φ p ( B) AR polynomial of B of order p, p Non-seasonal differencing operator = 1 B

Transcription

1 ARIMA Noion The ARIMA roceure coues he reer esies for given sesonl or non-sesonl univrie ARIMA oel. I lso coues he fie vlues, forecsing vlues, n oher rele vribles for he oel. The following noion is use hroughou his cher unless oherwise se: y (=, 2,..., N) N ( =, 2,..., N) q P Q D s Univrie ie series uner invesigion Tol nuber of observions Whie noise series norlly isribue wih en zero n vrince σ 2 Orer of he non-sesonl uoregressive r of he oel Orer of he non-sesonl oving verge r of he oel Orer of he non-sesonl ifferencing Orer of he sesonl uoregressive r of he oel Orer of he sesonl oving-verge r of he oel Orer of he sesonl ifferencing Sesonliy or erio of he oel φ ( B) AR olynoil of B of orer, φ ( B) = ϕb ϕ2b 2... ϕ B θ q ( B) MA olynoil of B of orer q, θ ( B) = ϑb ϑ2b 2... ϑ B q q q Φ P ( B) SAR olynoil of B of orer P, Φ ( B) = ΦB Φ2B 2... Φ B P P P Θ Q ( B) SMA olynoil of B of orer Q, Θ ( B) = ΘB Θ2B 2... Θ B s Q Q Q Non-sesonl ifferencing oeror = B Sesonl ifferencing oeror wih sesonliy s, = B B Bckwr shif oeror wih By = y n B s = s 44

2 ARIMA 45 Moels A sesonl univrie ARIMA(,,q)(P,D,Q) s oel is given by φ ( B) Φ ( B)[ y µ ] = θ ( B) Θ ( B) =, K, N () P s D q Q where µ is n oionl oel consn. I is lso clle he sionry series en, ssuing h, fer ifferencing, he series is sionry. When NOCONSTANT is secifie, µ is ssue o be zero. When P= Q = D =0, he oel is reuce o (non-sesonl) ARIMA(,,q) oel: φ ( B)[ y µ ] = θ ( B) =, K, N (2) q An oionl log scle rnsforion cn be lie o y before he oel is fie. In his cher, he se sybol, y, is use o enoe he series eiher before or fer log scle rnsforion. Ineenen vribles x, x 2,, x cn lso be inclue in he oel. The oel wih ineenen vribles is given by φ ( B) Φ ( B)[ ( y c x ) µ ] = θ ( B) Θ ( B) P s D i i q Q or Φ( B)[ ( B)( y c x ) µ ] = Θ( B) (3) where Φ( B) = φ ( B) Φ P ( B) ( B) = s D Θ( B) = θ q ( B) Θ Q ( B) i i n ci, i = 2K,,,, re he regression coefficiens for he ineenen vribles.

3 46 ARIMA Esiion Bsiclly, wo ifferen esiion lgorihs re use o coue xiu likelihoo (ML) esies for he reers in n ARIMA oel: Melr s lgorih is use for he esiion when here is no issing in he ie series. The lgorih coues he xiu likelihoo esies of he oel reers. The eils of he lgorih re escribe in Melr (984), Perln (980), n Morf, Sihu, n Kilh (974). A Kln filering lgorih is use for he esiion when soe observions in he ie series re issing. The lgorih efficienly coues he rginl likelihoo of n ARIMA oel wih issing observions. The eils of he lgorih re escribe in he following lierure: Kohn n Ansley (986) n Kohn n Ansley (985). Dignosic Sisics The following efiniions re use in he sisics below: N Nuber of reers R S T q P Q N = q+ P+ Q+ + wihou oel consn wih oel consn SSQ Resiul su of squres SSQ = ee, where e is he resiul vecor $σ 2 Esie resiul vrince $σ 2 SSQ =, where f = N N f SSQ' Ajuse resiul su of squres N SSQ' =SSQfΩ /, where Ω is he heoreicl covrince rix of he observion vecor coue MLE

4 ARIMA 47 Log-Likelihoo SSQ' N ln( 2π ) L= Nln( σ$ ) 2σ$ 2 2 Akike Inforion Crierion (AIC) AIC = 2L + 2N Schwrz Byesin Crierion (SBC) SBC = 2L + lnbg N N Genere Vribles Preice Vlues Forecsing Meho: Coniionl Les Squres (CLS or AUTOINT) In generl, he oel use for fiing n forecsing (fer esiion, if involve) cn be escribe by Equion (3), which cn be wrien s y D( B) y = Φ( B) µ + Θ( B) + c Φ( B) ( B) x i where f Bf f f Φ f DB = Φ B B Φ µ = µ i

5 48 ARIMA Thus, he reice vlues (FIT) re coue s follows: f = = i FIT y$ D( B)$ y Φ( B) µ Θ( B)$ c Φ( B) ( B) x i (4) where $ = y y$ n Sring Vlues for Couing Fie Series. To sr he couion for fie vlues using Equion (4), ll unvilble beginning resiuls re se o zero n unvilble beginning vlues of he fie series re se ccoring o he selece eho: CLS. The couion srs he (+sd)-h erio. Afer secifie log scle rnsforion, if ny, he originl series is ifference n/or sesonlly ifference ccoring o he oel secificion. Fie vlues for he ifference series re coue firs. All unvilble beginning fie vlues in he couion re relce by he sionry series en, which is equl o he oel consn in he oel secificion. The fie vlues re hen ggrege o he originl series n roerly rnsfore bck o he originl scle. The firs +sd fie vlues re se o issing (SYSMIS). AUTOINIT. The couion srs he [++s(d+p)]-h erio. Afer ny secifie log scle rnsforion, he cul ++s(d+p) beginning observions in he series re use s beginning fie vlues in he couion. The firs ++s(d+p) fie vlues re se o issing. The fie vlues re hen rnsfore bck o he originl scle, if log rnsforion is secifie. Forecsing Meho: Unconiionl Les Squres (EXACT) As wih he CLS eho, he couions sr he (+sd)-h erio. Firs, he originl series (or he log-rnsfore series if rnsforion is secifie) is ifference n/or sesonlly ifference ccoring o he oel secificion. Then he fie vlues for he ifference series re coue. The fie vlues re one-se-he, les-squres reicors clcule using he heoreicl uocorrelion funcion of he sionry uoregressive oving verge (ARMA) rocess corresoning o he ifference series. The uocorrelion funcion is coue by reing he esie reers s he rue reers. The fie vlues re hen ggrege o he originl series n roerly rnsfore bck o he originl scle. The firs +sd fie vlues re se o issing (SYSMIS). The eils of he les-squres reicion lgorih for he ARMA oels cn be foun in Brockwell n Dvis (99).

6 ARIMA 49 Resiuls Resiul series re lwys coue in he rnsfore log scle, if rnsforion is secifie. ( ERR) = y ( FIT ) =2K,,, N Snr Errors of he Preice Vlues Snr errors of he reice vlues re firs coue in he rnsfore log scle, if rnsforion is secifie. Forcsing Meho: Coniionl Les Squres (CLS or AUTOINIT) ( SEP) = σ$ = 2K,,, N Forecsing Meho: Unconiionl Les Squres (EXACT) In he EXACT eho, unlike he CLS eho, here is no sile exression for he snr errors of he reice vlues. The snr errors of he reice vlues will, however, be given by he les-squres reicion lgorih s byrouc. Snr errors of he reice vlues re hen rnsfore bck o he originl scle for ech reice vlue, if rnsforion is secifie. Confience Liis of he Preice Vlues Confience liis of he reice vlues re firs coue in he rnsfore log scle, if rnsforion is secifie: ( LCL) = ( FIT) α / 2, f ( SEP) = 2,, K, N ( UCL) = ( FIT) + α / 2, ( SEP) = 2,, K, N f b where α / 2, f is he α / 2g-h ercenile of isribuion wih f egrees of freeo n α is he secifie confience level (by eful α = 005. ). Confience liis of he reice vlues re hen rnsfore bck o he originl scle for ech reice vlue, if rnsforion is secifie.

7 50 ARIMA Forecsing Forecsing Vlues Forcsing Meho: Coniionl Les Squres (CLS or AUTOINIT) The following forecsing equion cn be erive fro Equion (3): y$ () l = D( B)$ y+ l + Φ( B) µ + Θ( B)$ + l + ciφ( B) ( B) x, i+ l (5) where f f f f f DB = Φ B B, Φ Bµ = Φ µ y$ () l enoes he l-se-he forecs of y + l he ie. y$ $ + l i + l j = = R S T R S T y+ l i if l i y$ ( l i) if l > i y+ l i y$ + l i () if l i 0 if l > i Noe h y$ () is he one-se-he forecs of y + ie, which is excly he reice vlue $y + s given in Equion (4). Forecsing Meho: Unconiionl Les Squres (EXACT) The forecss wih his oion re finie eory, les-squres forecss coue using he heoreicl uocorrelion funcion of he series. The eils of he lessqures forecsing lgorih for he ARIMA oels cn be foun in Brockwell n Dvis (99).

8 ARIMA 5 Snr Errors of he Forecsing Vlues Forcsing Meho: Coniionl Les Squres (CLS or AUTOINIT) For he urose of couing snr errors of he forecsing vlues, Equion () cn be wrien in he for of ψ weighs (ignoring he oel consn): ϑq( B ) Θ Q ( B ) i y = B ib B P B = ψ( ) = φ ψ ( ) Φ ( ) 0 i, ψ 0 = (6) Le y$ () l enoe he l-se-he forecs of y + l ie. Then 2 2 se[ y$ ()] l = { + ψ + ψ ψ } σ$ l Noe h, for he reice vlue, l =. Hence, ( SEP) = σ $ ny ie. Couion of ψ Weighs. ψ weighs cn be coue by exning boh sies of he following equion n solving he liner equion syse esblishe by equing he corresoning coefficiens on boh sies of he exnsion: φ ( B) Φ ( B) ψ( B) = θ ( B) Θ ( B) P s D q Q An exlici exression of ψ weighs cn be foun in Box n Jenkins (976). Forecsing Meho: Unconiionl Les Squres (EXACT) References As wih he snr errors of he reice vlues, he snr errors of he forecsing vlues re byrouc uring he les-squres forecsing couion. The eils cn be foun in Brockwell n Dvis (99). Box n Jenkins (976) Brockwell n Dvis (99)