THÈSE Estimation, validation et identification des modèles ARMA ...

Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 116
where
which entails that
Thus we have
Iij,h(1) = Ee1t−he1t−j−he1te1t−iσ −4
11 α i−1
01 α j−1
01 ,
Iij,h(2) = Ee1t−he1t−j−he1te2t−iσ −4
11 αi−1
01 αj−1
02 ,
Iij,h(3) = Ee1t−he1t−j−he2te1t−iσ −2
11 σ −2
22 α i−1
01 α j−1
01 ,
Iij,h(4) = Ee1t−he1t−j−he2te2t−iσ −2
11 σ−2
22 αi−1
01 αj−1
02 ,
Iij,h(5) = Ee1t−he2t−j−he1te1t−iσ −4
11 α i−1
02 α j−1
01 ,
Iij,h(6) = Ee1t−he2t−j−he1te2t−iσ −4
11 αi−1
02 αj−1
02 ,
Iij,h(7) = Ee1t−he2t−j−he2te1t−iσ −2
11 σ −2
22 α i−1
02 α j−1
01 ,
Iij,h(8) = Ee1t−he2t−j−he2te2t−iσ −2
11 σ−2
22 αi−1
02 αj−1
02 ,
Iij,h(9) = Ee2t−he1t−j−he1te1t−iσ −2
11 σ−2
22 αi−1
01 αj−1
01 ,
Iij,h(10) = Ee2t−he1t−j−he1te2t−iσ −2
11 σ−2 22 αi−1 01 αj−1 02 ,
Iij,h(11) = Ee2t−he1t−j−he2te1t−iσ −4
22 αi−1 01 αj−1 01 ,
Iij,h(12) = Ee2t−he1t−j−he2te2t−iσ −4
22 αi−1 01 αj−1 02 ,
Iij,h(13) = Ee2t−he2t−j−he1te1t−iσ −2
11 σ−2 22 αi−1 02 αj−1 01 ,
Iij,h(14) = Ee2t−he2t−j−he1te2t−iσ −2
11 σ −2
22 α i−1
02 α j−1
02 ,
Iij,h(15) = Ee2t−he2t−j−he2te1t−iσ −4
22 αi−1 02 αj−1 01 ,
Iij,h(16) = Ee2t−he2t−j−he2te2t−iσ −4
22 α i−1
02 α j−1
02 ,
I = 4
+∞
vecI = 4
+∞
i,j=1 h=−∞
⎛
⎜
⎝
+∞
+∞
i,j=1 h=−∞
(Iij,h(1),...,Iij,h(16)) ′ .
Iij,h(1) Iij,h(5) Iij,h(9) Iij,h(13)
Iij,h(2) Iij,h(6) Iij,h(10) Iij,h(14)
Iij,h(3) Iij,h(7) Iij,h(11) Iij,h(15)
Iij,h(4) Iij,h(8) Iij,h(12) Iij,h(16)
In the particular case where et is a martingale difference, the expression of I simplifies.
We then have
Iij,h(2) = ··· = Iij,h(5) = Iij,h(7) = ··· = Iij,h(10) = Iij,h(12) = ··· = Iij,h(15) = 0
and for h = 0, when i = j we have
Iii,0(1) = Ee 2 1t e2 1t−i σ−4
11 α2(i−1)
01 , Iii,0(6) = Ee 2 1t e2 2t−i σ−4
11 α2(i−1)
02 ,
Iii,0(11) = Ee 2 2t e2 1t−i σ−4
22 α2(i−1)
01 , Iii,0(16) = Ee 2 2t e2 2t−i σ−4
22 α2(i−1)
02 ,
⎞
⎟
⎠ .