A Weighted Moving Average Process for Forcasting

Transcription

1 Joual of Mode Applied aisical Mehods Volume 6 Issue Aicle A Weighed Movig Aveage Pocess fo Focasig hou Hsig hih Uivesiy of ouh Floida, sshih@mailusfedu Chis P Tsokos Uivesiy of ouh Floida, pofcp@casusfedu Follow his ad addiioal woks a: hp://digialcommoswayeedu/jmasm Pa of he Applied aisics Commos, ocial ad Behavioal cieces Commos, ad he aisical Theoy Commos Recommeded Ciaio hih, hou Hsig ad Tsokos, Chis P (007) "A Weighed Movig Aveage Pocess fo Focasig," Joual of Mode Applied aisical Mehods: Vol 6: Iss, Aicle 6 Available a: hp://digialcommoswayeedu/jmasm/vol6/iss/6 This Regula Aicle is bough o you fo fee ad ope access by he Ope Access Jouals a DigialCommos@Wayeae I has bee acceped fo iclusio i Joual of Mode Applied aisical Mehods by a auhoized admiisao of DigialCommos@Wayeae

2 Joual of Mode Applied aisical Mehods Copyigh 007 JMAM, Ic Novembe, 007, Vol 6, No, /07/$9500 A Weighed Movig Aveage Pocess fo Foecasig hou Hsig hih Chis P Tsokos Uivesiy of ouh Floida A foecasig model fo a osaioay sochasic ealizaio is poposed based o modifyig a give ime seies io a ew k-ime movig aveage ime seies The sudy is based o he auoegessive iegaed movig aveage pocess alog wih is aalyical cosais The aalyical pocedue of he poposed model is give A sock XYZ seleced fom he Foue 500 lis of compaies ad is daily closig pice cosiue he ime seies Boh he classical ad poposed foecasig models wee developed ad a compaiso of he accuacy of hei esposes is give Key wods: ARIMA, movig aveage, sock, ime seies aalysis Ioducio Time seies aalysis ad modelig plays a vey impoa ole i foecasig, especially whe ou iiial sochasic ealizaio is osaioay i aue ome of he ieesig ad useful publicaios elaed o he subjec aea ae Akaike (974), Baejee e al (993), Box e al (994), Bockwell ad Davis (996), Dickey ad Fulle (979), Dickey e al (984), Dubi ad Koopma (00), Gade e al (980), Havey (993), Joes (980), Kwiakowski e al (99), Roges (986), aid ad Dickey (984), akamoo e al (986), humway ad offe (006), Tsokos (973), Wei (006) The pupose of his sudy is o begi wih a give ime seies ha chaaceizes a ecoomic o ay ohe aual pheomeo ad as usual, is osaioay Box ad Jekis (994) developed a popula ad useful classical pocedue o develop foecasig models ha have bee show o be effecive I his aicle, hou Hsig hih ecely eceived he Ph D i aisics fom Uivesiy of ouh Floida hih s eseach maily coceaes o ime seies foecasig addess: sshih3@ampabaycom Chis P Tsokos is Disiguished Uivesiy Pofesso of Mahemaics ad aisics a he Uivesiy of ouh Floida He is he auho of moe ha 50 eseach publicaios addess: pofcp@casusfedu a pocedue fo developig a foecasig model ha is moe effecive ha he classical appoach is ioduced Fo a give saioay o { x osaioay ime seies,, geeae a k-day movig aveage ime seies,, ad he developmeal pocess begis Ceai basic coceps ad aalyical mehods ae eviewed ha ae esseial i sucuig he poposed foecasig model The eview is based o he auoegessive iegaed movig aveage pocesses The accuacy of he poposed foecasig model is illusaed by selecig fom he lis of Foue 500 compaies, compay XYZ, ad cosideig is daily closig pices fo 500 days The classical ime seies model fo he subjec ifomaio alog wih he poposed pocess was developed A saisical compaiso based o he acual ad foecasig esiduals is give, boh i abula ad gaphical fom The Poposed Foecasig Model: k-h Movig Aveage I is o appopiae o build a ime seies model wihou cofomig o ceai mahemaical cosais, such as saioaiy of a give sochasic ealizaio Almos always, he ime seies ha is give is osaioay i aue ad he, he ex sep is o educe i io { x beig saioay Le be he oigial ime seies The diffeece file is give by 69

3 60 A WEIGHTED MOVING AVERAGE PROCE FOR FORECATING Pice Time Figue Daily Closig Pice fo ock XYZ Pice Oigial Daa Classical ARIMA Time Figue Compaisos o Classical ARIMA Model V Oigial Time eies fo he Las 00 Obsevaios

4 HIH & TOKO 6 d ( B) () j B x = x j whee, ad d is he degee of diffeecig of he seies The pimay use fo he k-h movig aveage pocess is fo smoohig a ealized ime seies I is vey useful i discoveig a shoem, log-em eds ad seasoal compoes of a give ime seies The k-h movig aveage pocess of a ime seies { x is defied as follows: k = x k + + j k j = 0 y () whee = k, k +,, As k iceases, he umbe of obsevaios k of deceases, ad ges { x close o he mea of as k iceases Whe k =, educes o oly a sigle obsevaio, ad equals μ, ha is y = x j j = = μ (3) The, develop he poposed model by { x asfomig he oigial ime seies io by applyig () Afe esablishig he ew ime seies, usually osaioay, begi he pocess of educig i io a saioay ime seies Kwiakowski, e al (99) ioduced he KP Tes o check he level of saioaiy of a ime seies Apply he diffeecig ode d o he ew ime seies fo d = 0,,,, he veify he saioaiy of he seies wih he KP es uil he seies become saioay Theefoe, oe ca educe he osaioay ime seies io a saioay oe afe a pope umbe of diffeecig The poceed he model buildig pocedue of developig he poposed foecasig model Afe choosig a pope degee of diffeecig d, poceed wih he model buildig pocess by assumig diffee odes fo he auoegessive iegaed movig aveage model, ARIMA(p,d,q), also kow as Box ad Jekis mehod, whee (p,d,q) epese he ode of he auoegessive pocess, he ode of diffeecig ad he ode of he movig aveage pocess, especively The ARIMA(p,d,q) is defied as: d φ p( B)( B) y = θq( B) (4) whee φ is he ealized ime seies, p ad θq ae he weighs o coefficies of he AR ad MA ha dive he model, especively, ad is φ p θ he adom eo Wie ad q as ad φ ( B ) = (5) p p ( φb φb φpb ) θ ( B ) = (6) q q ( θb θb θqb ) omeimes i is difficul o make a decisio i selecig he bes ode of he ARIMA(p,d,q) model whe hee ae seveal models ha all adequaely epese a give se of ime seies Hece, Akaile s ifomaio cieio (AIC) (974), plays a majo ole whe i comes o model selecio AIC was ioduced by Akaike i 973, ad i is defied as: AIC(M)= (7) - l [maximum likelihood] + M, whee M is he umbe of paamees i he model ad he ucodiioal log-likelihood fucio suggesed by Box, Jekis, ad Reisel (994), is give by φμθσ = (8) l L(,,, )

5 6 A WEIGHTED MOVING AVERAGE PROCE FOR FORECATING Table Basic Evaluaio aisics Pice Time Figue 3 Time eies Plo of he Residuals fo Classical Model ( φμθ,, ) l πσ σ whee ( φ, θ ) is he ucodiioal sum of squaes fucio give by = ( φμθ,, ) = (9) [ E( φ, θ, y)] The quaiies φ, μ, ad θ ha maximize (8) ae called ucodiioal maximum l L( φ, θ, σ ) likelihood esimaos Because ivolves he daa oly hough ( φ, θ ), hese ucodiioal maximum likelihood esimaos ae equivale o he ucodiioal leas squaes esimaos obaied by miimizig ( φ, θ ) I pacice, he summaio i (9) is appoximaed by a fiie fom E(,,, y) whee φ μ θ is he codiioal expecaio of give φ, θ, y = M ( φμθ,, ) = [ E( φ, θ, y)] (0)

6 HIH & TOKO 63 Pice Oigial Daa New eies Time Figue 4 Thee Days Movig Aveage o Daily Closig Pice of ock XYZ Vs he oigial ime seies Pice Oigial Daa Poposed Model Time Figue 5 Compaisos o Ou Poposed Model V Oigial Time eies fo he Las 00 Obsevaios

7 64 A WEIGHTED MOVING AVERAGE PROCE FOR FORECATING Table Acual ad Pediced Pice N Acual Pice Pediced Pice Residuals whee M is a sufficiely lage iege such ha he backcas iceme E( φ, θ, y) E( φ, θ, y) is less ha ay abiay pedeemied small value fo ( M +) This expessio implies ha E( φ, θ, y) μ E(,,, y) ; hece, φ μ θ is egligible fo ( M +) Afe obaiig he paamee esimaes φ, μ, ad θ σ, he esimae of σ ca he be calculaed fom ( φ, θ ) σ = () Fo a ARMA(p,q) model based o obsevaios, he log-likelihood fucio is l L = () l πσ ( φμθ,, ) σ Poceed o maximize () wih espec o he paamees φ, θ, σ ad, fom (), l L = (3)

8 HIH & TOKO 65 Pice Time Figue 6 Time eies Plo fo Residuals fo Ou Poposed Model Table 3 Basic Evaluaio aisics l σ ( l π) Because he secod em i expessio (3) is a cosa, educe he AIC o AIC(M) = lσ + M (4) Thus, we a appopiae ime seies model is geeaed ad he saisical pocess wih he smalles AIC ca be seleced The model ideified will possess he smalles aveage mea squae eo The developme of he model is summaized as follows { x Tasfom he oigial ime seies io a ew seies Check fo saioaiy of he ew ime seies by deemiig he ode of diffeecig d, whee d = 0,,, accodig o KP es, uil saioaiy is achieved Decide he ode m of he pocess, fo his case, le m = 5 whee p + q = m Afe (d, m ) is seleced, lis all possible se of (p, q) fo p + q m Fo each se of (p, q), esimae he paamees of each model, ha is, φ φ,, φ, θ, θ,,, p θ q Compue he AIC fo each model, ad choose he oe wih smalles AIC

9 66 A WEIGHTED MOVING AVERAGE PROCE FOR FORECATING Table 4 Acual ad Pediced Pice N Acual Pice Pediced Pice Residuals Accodig o he cieio meioed above, he ARIMA(p,d,q) model ca be obaied ha bes fi a give ime seies, whee he φ φ,, φ, θ, θ,, θ coefficies ae, p q Usig he model ha we developed fo ad subjec o he AIC cieia, we foecas values of ad poceed o apply he back-shif opeao o obai esimaes of he oigial { x pheomeo, ha is, x = (5) k+ ky x x x The poposed model ad he coespodig pocedue discussed i his secio shall be illusaed wih eal ecoomic applicaio ad he esuls will be compaed wih he classical ime seies model The poposed model ad he coespodig pocedue discussed i his secio shall be illusaed wih eal ecoomic applicaio ad he esuls will be compaed wih he classical ime seies model

10 HIH & TOKO 67 Table 5 Basic Compaiso o Classical Appoach Vs Ou Poposed model Classical Poposed Fis, a ime seies foecasig model is developed of he give osaioay daa usig he odiay Box ad Jekis mehodology ecodly, he daa ae modified, Figue, o develop he poposed ime seies foecasig model A compaiso of he wo models will be give The geeal heoeical fom of he ARIMA(p,d,q) is give by d φ p ( B)( B) x = θ q ( B) (6) Followig he Box ad Jekis mehodology (994), he classical foecasig model wih he bes AIC scoe is he ARIMA(,,) Tha is, a combiaio of fis ode auoegessive (AR) ad a secod ode movig aveage (MA) wih a fis diffeece file Wie i as ( 963 B)( Bx ) = (7) ( ) B B Afe expadig he auoegessive opeao ad he diffeece file, ( 963 B 963 B ) x ( 053 B+ 058 B ) + = (8) ad ewie he model as by leig x = 963 x 963x + (9) = 0, hee is he oe day ahead foecasig ime seies of he closig pice of sock XYZ as x 963 x 963x = (0) Usig he above equaio, gaph he foecasig values obaied by usig he classical appoach o op of he oigial ime seies, as show by Figue The basic saisics ha eflec he accuacy of model (0) ae he mea, vaiace, sadad deviaio ad sadad eo of he esiduals Figue 3 gives a plo of he esidual ad Table gives he basic saisics Fuhemoe, esucue he model (0) wih = 475 daa pois o foecas he las 5 obsevaios oly usig he pevious ifomaio The pupose is o see how accuae ou foecas pices ae wih espec o he acual 5 values ha have o bee used Table gives he acual pice, pediced pice, ad esiduals bewee he foecass ad he 5 hidde values The aveage of hese esiduals is = Poceed o develop he poposed foecasig model The oigial ime seies of sock XYZ daily closig pices is give by Figue The ew ime seies is beig ceaed by k = 3 days movig aveage ad he aalyical fom of is give by

11 68 A WEIGHTED MOVING AVERAGE PROCE FOR FORECATING y x + x 3 + x = () Figue 4 shows he ew ime seies { x alog wih he oigial ime seies, ha will be used o develop he poposed foecasig model The bes model ha chaaceizes he behavio of is ARIMA (,,3) Tha is, ( 896 B 0605 B )( B) y 3 ( B 0056 B B ) = () Expadig he auoegessive opeao ad he fis diffeece file, we have 3 ( 896 B B B ) y = (3) 3 ( B 0056 B B ) Thus, wie (3) as y = (4) 896 y 8356 y 0605y The fial aalyical fom of he poposed foecasig model ca be wie as y = (5) 896 y 8356 y 0605y Usig he above equaio, a plo of he developed model (5), showig a oe day ahead foecasig alog wih he ew ime seies,, is displayed by Figue 5 Noe he closeess of he wo plos ha eflec he qualiy of he poposed model imila o he classical model appoach ha we discussed ealie, use he fis 475,,, obsevaios y y475 y o foecas 476,,, The, use he obsevaios y y476 o y foecas 477, ad coiue his pocess uil foecass ae obaied fo all he obsevaios,,,, ha is, 476 y477 y500 Fom equaio (), he elaioship ca be see bewee he { x foecasig values of he oigial seies ad he foecasig values of 3 days movig aveage seies, ha is, = 3 y x x x (6),,, Hece, afe 476 y477 y500 is esimaed, use he above equaio, (6), o solve he { x foecasig values fo Figue 6 is he esidual plo geeaed by he poposed model, ad followed by Table 3, ha icludes he basic evaluaio saisics Boh of he above displayed evaluaios eflec o accuacy of he poposed model The acual daily closig pices of sock XYZ fom he 476h day alog wih he foecased pices ad esiduals ae give i Table 4 The esuls give above aes o he good foecasig esimaes fo he hidde daa Compaiso of he Foecasig Models I his secio, he wo developed models ae compaed The classical pocess is give by x 963 x 963x = (7) I he poposed model, he followig ivesio is used o obai he esimaed daily closig pices of sock XYZ, ha is, y = (8) 896 y 8356 y 0605y i cojucio wih 3

12 HIH & TOKO 69 = 3 y x x x (9) Table 5 is a compaiso of he basic saisics used o evaluae he wo models ude ivesigaio The aveage mea esiduals bewee he wo models show ha he poposed model is oveall appoximaely 54% moe effecive i esimaig oe day ahead he closig pice of Foue 500 sock XYZ Coclusio Based o he aveage mea esiduals he poposed model was sigificaly moe effecive i such em of pedicig of he closig daily pices of sock XYZ Refeeces Akaike, H (974) A New Look a he aisical Model Ideificaio, IEEE Tasacios o Auomaic Cool, AC-9, Baejee, A, Dolado, J J, Galbaih, J W, & Hedy, D F (993) Coiegaio, Eo Coecio, ad he Ecoomeic Aalysis of No-aioay Daa, Oxfod Uivesiy Pess, Oxfod Box, G E P, Jekis, G M, & Reisel, G C (994) Time eies Aalysis: Foecasig ad Cool, 3 d ed, Peice Hall, Eglewood Cliffs, NJ, Box, G E P, Jekis, G M, & Reisel, G C (994) Time eies Aalysis: Foecasig ad Cool, 3 d ed, Peice Hall, Eglewood Cliffs, NJ, 4-47 Bockwell, P J, & Davis, R A (996) Ioducio o Time eies ad Foecasig, pige, New Yok, ecios 33 ad 83 Dickey, D A, & Fulle, W A (979) Disibuio ad he Esimaos fo Auoegessive Time eies Wih a Ui Roo, Joual of he Ameica aisical Associaio, Vol 74, No 366, Dickey, D A, Hasza, D P, & Fulle, W A (984) Tesig fo Ui Roos i easoal Time eies, Joual of he Ameica aisical Associaio, Vol 79, No 386, Dubi, J, & Koopma, J (00) Time eies Aalysis by ae pace Mehods, Oxfod Uivesiy Pess Gade, G, Havey, A C, & Phillips, G D A (980) Algoihm A54 A algoihm fo exac maximum likelihood esimaio of auoegessive-movig aveage models by meas of Kalma fileig, Applied aisics, 9, 3-3 Havey, A C (993) Time eies Models, d Ediio, Havese Wheasheaf, secios 33 ad 44 Joes, R H (980) Maximum likelihood fiig of ARMA models o ime seies wih missig obsevaios, Techomeics, 0, Kwiakowski, D, Phillips, P C B, chmid, P, & hi, Y (99) Tesig he Null Hypohesis of aioaiy agais he Aleaive of a Ui Roo, Joual of Ecoomeics, 54, Roges, A J (986) Modified Lagage Muliplie Tess fo Poblems wih Oe-ided Aleaives, Joual of Ecoomeics, Noh- Hollad, 3, aid, E, & Dickey, D A (984) Tesig fo Ui Roos i Auoegessive- Movig Aveage Models of Ukow Ode, Biomeika, 7, akamoo, Y, Ishiguo, M, & Kiagawa, G (986) Akaike Ifomaio Cieio aisics, D Reidel Publishig Compay humway, R H, & offe, D (006) Time eies Aalysis ad Is Applicaios: wih R Examples, d ed, pige, New Yok Tsokos, C P (973) Foecasig Models fom No-aioay Time eies-ho Tem Pedicabiliy of ocks, Mahemaical Mehods i Ivesme ad Fiace, Noh Hollad Publishig Co, Wei, W W (006) Time eies Aalysis: Uivaiae ad Mulivaiae Mehods, d ed, Peaso Educaio, Ic