Modeling and Forecasting CPC Prices

Transcription

1 CS-BIGS 3(): CS-BIGS hp:// Modelig ad Forecasig CPC Prices Delphie Blake Uiversié d Avigo e des Pays de Vaucluse, Frace Deis Bosq Uiversié Pierre e Marie Curie, Paris, Frace his paper deals wih modelig ad forecasig coke peroleum calciaio (CPC) prices. We firs cosider some empirical echiques (polyomial regressio, ol ad Wiers smoohig) ad compare hem wih he more geeral Box ad Jekis mehod. We also use oparameric predicors. he case sudy is accessible o readers wih a iermediae level of saisics. Prior exposure o Box-Jekis echiques is useful bu o sricly ecessary.. Iroducio he origi of he curre sudy was a reques cocerig a privaizaio projec of a compay, requirig is valuaio. Curre marke codiios for CPC are ivolved i he fuure urover of his projec. CPC (Coke Peroleum Calciaio) is derived from a by-produc of oil ad is used i alumium ad iaium alloys. I his paper, we sudy he srucure ad predicio of CPC prices (i dollars) from quarerly daa (deoed from Q o Q4) bewee 985-Q ad 8-Q4 (figure ). Due o he irregular variabiliy of hese daa, he problem is somewha iricae. I a firs repor, a exper used simple liear regressio ad obaied o very plausible resuls. he goal of his paper is o compare liear regressio (LR) wih more efficie mehods: parabolic regressio (PR), ol ad Wiers filerig (W), he Box ad Jekis mehod (BJ) ad fially, oparameric predicio (NP). For his purpose, we have used he sofware R developed by he R Developme Core eam (8). Figure : CPC prices from 985-Q o 8-Q4 I is oeworhy ha LR, PR ad W may be cosidered as special cases of BJ (see Secio ). hus, o surpris-

2 - 8 - Modelig ad Forecasig CPC Prices / Blake & Bosc igly, BJ appears as more efficie ha he former echiques. he o-homogeeiy of he daa led us o divide hem io hree pars: 985-Q o 995-Q4 where a seasoal compoe appears, 996-Q o 7-Q4 where he red is parabolic ad 8 daa ha ca be cosidered as ouliers (possibly due o he ecoomic crisis). I his paper, our aim is o cosruc forecass, especially for 9, o akig io accou he 8 exoic daa (sice hey are o represeaive of ypical codiios upo which o build a evaluaio of he compay). he ex secio deals wih polyomial regressio ad W. We specify he lik of hese mehods wih BJ ad explai why LR ad PR are o suiable for CPC sudy. he hird par is devoed o BJ. We obai wo differe models: oe for 985-Q o 995-Q4, aoher oe for 996-Q o 7-Q4. Fially, he NP mehod is cosidered i Secio 4.. Empirical Mehods he liear regressio (LR) model has he form: X = a + a + ε, Ζ where ( X ) is he observed process, α ad α are real coefficies ad ( ε ) is a whie oise: Eε = σ >, Eε =, E( εε s ) = ; s, Ζ, s. We firs show ha, i some sese, he LR model is a special ARIMA (Auo Regressive Iegraed Movig Average) process (ARIMA heory appears i Brockwell ad Davis (99) amog oher refereces). Se Y = X X a, Ζ () he Y = ε ε, Ζ () ad ( Y ) is a MA(), hece ( X ) is a o-ceered ARI- MA(,,). I addiio, despie he fac ha he polyomial of degree oe which appears i () has a ui roo, ( ε ) is he iovaio process of ( Y ). I order o prove ha asserio, we firs oe ha () implies ε = Y + + Y j + ε j, j. (3) he, usig (3) for j =,, k, oe obais k k j ε = ( ) Y j + ε j j= k k j= ad sice, for fixed, i follows ha k σ ε j k j k = E( ) =, k k j= j ( ) Y k L j k ε L where sads for covergece i mea square. Cosequely, ε M, he closed liear space geeraed by Ys, s. Noig ha ε is orhogoal o M, we coclude ha ( ε ) is he iovaio of ( Y ) ad ha ε is he orhogoal projecio of Y o M, ha is he bes liear predicor of Y give Y, s. Now he LR is o coveie for daa wih irregular variaios like CPC sice i oly compues a red, assumig ha his red is liear, ad does o ake io accou he correlaio bewee he X s. his limiaio appears i Figure ad 3 where he explaied variaces R are respecively 7% ad 8%! he parabolic regressio (PR) model is wrie as X = a + a+ a + ε, Ζ. (4) I order o give a ARIMA ierpreaio of PR, we differeiae o obai X X = a( ) + a + ε ε a secod differeiaio leads o he relaio X X + X = a + ε ε + ε. If we pu Y = X X + X a i follows ha ( Y ) is a MA() ad ( X ) becomes a oceered ARIMA(,,). Agai he polyomial associaed wih ( Y ) has a (double) ui roo ad ( ε ) is he iovaio of ( Y ). I order o prove ha claim, we se E = ε ε, Ζ he we have Y = E E ad, usig a similar mehod as above, we obai he relaio E M ad k k j ε = ( ) E j + ε j. j= k k j= Leig k edig o ifiiy gives ε M ad ε M, he proof is herefore complee. PR suffers for he same limiaio as LR bu fis CCP prices from 996 o 7 beer wih a R of 93% (Figure s

3 - 8 - Modelig ad Forecasig CPC Prices / Blake & Bosc 3). Noe however ha he R is o a compleely saisfacory crierio of efficiecy, we refer o Mélard (99) for a comprehesive discussio. he ol ad Wiers mehod is more sophisicaed because i also akes io accou a possible seasoaliy of he daa. For a addiive seasoal model wih period legh p ad if X ˆ ( ) deoes he predicio of X + give he daa X,... X, oe has he addiive ol- Wiers f orecas : Xˆ ( ) = a +b + S ++ ( )mod p where a, b ad S are recursively give by a = α (X - S ) + ( - α ) (a + b ) -p - - b = β( a - a -) + ( - β) b - S = γ( X - a ) + ( -γ ) S-p ad where he smoohig parameersα, β adγ are ake i [,]. heir opimal values are deermied by miimizig he squared oe-sep predicio error. he fucios a, b ad S are iiialized by performig a simple decomposiio i red ad seasoal compoe ad usig movig averages o he firs periods. his mehod is opimal for a SARIMA(,,)(,,) P model (here he riple (,,) refers o he AR, differecig ad MA orders for he series, he riple (,,) o he same orders for he seasoal compoes ad P o he legh of he seasoal period, see, for example, Bosq ad Lecoure, 99). he resuls obaied by his mehod are quie saisfacory for boh periods (see Figure ad 3). Figure : CPC predicios buil wih daa of 985-Q o 995-Q4 Leged: Black lie ad crosses: observed CPC prices (985-I o 995-IV) Black circles: observed CPC prices (996-Q o 997-Q4) Red: modelized values by BJ Blue: modelized values by W Gree: modelized values by PR Cya: modelized values by LR Doed: predicios by BJ, W, PR ad LR from 996-Q o 997-Q4 3. he BJ Mehod We have see ha he previous echiques are associaed wih various SARIMA models. he i is aural o use he BJ mehod for modelig ad forecasig our daa. For he period we obai a SARIMA (,,) 4 (,,) : he seasoaliy period is oe year (four quarers) ad here is o red (see Figure ). owever, he forecass for 997 ad 998 are o compleely saisfacory, idicaig a probable chage of model. For he ew model is ARIMA (,,) (see Figure 3). hus, he seasoaliy has disappeared ad he red is parabolic (I=). he forecass ad he predicio iervals (a cofidece level.95) are preseed i Figure 4. Recall ha our mai ieres is o predic 9 posulaig a reur o a quieer siuaio. Resuls obaied by LR are o icluded i hese predicio iervals bu PR appears as a upper limi of hem. I addiio, we oe ha he model remais he same by cosiderig oly he daa from 996- Figure 3: CPC predicios buil wih daa of 996-Q o 7-Q4 Leged: Black lie ad crosses: observed CPC prices (996-Q o 7- Q4) Black circles: observed CPC prices (8-Q o 8-Q) Red (resp. blue, gree ad cya): modelized values by BJ (resp by W, by PR ad LR) Doed: predicios by BJ, W, PR ad LR for 8-Q ad 8-Q

4 - 8 - Modelig ad Forecasig CPC Prices / Blake & Bosc Figure 4: Predicio iervals for CPC by BJ mehod Leged: Black lie: observed CPC prices (996-Q o 7-Q4) Black crosses: 8-Q ad 8-Q observed CPC prices I red: modelized values by BJ I gree: modelized values by PR I cya: modelized values by LR Doed: predicios by PR ad LR for 8-Q o 9-Q4 Yellow zoe : predicio ierval by BJ for 8-Q o 9-Q4 a cofidece level.95 Q o 6-Q4 leadig o sharp forecass of he observed year 7. O he corary, if oe adds he wo firs daa pois of 8, he model becomes a ARIMA (,,) bu wih a lesser fi. 4. he Noparameric Mehod he BJ mehod posulaes ha he uderlyig model is of SARIMA ype. ha assumpio beig somewha arbirary i is ofe coveie o employ a oparameric mehod (avoidig he esimaio of a possibly impora umber of parameers). his mehod is, i some sese, objecive sice he uderlyig model oly appears hrough regulariy codiios. If Y is a saioary ad Markovia process observed a isas,...,, he oparameric predicor of Y + (where represes he horizo) akes he form: Yˆ + = ˆr ( Y ) where rˆ ( x) is a oparameric esimaor of he regres- sio EY ( / Y = x) based o he daa ( Yi+, Yi) for i =,...,. For example, oe may choose he popular kere l mehod wih he Nadaraya-Waso (NW) esima- of ha predicor, a horizo, or. Cosrucio may be described as follows: suppose ha ( Yi, Y i + ) has a desiy f ( x, y) which does o deped o i, he E Y + / Y = x) = yf( x, y)d y f( x, y)dy. (5) ( i i Now a simple hisogram ype esimaor of f is (, ) = ( ) ( h h i h h i+ ( ) h [ x, x+ ] [ y, y+ ] i= f x y Y Y ) wih badwidh h such ha lim h =. A smooh versio of f has he form x Yi y Yi+ f ( x, y ) = ( ( ) K ) K ( ) h i= h h where K is a sricly posiive symmeric coiuous prob- ( f abiliy desiy correspodig o he choice K = ). Replacig f by f i (4) leads o he, if [, ] i = Yi+ i= i= x Y x Y ˆ i r ( x) K( ) K( ) h h ( Y ) is Markovia, he NW predicor of Y ˆ+ = Y Yi Y Yi Yi+ K K i= h i= h ( ) ( ). Y + is Predicio resuls for sochasic processes by kerel appear i Bosq ad Blake (7). Based o he BJ model, we cosruc he oparameric predicors o he wice differeia ed daa from 996-Q o 7-Q4 which ca be cosidered as a saioary process. he resuls are deailed i Aex A5. 5. Coclusio We have see ha he LR, PR ad W mehods are all associaed wih SARIMA models. Sice he BJ mehod selecs he bes SARIMA model, i is aural o cosider he BJ forecass as opimal. Fially, i is ieresig o oe ha he oparameric predicios are close o he BJ oes. Ackowledgmes We are graeful o S. Abou Jaoudé for havig raised his problem o us ad for his useful suggesios.

5 Modelig ad Forecasig CPC Prices / Blake & Bosc REFERENCES Bosq, D. ad Blake, D. 7. Iferece ad predicio i large dimesios. Wiley-Duod, Chicheser. Bosq, D. ad Lecoure, J.P. 99. Aalyse e prévisio des séries chroologiques. Masso, Paris (i Frech). Brockwell, P.J., ad Davis, R.A. 99. ime series: heory ad mehods, d ediio. New-York: Spriger-Verlag. ydma, R.J. 9. Forecas: forecasig fucios for ime series. R package versio.4. hp:// Mélard, G. 99. Méhodes de prévisio à cour erme. Ellipses, Bruxelles (i Frech). R Developme Core eam 8. R: A laguage ad evi- Ausria. hp:// for saisical compuig R Foudaio for Saisical Compuig, Viea, projec.org. Appedices A. Liear regressio Firs, le us examie he residuals for boh periods uder cosideraio: he ail values idicaig ha residuals have heavy ails. he obaied coefficies of deermiaio R are respecively abou.7 ad.8 (highlighig he lack of adequacy of he liear regressio model). Cocerig he firs period, we obai ha he regressio coefficies are o saisically sigifica a level α=.5. For 996-Q o 7-Q4, he regressio lie is: where represes he cosidered year (for example = for 996-Q). he forecass for 8-9 are: able A. Forecass by Liear Regressio Calculaed from 996-Q o 7-Q4 Qr Qr Qr3 Qr For he period (resp ), he mea- predicio error (he me a of square d iffereces square bewee observed ad fied values) is large: 9.7 (resp ). A. Parabolic regressio Cocerig he firs period, he same diagosics ca be performed o he obaied residuals; he muliple coeffi- of regressio R is sill poor (of order.). For cie 996-Q o 7-Q4, residuals are more saisfacory ad, ow, he R is of order.93. Figure A: Residuals of he liear regressio calculaed from 985-Q o 995-Q4 (lef) ad from 996-Q o 7- Q4 (righ). he residual plos are quie usaisfacory ad prese a red. he ormal QQ-plos prese also irregulariies i Figure A: Residuals of he parabolic regressio calculaed from 985-Q o 995-Q4 (lef) ad from 996-Q o 7- Q4 (righ).

6 Modelig ad Forecasig CPC Prices / Blake & Bosc I he las case, he obaied parabolic regressio equaio is: where sill represes he cosidered year. he forecass for 8-9 are: able A. Forecass by parabolic regressio calculaed from 996-Q o 7-Q4 Qr Qr Qr3 Qr Fially, cocerig h e period , he mea- predicio error is equal o while for 996- square 7, i is equal o 49.7 (which is a accepable value compared o hose obaied below for W ad BJ mehods). A3. ol-wiers mehod For 985-Q o 995-Q4, he ol-wiers addiive model is wihou red bu wih a seasoal compoe. he adjused smoohig values are α=, β= ad γ=. Sarig values for he iercep a ad he seasoal compoes s, s, s 3, s 4 are: a=75.535, s =.85, s = , s 3 = ad s 4 = he mea-square predicio error for his model is 8.8 (beer ha for polyomial regressio). able A4: Coefficies ad heir sadard deviaios for he period 985-Q/995-Q4. ar ar ma ma Coefficie SD sar sma iercep Coefficie SD As expeced, he mea square predicio error is abou 4.68, beer ha for he previous ly esed models. he qualiy of he fi is measured hrough he followig diag- osics which show ha he error erm of he model ca be cosidered as whie oise (see Figure A3). Cocerig he period 996-Q/7-Q4, oe obais a ARIMA(,,) wih esimaed coefficies give i able A5. he mea-square predicio error is ow abou 3.97 (isead of 3.7 for he ol-wiers mehod). able A5. Coefficies ad heir sadard deviaios for he period 996-Q/7-Q4 ar ar ma Coefficie SD Cocerig he period 996-7, he ol-wiers adjusme performs beer ha he parabolic oe wih a mea-square predicio error of order 3.7 (isead of 49.7). I his case, he adjused model has a red ad smoohig coefficies are α=.75854, β= ad γ=. Sarig values for a, b ad s, s, s 3, s 4 ow are: a= , b=3.847, s = , s =.69, s 3 = ad s 4 = Forecass for 8 ad 9 are give i he followig able. able A3. Forecass by ol-wiers calculaed from 996-Q o 7-Q4 Qr Qr Qr3 Qr A4. Box-Jekis mehod Figure A3: BJ residuals. Period 985-Q/995-Q4 For 985 o 995, he adjused model by he BJ mehod is a SARIMA(,,)(,,) 4 wih o-zero mea (usig he R package forecas, ydma, 9). he compued coefficies ad heir empirical sadard deviaio are give i able A4.

7 Modelig ad Forecasig CPC Prices / Blake & Bosc Figure A4: BJ residuals. Period 996-Q/7-Q4 Agai, he error erm of he model ca be cosidered as whie oise (see Figure A4). Fially we give i able A6, he predicio ierval a cofidece level 8% (resp. 95%): [Lo 8, i 8] (resp. [Lo 95, i 95]) while Forecas correspods o he fied value. able A6. Forecass ad heir predicio iervals for 8-9 Poi Forecas Lo 8 i 8 Lo 95 i 95 8: Q Q Q Q : Q Q Q Q process, we fid ha he bes liear predicor of Y + give Y, is.74y.y +.7Y.5Y 3 + Noig he impora role played by Y, i is coveie o use he couples ( Yi, Y i ) + for compuig he NW predicor. his allows us o avoid he well-kow oparameric curse of dimesioaliy. Values obaied for Yˆ ˆ 8 Q, Y8 Q, Yˆ 8 Q3 ad Yˆ 8 Q4 are give i able A7. Predicors of Xˆ ˆ 8 Q, X8 Q, ˆX 8 Q3 ad ca Xˆ 8 Q 4 be compued from he equaios: Xˆ ˆ 8 Q = Y8 Q X7 Q4 + X7 Q3 Xˆ ˆ ˆ 8 Q = Y8 Q X8 Q + X7 Q4 Xˆ ˆ ˆ ˆ 8 Q3 = Y8 Q3 X8 Q + X8 Q Xˆ ˆ ˆ ˆ 8 Q4 = Y8 Q4 X8 Q3 + X8 Q. We compue he Nadaraya-Waso esimaor by choosig a sadard ormal kerel. Fially, he badwidh h is empirically adjused by miimizig he predicio error for he observed year 7. his mehod leads o forecass ha are slighly lower ha values obaied by he BJ mehod (see able A5) bu sill wihi he predicio iervals. able A7. Forecass for 8 by NW predicor 8-Q 8-Q 8-Q3 8-Q4 Y ˆ ˆX A5. Noparameric kerel predicio Comig back o CPC prices, we begi by ierpreig he BJ resuls of Aex A4. Sice I = for 996-Q/7- Q4, we cosruc our oparameric predicor o he wice differeiaed daa ( Y ) which ca be cosidered as saioary: Y = X X + X where is varyig from 996-Q3 o 997-Q4 (so ha =46). Moreover, usig able A4 ad iverig he Correspodece: delphie.blake@uiv-avigo.fr