MPRA

More documents

Recommendations

Info

the remaining parameters of the intercept matrix in the second step. The strong consistency of the EbEE is showed under the assumptions that are weaker than the ones for the VTE of the same model. Under the mixing-conditions, the asymptotic distribution of the estimators of the parameters is normal. The main motivation for using the EbE method in application is the important gains in computational time and our experiments show that the reduction of computational time compared to the VT estimation can be eective. 6 Proofs Proof of Theorem 1. Let Q (k) n (θ (k) ) = 1 n n∑ l kt (θ (k) ), l kt (θ (k) ) = log σkt(θ 2 (k) ε 2 kt ) + σkt 2 (θ(k) ) . t=1 To prove the theorem of the consistency of the EbEE, the following results have to be shown i) lim n→∞ ∣ ∣∣Q (k) sup n (θ (k) ) − θ (k) ∈Θ (k) n (θ (k) ) ∣ = 0 a.s. ˜Q (k) ii) σ 2 kt (θ(k) 0 ) = σ 2 kt (θ(k) ) a.s. i θ (k) 0 = θ (k) . iii) El kt (θ (k) 0 ) < ∞ and if θ (k) ≠ θ (k) 0 then El kt (θ (k) 0 ) < El kt (θ (k) ). iv) There exists a neighborhood V(θ (k) ) of any θ (k) ≠ θ (k) 0 such that lim inf n→∞ inf θ ∗ ∈V(θ (k) ) ˜Q (k) n (θ ∗ ) > lim sup n→∞ ˜Q (k) n (θ (k) 0 ), a.s. The condition ∑ m k=1 b2 k < 1 of Assumption A5 and the compactness of Θ(k) imply that By a simple recursion, we have sup b 2 k < 1. (20) θ (k) ∈Θ (k) ∣ ∣ sup ∣∣˜σ 2 kt (θ (k) ) − σkt(θ 2 (k) ) ∣ = sup (b 2 k) t ∣∣˜σ 2 k0 (θ (k) ) − σk0(θ 2 (k) ) ∣ < Kρ t a.s. θ (k) ∈Θ (k) θ (k) ∈Θ (k) where, here and the sequel of the paper, K and ρ denote generic constants whose the exact values are not important and 0 < ρ < 1. 17
The proofs of i), iii) and i) are very similar to the ones given in Francq and Zakoïan (2010) for the standard GARCH without covariates and are omitted. Here I only thus show the point ii). Let L be the back-shift operator, i.e. L(u t ) = u t−1 . By Assumption A5, the polynomial 1 − b 2 k L is invertible for any θ(k) ∈ Θ (k) . Assume that σ 2 kt (θ(k) 0 ) = σ 2 kt (θ(k) ) a.s. We have ( m ) 2 ( ∑ m ) 2 ∑ (1 − (b 0 k) 2 L) −1 a 0 klε l,t−1 − (1 − b 2 kL) −1 a kl ε l,t−1 l=1 + (1 − (b 0 k) 2 L) −1 ( r∑ s=1 l=1 c 0 klx s,t−1 ) 2 − (1 − b 2 kL) −1 ( r∑ s=1 c ks x s,t−1 ) 2 =(1 − b 2 k) −1 ω kk − (1 − (b 0 k) 2 ) −1 ω 0 kk a.s. Then where a (k) ill ′ ∞∑ m∑ i=0 l,l ′ =1 a (k) ill ′ ε l,t−i−1 ε l ′ ,t−i−1 + ∞∑ r∑ j=0 s,s ′ =1 c (k) jss ′ x s,t−j−1 x s ′ ,t−j−1 = c a.s. (21) = (b 0 k )2i a 0 kl a0 kl ′ − (b k) 2i a kl a kl ′, c (k) jss ′ = (b 0 k )2i c 0 ks c0 ks ′ − (b k) 2i c ks c ks ′ and c = (1 − (b 0 k )2 ) −1 ω kk − (1 − b 2 k )−1 ω 0 kk . If b0 k ≠ b k or there exists l ∗ such that a 0 kl ∗ ≠ a kl ∗ then ε 2 l ∗ ,t−1 is a linear combination of the x s,ux s ′ ,u, s, s ′ = 1, . . . , r, u < t, the ε l,v ε l ′ ,v, l, l ′ = 1, . . . , m, v < t−1 and the ε l,t−1 ε l ′ ,t−1, (l, l ′ ) ≠ (l ∗ , l ∗ ) which is impossible by Assumption A7. Therefore b 0 k = b k and a 0 kl = a kl for all l = 1, . . . , m and (21) becomes ∞∑ r∑ j=0 s,s ′ =1 c (k) jss ′ x s,t−j−1 x s ′ ,t−j−1 = c a.s. Similarly, if there exists s ∗ such that c 0 ks ∗ ≠ c ks ∗, x2 s ∗ ,t−1 is a linear combination of x u,t−j x u ′ ,t−j, u, u ′ = 1, . . . , r, j > 1 and x s,t−1 x s ′ ,t−1, (s, s ′ ) ≠ (s ∗ , s ∗ ) which contradicts Assumption A8. Therefore we have c 0 ks = c ks for all s = 1, . . . , r. Hence ii) is proved. ✷ For the proof the asymptotic distribution in the Theorem 2, we need the following lemmas Lemma 1 Under assumptions of Theorem 2, there exists a neighborhood V(θ (k) 0 ) of θ (k) 0 18
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 and 10: ⎛ ⎞ ⎛ ⎞ where Ω = ⎝ ω 1
Page 11 and 12: where ⎛ Γ = ⎜ ⎝ ⎞ −J −
Page 13 and 14: Table 1: Sampling distribution of t
Page 15 and 16: Figure 3: Kernel density estimator
Page 17: Table 2: Sampling distribution of t
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

MPRA

Create successful ePaper yourself

Delete template?

Save as template?