L T n Lu t

Cc m hnh d bo n gi n

CHNG 13

D

BO V I M HNH ARIMA

D bo ti chnh d ng nh l m t khoa h c gp ph n lm cho chim tinh h c tr nn ng tin c y hn. Burton Malkiel, A Random Walk Down Wall Street (1985)1

M C TIU

B C C

Tm hi u m t s qu trnh ng u nhin Qu trnh ng u nhin c bi t nh nhi u tr ng, b c ng u M t s qu trnh ng u nhin c bi t Tnh d ng nhin M hnh d bo trn bi n chu i th i gian Hi u khi ni m tnh d ng c a qu trnh d ng khng c tnh ma ng u nhin. T c kh nng ki m M hnh d bo trn bi n chu i th i gian c nh c tnh d ng c a bi n chu i th i tnh ma gian v bi n i s li u n c tnh M hnh ARIMA d ng Phng php lu n Box-Jenkins C kh nng nh n d ng v xy d ng m hnh d bo trn s li u chu i th i M hnh ARIMA cho bi n chu i th i gian c tnh ma (SARIMA) gian d ng Tm t t nh n d ng m t s m hnh ARIMA N m phng php lu n Box-Jenkins ng v i ph n trong vi c xy d ng m hnh ARIMA Xy d ng m hnh ARIMA t m m Autobox cho s li u chu i th i gian khng d ng H n ch c a cc m hnh ARIMA Xy d ng ARIMA v i Eviews

Theo Ernst R.Berndt (1991). The Practice of Econometrics: Clasical and Contemporay. Addison-Wesley Publishing Company

1

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L T n Lu t

Cc m hnh d bo n gi n

13.1. Qu trnh ng u nhin Quan i m xy d ng m hnh d bo mang tnh th ng k c a chng ny l gi tr c a bi n s c n d bo tun theo m t qu trnh ng u nhin theo m t quy lu t no . Nhi m v c a nh d bo l xy d ng m t m hnh m t b n ch t qu trnh ng u nhin t o ra gi tr c a bi n s . Khc v i k thu t h i quy, tnh ng u nhin ch khng ph i quan h nhn qu ng vai tr quan tr ng trong qu trnh ny. 13.2. M t s qu trnh ng u nhin c bi t 13.2.1. Nhi u tr ng (White noise) Nhi u tr ng l m t qu trnh ng u nhin c trung bnh b ng 0, phng sai khng i v t t c cc h s t tng quan b ng 0. Nh l i cc gi thi t c a m hnh h i quy tuy n tnh c i n, n u tho mn cc gi thi t ny th ph n d c a h i quy tun theo nhi u tr ng.White noise10

8

6

4

2

0 1 -2 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

-4

-6

-8

-10

Hnh 13.1. M t nhi u tr ng theo phn ph i th ng c l ch chu n b ng 5. Khi m t bi n s c xc nh l m t nhi u tr ng th chng ta khng th xy d ng m hnh d bo m c d c th xy d ng m hnh xc su t m ph ng n. 13.2.2. B c ng u nhin (Random Walk) Tn g i b c ng u nhin th hi n tnh khng th on tr c c, qu trnh ny c v nh b c chn lo ng cho ng c a ng i say r u. Trong qu trnh tun theo b c ng u nhin, s thay i c a bi n s Y c rt ra m t cch c l p t m t phn ph i c trung bnh b ng 0. V y Yt = Yt 1 + t (13.1) V i E ( t) = 0 v E ( t s ) = 0, t s D bo cho th i o n t l 2

L T n Lu t

Cc m hnh d bo n gi n

Yt = E (Yt ) = E (Yt 1 + t ) = Yt 1 + E ( t ) = Yt 1D bo cho th i o n t+1 l Yt +1 = E (Yt + t +1 ) = E (Yt ) + E ( t +1 ) = Yt 1 V y d bo cho bao nhiu th i o n t i cng l Yt 1 . Tuy nhin, sai s c a d bo tng d n Sai s d bo cho th i k t+1 et +1 = Yt +1 Yt +1 = Yt + t +1 Yt = t +1 Phng sai c a c l ng var(et +1 ) = E ( t2+1 ) = 2 Sai s d bo cho th i k t+2 et + 2 = Yt + 2 Yt + 2 = Yt + t +1 + t + 2 Yt = t +1 + t + 2 Phng sai 2 2 2 var(et + 2 ) = E ( t +1 + t + 2 ) = E ( t +1 ) + E ( t + 2 ) + 2 E ( t +1 t + 2 ) = 2 2

(13.2) (13.3)

(13.4) (13.5) (13.6)

(13.7) V y m c d gi tr d bo khng i nhng sai s c a d bo tng d n. Tng t nh nhi u tr ng, khng th xy d ng m hnh d bo trn chu i tun theo b c ng u nhin. Gi s gi c a m t lo i c phi u Y phin u tin l 100 (ngn VND/c phi u) v tun theo b c ng u nhin v i nhi u tr ng tun theo phn ph i th ng v i l ch chu n l 5. V i ba l n m ph ng (ba k ch b n) gi 10 phin giao d ch c a c phi u say ny chng ta nh n c k t qu nh sau.The Drunken Stock130 120 110 100

(

)

'000 VND

90 80 70 60 50 0 1 2 3 4 5 6 7 8 9 10KB1 KB2 KB3

Phin

Hnh 13.2. Qu trnh b c ng u nhin 13.2.3. B c ng u nhin v i thnh ph n xu h ng

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L T n Lu t

Cc m hnh d bo n gi n

Bi n s hon ton tun theo b c ng u nhin t g p trong th c t , tnh hu ng chng ta g p th ng xuyn hn l bi n s c tnh xu h ng. Yt = Yt 1 + g + t (13.8) V i g l m c tng tr ng k v ng c a m t th i o n. Tr l i v d v gi c phi u, gi s c phi u Y tng trung bnh 2 ngn VND m t phin. Chng ta c th m ph ng qu trnh tng gi c a Y qua 10 phin nh sau. M c d chng ta vn c th d bo cho qu trnh b c ng u nhin c tnh xu h ng nhng phng sai c a d bo v n l n nh qu trnh b c ng u nhin thu n tu.

140 130 120 110

'000 VND

100 90 80 70 60 50 0 1 2 3 4 5 6 7 8 9 10

Phin

Hnh 13.3. Qu trnh b c ng u nhin c tnh xu h ng 13.3. Tnh d ng 13.3.1. S li u d ng v khng d ng Cc k thu t d bo trong chng ny d a trn gi thi t l qu trnh ng u nhin t o ra bi n s chu i th i gian c n d bo l b t bi n theo th i gian hay d ng. Khi bi n s tun theo qu trnh d ng th c th bi u di n chu i th i gian trong qu kh v tng lai b ng m t m hnh i s n gi n. M t bi n chu i th i gian d ng ph i c ba tnh ch t (1) Trung bnh khng (2) Phng sai khng i theo th i gian: E (Yt ) = Y , t2 i theo th i gian: var(Yt ) = E (Yt Y ) 2 = Y = 2 , t

[

]

(3) H s t tng quan khng

i theo th i gian: rk =

E[(Yt Y )(Yt k Y )]

2

= k , t

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L T n Lu t

Cc m hnh d bo n gi n

M t bi n s chu i th i gian ch c n khng c m t trong ba tnh ch t trn th c g i l chu i th i gian khng d ng. 13.3.2. Ki m (1) Nh n nh tnh d ng nh b ng th

N u s li u c tnh xu h ng th n khng d ng, v y n u m t cch tr c quan chng ta th y tnh xu h ng di h n th bi n chu i th i gian khng d ng. Ngoi ra trn th chng ta cng c th d dng nh n ra phng sai khng ng nh t. Ring tnh t tng quan th kh nh n bi t hn. th hnh 13.1 bi u di n t gi la M t 1-2002 n 92007. Chng ta th y trung bnh tng theo th i gian. V y t gi la M khng d ng.16400

16200

16000

15800

15600

15400

15200

15000

14800

14600

14400 Nov-02 Nov-03 Nov-04 Nov-05 Sep-02 Nov-06 Sep-03 Jul-02 Sep-04 Jul-03 Sep-05 Jul-04 Sep-06 Jan-02 Jul-05 May-02 May-03 May-04 May-05 May-06 May-07 Sep-07 Jan-03 Jan-04 Jul-06 Jan-05 Jan-06 Mar-02 Mar-03 Mar-04 Mar-05 Mar-06 Jan-07 Mar-07 Jul-07

Hnh 13.4. T gi VND/USD Ngu n: Ch s gi USD c a T ng C c Th ng K v tnh ton hon nguyn gi g c c a tc gi Hnh 13.2 bi u di n t l tng gi lng th c-th c ph m hng thng t 1-2002 n 92007. Chu i ny xem nh d ng v trung bnh nhng c v nh n vi ph m tnh d ng v phng sai v c u trc t tng quan. Ngoi ra chng ta c th th y tnh ma r r t. Gi lng th c-th c ph m tng cao vo thng 2 (t t c truy n) sau s t gi m trong thng 3, quy trnh ny l p i l p l i hng nm.

5

L T n Lu t

Cc m hnh d bo n gi n

6.00%

5.00%

4.00%

3.00%

2.00%

1.00%

0.00% Aug-02 Aug-03 Aug-04 Aug-05 Aug-06 Apr-02 Aug-07 Apr-03 Apr-04 Apr-05 Apr-06 Feb-02 Jun-02 Dec-02 Feb-03 Jun-03 Dec-03 Feb-04 Jun-04 Dec-04 Feb-05 Jun-05 Dec-05 Feb-06 Jun-06 Dec-06 Feb-07 Apr-07 Oct-02 Oct-03 Oct-04 Oct-05 Oct-06 Jun-07

-1.00%

-2.00%

-3.00%

Hnh 13.5. T l tng gi lng th c-th c ph m Ngu n: T ng C c Th ng K Trong th c hnh, tnh d ng v trung bnh l i u ki n quan tr ng nh t xy d ng cc m hnh d bo ng n h n, v phng php th cho chng ta k t lu n nhanh v tnh d ng c a chu i. (2) Php ki m Ki m nh nghi m n v (Unit root test) nh nghi m n v (Unit root test) theo Dickey-Fuller nh sau2

Gi s bi n Y tun theo qu trnh t h i quy b c 1

Yt = + Yt 1 + tv i v l cc tham s v t l nhi u tr ng.

(13.9)

i u ki n c n chu i d ng l 1 < < 1 . N u > 1 th chu i c xu h ng bng n (gi tr tng r t nhanh), hay chu i khng d ng. N u = 1 th chu i theo qu trnh b c ng u nhin c xu h ng, hay chu i cng khng d ng. V y gi thuy t khng cho ki m nh tnh d ng l = 1 v i thuy t l < 1 . T (13.9) rt ra h i quy cho ki m nh l

Yt = + ( 1)Yt 1 + t Yt = + Yt 1 + tN u Y tun theo qu trnh t h i quy b c cao hn th s d ng h i quy ki m2

(13.10) nh

Theo Brooks (2002), Introductory Econometrics for Finance, Cambridge University Press, trang 375.

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L T n Lu t

Cc m hnh d bo n gi n

p

Yt = + Yt 1 + i Yt i + ti =1

(13.11)

H0: H1:

= 0 , Chu i khng d ng < 0 , Chu i c tnh d ngnh l (13.12) se( )

N u < 0 th chu i c tnh d ng. V y tr th ng k ki mADF =

ADF r t gi ng tr th ng k t trong h i quy thng th ng, nhng gi tr t i h n c a tr th ng k ny khng th tra theo b ng th ng k t-Student m c xc nh b ng th c nghi m. Diskey-Fuller tnh ton v McKinnon hi u ch nh gi tr t i h n cho cc tr ng h p c th . Eviews bo co gi tr t i h n Mc Kinnon cho t ng php ki m inh c th . N u tr th ng k ADF nh hn gi tr t i h n Mc Kinnon th bc b H0, hay chu i th i gian c tnh d ng. Thao tc ki m nh trn Eviews nh sau: / View/ Unit Roots Test/ Augmented Diskey-Fuller/ Lags: p Chng ta th c hi n ki m nh trn 2 bi n t gi VND/USD v t l tng gi lng th cth c ph m trong v d ph n trn.

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L T n Lu t

Cc m hnh d bo n gi n

Hnh 13.6. K t qu ki m

nh Augmented Diskey-Fuller cho bi n VND/USD

K t qu ki m nh cho ADF = -1.515> -2.9 = ADF=5% cho k t lu n l khng bc b c bi n t gi VND/USD khng c tnh d ng v i m c ngha 95%. K t qu ki m nh cho ADF = -3.859< -2.9 = ADF=5% cho k t lu n l bi n tng gi lng th c-th c ph m INFL_FOODS c tnh d ng v i m c ngha 5%.

Hnh 13.7. K t qu ki m 13.3.3. Bi n

nh Augmented Diskey-Fuller cho bi n INFL_FOODS

i s li u khng d ng thnh d ng c bi n i thnh d ng b ng cch l y sai (13.13) (13.14)

M t bi n chu i th i gian khng d ng c th phn b c 1 I (1)t = d (Yt ) = Yt Yt 1

N u sai phn b c 1 cha d ng th ti p t c l y sai phn b c cao hn I ( 2)t = d 2 (Yt ) = d (Y )t d (Yt 1 )

Thng th ng chu i sai phn b c 1 c tnh d ng. V d chng c ch s gi tiu dng CPI c xu h ng tng nhng sai phn c a n l i c tnh d ng (hnh 13.8).

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L T n Lu t

Cc m hnh d bo n gi n

150

4 3

1402

130

1 0 -1

120

110-2

100 2002

2003

2004

2005 CPI

2006

2007

-3 2002

2003

2004

2005 DCPI

2006

2007

Ch s gi tiu dng (12/2001=100) M c tng CPI ( DCPIt=d(CPIt) Hnh 13.8. Ch s gi tiu dng v sai phn c a n Trong th c t th sai phn c a CPI (m c tng tuy t i) khng c ngha kinh t b ng khi ni m l m pht (m c tng tng i) d (CPI t ) INFLATION t = (13.15) CPI t0.03 0.02 0.01 0.00 -0.01 -0.02 -0.03 2002

2003

2004

2005

2006

2007

INFLAT IO N

Hnh 13.9. L m pht c tnh d ng 13.3.4. Hm t tng quan v hm t tng quan ring ph n (1) Hm t tng quan (AC-Autocorrelation) Hm t tng quan v i tr kE [(Yt )(Yt k )]

k =

E (Yt ) 2 E (Yt k ) 2

[

][

]

(13.16)

i v i chu i th i gian d ng, E (Yt ) 2 = E (Yt k ) 2 = 2 . V y

[

] [

]

k =

E [(Yt )(Yt k )]

2

(13.17)

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L T n Lu t

Cc m hnh d bo n gi n

Trong th c hnh chng ta ch tnh Autocorrelation)n

c hm t

tng quan m u (SAC-Sample

(Y Y )(Yt t = k +1

t k

Y ) (n k )2

rk =

(Y Y )t t =1

n

n

(13.18)

Trong tr ng h p t t c cc h s t tng quan u b ng 0 v i cc k>0, chng ta c tr ng h p g i l qu trnh nhi u tr ng, h qu l cc h th p s c m nh c a cc m hnh d bo d a trn tnh ch t c a chu i th i gian. Tr th ng k QLjung-Box cp

nh ngha nh sau.rj2

QLB , p = n (n + 2)j =1

(T j )

(13.19) n b c p.

H0: 1 = 2 = K = p = 0, khng t n t i t tng quan

D i gi thuy t khng, QLB , p tun theo phn ph i Chi-bnh phng v i p b c t do. Quy t c quy t 5%. 2 N u QLB , p p ,95% th khng th bc b H0 hay khng t n t i t tng quan

nh n b c p v i m c ngha

2 N u QLB , p > p ,95% th bc b H0 hay c t n t i t tng quan

nb cpv i nh. Ch

m c ngha 5%. Eviews tnh ton s n gi tr P c a QLB , p nn cng ti n l i cho thao tc ki m l n u c th bc b H0 n b c p th khng c n xt ti p n b c (p+1). Inflation l s li u l m pht c a Vi t Nam t 1-1995 n 12-2005.

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L T n Lu t

Cc m hnh d bo n gi n

0.04 0.03 0.02 0.01 0.00 -0.01 -0.02 95

96

97

98

99

00

01 INF L

02

03

04

05

Hnh 13.10. L m pht hng thng c a Vi t Nam giai o n 1-1995

n 12-2005

nh t tng quan trong Eviews nh sau: Infl/ View/ Thao tc ki m Correlogram/Level/ Lags to includ: 12/OK

Hnh 13.11. Ki m

nh tnh t tng quan c a Infl (Y)

Ngay b c tr th 1 c Prob(Q-Stat) < 5%, v y bi n l m pht c t tng quan v i tr 1 khc 0 v i m c ngha 5%. (2) Hm tng quan ring ph n (PAC-Partial Autocorrelation) Hm t tng quan ring ph n tr k o l ng tng quan gi a cc gi tr Y cch nhau k th i o n sau khi lo i b tng quan gi a cc b c tr trung gian. Eviews o l ng h s tng quan ring ph n b ng cng th c sau

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L T n Lu t

Cc m hnh d bo n gi n

rkk = rkk 1

,k = 1 ,k > 1 (13.20)

rk rk 1, j rk j rkk =j =1 k 1

1 rk j , j rjj =1

V i rk l hm t tng quan v i

tr k. ch nh m hnh ARIMA, m hnh quan

AC v PAC l cc tr th ng k quan tr ng tr ng nh t c a chng ny.

13.2. M hnh d bo trn bi n chu i th i gian d ng khng c tnh ma 13.2.1. M hnh t h i quy (AR)M hnh t h i quy b c p c d ng AR(p): Y t= + 1Yt 1 + K + pYt p + t V i i l cc tham s cha bi t o l ng nh h ng c a Yt-i. i u ki n qu trnh AR(p) h i t l 0< i