THE STOCHASTIC SEASONAL BEHAVIOR OF THE NATURAL GAS PRICE (*)

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1 HE SOCHASIC SEASONA BEHAVIOR OF HE NAURA GAS PRICE Andrés García irans a, Javir Población b and Grgorio Srna c a F. amáicas, Univrsidad d Ovido, Calvo Solo s/n, 337, Ovido, Spain. -mail: andrs_g@lcabl.s l: // b D.G.A. Suprvisión, Banco d España, C/ Alcalá 48, 84, adrid, Spain. -mail: avir.poblacion@bd.s l: // c Faculad d Cincias Jurídicas y Socials. Univrsidad d Casilla-a ancha. Cobrizo d San Pdro árir, s/n, 457 oldo, Spain. -mail: Grgorio.Srna@uclm.s l: Absrac In his papr i has bn dvlopd a gnral nm-facor modl of h sochasic bhavior of commodiy prics ha considrs h sasonaliy as a sochasic facor, wih n non-sasonal facors, dscribd in h liraur, and m sasonal facors. W apply hr, four and fiv-facor modls o Hnry Hub naural gas fuurs conracs radd a NYEX. h Kalman filr mhodology is usd o sima h paramrs of h modls. Using h simad paramrs, w analys h modls goodnss of fi o h rm srucur of fuurs prics and volailiis. I is found ha modls allowing for sochasic sasonaliy ouprform sandard modls wih drminisic sasonaliy. Ky Words: Sochasic Calculus, Sasonaliy, Commodiy Prics, Kalman Filr, NYEX, Naural Gas. JE Cods: C3, C5, C6, G3. his vrsion : Ocobr 7 his papr is sol rsponsibiliy of is auhors and h viws rprsnd hr do no ncssarily rflc hos of h Banco d España. Corrsponding Auhor.

2 . Inroducion In rcn ims, boh acadmics and praciionrs hav bn paying anion o h valuaion and hdging of commodiy coningn claims and o h procdurs for valuaing naural rsourcs invsmn procs, spcially o h rul for drmining whn i is opimal o invs. h sochasic bhavior of commodiy prics plays a cnral rol in his ara. Early sudis on h sochasic bhavior of commodiy prics assumd ha spo prics follow a gomric Brownian moion s for xampl Brnnan and Schwarz, 985; Paddoc al., 988, among ohrs. Howvr, h gomric Brownian moion hypohsis implis a consan ra of growh in h commodiy pric and a consan volailiy of fuurs pric rurns, which ar no ralisic assumpions. In pracic i is found ha commodiy prics show manrvrsion and h volailiy of fuur pric rurns is a dcrasing funcion of im. Consqunly, in rcn yars svral auhors, such as aughon and Jacobi 993 and 995, Ross 997 or Schwarz 997, hav considrd ha a man-rvring procss is mor appropria o modl h sochasic bhaviour of commodiy prics. Unforunaly, hs onfacor man-rvring modls ar no vry ralisic sinc hy gnra a volailiy of fuurs rurns which gos o zro as h im o mauriy of h fuur conrac approachs infiniy. Evn mor, hs modls ha considr a singl sourc of uncrainy ar no vry ralisic sinc hy imply ha fuurs prics for diffrn mauriis should b prfcly corrlad, which dfis xising vidnc. ooing for mor ralisic rsuls, muli-facor modls hav bn dvlopd Schwarz, 997; Schwarz and Smih, ; Corazar and Schwarz, 3; Corazar and Narano, 6, among ohrs. All hs muli-facor modls assum ha h spo pric is h sum of shor-rm and long-rm componns. ong-rm facors accoun for h long-rm dynamic of commodiy prics, which is assumd o b a random wal, whras h shor-rm componns accoun for h man-rvrsion componns in h commodiy pric. os of hs aricls ar focusd on oil prics. h numbr of paprs addrssing h sudy of naural gas prics is sill scarc. Howvr naural gas rprsns almos h fourh par of h

3 world nrgy consumpion, wih similar figurs o coal and only bhind oil. World naural gas consumpion is abou 45 millions barrls of oil quivaln pr day whil world oil consumpion is abou 8 millions barrls pr day. World provd rsrvs ar mor or lss h sam for naural gas and oil which ar roughly on rillion barrls of oil quivaln for ach on. Almos h hird par of naural gas world consumpion is locad in h Unid Sas. In his counry gas naural rprsns h 5% of h consumd nrgy and his prcnag is growing vry fas. ha is h rason why h mos dvlopd gas naural mars ar locad in h Unid Sas. h lac of conomical ransporaion mas h naural gas pric subsanially diffrn along h counry. h mos liquid and famous naural gas mar is locad in ouisiana, nar o h xas bordr, and is nam is Hnry Hub. his lac of conomical ransporaion and h limid sorabiliy of naural gas ma is supply unabl o chang in viw of sasonal variaions of dmand. hrfor, naural gas prics ar srongly sasonal. On of h clars ways o visualiz his sasonaliy is hrough h forward curv. In Figur i is possibl o apprcia ha Hnry Hub naural gas prics ar xpcd o b highr during winr monhs and lowr during summr monhs. I is also possibl o noic ha in spo pric hisorical sris h highs prics hav bn rachd in winr whil h lows prics appar in summr. hr ar sudis ha a ino accoun h sasonal bhavior of som commodiy prics, such as ucia and Schwarz or Sornsn among ohrs, bu, o h bs of h auhors nowldg, sasonaliy has nvr bn considrd as a sochasic facor. As poind ou by Schwarz 997, h sochasic procss assumd for h commodiy pric is imporan no only for drivaivs valuaion purposs, bu also for h valuaion of naural rsourc invsmn procs, spcially for h rul for drmining whn i is opimal o invs. In his papr, i has bn dvlopd a gnral nm-facor modl ha considrs sasonaliy as a sochasic facor. his gnral nm-facor modl assums ha h log-spo pric is h sum of n and m sochasic facors n non-sasonal and m sasonal. h nonsasonal facors ar h facors of h modls mniond abov. h sasonal facors ar rigonomric componns gnrad by sochasic procsss. hn his gnral modl has bn 3

4 paricularizd for m and n,, 3, hus, hr, four and fiv-facor modls hav bn obaind o xplain h sochasic bhavior of Hnry Hub naural gas prics. h Kalman filr mhodology has bn usd o sima h paramrs of h modls basd on h naural gas in Hnry Hub fuurs conracs radd a h Nw Yor rcanil Exchang NYEX. And finally, using h simad paramrs, i is analysd h modls goodnss of fi o h spo pric dynamic and h rm srucur of fuurs prics and volailiis. Inrsingly, i is found ha modls allowing for sochasic sasonaliy ouprform sandard modls wih drminisic sasonaliy. his papr is organizd as follows. Scion dals wih sasonaliy in Hnry Hub naural gas prics. h modl allowing for sochasic sasonaliy is dvlopd in Scion 3. h simaion mhodology is discussd in Scion 4. Scions 5 and 6 prsn h daa and h mpirical rsuls rgarding h simaion of h modls. h goodnss of fi of h modls rgarding h spo pric, forward curv and volailiy of fuur rurns simaions is conaind in scion 7. Finally, scion 8 concluds wih a summary and discussion.. Sasonaliy in Hnry Hub Naural Gas Prics In Figur i sms clar ha h naural gas in Hnry Hub pric is sasonal wih a on yar priod. A vry simpl algorihm can b dvlopd o undrsand i mor clarly. S b h spo pric and Y h cnrd moving avrag in a yar of S dfind as follows. If {S },,3, is h spo pric im sris wih monhly frquncy, hn Y.5S 6 S 5 S 4 S S -5.5S -6 /. us dfin τ S /Y, which is a masur of how big is h spo pric in monh wih rspc o h prics in on-yar priod cnrd in his monh. If h pric in his monh is highr han h pric in h prvious and following monhs, hn τ will b grar han on, if no, τ will b lss han on. us also dfin i m as h avrag of τ for monh m m January, Fbruary,, Dcmbr and r m, h scaling facor for monh m, as r m im / i i... i. I is obvious ha r r r, and i is also asy o show ha if r m is grar han on h prics in monh m ar highr han h avrag pric and if r m is lss han on h prics in monh m ar lss han h avrag pric. 4

5 As can b sn in Figur, h spo pric and h forward curv scaling facors prsn h sam parn: In h winr monhs hy ar highr han on and in h summr monhs hy ar lss han on. his is clar vidnc of sasonaliy in h pric of naural gas in Hnry Hub. A mor sophisicad analysis can b implmnd o compl his rsul. h spcrum of a saionary procss is dfind as h Fourir ransformaion of h procss auocovarianc funcion. I can b provd Wi 5 ha drminisic sasonaliy appars in h spcrums as sharp pas acually Dirac dla funcions in svral frquncis whras sochasic sasonaliy shows a sofr parn. his implis ha, in daa analysis, a sharp spi in h sampl spcrum may indica a possibl drminisic cyclical componn, whil broad pas ofn imply a nondrminisic sasonal componn. Of cours, rsuls should b an wih som car as simaion rrors and aliasing ffcs can a plac, confusing drminisic and sochasic idal parns. I can b provd Wi 5 ha in a gnral saionary ARAp,q modl, ψ p S θ q, whr is h lag funcion and is whi nois wih varianc, h spcrum is givn by: f θ iw q w iw π ψ p. If i is assumd ha h Hnry Hub spo naural gas pric follows an AR wih yarly sochasic sasonaliy, ha is, -φ-φ S, is spcrum should b: f w π Φ Φ cosw cos w hus, for Φ>, ohr han a pa a w, h spcrum also xhibis pas and roughs a sasonal harmonic frquncis w π/ and w π-/ rspcivly,, 3, 4, 5 and 6. h Hnry Hub spo naural gas pric spcrum is dpicd in Figur 3. I sms ha, mor or lss, h spcrum xhibis pas and roughs a hos frquncis. hrfor, his analysis lads o hin ha naural gas in Hnry Hub pric has a yarly sochasic sasonal componn. 3. A modl for sochasic sasonaliy As mniond abov, in his scion w prsn h gnral nm-facor modl. his modl assums ha h log-spo pric X is h sum of nm sochasic facors n non-sasonal and m sasonal. X n i m i 5

6 h non-sasonal facors and i ar h sam ha Corazar and Narano 6 us in hir papr. hir sochasic diffrnial quaions SDE ar: d μ d d κ d i,, 3,..., n- 3 i i i i i whr μ, κ i, and i ar posiiv consans and and i ar corrlad Brownian moions incrmns. Each sasonal facor is modlld hrough a rigonomric componn. h rigonomric SDE is complx: da i a d Q whr a is a complx facor a i, Q is a complx numbr Q Q iq and a W a a complx Brownian moion W a W iw, providd ha d and d bing uncorrlad and wih h sam varianc Osndal 99. o g i, a ncssary and sufficin condiion is o assum ha and ar uncorrlad. A proof of his fac can b found in appndix A. Assuming i, in appndix A i is also provd ha h argumn of Q Q xprssd in polars is Q iθ has no ffc in h modl onc xprssd using only ral numbrs, ha is qualling ral componn wih ral componn and imaginary componn wih imaginary componn in h prvious quaion. ha is h rason why θ is chosn qual o zro and, consqunly, Q. hrfor h las SDE can b wrin as: da i a d a a Equalling ral componn wih ral componn and imaginary componn wih imaginary componn in h prvious quaion i follows wo ral SDEs for ach sasonal facor: d d 4 d d,, 3,..., m 5 whr W and W ar uncorrlad. o assss drivaivs conracs h ris-nural vrsion of h modl has o b usd. h SDEs for h facors undr h quivaln maringal masur can b xprssd as: μ λ d d i κ i i λ i d ii d i,, 3,..., n- 3 λ d d 4 d λ d,, 3,..., m 5 6

7 whr λ, λ i, λ y λ ar h ach facor ris-prmiums and W, W, W and W ar h ach facor Brownian moions undr h quivaln maringal masur. I is admid any corrlaion srucur among Brownian moions wih h rsricion xplaind abov W and W ar uncorrlad. For ach sasonal facor, boh componns corrsponding o ral and imaginary pars in a complx procss should hav qual varianc and b uncorrlad. ' Z b h vcor of all facors. n m m h ris-nural SDE of Z can b xprssd as: whr dz b AZ d Ω Z Z is a vcor of indpndn Brownian moions, and hrfor VardZ R ΩΩ Ω is h ranspons marix of Ω wih h rsricion xplaind abov, b μ λ λ λ λ λ λ λ n m m A and: Undr his noaion X cz, whr n m. m n m c. I is asy o prov ha h uniqu soluion of ha problm is Osndal, 99: Z A A s A s Z Ω b ds Zs 6 I is clar ha, undr h ris-nural masur, givn Z, varianc 3 : Z is Gaussian, wih man and his mhodology is gnral, i can b usd in all ind of problms, i dos no mar which is b, A and R. Evn in h cas ha b, A and R wr funcion of, if A and As ds commu, h soluion of ha problm is 6. 3 E[] and Var[] ar h man and varianc undr h ris nural masur. 7

8 E [ Z ] A Z As b ds 7 As X c Z A As As A [ Z ] R ds Var n i m. 8 i hn, undr h ris-nural masur, X is also Gaussian wih man and varianc: E X ce 9 Z X cvar Z c Var And his provids a valuaion schm for all sor of commodiy coningn claims as financial drivaivs on commodiy prics, ral opions, invsmn dcisions and ohr mor. In paricular, h pric of a fuur conracd a im wih mauriy a im F E S I xp E X I Var X is: [ ] [ ] [ ], availabl a im. I can b xprssd as: A [ c Z g ] I, whr I is h informaion F, xp whr A As A As A A g c b ds c R ds c, which is a drminisic funcion. h squard volailiy of a fuurs conrac radd a im wih mauriy a im is dfind as 6 : Var lim h [ log F log F ] h, h,. I is asy o prov ha i is h xpcd valu of h squar of h cofficin of h Brownian moion in h xpansion: d log F F μ ds,, whr W F is a scalar canonical Brownian moion. hrfor, aing logarihms and diffrnials on boh sids of Equaion, i follows ha: d A A A log F c dz c [ b AZ ] d c Ω, 6 h sam rsuls ar going o b obaind if h volailiy is dfind as Var [ log F log F ] 8 h NYEX is h biggs mar for naural gas. lim h h, h h,. 8

9 So, h fuurs squard volailiy is simply: A A c R c Nx w ar going o pariculariz h gnral modl prsnd abov for n,,3 and m. ha is o say, w prsn hr, four and fiv-facor modls in ordr o xplain h sochasic bhavior of Hnry Hub naural gas prics. Du o h analysis dvlopd in Scion, hs paricular modls will hav only on sasonal facor and i is xpcd ha h simad phas φ is on yar π. h non-sasonal par will b h sam as in Schwarz 997 for h hr-facor modl, Schwarz-Smih for h four-facor modl and Schwarz-Corazar 3 for h fivfacor modl. Du o h sasonal facor, h Schwarz 997 on-facor modl bcoms a hr-facor modl, h Schwarz-Smih wo-facor modl bcoms a four-facor modl and h Schwarz-Corazar 3 hr-facor modl bcoms a fiv-facor modl. 3.. h hr-facor odl In his modl i is assumd ha h log-spo pric X is h sum of wo sochasic facors: a shor-rm componn and a sasonal componn, and a drminisic facor: h longrm componn. X h hird sochasic facor is h ohr sasonal facor which complmns. h SDE of hs facors ar: μ d 3 d d κ d 4 d d 5 h ris-nural SDE ar: d d 6 μ d 3 d ± κ d d λ 7 d ± λ d ± d λ d 8 9 Applying h rsul in quaion 6, h h log-pric of a fuurs conrac wih mauriy a im radd a im is: 9

10 whr: A [ F X, ] cos sn A ln / μ λ λ λ sn λ cos / { sn cos } cos sn { } /4 I is imporan o no ha h rigonomric rms in h xprssions abov ry o capur h sasonaliy in h forward curv. Paricularizing quaion i is drmind h squard volailiy of fuurs rurns implid by his modl: cos sn F 3 As can b sn in h prvious quaion, on inrsing fac of his modl wih rspc o h on facor modl in Schwarz 997 is h fac ha in his cas h volailiy of fuurs rurns dos no gos o zro as h im o mauriy of fuurs conrac approachs o infiniy, which is an undsirabl propry. I happns bcaus h sasonal facors ar long-rm facors. Ohr inrsing fac o a ino accoun is h sasonaliy in h volailiy of fuurs rurns. In his modl his sasonaliy disappars whn h im o mauriy of fuurs conrac approachs o infiniy. I happns bcaus his sasonaliy coms from h corrlaion bwn h sasonal facors and h non-sasonal facors and in his modl hr is only on nonsasonal sochasic facor which is a shor-rm facor. In following scions his facs ar going o b discussd again. 3.. h Four-Facor odl In his modl h log-spo pric X is h sum of hr sochasic facors: a long-rm componn, a shor-rm componn and a sasonal componn. X 3 h fourh sochasic facor is h ohr sasonal facor which complmns. h SDE of hs facors ar: d μ d 4 d κ d 5 d d 6 d d 7 h quaions 4 and 5 ar idnical o quaions and in Schwarz and Smih.

11 h ris-nural SDE ar: d μ ' d 8 κ λ d λ d λ d d 9 d d whr μ μ - λ is h ris-nural drif. 3 3 As bfor, h h log-pric of a fuurs conrac wih mauriy a im radd a im can b calculad applying h rsul in quaion 6: A whr: [ F X, ] cos sn A ln μ '.5.5 λ / λ λ sn λ F 4 / { cos } { cos sn } sn cos And h squard volailiy of fuurs rurns is: cos sn cos / sn.5 / { } 33 cos sn 34 Again givn ha in his modl hr is a long-rm sochasic facor, h sasonaliy in h volailiy of fuurs rurns dos no disappar whn h im o mauriy of fuurs conrac approachs o infiniy h Fiv-Facor odl In his modl h log-spo pric X is h sum of four sochasic facors: a long-rm componn, wo shor-rm componns and and a sasonal componn. X 35 h fifh sochasic facor is h ohr sasonal facor which complmns. h SDE of hs facors ar: d μ d 36 d κ d 37

12 d d κ 38 d d 39 d d 4 In Appndix B i can b sn ha h non-sasonal par of his modl is quivaln o h hr facor modl proposd in Corazar and Schwarz 3. h ris-nural SDE ar: d d μ ' 4 d d λ κ 4 d d λ κ 43 d d λ 44 d d λ 45 whr μ μ - λ is h ris-nural drif. Onc again, h h log-pric of a fuurs conrac wih mauriy a im radd a im can b calculad applying h rsul in quaion 6: [ ] cos, ln 5 A sn X F 46 whr: { } { } { } { } { } 5 cos cos cos cos /.5 / /.5 cos / / / cos / /.5.5 ' λ λ λ λ λ μ sn sn sn sn sn sn A 47 And h squard volailiy of fuurs rurns is: cos cos cos 5 sn sn sn F 48

13 4. Esimaion mhodology 4.. h Kalman Filr As sad in prvious sudis, on of h main difficulis in simaing h modl s paramrs is h fac ha h facors or sa variabls ar no dircly obsrvabl and mus b simad from spo and/or fuurs prics. Inuiivly, h non-sasonal facors long rm and shor-rm facors ar going o b simad basd on h rlaionship bwn long-mauriy fuurs and shor-mauriy fuurs or spo prics and h sasonal facors ar going o b simad hrough h rlaionship bwn fuurs conracs mauring in diffrn monhs. h formal way o do his is hrough h Kalman filring mhodology. his mhodology nabls h calculaion of h lilihood of a daa sris givn a paricular s of modl paramrs and a prior disribuion of h variabls which prmis h simaion of h paramrs using maximum lilihood chniqus. Daild accouns of Kalman filring ar givn in Harvy 989. h Kalman filr mhodology is a rcursiv mhodology ha simas h unobsrvabl im sris, h sa variabls or facors Z, basd on an obsrvabl im sris Y which dpnds on hs sa variabls. h rlaionship bwn h obsrvabl im sris and h sa variabls is dscribd hrough h masurmn quaion: Y d Z η,, N 49 whr Y n n x n h, d R, R, Z R h is h numbr of sa variabls, or facors, in h n modl and η R is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix H. h voluion of h sa variabls is dscribd hrough h ransiion quaion: whr c R Z c Z ψ,, N 5 h h x h h, R and ψ R is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix Q. In h radiional vrsion of his mhodology, wo condiions nd o b fulfilld. Firs, no missing poins in h daa s. Scond, lngh of vcor Y mus b indpndn of. Howvr, an improvd vrsion of his mhodology has bn dvlopd in Corazar and Narano 6 o handl wih incompl daa ss and vcors Y whos lngh dpnds on. h problm using his mhodology wih a daa s wih a lo of missing poins in h fuurs conracs wih highr mauriis is h unbalanc in h rlaionship bwn long and shor ffcs and, in h cas handl in his aricl, h unbalanc in h rlaionship bwn fuurs conracs whos mauriy occurs in diffrn monhs which accouns for h sasonal ffcs. As dvlopd blow, aing ino accoun hs considraions and ohr concrns rlad wih daa liquidiy, 3

14 h usd daa s has no missing poins and all vcors has h sam lngh, hrfor in his aricl h radiional vrsion of h Kalman filr mhodology is usd. In Corazar-Schwarz 3 an alrnaiv o h Kalman filr mhodology has bn dvlopd o sima h modl paramrs and h sa variabls. his chniqu is a simpl on which only nds a spradsh o b implmnd. 4.. Discrizaion o sima h paramrs of h modls hrough h Kalman filr mhodology, or hrough h Corazar and Schwarz 3 chniqu, w nd a discr-im vrsion of h modls. As sad in Scion 3, h soluion o h gnral problm of his aricl is 6 and Z is gaussian wih man and varianc givn by xprssions 7 and 8 rspcivly. hus, if h diffrnc bwn h currn priod and h iniial priod is on priod im and considring h ral SDEs insad of h ris-nural ons, Z follows h discr procss: A whr c As Z bds R c Z ψ,, N 5 h, A h h R and ψ R is a vcor of srially uncorrlad Gaussian disurbancs wih zro man and covarianc marix A As As A R ds Q. his quaion will b calld, following sandard convnions in h liraur, h ransiion quaion of ach modl. h masurmn quaion is us h xprssion of h log-fuurs prics Y in rms of h facors Z by adding srially uncorrlad disurbancs wih zro man η o a ino accoun masurmn rrors drivd from bid-as sprads, pric limis, non-simulaniy of obsrvaions, rrors in h daa, c. o avoid daling wih a gra amoun of paramrs, h covarianc marix H will b assumd diagonal and all is diagonal lmns ar h sam. his simpl srucur for h masurmn rrors is imposd so ha h srial corrlaion and cross corrlaion in h log-prics is aribud o h variaion of h unobsrvabl sa variabls. h masurmn quaion will b xprssd as: Y d Z η,, N 5 h spcific ransiion and masurmn quaions for h paricular modls considrd in his papr hr, four and fiv facor modls ar drivd in Appndix C. 4

15 5. Daa h daa s mployd in h simaion procdur consiss of wly obsrvaions of Hnry Hub naural gas fuurs prics radd a NYEX 8. I has bn dfind four daa ss for h simaion procdur. In all cass nin fuurs conracs i.. n 9 hav bn usd. h hr firs ons daa ss ar mad of conracs F, F5, F9, F4, F8, F, F7, F3 and F35 whr F is h conrac closs o mauriy, F is h scond conrac closs o mauriy and so on. h firs on conains obsrvaions from 9//997 o /6/6, which implis 438 quoaions of ach conrac, h scond on from 9//997 o /6/, which implis quoaions of ach conrac, and h hird on from /3/ o /6/6, which implis 6 quoaions of ach conrac. h las daa s nails conracs F, F8, F5, F, F9, F36, F43, F5 and F57 from /3/ o /6/6. As xplaind in Schwarz 997, sinc fuurs conracs hav a fixd mauriy da, h im o mauriy changs as im progrsss, bu rmaining in a narrow im inrval. ha is h rason why, as in Schwarz 997, i is assumd ha h im o mauriy dos no chang wih im and i is qual o on monh for F, o wo monhs for F and so on. hr ar currnly 7 conracs radd for diffrn mauriis ranging from o 7 monhs. Howvr, unil Dcmbr h maximum mauriy radd a NYEX was only 36 monhs. In his da nw conracs wr inroducd o includ mauriis up o 7 monhs, bu h liquidiy of hs nw conracs is rlaivly low and in rcn ims hr is no quoaion for h conracs wih h las mauriis in som das. Bfor Spmbr 997 hr wr also liquidiy problms wih h conracs wih h las mauriis in hos ims and, o b prcis, hr is no quoaion for h las mauriis ons in hos ims from F o F36 in many das. As sad abov, i is xpcd ha h sasonal facor in h paricular modls has on yar priod. hus, h numbr of fuurs conracs, as long as conracs wih diffrn mauriis, ndd in h prsn sudy is considrabl highr han in prvious sudis.. Spcifically, w nd fuurs conracs wih mor han on yar o mauriy, conracs wih diffrn mauriis and as much conracs as possibl o accoun for i. Fuurs conracs wih long-rm and shorrm mauriis ar also ncssary o sima proprly h paramrs of h non-sasonal facors long-rm and shor-rm facors. orovr, h highr h numbr of conracs usd in h simaion, h mor prcis h simas of h paramrs. Nvrhlss, hr ar also ohr considraions o a ino accoun. As xplaind abov, in ordr o sima proprly h rlaionship bwn long and shor-rm ffcs and h paramrs in h sasonal procss, i is dsirabl ha h daa s includs mor or lss h sam quoaions for all fuurs conracs.. orovr, o accoun for srucural changs in h naural gas pric dynamics i is dsirabl o considr diffrn daa ss wih diffrn sampl priods. 5

16 aing ino accoun all hs argumns, i has bn dfind h ss of fuurs conracs spcifid abov. Nonhlss, h simaion has bn rpad using diffrn daa ss daa ss wih mor fuurs conracs and wih fuur conracs wih ohr mauriis and h rsuls ar mor or lss h sam han hos prsnd in his aricl. NYEX quoaions ar mar prics, ha is, NYEX is an xchang whos prics ar formd from bid and as ordrs which marrid. Howvr, hr ar no Hnry Hub naural gas spo prics. Bloombrg provids naural gas in Hnry Hub spo prics asing h paricipans in h mar for hir bs simaion of i and applying an inrnal procdur. hrfor, h spo prics ar no proprly mar prics. abl I conains h man and h sandard dviaion of all NYEX daa sris mployd in his sudy. In all cass, uni ar $/Bu Empirical Rsuls 6.. h hr-facor odl Assuming ha h varianc-covarianc marix of η is diagonal and all diagonal lmns ar h sam, hr ar lvn paramrs o b simad in h hr-facor modl: μ,, φ,,,,, λ, λ, λ and η. abl II prsns h rsuls for h hr-facor modl applid o h four daa ss dscribd abov. On inrsing issu from hs rsuls is h fac ha in all cass h sasonal priod is on yar, φ is mor or lss π, and h sandard dviaion of h sasonal facor is significanly diffrn from zro. his implis ha h sasonaliy in h Hnry Hub naural gas pric is sochasic wih on yar priod, which is consisn wih h rsuls from Scion. h spd adusmn is highly significan which implis, li in h cas of oil Schwarz, 997, man rvrsion in h naural gas pric. h mar prics of ris, h λ s, ar no significanly diffrn from zro in mos of h cass. h long-rm rnd μ is posiiv and significanly diffrn from zro in all cass, which implis long-rm growh in h naural gas pric. As can b sn in h abl, i has bn obaind mor or lss h sam rsuls wih h firs hr daa ss. h rsuls wih h las daa s ar slighly diffrn. Spcifically, hr is significanly lss man rvrsion, h volailiy of fuur rurns, calculad subsiuing h 9 Bu mans millions of Briish hrmal unis and is an nrgy masur. US $ pr Bu is h accpd way o rprsn h naural gas pric in YEX, naural gas quos in US $ pr Bu. h nrgy conaind in a crud oil barrl dpnds on h ind of h oil, bu broadly spaing an oil barrl conains somhing lss han six Bu of nrgy. In ordr o sav spac only h comparison of h rsuls obaind wih h Schwarz and Smih wo-facor modl wih drminisic sasonaliy and our four-facor modl wih sochasic sasonaliy ar prsnd, alhough similar conclusions ar obaind wih ohr modls. 6

17 paramrs is quaion 3, is also significanly lowr, h pric of ris for h sor-rm facor is highly significan and h long-rm rnd μ is also diffrn. h fourh daa s conains fuurs conracs wih highr mauriis han hos of h firs hr daa ss. i in h oil cas, Schwarz 997, i has imporan implicaions in valuaion and hdging naural gas coningn claims or in invsmn dcisions, sinc i is ncssary o accoun wih h paymns rm srucur o choos h fuurs conracs o sima h modl paramrs. his fac is also a firs sign ha mor srucur i is ndd. As sad abov, h sasonal facor is a long-rm facor, bu, as i will b discussd blow, i is unabl o capur all sochasic long-rm ffcs prsn in h naural gas pric. A las on sochasic long-rm facor is ndd. 6.. h Four-Facor odl Assuming ha h varianc-covarianc marix of η is diagonal and all diagonal lmns ar h sam, hr ar sixn paramrs o b simad in h four-facor modl: μ,, φ,,,,,,,,,, μ, λ, λ, λ and η. abl III prsns h rsuls for h four-facor modl applid o h daa ss dscribd abov. As in h prvious modl, in all cass h sasonal priod is on yar and h sandard dviaion of h sasonal facor is significanly diffrn from zro alhough is magniud is lowr han in h hr-facor modl. his is du o h fac ha in h hr-facor modl h sasonal facor capurs sochasic long-rm ffcs which ar capurd by h sochasic longrm facor in h four-facor modl. h spd adusmn is highr han prviously, which implis mor man rvrsion, possibly bcaus h shor-rm facor in h hr-facor modl was capuring long-rm sochasic ffcs, and h paramrs of h nw sochasic facor, h long-rm on, ar also highly significan, which confirms h prvious prcpions ha a sochasic long-rm facor is ndd. h mar prics of ris, h λ s, ar no significanly diffrn from zro in mos of h cass, h long-rm rnd is posiiv and h long-rm rnd adusd by ris μ is ngaiv and in boh cass significanly diffrn from zro. In his cas h rsuls ar mor or lss h sam in all daa ss. h only diffrnc which rmains is ha in h fourh daa s, which uss fuurs wih highr mauriis, hr is lss man rvrsion, which is consisn wih oil prics Schwarz

18 6.3. h Fiv-Facor odl Assuming ha h varianc-covarianc marix of η is diagonal and all diagonal lmns ar h sam, hr ar wny hr paramrs o b simad in h fiv-facor modl: μ,,, φ,,,,,,,,,,,,,, μ, λ, λ, λ, λ and η. abl IV prsns h rsuls for h fiv-facor modl applid o h daa ss dscribd abov. h rsuls ar qui clos o h four-facor modl ons. In his modl hr ar wo shorrm facors whos paramrs ar highly significan, and in all cass on of hs facors has highr spd of adusmn han h ohr. ha mans ha hr ar wo yps of sochasic shor-rm ffcs, on h on wih highr wih srongr man rvrsion han h ohr h on wih smallr, and boh of hm significan. I is inrsing o no ha, in all cass, h pric of ris for h shor-rm facor wih srongr man rvrsion is highly significan. 7. Comparing h modls In his scion w will compar h rlaiv prformanc of h modls. Spcifically, w will analys h in-sampl and ou-of-sampl prformanc and h goodnss of fi for h spo pric, h forward curv and volailiy of fuurs rurns. In all cass h rsuls will b compard wih hos obaind wih h Schwarz and Smih wo-facor modl wih drminisic sasonaliy. h analysis of h goodnss of fi for h spo pric and h forward curv will b dvlopd using h firs daa s bcaus i is h biggr on whras h goodnss of fi for h volailiy will b dvlopd using h fourh on o accoun for volailiis of fuurs wih mauriis highr han 36 monhs. Nvrhlss, using whavr ohr daa s h rsuls ar broadly h sam. 7. Drminisic vrsus sochasic sasonaliy Bfor comparing h rlaiv prformanc of our hr modls, i could b usful o compar h rsuls obaind wih our modls wih sochasic sasonaliy wih hos obaind wih h sandard modls wih drminisic sasonaliy. abl V prsns h rsuls of h simaion of h Schwarz and Smih wo-facor modl wih drminisic ssonaliy, which will b compard wih hos obaind wih our fourfacor modl wih sochasic sasonaliy abl III. Firs of all i is inrsing o compar h valu of h Schwarz Informaion Cririon SIC obaind wih boh modls. If w dfin h SIC as ln q / ln, whr q is h numbr of simad paramrs, is h numbr 8

19 of obsrvaions and is h valu of h lilihood funcion using h q simad paramrs, hn h prfrrd modl is h on wih h highs SIC. I is found ha h valu of h SIC for our four-facor modl wih sochasic sasonaliy, shown a h boom of abl III, is highr han h corrsponding valu obaind wih h sandard modl wih drminisic sasonaliy, shown a h boom of abl V. A scond way of comparing h modls is hough hir prdiciv abiliy. h in-sampl prdiciv abiliy of h Schwarz and Smih wo-facor modl and our four-facor modl is prsnd in abl VI, using h whol daa s, i.. daa s. I is found ha, in gnral, h modl accouning for sochasic sasonaliy ouprforms h sandard modl wih drminisic sasonaliy. h advanags of h sochasic sasonaliy modl ovr h drminisic sasonaliy on ar vn mor clar whn w analys h ou-of-sampl prdiciv abiliy abl VII.h ou-of-sampl rsuls ar obaind valuing h conracs in daa s 3 wih h paramrs obaind simaing h modls wih daa s. 7. Spo Pric Following a procdur clos o h Kalman filr chniqu i has bn obaind facors simaions in ach im basd on h informaion availabl unil his im for ach modl and for h paramrs simad in ach daa s. h acual spo pric providd by Bloombrg and h spo pric simad by ach modl for h firs daa s and h dviaion from h acual spo pric of h spo prics simad by ach modl ar dpicd in Figur 4. In h firs char of Figur 4 w can apprcia ha h now classical wo-facor modl dos no sima as accuraly as h modls prsnd in his aricl h Hnry Hub naural gas spo pric. ha is bcaus h wo-facor modl dos no accoun for sasonaliy. In h scond char of Figur 4 i is possibl o noic ha h fiv-facor modl is h modl wih h bs prformanc, h four-facor modl h scond on and h hr-facor modl is h modl wih h wors prformanc of h modls prsnd in his aricl in simaing h acual spo pric. h sandard dviaion of h dviaion from acual spo pric is 9% for h fiv-facor modl, 9.6% for h four-facor modl, 3.% for h hr-facor modl and 9.% for h wo-facor modl. On inrsing fac, which has bn obsrvd in h prvious scion, is h diffrnc in h simaions obaind wih h hr-facor modl and wih h four and fiv-facor modls. h simaions of h las o modls ar much br han h hr-facor modl ons. I confirms ha a long-rm sochasic facor is ndd o undrsand h naural gas pric dynamics. h fiv-facor modl simaions ar br han h four-facor modl ons, bu boh of hm ar clos and highly accura. 9

20 7.3 Forward Curv Procding in a similar mannr as in h prvious sub-scion and using h firs daa s as wll, i is also possibl o obain simaions of h forward curv. h acual forward curv and h on simad by h modls for a randomly chosn da ar dpicd in Figur 5. As obaind prviously in h cas of h goodnss of fi for h spo pric, h modl which br fis o acual daa is h fiv-facor modl, h scond on is h four-facor modl and so on. I is also possibl o obsrv ha h wo facor modl dos no fi h acual daa bcaus i dos no accoun for sasonaliy. orovr, h simaions for h fiv and four-facor modls ar clos and ar much br han h hr-facor modl ons. 7.4 Volailiy of Fuurs Rurns h volailiy of fuurs rurns can b calculad by subsiuing h simad paramrs prsnd scion 6 in h corrsponding formulas dvlopd in scion 3. h volailiy of fuurs rurns simad by ach modl compard wih h acual on ar dpicd in Figur 6. In his cas i has bn usd h fourh daa s bcaus wih h firs on i is no possibl o calcula h acual volailiy for mauriis highr han 36 monhs bu h conclusion of h analysis will b h sam if h firs daa s ss wr usd. h commns in h prvious sub-scions also apply hr. On inrsing issu o no is ha h acual naural gas pric volailiy is sasonal, opposily o oil. As sad in scion 3, in h modls proposd in his aricl, naural gas pric volailiy is also sasonal. In h cas of h hr-facor modl his sasonaliy dcrass whn fuur mauriy grows, going o zro whn mauriy gos o infiniy, whras i dos no in h cas of h four and fiv-facor modls. ooing a Figur 6 i sms ha h sasonaliy in h acual volailiy dos no dcras, which a nw issu in favour of h four and fiv-facor modls and agains h hr-facor on. orovr, in h cas of h hr-facor modl i is no possibl o apprcia h sasonaliy vn for shor mauriy conracs, bcaus in his cas is much smallr han. 8. Conclusions os sudis on h sochasic bhavior of commodiy prics ar focusd on oil prics. h numbr of paprs addrssing h sudy of naural gas prics is sill scarc. Howvr naural gas rprsns almos h fourh par of h world nrgy consumpion. h lac of conomical ransporaion and h limid sorabiliy of naural gas ma is supply unabl o chang in viw of sasonal variaions of dmand. hrfor, naural gas pric is srongly sasonal. Analysing h naural gas pric spcrum, i sms highly probabl ha h sasonaliy in naural gas pric is a sochasic facor and no a drminisic on, Howvr, o h bs of h

21 auhors nowldg, sasonaliy has nvr bn considrd as a sochasic facor in prvious sudis. hrfor, in his aricl i has bn dvlopd a gnral nm-facor modl of h sochasic bhavior of commodiy prics ha considrs h sasonaliy as a sochasic facor. hn his gnral modl has bn paricularizd for m and n,,3, hus, hr, four and fiv-facor modls has bn obaind o xplain h sochasic bhavior of h Hnry Hub naural gas pric. h paramrs of h modls hav simad hrough h Kalman filr mhodology and using NYEX daa. On of h main conclusions of his sudy is h confirmaion of h fac ha h sasonaliy in h naural gas pric is sochasic and no drminisic as many sudis assum. I is also found ha h naural gas pric sasonal priod is on yar. h sasonal facors considrd in his papr hav a long-rm componn bu i is dmonsrad ha his long-rm componn is unabl o capur proprly h naural gas longrm dynamic. Consqunly, i is ncssary o us a random wal as a long-rm facor. hrfor h four and fiv-facor modls ar much br han h hr-facor on in xplaining h naural gas pric bhavior. h classical wo-facor modl fails in his as bcaus i dos no accoun for sasonaliy. Consqunly, h four and h fiv-facor modls prsnd in his aricl sm appropria o valu all naural gas coningn claims or invsmn procs. h fiv-facor modl is br han four-facor on alhough i nds mor srucur. Hnc, h us of h four or h fivfacor modls will dpnd on h prcision ndd. APPENDICES Appndix A: Sasonal Facors As sad abov, h sochasic diffrnial quaion SDE for ach sasonal facor is: da i a d Q whr a is a complx facor a i, Q a complx numbr Q Q iq and W a a complx Brownian moion W a W iw, providd ha a d and d bing uncorrlad and wih h sam varianc. Equalling ral componn wih ral componn and imaginary componn wih imaginary componn in h prvious quaion i follows wo ral SDEs: d d Q Q 4 d d Q Q 4

22 If W and W ar uncorrlad hn d and d ar also uncorrlad and wih h sam varianc which is Q Q, bing h complx numbr Q modul h sufficincy condiion. cov, In h ohr hand, d, d Q Q cov, Var d Q d Q d Q Q cov, and Var d Q d Q d Q Q cov,. o g cov d, d and Var d Var d i is ncssary ha cov, h ncssary condiion. now θ h phas. Q b xprssd in polar coordinas Q iθ whr is h modul and Dfining a iθ a, i is clar ha Wa iθ as or is quivaln: W a iθ W a. Equalling ral componn wih ral componn and imaginary componn wih imaginary componn in h las quaion hn: W cosθ snθ W W snθ cosθ W From h prvious quaion, aing ino accoun ha W and W ar uncorrlad, i follows ha: W Var W W Var W I bing I h x idniy marix. hrfor, h phas θ is indisinguishabl. Appndix B: Fiv facor modl rformulaion In h fiv facors modl h log-spo pric X is givn by: X h facors SDE can b xprssd as: d μ d d d d d 3 d

23 Dfining κ and v y i is clar ha: κ d κ κ κ κ y d κ dy κ d κ κ dv d d v d y v v μ and μ, in his cas: dv dv v d μ v d d d μ d 3 4 dx I is asy o s ha: X hrfor: μ v d y v 5 dy d d d d d d dv d μ d v y d X X whr X X y d 6 hus, h modl can b xprssd in rms of h nw variabls. h nw facors ar: X, y, v, and. h SDE of hs nw facors ar: dx v y d X X dy y d dv μ v d 3 d d 8 d d 9 And now i is obvious ha his modl is a gnralizaion of h hr-facor modl dvlopd in Corazar and Schwarz 3. Appndix C: ransiion and masurmn quaions h hr-facor odl ransiion quaion: Z c Z ψ,, N 49 3

24 4 whr Z, c μ, cos sin sin cos and { } { } { } { } sn sn sn sn Var cos cos cos cos / ψ asurmn quaion: Z d Y η,, N 5 whr n F F Y ln ln, 3 3 n A A d, cos cos n n sn sn n h Four-Facor odl ransiion quaion: Z c Z ψ,, N 5 whr Z, c μ, cos sin sin cos and { } { } { } { } sn sn sn sn sn sn Var cos cos / cos cos cos / cos / / ψ asurmn quaion: Z d Y η,, N 5 whr n F F Y ln ln, 4 4 n A A d, cos cos n n sn sn n h Fiv-Facor odl ransiion quaion: Z c Z ψ,, N 53

25 5 whr Z, c μ, cos sin sin cos and Var ψ is h sam ha in h ohr modls bu wih a nw column: { } { } { } { } cos cos cos cos / / / ψ sn sn sn sn Var asurmn quaion: Z d Y η,, N 54 whr n F F Y ln ln, 5 5 n A A d, cos cos n n sn sn n n Acnowldgmns W han Andrés Ubirna Gorricho, Juan anul arín Prio and Ángl ón Vall for hir hlp. W spcially han Rpsol YPF for is suppor. os of his rsarch was don whil Francisco Javir Población García was woring in Rpsol YPF. W also han paricipans in h XXX Simposio d Análisis Económico in urcia and h paricipans in h I Congrso d la Asociación Española para la Economía Enrgéica in adrid. Grgorio Srna acnowldgs h financial suppor providd by h inisrio d Educación y Cincia gran SEJ5-893-C- and Juna d Casilla-a ancha gran PAI5-74. Any rrors ha rmain ar, howvr, nirly h auhors own. REFERENCES Brnnan,.J. and E.S. Schwarz, 985, Evaluaing naural rsourc invsmns, Journal of Businss 58, Corazar, G. and E.S. Schwarz, 3, Implmning a sochasic modl for oil fuurs prics, Enrgy Economics 5, 5-8. Corazar G, Narano, 6, An N-Facor gaussian modl of oil fuurs prics, Journal of Fuurs ars, 6 3, 9-33.

26 Harvy, A.C Forcasing, Srucural im Sris odls and h Kalman Filr. Cambridg Univrsiy Prss, Cambridg, U.K. aughon, D.G. and Hnry D. Jacoby, 993, Rvrsion, iming opions, and long-rm dcision maing, Financial anagmn 33, 5-4. aughon, D.G. and Hnry D. Jacoby, 995, h ffcs of rvrsion on commodiy procs of diffrn lngh,. rigorgis, d. Ral Opions in Capial Invsmn: odls, Sragis and Applicaions, Pragr, Wspor, C, ucia, J. and E.S. Schwarz,, Elcriciy Prics and Powr drivaivs: Evidnc from h Nordic Powr Exchang, Rviw of drivaiv Rsarch 5:, 5-5. Osndal, B.,99, Sochasic Diffrnial Equaions. An Inroducion wih Applicaions, 3rd d. Springr-Vrlag, Brlin Hidlbrg. Ross, S.A., 997, Hdging long run commimns: Exrciss in incompl mar pricing, Banca on Econom, Nos 6, Schwarz, E.S., 997, h sochasic bhaviour of commodiy prics: Implicaion for valuaion and hdging, Journal of Financ 5, Schwarz, E.S. and J.E. Smih,, Shor-rm variaions and long-rm dynamics in commodiy prics, anagmn Scinc 46, Sornsn, C.,, odling sasonaliy in agriculural commodiy fuurs, Journal of fuurs mars 5, Wi, W.S., 5, h im sris analysis. Univaria and mulivaria mhods, h d., Addison Wsly. 6

27 FIGURE HENRY HUB NAURA GAS PRICES. WEEKY OBSERVAIONS FRO NYEX Pric of Naural Gas in Hnry Hub 8 Spo Pric Forward Curv 6 4 oc-89 ul-9 abr-9 n-9 oc-9 un-93 mar-94 dic-94 sp-95 un-96 mar-97 dic-97 sp-98 may-99 fb- nov- ago- may- fb-3 nov-3 ago-4 abr-5 n-6 oc-6 ul-7 abr-8 n-9 oc-9 ul- mar- $/Bu FIGURE SCAING FACORS WEEKY OBSERVAIONS FRO NYEX. Spo Pric Scaling Facors Jan Fb ar Apr ay Jun Jul Aug Sp Oc Nov Dc [Ponr la fcha d la curva forward]. Forward Curv Scaling Facors Jan Fb ar Apr ay Jun Jul Aug Sp Oc Nov Dc 7

28 FIGURE 3 SPECRU Naural Gas in Hnry Hub Spo Pric Spcrum Frcuncy 8

29 FIGURE 4 SPO PRICE ESIAION 5 Naural Gas in Hnry Hub Spo Pric $/Bu 5 wo-facor odl Schwarz-Smih hr-facor odl Four-Facor odl Fiv-Facor odl Bloombrg Daa 5 sp-97 dic-97 mar-98 un-98 sp-98 dic-98 mar-99 un-99 sp-99 dic-99 mar- un- sp- dic- mar- un- sp- dic- mar- un- sp- nov- fb-3 may-3 ago-3 nov-3 fb-4 may-4 ago-4 nov-4 fb-5 may-5 ago-5 nov-5 fb-6 Naural Gas in Hnry Hub Spo Pric: Dviaion from Acual Pric % 6% Dviaion % -% sp-97 dic-97 mar-98 un-98 sp-98 dic-98 mar-99 un-99 sp-99 dic-99 mar- un- sp- dic- mar- un- sp- dic- mar- un- sp- nov- fb-3 may-3 ago-3 nov-3 fb-4 may-4 ago-4 nov-4 fb-5 may-5 ago-5 nov-5 fb-6-6% wo-facor odl Schwarz-Smih hr-facor odl Four-Facor odl Fiv-Facor odl 9

30 FIGURE 5 FORWARD CURVE 5/7/ Forward Curv: 5/7/ 4,4 4, 4, $/Bu 3,8 3,6 3,4 wo-facor odl Schwarz-Smih hr-facor odl Four-Facor odl 3, Fiv-Facor odl Acual Daa 3, Fuur auriy onhs 3

31 FIGURE 6 VOAIIY OF FUURES REURNS 6% Volailiy of Fuurs Rurns Volailiy 5% 4% 3% % wo-facor odl Schwarz-Smih hr-facor odl Four-Facor odl Fiv-Facor odl NYEX Daa % % Fuur auriy onhs 3

32 ABE I HENRY HUB NAURA GAS FUURES PRICES. EAN AND SANDARD DEVIAION $/Bu Daa-S Daa-S Daa-S 3 Daa-S 4 an Sand. Dv. an Sand. Dv. an Sand. Dv. an Sand. Dv. F 4,57,53 3,9,5 5,99,57 F 5,99,57 F5 4,6,38 3,6, 6,,57 F8 5,96,7 F9 4,5,5 3,,5 5,94,57 F5 5,8,9 F4 4,4, 3,3,89 5,86,56 F 5,5,7 F8 4,8,96,97,8 5,63,56 F9 5,34,7 F 4,,85,93,78 5,5,55 F36 5,,55 F7 4,7,89,9,7 5,45,54 F43 5,7,35 F3 4,6,7,88,7 5,7,54 F5 5,8,43 F35 4,3,65,88,68 5,,53 F57 4,9,9 ABE II EPRICIA RESUS. HREE-FACOR ODE Daa-S Daa-S Daa-S 3 Daa-S 4 Numbr obs μ Φ λ λ λ η og-lilihood AIC SIC Sandard rrors in parnhss 3

33 ABE III EPIRICA RESUS. FOUR-FACOR ODE Daa-S Daa-S Daa-S 3 Daa-S 4 Numbr obs μ K Φ μ λ λ λ η og-lilihood AIC SIC Sandard rrors in parnhss 33

34 ABE IV EPIRICA RESUS. FIVE-FACOR ODE Daa-S Daa-S Daa-S 3 Daa-S 4 Numbr obs μ Φ μ λ λ λ λ η og-lilihood AIC SIC Sandard rrors in parnhss 34

35 ABE V EPIRICA RESUS. SCHWARZ AND SIH WO-FACOR ODE WIH DEERINISIC SEASONAIY Daa-S Daa-S Daa-S 3 Numbr obs μ K Φ μ λ λ λ η og-lilihood 988,8 86, AIC SIC

36 ABE VI IN-SAPE PREDICIVE ABIIY PANE A: SCHWARZ AND SIH WO-FACOR ODE WIH DE. SEAS. Bias Bias % Sd Sd % man Sd % sd RSE RSE % man RSE %RSE F,873,85655,7374 5,3749 4,4848,7469,5386,556 F5 -,563 -,758,583 4,995,7,68,4585,4345 F9 -,594 -,45, ,49457,6894,4996,353,3346 F4,658,9846,483 3,5686 9,545,435,359,99 F8,8,6344,37473,7558 8,787,37538,7599,6335 F,7588,5636,34945, ,48733,35759,6537,5379 F7,5,669,36535, ,95945,36599,734,639 F3 -,75 -,346,388, ,7367,38,899,774 F35 -,5 -,5535,36748,7867 9,644839,3685,79,683 PANE B: FOUR-FACOR ODE WIH SOCHASIC SEASONAIY F,7864,567,7977 5,953 4,377,745,5,493 F5 -,9 -,6585,574 3,73846,658,53538,37948,3595 F9 -,56 -,448,4567 3,3979 9,88676,4553,3567,3955 F4 -,39 -,735,3748, ,3454,375,684,5556 F8,7655,568,35478, ,3983,3694,6684,546 F,533,39453,346, ,8695,3458,563,455 F7 -,4 -,345,3555,636 8,64499,3557,63,58 F3,3,365,3943,9837,5,39554,997,868 F35 -,65 -,4664,3658, ,46374,36579,7739,

37 ABE VII OUR-F-SAPE PREDICIVE ABIIY PANE A: SCHWARZ AND SIH WO-FACOR ODE WIH DE. SEAS. Bias Bias % Sd Sd % man Sd % sd RSE RSE % man RSE %RSE F,6,9383,793 6, ,333,73847,6857,64534 F5 -,638 -,4598,668 5, ,436,6766,6679,5997 F9 -,84 -,66,5649 4, ,6853,5799,53648,5649 F4 -,44 -,989, ,4848 7,87,5958,4959,48 F8 -,4 -,359,4866 3, ,9585,443,4998,477 F -,43 -,46, ,5899 5,89,37674,368,357 F7,3337,396,435 4,8993 8,4734,438,444,447 F3,4,3,4573 4, ,558,466,44883,43797 F35,8478,798,449 4, 9,49,46334,4493,43935 PANE B: FOUR-FACOR ODE WIH SOCHASIC SEASONAIY F,579,46688,735 6,5347 8,78,776,6697,6987 F3 -,559 -,477,5487 5,4566 7,438,574,56,49933 F5 -,96 -,7389, ,4435 4,74753,53675,49656,4786 F7 -,49 -,3958,34 3,97334,7384,435,3956,38364 F -,598 -,5685,343 3, ,499,347,3983,3 F -,6 -,9743,36,983895,734,373,3389,3557 F4,333,9857,356 3, ,853,3575,3488,3346 F6,63,5567,3746 3,6938 6,9563,4686,39475,385 F8,345,3,376 3,6489 7,4384,398,37836,