ARIMA

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1 ARIMA

2 ARIMA I

3 Jim Slaer The Zulu Principle ARIMA ARIMA 95% 1 ARIMA 0,1,2 95% % 2 ARIMA 5,1,3 95% 3.7% 3.7% % 95% ARIMA ARIMA II

4 Absrac In his research, we use mehod of he Zulu Principle of Jim Slaer o selec he company. The daa of his research come from he Over-he-Couner during January 2003 o December 2008, and use he ARIMA o find he bes prediced model of his company. Moreover, his mehod may also apply in mahemaic exams scores o find he bes prediced model. And using he 95 percen confidence inerval and mean absolue percen error o measure he effec upon prediced of he model, which one is he bes prediced model. The empirical resul shows ha 1 The bes prediced model of S.Shine is ARIMA ( 0,1,2). I prediced value all fall in 95% confidence inerval, and mean absolue percen error is % 2 The bes prediced model of mahemaics examinaion scores is ARIMA ( 5,1,3). And, he prediced value all fall in 95% confidence inerval, and mean absolue percen error is 3.7%. The S.Shine is compared wih mahemaics examinaion scores, and find is error is smaller han mahemaics examinaion scores. Bu boh of hem are all fall in 95% confidence inerval. So, he prediced model of S.Shine is beer han mahemaics examinaion scores. Keywords: The Zulu Principle, ARIMA, Whie Noise, Augmened Dickey-Fuller III

5 ARIMA IV

6 2-2-1 ESACF p p ARIMA ( 0,1,2 ) p p p p ARIMA ( 5,1,3 ) p p V

7 ACF PACF ACF PACF ACF PACF VI

8 20 Jim Slaer auoregressive inegraed moving average 1

9 ARIMA ARIMA ARIMA ARIMA ARIMA ARIMA ARIMA ARIMA

10 2009 ARIMA equaing 3

11 4

12 ARIMA Jim Slaer Sunday Telegraph 68.9% 3.6% 1975 Slae Walker Securiies The Zulu Principle Black Leisure The Bale of Isandlwana 5

13 Price-Earnings/Growh Raio PEG % % % 5 =,

14 =, = 100%,

15 50% 40% = 100%, = %, / ( + ) 7.5% 10% 20% 5% 20% 8

16 ,000 1, ~ % 15% 2 15% 9

17 4% 20% 4% 20% 8 10

18 ARIMA auoregressive inegraed moving average ARIMA Box & Jenkins 1976 ARIMA auoregression AR( p) inegraed I moving average ( MA q) 2005 X shock a p { } X X = C + φ X + φ X + + φ X + a, p p Φ B ) X = C + a, p ( C φ, φ,..., φ ) ( 1 2 p 2 p Φ ( B) = 1 φ B φ B φ B B back-shif operaor p p a ~ WN 0, σ a X auoregressive model AR p AR( p) φ ( B) = p 11

19 X = X X 1 X, = 1 B ( 1 B) d ( 1 B) d d { X } = 1 ARIMA 1992 { } X X = µ + a θ a θ a, q q X = µ + θ B) a, q ( θ, θ,..., θ ) θ ( B) 1 θ B θ B q ( 1 2 q 2 q = 1 a ~ WN 0, σ a X moving average model MA q MA( q) inveribiliy φ ( B) = 0 θ < 1 auoregression moving average model ARMA p,q q q 12

20 X = C + φ X + φ X + + φ X + a θ a θ a θ a, p p q q φ B) X = C + θ ( B) a, p ( q φ, φ,..., φ ) θ, θ,..., θ ) C ( 1 2 p ( 1 2 q 2 a WN 0, σ X ~ a { } mixed auoregressive moving average ARMA p,q ARMA( p,q) ( ARMA p,q) φ ( B) = 0 p ϕ ( B ) = 0 ARIMA ARMA p,q d ARMA p,q ( p,d,q) p ( p) AR p ARIMA ARIMA( p,d,q) d ( B)( B) X θ ( B) a, 1 q φ = order 0 p q MA( q) ( φ1, φ2,..., φ p q a WN 0, σ d q ~ a B ) θ, θ,..., θ ) ( 1 2 q ( 1 2 p ) 2 p ϕ B = 1 ϕb ϕ B ϕ B φ p ( B ) = 0 p ( 1 2 q ) 2 q θ B = 1 θb θb θb θ ( B) = 0 q q ARIMA p,d,q q = 0 { } d X ( AR p) ARI( p, q) ARIMA( p, d,0) X p, d { } p = 0 d X ( MA q) X ( d, q) inegraed moving average ( IMA d,q) ARIMA( 0,d,q) 13

21 1992 saionary ime series E( X ) = µ Cov ( X r, X s ) = Cov ( X + r, X + s ) nonsaionary { } X k X k Cov( X, X + k ) ρ k = ρ( X, X + k ) =, σ σ auocorrelaion funcion ACF ρ ρ k X ACF sample auocorrelaion funcion X + k X + k k ρ κ = 1 n n k = 1 ( X 1 n n = 1 X )( X ( X + k X ) 2 X ) ˆ γ k = ˆ γ 0, γˆ k + k covariaion ˆ γ 0 k ρ k s ( ˆ ρ ) = (1 + 2 ˆ k ρi ) n i= 1 H H 0 1 : ρ k = 0, : ρ 0 0 k H X X + k X X + k X + 1, X + 2,..., X + k 1 X X + k 14

22 Cov( X, X + k X + 1, X + 2,..., X + k 1 ) ρ kk = φkk =, σ ( X ) σ ( X ) parial auocorrelaion funcion PACF ρ + k whie noise X i.i.d 2 Gaussian process ( a ~ WN 0 σ ) kk, 2 0 σ { } a { } X a X = a i. i. d ~ WN ( 2 0, ), a a σ a X 0 0 zero-mean whie noise simple whie 1995 inegraed order augmened dickey-fuller ADF Fuller 1976 Dickey & Fuller 1979, 1981 Ordinary Leas Squares OLS 2005 X = φ X + a, X = α + φ X + a,

23 X = α + φ X + β + a, α (drif) (rend) φ 1 a 2 ( 0, ) iid ~ a a σ F F F φ < 1 φ = φ > 1 1 AR(1) p AR iid ~ a 2 a 0, σ Said & Dickey 1984 lag ARAM p X p 1 X 1 + β j X j+ 1 + a, j= 2 = φ X p 1 X 1 + β j X j+ 1 + a, j= 2 = α + φ X = α + φ X + α + β X + a j j+ 1 j= 2 Η φ 0 Η φ 0 0 : 1 = 1 : 1 < p 16

24 Tsay & Tiao 1984 ARMA p,q ieraed auo regression ARMA p,q AR exended sample auocorrelaion funcion ESACF ρˆ ( m) j p q 2003 { X } ARMA( p,q) Y m = X ˆ φ i X i, ( j ) ( j ) i= 1 φˆ j ( AR m) i j i m j ρˆ Y j { } p q j = qq, + 1, L ( j Y ) { } m Tsay & Tiao 1984 ρˆ ρ 0, q p p p ρ = 0, j = q+ 1, q+ 2, L j p p ρ 0, 0 m p< j q, jm p p ρ c( m p, j q), 0 j q m p. jm j MA q m ESACF ρˆ m + 1, j j k = 0,1,2,L j = 1, 2, L p, q ARMA 17

25 2-2-1 ESACF { } ( m) ESACF ( ARMA p,q) ρˆ ESACF m j AR ESACF 1 ARMA( p,q) ESACF j 0 p, q 2 0 X X Mean Absolue Persen Error X 100 MAPE = l l a X n+ = 1 n+ %, a = X X n l 18

26 ARIMA ARIMA 2004 Campbell 1991 Vecor auoregressive model, VAR Marginal-Likelihood crieria

27 1995 Seemingly Unrelaed Regression model, SUR ARIMA ARIMA ARIMA ARIMA ARIMA(3,1,0) 20

28 2006 Wood & Dasgupa 1996 ARIMA (, ) (, ) ARIMA ( 210, ) ARIMA 01, 2 ARIMA 211,, ARIMA 01,, 2 ARIMA 01,, 2 ARIMA ARIMA ARIMA 2003 ARIMA 21

29 12 ( p, 10) ARIMA, ( 210) ARIMA,, ARIMA 110,, ARIMA ( 210, ), ARIMA 210,, ARIMA 110,, ARIMA 310,, ARIMA ARIMA MAPE 7% ARIMA ARIMA ARIMA 22

31 ARIMA ARIMA d q p 1999 ESACF Η : 0 Η φ 0 Η φ 0 0 : 1 = 1 : 1 < 24

32 Η : φ 0 Η : φ 0 φ p 0 = 1 α = 0.05 p µ µ µ e = X Xˆ = 1,2, L, n 2 1. e WN 0, σ 2. 2 ~ a 2 e ~ IID 0, σ a 3. e ~ N 0, σ a 1995 H H 0 1 p α = 0.05 p α =

33 OTC 547 SAS ( 9.1 ) ARIMA 100 ARIMA SAS ( 9.1 ) 26

34 11 ARIMA ARIMA ARIMA Ligh Emiing Diode LED

35 % * * * * * % % % % % % % *

37 = %, % 45.88% 27.27%

38 7 7 Liquid Crysal Display LCD LED % 7 471% 226% 625% 100%

39 32 150% 7 200% 40% 7 30% 12% 16% 19% 2007

40 X X X X X X X X X X X X X

41 % 26.23% 7.48% % X % X % % % % %

42 X % 7 40% 40% 28.59% 26.94% % % 10.35% 35

43 20% 10% 15.65% 14.25% 11.15% 10.02% 0.43% 0.04% LED 10 LED IC % DC/DC MBI6651 IC MR LED LED LED LED Thermalake DIY 36

44 QFan QFan Plus PF CPU NVIDIA Enhusias Sysem Archiechure ESA PC IC LCD LCD LCD LCD-TV 37

45 NB Nebook % % 43.7% % LED IC ISO9001 CE GMP FDA510(k) CMDCAS TICON 38

46 ASP 50% 2008 H % %

47 POSIFLEX 20 Poin on Sale POS ISO14001 OHSAS REACH-SVHC %

48 % 10.35% 4% % 0.43%

49 4-2-1 X X

50 Y = ln X 43

51 Y = ln X ACF PACF 44

52 4-2-4 ACF PACF ACF PACF X +1 X p α = 0.05 H 0 45

53 ACF 46

54 4-2-8 PACF ACF ACF 2 24 X PACF 2 24 X X ARIMA ESACF 1,1,1 ARIMA 0,1,2 47

55 4-2-4 p p ARIMA ( p,d,q) α Pr ChiSq ARIMA ( 1,1,1) ARIMA ( 0,1,2) * Pr ChiSq µ θ φ µ θ θ * p α = 0.05 ARIMA 1,1,1 ARIMA 0,1,2 ( 1,1,1) 2 ARIMA θ p = 0.05 ARIMA 0,1,2 α θ ARIMA ( 0,1,2 ) * ARIMA ( 0,1,2 ) ARIMA ( 0,1,2 ) α Pr ChiSq p p Pr ChiSq µ θ * θ p α = 0.05 θ 2 ARIMA 0,1, (2) ARIMA 0,1,2 X ( ) a, = a

1 5 2009 1 21 4-2-9 4-2-6 X Y Y X Y % 1 108 4.682131 4.6686 106.55 0.28938 2 109 4.691348 4.6787 107.63 0.26998 3 109 4.691348 4.682 108 0.19951 4 105 4.65396 4.6853 108.34 0.67328 5 107 4.672829 4.

56 X Y Y X Y %

57 15 95% % 2% MAPE % 0 X X

58 4-3-1 X

59 4-3-2 ACF PACF ACF 0 X 3 X PACF 3 9 X

60 p = α = 0.05 p H 0 ESACF ARIMA 513, (, ) ARIMA ( 611,, ) ARIMA ( 711, ), = 0.05 α ARIMA ( 513,, ) ARIMA ( 611,, ) ARIMA ( 711, ) (, ), ARIMA 611, ARIMA 711,, p ARIMA 513,, θ θ 2 3 φ 2 φ θ θ 2 3 φ 2 φ 5 p ARIMA 513,, X ARIMA 513,, ( ) X + ( ) X + a ( ) a 0. a, = % % 95% 53

61 4-3-2 p p ARIMA ( p,d,q) α Pr ChiSq Pr ChiSq µ ( 513) ARIMA,, ( 611) ARIMA,, ( 711, ) ARIMA, * θ θ 0.001* 2 θ * 3 φ φ * φ φ φ * µ θ φ φ φ φ φ φ µ θ φ φ φ φ φ φ φ

4-3-3 ARIMA ( 513, ) ( 513, ) ARIMA, *, p p α Pr ChiSq 0.05 0.4601 Pr ChiSq µ 0.9962 θ 2 3 0.0001* θ 0.0001* φ 2 φ 5 0.0001* 0.0001* 4-3-4 4-2-4 X X % 1 87 87.8983 1.03257 2 91 90.

62 4-3-3 ARIMA ( 513, ) ( 513, ) ARIMA, *, p p α Pr ChiSq Pr ChiSq µ θ * θ * φ 2 φ * * X X %

63 56

64 ARIMA( 0,1,2) % 95% Y = ln X ACF PACF 0 ACF PACF 2 X 2 24 ACF PACF ESACF ( 1,1,1) ARIMA ARIMA( 0,1,2) ARIMA( 1,1,1) ARIMA 0,1,2 θ θ ARIMA 0,1, % % 2 57

65 ARIMA( 5,1,3) 3.7% 95% ACF X PACF X X 3 X ACF PACF (, ) (, ) ARIMA ( 7 11, ) ESACF ARIMA 513, ARIMA 611, 3 9, ARIMA 513,, (, ) (, ) ARIMA ( 513, ) ARIMA 611, ARIMA 7 11,, θ 2 θ 3 φ 2 φ 5 ARIMA 513,, ARIMA 513,, ARIMA 513,, 3.7% 95% 3.7% % 95% ARIMA 58

66 0 ARIMA 3 5 ARIMA 59

67 hp://w.sock.yahoo.com/news_conen/url/d/a/090311/3/1e0xn.hml Q2 hp://w.sock.yahoo.com/news_conen/url/d/a/090422/2/1gh7j.hml ,1(2) ( ) 2003,19(2) hp://

68 NoID=50596&MMMediaType=ci_cusomer&source=rss hp://pchome.syspower.com.w/socknews/?newsid= &ca= ANFIS EPS 2.07 hp://w.sock.yahoo.com/news_conen/url/d/a/080430/1/xide.hml 2007 hp://diary.blog.yam.com/enrichlife Money Box, G. E. P. and Jenkins, G. M. (1976). Time series analysis forecasing and conrol. San Francisco: Holden-Day. Dickey, D. A. & Fuller, W. A. (1979). Disribuion of he Esimaors for Auoregressive Time Series wih a Uni Roo. Journal of he American Saisical Associaion, 74, Dickey, D., & Fuller, W. A. (1981). Likelihood raio saisics for auoregressive ime series wih a uni roo. Economerica, 49,

69 Fuller, W. A. (1976), Inroducion o saisical ime series, New York John Wiley & Sons. Said, S. E. & Dickey, D. A. (1984). Tesing for Uni Roos in Auoregressive Moving Average Models of Unknown Order. Biomerika, 71, Tsay, R. S. & Tiao, G. C. (1984). Consisen esimaes of auoregressive parameers and exended sample auocorrelaion funcion for saionary and nonsaionary ARIMA models. J. Amer. Sais. Assoc., 79, Wood, D., & Dasgupa, B. (1996). Classifying rend movemens in he MCSI U.S.A. capial index A comparison of regression, ARIMA and Neural Nework mehod. Compuer and Operaions Research, 23,

71 1-2 Y = X

72 ESACF 1-7 ARIMA( 1,1,1) 1-8 ARIMA( 0,1,2) 65

73 1-9 ARIMA( 1,1,1) 1-10 ARIMA( 0,1,2) 1-11 ARIMA ( 01, 2), 1-12 ARIMA ( 01, 2), 66

74 %

76 2-4 ESACF 2-5 ARIMA ( 513, ), 2-6 ARIMA ( 611, ), 2-7 ARIMA ( 711, ), 69

77 2-8 ARIMA ( 513, ), 2-9 ARIMA ( 611, ), 2-10 ARIMA ( 711, ), 2-11 ARIMA ( 513, ), 70

78 2-12 ARIMA ( 513, ), %

79 72

报告总结

报告总结 CFEF RR/04/0 CAViaR RR/04/0 004 CAViaR 00080 VaR VaR Engle Manganelli 999 VaR CAViaR Chow CAViaR B Engle Manganelli CAViaR Absrac: Value-a-Risk (VaR) has become a sandard ool o measure marke risk widely