MPRA

Under Assumption A11 and using the same argument as in the reference, we have for
some ν and δ > 0
(
)
∥ H 1/2
0t,s ⊗ H 1/2
0t,s vec(η t η ′ t − I m ) ∥ < ∞.
2+ν
Note that D −1
0t H 0t D −1
0t is the conditional correlation matrix of ε t . Because D −1
0t H 0t D −1
( ) ( ) ′,
D −1
0t H 1/2
0t D −1
0t H 1/2
0t all the elements of the matrix D
−1
0t H 1/2
0t are smaller than one.
( ) ( )
It is thus easy to show that all the ones of the matrix D −1
0t,sH 1/2
0t,s ⊗ D −1
0t,sH 1/2
0t,s are
also smaller than one. Using the Holder inequality, we then have
( ) ( )
∥
∥∆ ts T m D −1
0t,sH 1/2
0t,s ⊗ D −1
0t,sH 1/2
0t,s vec(η t η ′ t − I m ) ∥
2+ν
∥ ( ) ( )
∥∥Tm
≤ ‖∆ ts ‖ (2+ν)(1+1/δ)
D −1
0t,sH 1/2
0t,s ⊗ D −1
0t,sH 1/2
0t,s vec(η t η ′ ∥
t − I m )
∥
(2+ν)(1+δ)
0t =
≤ K ‖∆ ts ‖ (2+ν)(1+1/δ)
‖vec(η t η ′ t − I m )‖ (2+ν)(1+δ)
< ∞,
where the last inequality follows from (20) and Assumption A11. It implies that ‖Y t,s ‖ 2+ν <
∞. The process (Y t,s ) t , for s xed, is thus strongly mixing under Assumption A11. Applying
the central limit theorem of Herrndorf (1984), we get
⎛
∑ n
1
√ ⎝
t=1 Υ 0tvec ( )
⎞
η t η ′ t − I m
n
∑
⎠ → d N (0, Σ) .
n
t=1 vech(x tx ′ t − E(x t x t ))
The asymptotic distribution of Theorem 2 thus follows from the Slutzky theorem.
✷
The proof of Theorem 3
The strong consistency of ̂ξ n is obviously obtained.
Now we turn to its asymptotic normality.
Using the elementary relation vec(ABC) = (C ′ ⊗ A)vec(B), we have
vec(Ân ̂Σ
)
εn  ′ n − A 0 Σ ε0 A ′ 0) =vec
(Ân ̂Σεn (Â′ n − A ′ 0) + vec
(Ân ( ̂Σ
)
εn − Σ ε )A ′ 0
(
)
+ vec (Ân − A 0 )Σ ε A ′ 0
=
{I m ⊗ (Ân ̂Σ
}
εn ) + ((A 0 Σ ε ) ⊗ I m )M mm vec(Â′ n − A ′ 0)
+ (A 0 ⊗ Ân)D m vech( ̂Σ εn − Σ ε ).
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