1) by (4.2). Thus

(4.3) becomes

Pre and post multiplying (4.4) respectively by ξ⊥ and α₂, we see that

α θ θβ 2 ψ = g (1) Ip-r-s

Then ∖ψ∖ = 0, ∖α'2θβ2∖ = 0 and ψ = g(1)(αθθβ2)^-1 follow from g(1) =
0 and the I(2) condition is satisfied.

Assume now the I(2) condition holds and suppose d = m — a > 2;

(4.5) becomes

α θ θβ 2 ψ = 0 p-r-s

and ψ = 0_p-r-s contradicts ∖α2θβ2 ∖ = 0 and implies m — a = 2. ■

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