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

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Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 114 when k ′ = 2(k − 1) + j ′ 2 and of the form Ee1t−he1t−j−he2tej ′ 2 t−i when k ′ = 8 + 2(k − 1)+j ′ 2 . Similarly we have M12,ij,h = M24,ij,h = M36,ij,h = M48,ij,h where the (k,k ′ )-th element is of the form Ee1t−he2t−j−he1tej ′ 2 t−i when k ′ = 2(k −1) +j ′ 2 and of the form Ee1t−he2t−j−he2tej ′ 2 t−i when k ′ = 8+2(k−1)+j ′ 2 . We also have M19,ij,h = M2 11,ij,h = M3 13,ij,h = M4 15,ij,h where the (k,k ′ )-th element is of the form Ee2t−he1t−j−he1tej ′ 2t−i when k ′ = 2(k − 1) + j ′ 2 and of the form Ee2t−he1t−j−he2tej ′ 2t−i when k ′ = 8 + 2(k − 1) + j ′ 2 . We have M1 10,ij,h = M2 12,ij,h = M3 14,ij,h = M4 16,ij,h where the (k,k ′ )-th element is of the form Ee2t−he2t−j−he1tej ′ 2t−i when k ′ = 2(k −1) +j ′ 2 and of the form Ee2t−he2t−j−he2tej ′ 2 t−i when k ′ = 8+2(k −1)+j ′ 2 ⎛ I2 ⊗λ ′ h = ⎜ ⎝ α h−1 1 0 0 0 02×4 α h−1 2 0 0 0 0 α h−1 1 0 0 02×4 0 α h−1 2 0 0 0 0 α h−1 1 02×4 0 0 α h−1 0 0 2 0 0 α h−1 02×4 1 . We have the 16×4 matrix 0 0 0 0 α h−1 and let the 256×16 matrix Fij = (I2 ⊗λ ′ i)⊗ I2 ⊗λ ′ j defined, by Fij(1,1) = Fij(5,2) = Fij(9,3) = Fij(13,4) = Fij(65,5) = Fij(69,6) = Fij(73,7) = Fij(77,8) = Fij(129,9) = Fij(133,10) = Fij(137,11) = Fij(141,12) = Fij(193,13) = Fij(197,14) = Fij(201,15) = Fij(205,16) = α i−1 1 αj−1 1 , Fij(52,1) = Fij(56,2) = Fij(60,3) = Fij(64,4) = Fij(116,5) = Fij(120,6) = Fij(124,7) = Fij(128,8) = Fij(180,9) = Fij(184,10) = Fij(188,11) = Fij(192,12) = Fij(244,13) = Fij(248,14) = Fij(252,15) = Fij(256,16) = α i−1 2 αj−1 2 , Fij(4,1) = Fij(8,2) = Fij(12,3) = Fij(16,4) = Fij(68,5) = Fij(72,6) = Fij(76,7) = Fij(80,8) = Fij(132,9) = Fij(136,10) = Fij(140,11) = Fij(144,12) = Fij(196,13) = Fij(200,14) = Fij(204,15) = Fij(208,16) = α i−1 1 α j−1 2 , Fij(49,1) = Fij(53,2) = Fij(57,3) = Fij(61,4) = Fij(113,5) = Fij(117,6) = Fij(121,7) = Fij(125,8) = Fij(177,9) = Fij(181,10) = Fij(185,11) = Fij(189,12) = Fij(241,13) = Fij(245,14) = Fij(249,15) = Fij(253,16) = α i−1 2 αj−1 1 , 2 ⎞ ⎟ ⎠
Chapitre 3. Estimating the asymptotic variance of LSE of weak VARMA models 115 and 0 elsewhere. We also define the 16 × 16 matrix Λij,h = Mij,hFij, with obvious notation, by Λij,h(1,1) = Λij,h(3,2) = Λij,h(9,5) = Λij,h(11,6) = Ee1t−he1t−j−he1te1t−iα i−1 01 αj−1 01 , Λij,h(1,3) = Λij,h(3,4) = Λij,h(9,7) = Λij,h(11,8) = Ee1t−he1t−j−he2te1t−iα i−1 01 α j−1 01 , Λij,h(2,1) = Λij,h(4,2) = Λij,h(10,5) = Λij,h(12,6) = Ee1t−he1t−j−he1te2t−iα i−1 01 αj−1 02 , Λij,h(2,3) = Λij,h(4,4) = Λij,h(10,7) = Λij,h(12,8) = Ee1t−he1t−j−he2te2t−iα i−1 01 α j−1 02 , Λij,h(1,9) = Λij,h(3,10) = Λij,h(9,13) = Λij,h(11,14) = Ee2t−he1t−j−he1te1t−iα i−1 01 αj−1 01 , Λij,h(1,11) = Λij,h(3,12) = Λij,h(9,15) = Λij,h(11,16) = Ee2t−he1t−j−he2te1t−iα i−1 01 αj−1 01 , Λij,h(2,9) = Λij,h(4,10) = Λij,h(10,13) = Λij,h(12,14) = Ee2t−he1t−j−he1te2t−iα i−1 01 αj−1 02 , Λij,h(2,11) = Λij,h(4,12) = Λij,h(10,15) = Λij,h(12,16) = Ee2t−he1t−j−he2te2t−iα i−1 01 αj−1 02 , Λij,h(5,1) = Λij,h(7,2) = Λij,h(13,5) = Λij,h(15,6) = Ee1t−he2t−j−he1te1t−iα i−1 02 αj−1 01 , Λij,h(5,3) = Λij,h(7,4) = Λij,h(13,7) = Λij,h(15,8) = Ee1t−he2t−j−he2te1t−iα i−1 02 α j−1 01 , Λij,h(6,1) = Λij,h(8,2) = Λij,h(14,5) = Λij,h(16,6) = Ee1t−he2t−j−he1te2t−iα i−1 02 αj−1 02 , Λij,h(6,3) = Λij,h(8,4) = Λij,h(14,7) = Λij,h(16,8) = Ee1t−he2t−j−he2te2t−iα i−1 02 α j−1 02 , Λij,h(5,9) = Λij,h(7,10) = Λij,h(13,13) = Λij,h(15,14) = Ee2t−he2t−j−he1te1t−iα i−1 02 αj−1 01 , Λij,h(5,11) = Λij,h(7,12) = Λij,h(13,15) = Λij,h(15,16) = Ee2t−he2t−j−he2te1t−iα i−1 02 α j−1 01 , Λij,h(6,9) = Λij,h(8,10) = Λij,h(14,13) = Λij,h(16,14) = Ee2t−he2t−j−he1te2t−iα i−1 02 αj−1 02 , Λij,h(6,11) = Λij,h(8,12) = Λij,h(14,15) = Λij,h(16,16) = Ee2t−he2t−j−he2te2t−iα i−1 02 αj−1 02 , and 0 elsewhere. We then deduce that Λij,hvec vecΣ −1 −1 ′ e0 vecΣe0 = (Iij,h(1),...,Iij,h(16)) ′ ,
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UNIVERSITÉ CHARLES DE GAULLE - LIL
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Résumé Dans cette thèse nous él
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Abstract The goal of this thesis is
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Table des matières Résumé ii Abs
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4.6 Limiting distribution of the po
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4.12 Empirical size (in %) of the s
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Chapitre 1 Introduction Lorsque l
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Chapitre 1. Introduction 3 Pour les
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Chapitre 1. Introduction 5 VARMA(p0
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Chapitre 1. Introduction 7 obtenir
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Chapitre 1. Introduction 9 les outi
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Chapitre 1. Introduction 11 1.1.1 E
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Chapitre 1. Introduction 13 Propri
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Chapitre 1. Introduction 15 H7 : Po
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Chapitre 1. Introduction 17 Défini
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Chapitre 1. Introduction 19 où λ
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Chapitre 1. Introduction 21 Remarqu
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Chapitre 1. Introduction 23 où Eij
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Chapitre 1. Introduction 25 La dém
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Chapitre 1. Introduction 27 Théor
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Chapitre 1. Introduction 29 La sél
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Chapitre 1. Introduction 31 La dém
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Chapitre 1. Introduction 33 de BP d
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Chapitre 1. Introduction 35 Théor
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Chapitre 1. Introduction 37 Les exp
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Chapitre 1. Introduction 39 indispe
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Chapitre 1. Introduction 41 Lemme 1
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Chapitre 1. Introduction 43 En util
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Chapitre 1. Introduction 45 I11 = 2
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Chapitre 1. Introduction 47 ceux-ci
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Chapitre 1. Introduction 49 Remarqu
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Références bibliographiques Ahn,
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Chapitre 1. Introduction 53 Hannan,
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Chapitre 2 Estimating structural VA
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Chapitre 2. Estimating weak structu
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Page 75 and 76: Chapitre 2. Estimating weak structu
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Chapitre 5. Model selection of weak
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