MPRA

denoted by tr(A), the determinant is denoted by det(A) and the spectral radius of A is
denote by ρ(A), i.e., ρ(A) is the maximum among the absolute values of the eigenvalues of
A. The operator vec stacks all columns of a matrix into a column vector, vech denotes the
operator that stacks only the lower triangular part including the diagonal of a symmetric
matrix into a vector, and vech 0 is the operator which stacks the sub-diagonal elements
(excluding the diagonal) of a matrix. The (mn × mn) commutation matrix M mn is
dened such that, for any (m × n) matrix A, M mn vec(A) = vec(A ′ ). D m and L m denote
the duplication matrix and elimination matrix dened such that , for any symmetric
(m × m) matrix A, vec(A) = D m vech(A) and vech(A) = L m vec(A). Denote T m be a
m × m(m + 1)/2 matrix such that T m vec(A) = diag(A) for any m × m matrix A. Let
also P m be a ( m(m−1)
2
× m 2 ) matrix such that vech 0 (A) = P m vec(A), for any symmetric
(m × m) matrix A. The Kronecker product of A and B is dened by A ⊗ B = {a ij B}.
The Euclidean norm of the matrix, or vector A, is dened as ‖A‖ = √ tr(A ′ A), and the
spectral norm is dened as ‖A‖ sp = √ ρ(A ′ A).
2 The model and EbE estimation
2.1 The model
Let ε t = (ε 1t , · · · , ε mt ) ′ denote a (m × 1) vector of random variables and let x t =
(x 1t , · · · , x rt ) ′ be a r-dimentional vector of exogenous variables. Assume the existence
of the (m × m) positive denite matrix H t such that
E(ε t |F t−1 ) = 0, E(ε t ε ′ t|F t−1 ) = H t , (1)
where F t = σ{ε u , x ′ u; u, u ′ ≤ t} implies the information set at time t. Note that H t is
the conditional covariance of ε t given F t−1 the information set until time t − 1.
We consider the following model
⎧
⎨ ε t = H 1/2
t η t
(2)
⎩ H t = Ω + Aε t−1 ε ′ t−1A ′ + BH t−1 B + Cx t−1 x ′ t−1C ′
where A = (a kl ) 1≤k,l≤m , B = diag(b 1 , · · · , b m ), C = (c kl ) 1≤k≤m,1≤l≤r and Ω = (ω kl ) 1≤k,l≤m
is a positive denite (m × m) matrix.
4