Ni dungKin thc c sM hnh ARIMAPhng php Box-Jenkins

o mc tng quan gia 2 bin Yt v Yt-k

Tp hp cc gi tr tng quan to thnh hm t tng quan ACF.

Hm t tng quan ACF

M hnh nhiu trng

Trong :c l hng s thi ket l sai s ngu nhinE(et) = 0; Var(et) = 2; Cov(et , et-p) = 0

Phn phi mu ca t tng quan

Nu chui thi gian l nhiu trng th h s t tng quan c phn phi xp x phn phi chun N(0,1/n)Nu chui thi gian nhiu trng th vi tin cy 95%, cc h s tng quan nm trong khong 1.96/

Kim nh s bng 0 khng ng thi ca cc h s t tng quanH0: r1 = r2 = = rk = 0H1 c rp 0 (p [1, k])

Kim nh Portmanteau

Vi:h: s k trn: s quan strk: h s t tng quanThng k:Kim nh Box-Pierce

Vi:h: s k trn: s quan strk: h s t tng quanThng k:Kim nh Ljung-Box

Q v Q* c phn phi 2 bc t do h-m, vi m l s tham s trong m hnh.Nu Q (hay Q*) ln hn 2(h-m) th chui d liu khng nhiu trng.

o lng mi quan h gia Yt v Yt-k khi tt c cc bin khc gi nguyn.H s t tng quan ring bc k c tnh t m hnh hi quy:

H s t tng quan ring k l c lng ca bk t m hnh hi quy trn.H s t tng quan ring

Hm t tng quan ring (PACF)Tp hp cc gi tr t tng quan ring to thnh hm t tng quan ring PACF.Nu chui thi gian l nhiu trng th h s t tng quan ring c phn phi xp x phn phi chun N(0,1/n)Nu chui thi gian nhiu trng th vi tin cy 95%, cc h s tng quan nm trong khong 1.96/

Nhn bit ma vMa v khi. c s lp li ca mt hin tng trong mt khong thi gian nht nh.Nhn bit ma v qua h s t tng quan hay t tng quan ring ln.

Tnh dng ca chui thi gianChui thi gian c tnh dng nu k vng, phng sai v hip phng sai khng i theo thi gian

Chui thi gian khng dng

Chui thi gian dng

Kim nh tnh dngNu phi ly sai phn d ln chuyn chui thi gian Yt khng dng thnh dng th Yt c gi l c cha d n v gc (unit roots).Khi d = 1: Yt = Yt Yt-1 l dngKhi d = 2: Yt = Yt - Yt-1 l dng

Kim nh Dickey-FullerXt m hnh: Yt = Yt-1 + etGi thit:H0: = 1 (chui cha n v gc, khng dng)H1: < 1 (chui dng)

Xt m hnhYt = ( -1)Yt-1 + et = Yt-1 + et H0: =0 (chui khng dng)H1: < 0 (chui dng)Nn so snh vi gi tr trong bng Fuller.

Khng c xu hng, trung bnh bng 0:

p l s k tr, thng cho p=3Khng c xu hng, trung bnh khc 0:

C xu hng:

Kim nh Dickey-Fuller m rng

Kh tnh khng dng trong chui thi gian bng phng php sai phn:Yt = Yt- Yt-1

Kh khng dng bng sai phnLy sai phn tng dn n khi c chui dng:Yt = Yt- Yt-1

M hnh bc i ngu nhin

Vi et l nhiu trngThng dng phn tch gi c phiuM hnh bc ngu nhin

Vi chui thi gian c tnh ma v v khng dng, c th ly sai phn theo maSai phn theo ma vi s l s k trong ma: Yt = Yt Yt-sS liu theo thng: Yt = Yt Yt-12S liu theo qu: Yt = Yt Yt-4Sai phn ma v

Ton t li (backshift)Gi B l ton t liLi 1 thi k:BYt = Yt-1Li 2 thi k:B(BYt)= B2Yt = Yt-2Li k thi k:B(BYt)= BkYt = Yt-k

Biu din sai phn bng ton t liSai phn cp 1:Yt=Yt - Yt-1 =Yt BYt= (1 B)YtSai phn cp 2:Yt= Yt - Yt-1 = (Yt - Yt-1) - (Yt-1 - Yt-2) =Yt - 2Yt-1 - Yt-2 = (1-2B-B2)Yt = (1-B)2Yt Tng qut cho sai phn cp d: (1-B)dYt

Biu din sai phn ma v bng ton t liSai phn ma v sau ly sai phn cp 1:(1-B)(1-Bs)Yt=(1-B-Bs+Bs+1)Yt = Yt - Yt-1 - Yt-s + Yt-s-1

M hnh t hi quyXt m hnh hi quy:Yt=c+1Yt-1+2Yt-2+ ... +pYt-p+etM hnh ny c gi l m hnh t hi quy bc p: AR(p) hay ARIMA(p,0,0)S dng ton t dch chuyn li:Yt - 1Yt-1 - 2Yt-2 - ... -1pYt-p= c+et(1-1B- 2B2- ... -pBp)Yt= c+et

ACF v PACF ca AR(1)

ACF v PACF ca AR(2)

Xt m hnh hi quy:Y=c -1et-1 - 2et-2 - ... - qet-q + etM hnh ny c gi l m hnh trung bnh trt bc q: MA(q) hay ARIMA(0,0,q)Lu : khi nim ny khc vi khi nim trung bnh trt chng 3.S dng ton t dch chuyn li:Yt = c+(1- 1B- 2B2- ... - qBq)etM hnh trung bnh trt

ACF v PACF ca MA(1)

ACF v PACF ca MA(2)

Mo hnh ARIMA la ket hp cua mo hnh t hoi quy (AR) va mo hnh trung bnh trt (MA)ARIMA(p,d,q)p: bac cua t hoi quyd: bac cua sai phanq: bac cua trung bnh trtARMA(p, q) = ARIMA(p,0,q)M hnh ARIMA

ARIMA(1,0,1): (1-1B)Yt = c+(1-1B)et M hnh ARIMA bng ton t liTrien khai chi tiet: Yt - 1BYt = c + et -1Bet Yt - 1Yt-1 = c + et -1et-1 Yt = c + 1Yt-1 +et -1et-1

ARIMA(1,1,1): (1-1B)(1-B)Yt = c+(1-1B)et AR(1)Sai phn cp 1MA(1)Trien khai chi tiet: (1-1B) Yt = c+et -1Bet Yt -1BYt = c+et -1Bet Yt -1Yt-1 = c+et -1et-1 Yt = c+ 1Yt-1 +et -1et-1

ACF v PACF ca ARMA(1,1)

M rng m hnh ARIMA biu din ma v (s l s k trong ma):ARIMA (p,d,q) (P,D,Q)sBiu din ma v bng m hnh ARIMAThng phn khng ma vThng phn ma v

Xt m hnh:ARIMA (1,1,1) (1,1,1)4Biu din bng ton t li: (1-1B) (1-1B4) (1-B) (1-B4)Yt = (1-1B) (1-B4)et Biu din ma v bng ton t liAR(1) khng maAR(1) maSai phn khng maSai phn maMA(1) khng maMA(1) ma

Trin khai chi titVe trai:

Ve phai:

Mo hnh chi tiet:

Mt s v dARIMA(3,1,0):ARIMA(0,1,1)(0,1,1)12:ARIMA(2,1,1)(1,1,1)4: ?

Bc 1a: Chuyn v chui dng Xet tnh dng:- Ve o th thi gian, ACF, PACF- Chuoi dng neu cac quan sat nam xung quanh mot mc trung bnh, hay ACF va PACF tien ve 0.

Neu chui khong dng:- Lay sai phan e chuyn sang dng.Neu chui dng:- Xet yeu to mua vu.- Nhan dang mo hnh se c s dung.

Ma v tr 12. p dng sai phn 12 k: Yt = Yt Yt-12

Nhieu khi can x dung ca sai phan bnh thng va sai phan mua vu do chuoi thi gian cha ca yeu to xu hng lan mua vu.Trc khi lay sai phan, ta cung co the chuyen dang bang cac ham toan hoc e e lam on nh phan bien thien.

Lng khch i my bay tng theo thi gian (xu hng) v thay i theo ma (ma v). Thay i trong cc ma ln theo thi gian (bien thin thay i).

Cn chuyn dng trc khi ly sai phn

Mt s quy tcNeu ch co xu hng va o bien thien tang theo gia tr trung bnh, ap dung ham chuyen logarit.Neu co xu hng va mua vu, va anh hng mua vu tang dan theo gia tr trung bnh, ap dung ham chuyen logarit

Bc 1b: Nhn dng m hnhSo snh cc ACF va PACF ca d liu vi cc hnh mu l thuyt.ACF gim theo dng m, PACF gim t ngt => AR.ACF gim t ngt, PACF gim theo dng m => MA.ACF v PACF gim theo dng m => ARIMA

Khong co t tng quan co y ngha sau bc tre q, dung MA(q)Khong co t tng quan rieng co y ngha sau bc tre p, dung AR(p)S dng mo hnh ARIMA neu khong co cac dau hieu tren.

Bc 2: c lng m hnhDung OLS e c lng cac tham so cua mo hnh.Kiem nh cac tham so cua mo hnh bang thong ke t.

c lng trung bnh bnh phng ca phn d (residual mean square error):

Bc 3: Kim tra m hnhKim tra xem cc phn d c tha mn cc gi thuyt ca hi quyCc h s t tng quan ca phn d c nm trong khong 1.96/n

Dng thng k Q:

Nu p-value nh (p_value < 0.05), th m hnh khng ph hp.Kim nh Ljung-Box

Kim nh bng hm LGia tr hp ly cua ham log-likehood (L): L cang ln cang phu hpE-Views cho ham L nh sau:

Tieu chuan AICTieu chuan AIC (Akaike info criterion) cang nho cang phu hpE-Views cho ham AIC nh sau:

Tieu chuan SchwarzTieu chuan Schwarz (Schwarz criterion)SC cang nho mo hnh cang phu hp.E-Views cho ham Schwarz nh sau:

Bc 4: D boDng m hnh c lng d bo Khi c s thay i ln trong d liu cn phi c lng li hoc xy dng m hnh mi.