MPRA

To ensure the positivity of the volatilities, we assume that ω kk > 0. In view of (3), the
process (η ∗ t ) can be called the vector of equation by equation (EbE) innovations of (ε t ).
Let a 0 k = (a0 k1 , ..., a0 km )′ and c 0 k = (c0 k1 , . . . , c0 kr )′ be, respectively, the k-th row vectors of
the matrices A 0 and C 0 . Then the vector of unknown parameters involved in the k-th
equation (4) can be denoted by θ (k)
0 = (ω 0 kk , a0′ k , (b0 k )2 , c 0′
k )′ ∈ R d , d = m + r + 2. It is clear
that an identiability condition must be required such that σkt 2 is invariant to a change of
sign of the vectors a 0 k , c0 k and b0 k . Without lost of generality, we can assume that a0 k1 > 0,
b 0 k > 0 and c0 k1 > 0, for k = 1, . . . , m.
Let θ (k) = (ω kk , a ′ k , b2 k , c′ k )′ = (ω kk , a k1 , ..., a km , b k , c k1 , . . . , c kr ) ′ be a generic parameter
vector of the parameter space Θ (k) which is an any compact subset of
(0, +∞) 2 × R m−1 × [0, 1) × (0, +∞) × R r−1 .
2.2 Equation-by-equation estimation of parameters
Let ε 1 , . . . , ε n be observations of a process satifying the semi-diagonal BEKK-X representation
(2) and x 1 , . . . , x n be observations of a process of the explanatory variables.
For all θ (k) ∈ Θ (k) , we recursively dene ˜σ kt 2 (θ(k) ) for t = 1, . . . , n by
( m
) 2 ) 2
∑
˜σ kt(θ 2 (k) ) = ω kk + a kl ε l,t−1 + b 2 k˜σ k,t−1(θ 2 (k) ) + c ks x s,t−1 (5)
l=1
( r∑
s=1
with the arbitrary initial values ˜ε 0 , ˜σ 0 and x 0 . Let
˜Q (k)
n (θ (k) ) = 1 n∑
˜l kt (θ (k) ), ˜lkt (θ (k) ) = log ˜σ 2
n
kt(θ (k) ) +
t=1
The EbE estimator, denoted by
measurable solution of the following equation
Let θ 0 =
̂θ (k)
n
ε 2 kt
˜σ 2 kt (θ(k) ) .
(k) ˆθ n , of the true parameter vector θ (k)
0 is dened as a
= arg min
θ (k) ∈Θ (k)
˜Q(k) n (θ (k) ). (6)
) ′. (θ (1)′
0 , . . . , θ (m)′
0 Note that θ0 includes the diagonal elements of Ω 0 and
all components of the matrices A 0 , B 0 and C 0 . This parameter vector belongs to the
parameter space Θ m = Θ (1) × · · · × Θ (m) , whose generic element is denoted by θ =
(
θ (1)′ , . . . , θ (m)′) ′
. The estimator of θ0 is given by ̂θ
) (̂θ(1)
n =
′
′
, . . . , which is the
collection of the equation by equation estimators.
6
n
̂θ
(m)′
n