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Let H t,s (ϑ) be such that, for s > 0, vec(H t,s (ϑ)) = s∑ (B ⊗2 ) ( k vec(Ω) + A ⊗2 vec(ε t−k−1 ε ′ t−k−1) + C ⊗2 vec(x t−k−1 x ′ t−k−1) ) , k=0 where A ⊗2 denotes the Kronecker product of a matrix A and itself. subspace such that for all ϑ ∈ Θ, H t (ϑ) ∈ S and for all s > 0, H t,s (ϑ) ∈ S. A12: There exists K > 0 such that Let also S be a ∥ ∥H 1/2 t (ϑ) − H ∗1/2 t (ϑ) ∥ ≤ K ‖H t (ϑ) − H ∗ t (ϑ)‖ for all H t (ϑ), H ∗ t (ϑ) ∈ S. Remark 3 Assumptions A11 and A12 are also required in Thieu (2016) for the BEKK- X model estimated by the variance targeting method. In order to state the asymptotic normality, we have to introduce the following notations. Let the (d × d) matrices J ks = E(∆ kt ∆ ′ st), k, s = 1, . . . m, where ∆ kt = 1 ∂σkt 2 (θ(k) 0 ) σkt 2 ∂θ (k) and let J = diag{J 11 , . . . , J mm } in bloc-matrix notation. Let also ∆ t = diag(∆ 1t , . . . , ∆ mt ), N 1 = L m (I m 2 − A ⊗2 0 − B ⊗2 0 ) −1 (I m 2 − B ⊗2 0 ) and N 2 = L m (I m 2 − A ⊗2 0 − B ⊗2 0 ) −1 C ⊗2 0 D r . We also dene the following matrices Σ 11 = Σ 22 = Σ 12 = ∞∑ h=−∞ ∞∑ h=−∞ ∞∑ h=−∞ cov ( Υ 0t vec (η t η ′ t) , Υ 0,t−h vec ( η t−h η ′ t−h)) , (9) cov ( vech(x t x ′ t), vech(x t−h x ′ t−h) ) , (10) cov ( vech(x t x ′ t), Υ 0,t−h vec ( η t−h η ′ t−h)) , (11) where Υ 0t = ⎛ ( ) ( ) ⎝ ∆ tT m D −1 0t H 1/2 0t ⊗ D −1 0t H 1/2 0t H 1/2 0t ⊗ H 1/2 0t ⎞ ⎠ . (12) Theorem 2 Under A1 - A12, as n → ∞, ⎛ ⎞ ̂θ √ n − θ 0 d n ⎜ ̂γ ⎝ εn − γ ε0 ⎟ → N (0, ΓΣΓ ′ ) , (13) ⎠ ̂γ xn − γ x0 9
where ⎛ Γ = ⎜ ⎝ ⎞ −J −1 0 0 ⎛ ⎞ 0 N 1 N 2 ⎟ ⎠ and Σ = ⎝ Σ 11 Σ 12 ⎠ . (14) Σ ′ 12 Σ 22 0 0 I r(r+1)/2 The original parameter vector is denoted by ξ 0 = ( (vech 0 (Ω 0 )) ′ , θ 0) ′ ′ ∈ R m(m−1)/2+md . (15) The estimator of ξ 0 can be given by ̂ξ ( n = ̂ω ′ n, ̂θ n) ′ ′, where ̂ωn = vech 0 ( ̂Ω n ) is the estimator of ω 0 = vech 0 (Ω 0 ). follows The strong consistency and the asymptotic distribution of the estimation is given as Theorem 3 If A1 - A8 hold, the estimator ̂ξ n of ξ 0 is strongly consistent: ̂ξn → ξ 0 a.s. as n → ∞. If, in addition, A9 - A12 hold, then √ n ⎛ ⎝ ̂ω n − ω 0 ̂θ n − θ 0 ⎞ ⎠ d → N (0, ΩΣΩ ′ ) , (16) where ⎛ ⎞ Ω = ⎝ Ω 1 ⎠ , Ω 2 ( ) Ω 1 = A ∗ B ∗ C ∗ E ∗ X ∗ Ψ, ( ) Ω 2 = I md 0 m(m+1)/2 0 r(r+1)/2 , (17) with A ∗ = −P m {I m ⊗ (A 0 Σ ε ) + ((A 0 Σ ε ) ⊗ I m )M mm } , B ∗ = −P m {I m ⊗ (B 0 Σ ε ) + (B 0 Σ ε ) ⊗ I m } , C ∗ = −P m {I m ⊗ (C 0 Σ x ) + ((C 0 Σ x ) ⊗ I m )M mr } , E ∗ = P m (I m 2 − B 0 ⊗ B 0 − A 0 ⊗ A 0 )D m , X ∗ = −P m (C 0 ⊗ C 0 )D r . 4 Numerical illustrations This section presents the results of Monte Carlo simulations studies aimed at examining the performance of the EbEE of the semi-diagonal BEKK-X model and comparing the nite sample properties of the EbEE and VTE. 10
Page 1 and 2: MPRA Munich Personal RePEc Archive
Page 3 and 4: matrix. Recent developments in the
Page 5 and 6: denoted by tr(A), the determinant i
Page 7 and 8: To ensure the positivity of the vol
Page 9: ⎛ ⎞ ⎛ ⎞ where Ω = ⎝ ω 1
Page 13 and 14: Table 1: Sampling distribution of t
Page 15 and 16: Figure 3: Kernel density estimator
Page 17 and 18: Table 2: Sampling distribution of t
Page 19 and 20: The proofs of i), iii) and i) are v
Page 21 and 22: For all δ > 0, there exists V(θ (
Page 23 and 24: By the similar arguments used to sh
Page 25 and 26: Under Assumption A11 and using the
Page 27 and 28: where Ψ is a d 1 ×d 2 matrix, d 1
Page 29: Hafner, C.M. and A. Preminger (2009

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