Modelli autoregressivi e carte di controllo ewma: effetto della correzione della stima dei parametri
Testo completo
(2)
(3)
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(6) . . . . ǣ . ǣǤ ǣ . ʹͲͲͺȀʹͲͲͻ.
(7) . . .
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(12) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͵ .
(13) ͳǣ
(14) ǯ
(15)
(16)
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(18) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ ͷǤͷ
(19) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻ ͷǤ ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ ͷǤȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷͺ ͷǤͺȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͶ ͷǤͻȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͺ ͷǤͼǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͽ .
(20) ʹǣ
(21)
(22)
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(27) ǯǣ
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(31) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͵͵ Ǥͷ ȋȌǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ Ǥ ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻ Ǥ ǣǤǤǤǤǤǤǤǤǤǤǤǤǤͺͶ Ǥͺ ǤǤǤǤǤǤͺͼ .
(32) ͵ǣ
(33) .
(34)
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(38) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͶͺ Ǥͷ ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͺ; Ǥ ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻͷ . ϭ.
(39)
(40) Ͷǣ
(41)
(42)
(43)
(44) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͷ ͺǤͷ ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͻͽ ͺǤ
(45) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͼͶ .
(46) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤ͵ .
(47) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͶ .
(48)
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(50) ǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤǤͺ͵ . Ϯ.
(51)
(52) .
(53) ǡ ̵ ǡ ̵ Ǥ ǯ ° ǯ ǡ ǡǡ °Ǥ
(54) ǡ ǡ Ǥ ǡ Ǥ ǡ °ǡ° ǡ Ǥǯ ǡ° ǡ ǡ î Ǥǡ ǡ Ǥ ǯ ° ǯ . ϯ.
(55) Ǥ ǯ ȋȌ ǡ Ǥ
(56) ̵ ǡ ȋȌȋȌǡ Ǥ ̵ ǡ Ǥͳ ǯǡ Ǥ ʹ ȋȌǡǡ Ǥ
(57) ͵ ǯ Ǥ
(58) Ͷ ǡ Ǥ ϰ.
(59) ͳ ǯ ͳǤͳ
(60)
(61) ǣ Ǧ ǡ ǡ ǡ Ö ǡ î ǡ ͳǤ ǡǡ °ǯǤ ǡ ǯ° ǣ ͳǤ ǯǦ ȋȌǢ ʹǤ ǯǦ ȋ ȌǤ
(62) ǡ ǯǦǡǯ° ǡ ǯǯ . ȋ ͳ ǡ…ǡ Ȍ = ∏ ȋ Ȍ =ͳ. ͳ. ǡ ° ǡ Ǥ. ϱ.
(63) ° ° ȋ ȌǤ ° ǡ Ǥ
(64) ǡǡ Ǥ ǡ Ǥ
(65) ǡ ǡ î îǯǦͳ ǢÖǡ ǡ Dz ±dzǤ ° ʹǡ Ǧ ǡ ± ǯ ǯ ǡ °Ǥ ǯ ° Ǥ ǡ ǡ Ö ǣ ° Ǥ ǡ ǯ ǡ ° Ǥ . ʹ.
(66) ǡ °ȋ°Ȍ Ǥ. ϲ.
(67) ǣ Ǧ xt ǡ ͳǤ ° f (..., xt -1 , xt , xt +1 ,...) Ǣ ʹǤ ° Ǣ xt ǡ ( xt , xt +1 ) ¿ Ǣ ¿ Ö î Ǣ ͵Ǥ ǡ ° ǡ ǡ E ( xt ) = µ t , Var ( xt ) = σ t , Cov ( xt , xt -1 ) = γ t ,t −1 ¿Ǣ 2. ͶǤ ǡ ° ȋ Ȍ Ǥ Ǥ Ö ǡǣ ͳǤ ȋ Ȍ ° ǡ Ö Ǣ ǡ ǯ° ǡ Ǥ ʹǤ ǡ ° . ϳ.
(68) ǡ ° ǡ ǯǤ ǣ Ǥ ǣ ȋ Ȍ Ǥ ǡ Ǣ ǡ ǯǡ Ǥ Ö Dzdz ǡ k. Wt = ( xt ,..., xt + k -1 ) Ǥ ° ǡ. ǡǡÖǤÖ Ö k. k. k. ǡ Wt ° Wt +1 ǡ Wt + 2 ¿Ǥ Ǥ
(69) ǡ ° Ǥǡ ǡʹǣ 2. Wt = ( xt , xt + k ) ǡ ͵Ǣ E ( xt ⋅ xt + k ) ǡ ǯȋ ȌǤ ǡ ǯǢ ǡ Ö Ǣ ǡ Ǥ
(70) ǡ ǡ ǣ ǡ ǡ ° ͵. 1 Ǥ. ϴ.
(71) °ǡ ° Ǥ ° ǡ ° Ǥ ǡ ǡ ǡ° Ǥ ǯ ǡ ° ǣ ° ¿ ¿ ǡ ǡ ± Ǥ
(72) ǡ ǡ ° ǯ ǯ ǯǤ ° ǤǡÖ ° Ǣ ǡ° ° Ǥ
(73) ° ȋ Ȍǣ 1 n ∑ Cov ( xt , xt -k ) = 0 n →+∞ n k =1 lim.
(74) ǡ ° ǡ° Ǥȋ Ȍ ǡ ϵ.
(75) ° ǡǯ ° ǡ ǡ ǯ Ǥǡǡ xt ρǡ ° ρÖ ±Ǥ
(76) ǡÖ ǯ° ° Ǥ ǡ ǯ ǡ ǯ ° ǡ Ǥ ǡ xt ρ ɐʹǤ
(77) ǡ ǡ γ k = E[( xt - µ )( xt -k - µ )] . Ǥ Ǣ ǡ ǡ γ ( k ) = γ k Ǥ ± ǯ Ͳ ° ° γ k = γ - k ǡ°Ǥ° ǡ ρ0 =. γk γ = k2 γ0 σ. ǡͲǡ ǡ ǯ . ϭϬ.
(78) Ǥ
(79) ǡ γ 1 ≠ 0 ǡ f ( x t xt -1 ) ≠ f ( x t ) . E ( x t xt -1 ) ≠ E ( x t ) Ǥ. ǯ x t -2 ǡ x t -3 Ǥ Ǧͳǡ I t -1 Ǥ . ϭϭ.
(80) ͳǤʹ .
(81) î Ǥ ǡ Ǥ ǡǡ ǡ ǯǤ ǡǡǤ ǡ Ǥ ǯ ȋ Ȍ Ǣ° ǡ x t x t ǡǤ
(82) ǡ Bx t = x t -1 Ǥ. ǯ ǡ n. B x t = xt -n Ǥ Ͳ αͳǤǯ° ǡ ǡ B (ax t + b ) = aBx t + b = ax t -1 + b Ǥ. î ° ǡ ǡ Ǥ ǯǤ ǯ ǡ ° ° ȋDz dzȌǤ °î Öǣǡ° . ϭϮ.
(83) Ǣ ȋ ° Ȍǡ ±Ǥ Ö î ǣ ǡ Ǧ εt ǡ ǣ E (ε t ) = µ E (ε t2 )=Var (ε t )=σ 2. γk = 0. . per k > 0. ° ǡ ǡ Ǣ ǡ ǡ ǯ ǯǤ ǡ Ǥ Ö ǡ ȋ ε t ǡ ε t +k Ȍǡ¿Ǥ ǡǡ° Ǥ
(84) ǡ ǡ ǡ ǯǤ
(85) ǯǤ . ϭϯ.
(86) ͳǤ͵ȋȌ. ǡ ȋ Ȍǡ° q yt = ∑ θi ⋅ε t -i = C (B )ε t i =0. ȋȌ ° ǯ εt ° Ǥ ǡǡ C (0)=θ0 =1 ǤȋȌ° ǡ yt ° ȋȌǤ ǡ ǣ q q E (yt ) = E ∑ θi ⋅ε t -i = ∑ θi ⋅ E ε t -i = 0 i =0 i = 0. ͲǤ ǡ ǯ ǡ ǡǡ° ͲǤǡ°î ǡ xt E ( xt )= µt Ö yt = xt - µt Ǥ yt ° ǡ yt xt Ǥ. ǡ ǡ 2 q Var (yt )= E (yt )=E ∑ θi ⋅εt -i i =0 2. ϭϰ.
(87) ǡ ǣ 2. q q q ∑ θi ⋅ε t -i = ∑ θi2ε t2−i + ∑ ∑ θi θ j ε t -i ε t - j i = 0 j ≠i i =0 i =0. ǡ ° ǡ ± q q q q E (yt 2 )=E ∑ θi2ε t2−i = ∑ θi2E (ε t2−i )= ∑ θi2σ 2 =σ 2 ⋅ ∑ θi2 i =0 i =0 i =0 i = 0. q. 2 ∑ θi < +∞ ǡ °Ǥ
(88) ǡ i =0. ǣ ǯ ° q q E (yt yt +k )=E ∑ θi εt -i ∑ θ j εt - j +k i =0 j =0 . q = ∑ θ i i =0. q ∑ θ j E (εt -i εt - j +k ) j =0 . ǡ E (εt -i εt - j +k )=σ 2 ε ή Ͳ ǡ ± ǯ. ǣ 2. q. γ k = E(yt yt+k ) = σ ∑ θi θi+k i=0. θi =0 ιǤ ǣ •. ǯ ° ǡαͲǢ. ϭϱ.
(89) •. εǡ Ǥ. ȋȌǡ ǡ ° î î ° Ǥ ǯ Ö Ǣ ǡ ǡ ǯ ȋ Ȍ ° q. 2 ∑ θi < +∞ Ǥ i =0. ǯî ǡ ȋͳȌ ǣ ǯ yt ° ͳǤͳǤǡ θ =0 ° Ǥǡ θ î ȋ DzdzȌ ȋ ǯȌǤ ͳǤͳǣ ȋͳȌǦ θ =0 ȋȌ. . ϭϲ.
(90) ǡ î Ǥ î ȋͳȌǡ Ǥǡǯ ͳ ȋͳȌ° θ Ǥ 1+θ 2. ρ1 =. ° ͳǤʹǢÖ ρ1 °ͲǤͷǡ θ =1 Ǥ ǡ ǡǤ
(91) ǡ ͳǤ ǣ ȋͳȌǡ ȋǡ θ Ȍǫ ǡ ȋ Ȍ ǯ Ǥ ¿ǡ ǣ ° ȋͳȌǡ ° ǡ ǯ Ǥ
(92) ǣ ρˆ1. p →. θ 1+θ 2. . ± ° θ ǡ θ ǡ ǯ . ϭϳ.
(93) ͳǤʹǣȋͳȌǦ θ . θˆ 1+θˆ2. ρˆ1 =. °ǣ θˆ =. (. ). 1 1- 1-4 ρˆ12 Ǥ 2 ρˆ1.
(94) ǡ° ǯ ȏǦͳǡͳȐ ± ǡ Ǥ ǡ ¿Ǥ1 î ǡ Ö ǡ ǡ ȋȌǤ î ǡ . ϭϴ.
(95) Ǥ ° ǡ ǣ ǡ ± ǡ ° Ǥ°ǣ ǡ ǡ Dzdz ǡ ȋ ȌǤ 1 ǡ ǡ ǡ Ǥ . ϭϵ.
(96) ͳǤͶȋȌ. ǯ ° Ǥ ǡ ǡî ǡ ±ǯ° ǡ î Ǥ
(97) Ǥ y = ϕ1y1 + ... + ϕ p y p + ε t t. ° ǡ ǡ Ö ǡ ° Ǣ ǡ ǯ Ǥ
(98) ±ǡ ° ǯ ǯǡ ° ǯ ǯ Ǥ
(99) A(B )yt = ε t . ȋȌ° ȋͲȌαͳǤ ǡ î ǣ αͳ Ö yt = ϕyt-1 + εt. →. (1 − Bϕ ) yt = εt . . ϮϬ.
(100)
(101) ȋͳȌ ǣ ° ǡ Ǥ ɊǤ ǣ µ = E(yt ) = ϕ E (yt-1 ) + E( ε t ) = ϕµ . ǯ Ö ǣ Ɋ α Ͳǡ ° ɔǡ ɔαͳǡǯ° Ɋǡ ° Ǥ
(102) ǡ ± ȋȌαͲ ° ͳǤ ȋȌ ° Ǥ ǡ Ǥ ȁɔȁ δ ͳ ȋ Ȍǡ A(B) -1 = (1- Bϕ ) = 1+ Bϕ + B 2 ϕ 2 + ...= C(B) -1. . yt = (1+ Bϕ + B 2 ϕ 2 + ...) ⋅ ε t = C(B) ⋅ ε t . ° Ʌαɔǡ Ǥ ǡ Ǣ ɐʹǤ ǡ ǡ . Ϯϭ.
(103) 2 V = E(yt2 ) = E (ϕ yt-1 + εt ) = ϕ 2V + σ 2 + 2ϕ E(yt-1εt ) . ǯ ° Ͳǡ ȋȌǤ V=. σ2 1− ϕ2. .
(104) ǡ ǯ ȁɔȁ δ ͳ ǡ î ǯ Ǥ ȋͳȌ ǡ ° ȁɔȁεͳǤ ǡ ° ǡ ɐʹǡ °. ℑ Ǧͳ Ǥ° . ɐʹǡ°îɔ° ͳǣ î ° ǡ î Ǥ Ǧͷ ǯ î ° Ǥ ǣ ǯ Ͳ°ǡ Ǣǯ ͳ° γ k = E(yt yt-k ) = E[( ϕ yt-1 + εt )yt-k ] = ϕγ k-1 . γ k = ϕk. σ2 1−ϕ 2. . . ϮϮ.
(105) ǣ ρk = ϕ k . °ǯǣ ǡ ǡ îȋȌǡîȋȌ°ɔǡ ǯ ɔ Ǥ
(106) îǡ ǯ ǡ ɀ ° ͲǤ ȋȌǣ ͳǤ ǡ Ǣ ʹǤ Dz dzͲǡɊȀȋͳȌǡȋͳȌ° ȋȌαͳ ±α Ǥ ǯ ǯ ͳ° ȋȌ Ǥ
(107) ǡ ǯ Ǥ Ǥ . Ϯϯ.
(108) ͳǤͷȋȌ. Ǥ ȋǡȌ° A(B)yt = C(B)εt. . °ǯȋȌ°ǯȋȌǤ Ǥ
(109) ȋαͲ αͲ ȌǤ ȋȌ ͳ ǡÖ ǡǣ yt = A(B)-1C(B)εt = D(B) ⋅ εt . ȋȌ°εͲǤ ȋȌ° ± Ǥǡ ȋȌ ° ǡ ȋεͲȌ A(B)yt C(B)-1 = A(B) ⋅ yt = εt. .
(110) ǡ °Ǥ ȋǡȌ Ǥ ǯ ° ǯ ǡ ° ɊȀȋͳȌǤ ǫ
(111) ǡ ǡ Ö Ǥ ǯ°. Ϯϰ.
(112) °ǡǡǤ° ǡ ǣ ǡ ǯ ȋî ȌǤ ǡ Ǥ
(113) ǡ yt = F(B)εt . ȋȌ Ǥ Ö Ö ǯ ȋȌǢ ǡ Ö ȋȌȋȌ ȋȌ F(z) ≃. C(z) A(z) . ǯ ǡ A(B)yt = D(B) ε t ǯǡǣ A(B)yt = D(B) ε t *. *. εt =. A (B ) ⋅ F (B )ε t D (B ). . Ϯϱ.
(114)
(115) ε t °ǡǡǡ *. ǡÖ Ǥ ε t *. °î Ǥ
(116) ǡ ǯȋȌ ȋȌ ǡ ǡ Ǥǡ ε t Ǥ ǯ *. ǡ ǡ ǡ ǡ ǡǤ Ϯϲ.
(117) ͳǤ. Ǥ ȋ Ȍǡ Ǥ ° Ǥ ǡ ± °ǤÖ Ǥ ° ǡ Ǥ ǡ Ǥ Ǥ ǡ ° ǡ ȋͳǡǤǤǤǡȌǡ Ǥ ǡ ° ǣ 1 L(ψ) = f(x; ψ) = T ∑ 2π . -. 1 2. {. }. 1 exp - (x-k)' ∑-1 (x-k) 2. °ȋͳǡǤǤǤǡȌ Ǣ ∑ ǡ ψ Ǥǡǯ ∑ ° ǯ ȁ Ǧ ȁ ǡ ǡ ° Ǥ 1 ǡ Ǥ
(118) ǡ ǡ . Ϯϳ.
(119) ȋ Ǧ ȌǤ
(120) ǡ ǡ ȋɗȌǡ °Ǥ
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