MPRA

⎛ ⎞
⎛ ⎞
where Ω = ⎝ ω 11 ω 12
⎠ is a symmetric positive denite matrix, A = ⎝ a 11 a 12
⎠
ω 12 ω 22 a 21 a 22
⎛ ⎞
and C = ⎝ c 11 c 12
⎠, with a 11 > 0, a 21 > 0, c 11 > 0 and c 21 > 0. The volatility of the
c 21 c 22
rst component, ε 1t , of ε t is thus given by
σ 2 1t = ω 11 +a 11 ε 2 1,t−1+2a 11 a 12 ε 1,t−1 ε 2,t−1 +a 12 ε 2 2,t−1+c 11 x 2 1,t−1+2c 11 c 12 x 1,t−1 x 2,t−1 +c 12 x 2 2,t−1.
Assumption A7 precludes, for example, that x 1,t−1 = ε 1,t−1 for which the model is not
identiable.
Similarly, Assumption A8 rules out the existence of linear combination of a nite
number of the x s,t−i x s ′ ,t−i that is obviously necessary for the identiability.
Theorem 1 Under A1 - A8, the EbEE of θ (k)
0 is strongly consistent
̂θ (k)
n
→ θ (k)
0 , a.s. as n → ∞.
The following result is an immediate consequence of Theorem 1.
Corollary 1 Under the assumptions of Theorem 1, ̂ϑ n is a strongly consistent estimator
of ϑ 0 .
Now we turn to the asymptotic distribution of the estimation. We need some additional
assumptions.
A9: θ (k)
0 belongs to the interior of the parameter space Θ (k) , for k = 1, . . . , m.
A10: E‖η t ‖ 4(1+δ) < ∞, E||ε t || 4(1+1/δ) < ∞ and E||x t || 4(1+1/δ) < ∞ for some δ > 0.
A11: The process z t = (x ′ t, ε ′ t, η ′ t) ′ satises E‖ε t ‖ (4+2ν)(1+1/δ) < ∞, E‖x t ‖ (4+2ν)(1+1/δ) <
∞ and E‖η t ‖ (4+2ν)(1+δ) < ∞, moreover the strong mixing coecients, α z (h), of the
process (z t ) are such that
∞∑
{α z (h)} ν/(2+ν) < ∞ for some ν > 0 and δ > 0.
h=0
8