MPRA

Once the EbEE estimators Ân, ̂Bn and Ĉn of the matrices A 0 , B 0 and C 0 , respectively,
are obtained, the matrix Ω 0 can be fully estimated as follows
vech 0 ( ̂Ω n ) = vech
(̂Σεn 0 − Ân̂Σ εn  ′ n − ̂B
)
n̂Σεn ̂Bn − Ĉn̂Σ xn Ĉ ′ n , (7)
where ̂Σ εn = 1 ∑ n
t=1
n
ε tε ′ t and ̂Σ xn = 1 ∑ n
t=1
n
x tx ′ t are the empirical estimators of the
second order moment matrices Σ ε = E(ε t ε ′ t) and Σ x = E(x t x ′ t), respectively. The
estimation of the model (2) is thus nothing else than the estimation of
ϑ 0 = (θ ′ 0, γ ′ ε0, γ ′ x0) ′ , γ ε0 = vech(Σ ε ), γ x0 = vech(Σ x ).
Its estimator can be given by ̂ϑ n = (̂θ ′ n, ̂γ ′ εn, ̂γ ′ xn) ′ , where ̂γ εn = vech(̂Σ εn ) and ̂γ xn =
vech(̂Σ xn ).
3 EbE estimation inference
For the consistency of the estimator, the assumptions following will be made
A4: θ (k)
0 ∈ Θ (k) , Θ (k) is compact, for k = 1, . . . , m.
A5: ρ(A 0 ⊗ A 0 + B 0 ⊗ B 0 ) < 1 and ∑ m
k=1 b2 k < 1, for all θ(k) ∈ Θ (k) .
A6: There exists s > 0 such that E|ε kt | s < ∞ and E|x kt | s < ∞.
A7: For all l ∗ = 1, . . . , m, ε 2 l ∗ t
does not belong to the Hilbert space generated by the
linear combinations of the ε lu ε l ′ u's, the x sv x s ′ v's for u < t, v ≤ t, l, l ′ = 1, . . . , m,
s, s ′ = 1, . . . , r and the ε lt ε l ′ t for (l, l ′ ) ≠ (l ∗ , l ∗ ).
A8: For all s ∗ = 1, . . . , r, x 2 s ∗ t does not belong to the Hilbert space generated by the
linear combinations of the the x sv x s ′ v's for v < t, s, s ′ = 1, . . . , r and the x st x s ′ t for
(s, s ′ ) ≠ (s ∗ , s ∗ ).
Remark 2 Assumptions A7 and A8 are identication conditions. For simplicity, let us
consider (2) when m = 2, r = 2 and the conditional covariance matrix is given by
H t = Ω + Aε t−1 ε ′ t−1A ′ + Cx t−1 x ′ t−1C ′ , (8)
7