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 = π2l +Π(1)βα'∏(1), α2 = α⊥ξ⊥ and β2 = β⊥η⊥
is that
a = m — 2
in Πa(z) = (z — 1)aH(z) and H(1) = 0p. 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)Ip implies α⊥∏(1)β⊥ζ2 =
ζ2α⊥∏(1)β⊥ = 0p-r and thus ∖ζ2∖∖α⊥∏(1)β⊥∖ = 0; if ∖α⊥∏(1)β⊥∖ = 0
then ζ2 = 0p-r contradicts H(1) = 0p; 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
(4.2)
• 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 Ip = 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
(4.4)
∏(1)
2
a --- a
+ ∏(1) /3 α ' ∏(1)
--- a
H (1)+ ξη'β'⊥ H(1) = g (1) α.
Pre and post multiplying (4.4) respectively by ξ⊥ and α2, we see that
(4.5)
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 = 0p-r-s contradicts ∖α2θβ2 ∖ = 0 and implies m — a = 2. ■