DETERMINISTIC TREND / DETERMINISTIC SEASON MODEL. Professor Thomas B. Fomby Department of Economics Southern Methodist University Dallas, TX June 2008

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1 DTRMINISTIC TRND / DTRMINISTIC SASON MODL Professor Toms B. Fomby Deprmen of conomics Souern Meodis Universiy Dlls, TX June 008 I. Inroducion Te Deerminisic Trend / Deerminisic Seson DTDS model is one of e firs ime series models proposed o ndle rends nd sesonliy in economic nd business d. I populriy depended on e relive ese of esiming suc model by e meod of ordinry les squres nd e inerprebiliy of e model. Ineresingly, is model is specil cse of e Unobservble Componens model UCM wi fixed level 0 nd fixed slope 0 resuling in e deerminisic rend specificion 0, 0 fixed dummy sesonls 0, ssuming n uoregressive fcor, r r, nd no irregulr componen 0. Te fc e UCM encompsses e curren simple model we re going o enerin us goes o iglig ow fr ime series modeling s come over e ls severl decdes. Acully, s you will see below, we generlize e uoregressive erm o be n ARr process s compred o e AR erm ssumed in e UCM. Bu e poin is, essenilly, e curren model is specil cse of e UCM. II. Noion for e DTDS Model Assume e imes series nively of wriing e DTDS model s y y is observed monly. A firs blus, one mig ink D D 3D3 D,

2 r r were y is e rge vrible,,,..., T, D is sesonl dummy vrible kes e vlue of one if e - observion is observed in e - mon nd 0 oerwise, nd e errors follow n ARr process nd e errors re wie noise. However, closer look indices redundncy in e smple design implied by e prmerizion nd. In fc, e full se of sesonl dummy vribles D is pefecly colliner wi e inercep. To void e so-clled, D,, D, Dummy Vrible rp, we my ke one of ree cs: We cn drop one of sesonl dummies, sy e Jnury sesonl D. Tis implies e inercep,, is e Jnury inercep wile e coefficiens of e oer mons re incremens o e Jnury inercep. Le us cll is e relive o Jnury prmerizion. T is, e Februry inercep is, e Mrc inercep is, ec. Of course, e dropping of e Jnury dummy is rbirry nd 3 one could us s esily drop ny oer mon. For now we will sick wi dropping e Jnury dummy nd sick wi e lbel cosen for e prmerizion. We could drop e inercep in our model nd en e sesonl dummy coefficiens would represen e respecive inerceps of e mons. Tis, of course, requires e regression be run roug e origin nd in is cse some of e clssicl mesures of goodness-of-fi like e coefficien of deerminion nd e overll F-sisic re no longer pplicble loug rdiionl ypoesis esing cn sill be done in is conex. One mig cll is e ec seson s is own inercep prmerizion. We cn keep ll of e coefficiens in bove bu impose resricion on e sesonl dummy coefficiens voids e Dummy Vrible rp. One resricion is ofen imposed is seing e sum of e sesonl coefficiens equl o zero. Te dvnge of is prmerizion is e signs of e coefficiens revel e sronger mons ose wi posiive coefficiens versus e weker mons ose wi negive coefficiens. One mig cll is e zero sum consrin prmerizion.

3 As i urns ou, ll of ese prmerizions re equivlen o ec oer in e sense e coefficiens esimes one mig ge from using one prmerizion cn esily be rnsled ino e coefficiens esimes produced by eier of e oer wo prmerizions. Since e relive o Jnury prmerizion is esier o implemen in SAS, especilly wen using Proc Auoreg o esime e uocorrelion srucure, we will pursue is prmerizion exclusively in e following discussion nd wrie i s y D 3D3 D, 3 r r. 4 In erms of convenionl ddiive ime series decomposiion, e rend is represened by e pr of e model is e y inercep nd is e slope of e deerminisic rend line,, e pr D 3D3 D, is sesonl pr of e model, nd conins e irregulr pr,, plus e cyclicl pr, r r, of e model. III. xmining Some of e Deils of e DTDS model Le us look more closely e DTDS model 3 nd 4. In is form e model is liner rend model s compred o qudric rend model. If > 0 nd > 0, en, generlly speking, e y d is posiive ime 0, nd s posiive slope o i. Of course we could ve < 0 wic would imply e d is declining. In culiy, if e d s curvure in i we could, insed, model e rend s. Of course, we cn mke e coice beween e liner nd qudric rend forms of e d by closely inspecing e d nd, s we will ler see, using ess of ypoeses concerning. For now, we will ssure e rend in e d we re nlyzing is liner, ence we ssume rend for now. 3

4 W re e menings of e sesonl dummy vribles? Given e specificion we see Jnury's rend line is given by, Februry's rend line is given by,..., nd December's rend line is given by. Tus, e bse ' rend line is Jnury's rend line, wile e,, 3,..., denoe e incremenl inercep coefficiens disinguises e oer mons' rend lines in priculr e inerceps from e rend line for e Jnury mons of ec yer. One cn now see wy is prmerizion of e model is clled e ``Relive o Jnury'' prmerizion. Terefore, in idenifying e sesonl effecs bo relive nd bsolue we cn o compre e mgniudes of e Jnury inercep, e Februry inercep, ec.. If e coefficien is posiive, en e mon is ``sronger'' n Jnury, oerwise i is weker n Jnury. Sreng of sesonl effec en is relive o Jnury. Obviously, if ll of e ' s is posiive en, by deful, Jnury's sesonl effec is e wekes. Te verge sesonl effec is, of course, [ / [ ]/ /. 5 Ten e srong mons in erms of sesonl effec re ose wose rend inercep for Jnury nd for e oer mons re greer n wile e wek mons ve or inerceps re less n. If one wns o sndrdize e sesonl effecs, one could do so by forming e coefficiens *, * *,.,. 6 Tese coefficiens re suc eir signs elps us disinguis beween e posiive sesonlly srong mons nd e negive sesonlly wek mons. Also eir bsolue mginiudes cn be used for compring e sesonl effecs cross mons. For exmple, consider e sndrdized sesonl coefficiens 6 for e Plno Tx Revenue d s produced by e progrm Plno_Tes_Sesonliy.ss: 4

5 Tble Sndrdized sesonl effecs by mon. Tey sum o zero. Srong mons re posiive nd wek mons re negive. Teir mgniudes cn be compred. sum d d d3 d4 d5 d d7 d8 d9 d0 d d Te vrible sum represens e sum of e sndrdized coefficiens wic is equl o zero up o minue compuing rounding error. Obviously, from sesonl effec sndpoin, e sronger mons re, 5, 8, nd Februry, My, Augus, nd November wile e oer mons re weker. Te sronges mon is Februry wile e wekes mon is April. Bu w bou e cycle pr of is model? To undersnd ow e equion is cpuring e cyclicl pr of e d. Consider e AR version of 4, 4 were we require, < in order for e process o be sionry i.e. ve consn men, vrince nd covrince wi respec o ime. We simply use rer n o simplify e noion somew. In e Appendix we sow i 0 ii Vr iii Cov, 7 iv Corr, For exmple, in e specil AR model 0.8 e uocorrelion funcion of 5

6 e errors is represened grpiclly by e following grp , , , , , , Given is uocorrelion funcion, we cn see dcen errors, ve n uocorrelion coefficien of 0.8 bu errors nd re -periods pr ve smller correlion of beween em Te furer e errors re pr, e less e correlion is Bu ow does suc n error process model cycliciy in economic nd business d? Well, pr form e rend nd sesonl effec, if e d y is bove e rend plus sesonl effec i.e. > 0 en e subsequen y is likely o be bove e rend plus sesonl effec in e nex period i.e. > 0. Similrly if < 0 en ere is subsnil possibiliy of < 0. T is, given bove verge bevior of y in ime, ere is likely o be bove verge bevior of y in ime nd lso quie likely so for severl subsequen periods. Similrly, given below verge bevior of y in ime, ere likely o be below verge bevior of y in ime nd for severl periods erefer. Bu is is ypicl of e cycles we finds in mcroeconomic nd business d ssocied wi e overll business cycle of n economy. 6

7 IV. simion nd Predicion in e DTDS Model Suppose we wn o use model 3-4 for forecsing purposes. Firs, we ve o esime e coefficiens of e model. A firs pproximion cn be obined by e so-clled meod of Ordinry Les Squres o ge esimes of e prmeers,,,..., wic we lbel ˆ, ˆ, ˆ,..., ˆ. Te Ordinry Les Squres esimes re en derived by minimizing e sum of squred errors of e model, nmely, S T y ˆ ˆ ˆ D ˆ D, 8, Te Ordinry Les Squres errors of e model re ˆ ˆ ˆ ˆ ˆ y D D,. 9 In similr mnner we cn consider e fied ARr error model ˆ ˆ ˆ, 0 ˆ r r nd use Ordinry Les Squres squres o obin esimes of e uoregressive prmeers,,..., r in 0, sy, ˆ, ˆ,..., Squres esimes of ˆ, ˆ, ˆ,..., ˆ, ˆ,..., nd ˆ r. Finlly, wi e Ordinry Les ˆ, we cn forecs y one-sep-ed forecs using e minimum men squre error formul yˆ ˆ ˆ ˆ ˆ ˆ D, D, were ˆ ˆ ˆ ˆ r r is e forecsed error for ime period. Te oer forecss y, yˆ, cn be obined by using nd recursively. r ˆ 3 In erms of sofwre implemenion of is model, in SAS we cn use PROC FORCAST o produce e forecss in is wy. However, more efficien wy o esime e presen model is o use Generlized Les Squres o esime e regressive prmeers,,,..., long wi e error prmeers,,..., r. PROC AUTORG in SAS cn be used o ccomplis is sk. Te erm Generlized Les Squres derives is nme from e fc e vribles of originl model 3 re 7

8 rnsformed in suc wy so e resuling errors of e rnsformed equion sisfy e ssumpions of Ordinry Les Squres, nmely e errors of e rnsformed model re independenly nd ideniclly disribued wi zero men nd consn vrince. For exmple, if r nd e errors of e DTDS model follow n AR process, populr Generlized Les Squres rnsformion of e d is e so-clled Cocrne-Orcu rnsformion. Tis rnsformion clls for rnsforming e originl model 3 ino e Generlized Les Squres equion y * * * D D for,3,, T. 3 * *, * were * *,, D D D,,, D D. Noe * *,, e derived error erm is now wie noise s required by Ordinry Les Squres. See e ppendix for is derivion. To implemen Fesible Generlized Les Squres one firs esimes by Ordinry Les Squres producing T ˆ ˆ ˆ. 4 T ˆ nd en pplying Ordinry Les Squres o e rnsformed equion 3 wi ˆ subsiued in for. Fesible Generlized Les Squres cn be exended in similr fsion o e iger order ARr error process 4 nd SAS Proc Auoreg provides e mens for doing so. V. Predicion in e DTDS Model In generl e predicions produced by e e deerminisic rend/ deerminisic sesonl model 3-4 rever o e esimed rend line plus sesonl effecs s e forecs orizon,, pproces infiniy. Tis resul is mos esily sown for e cse were e model s AR errors. In is cse, e -ed forecs of y is ˆ ˆ ˆ ˆ yˆ D, D, 4 ˆ 8

9 were ˆ ˆ ˆ. Since < ˆ, we ve ˆ 0 s,nd e -sep ed forecs of e model pproces e rend plus sesonl pr ˆ ˆ D D. T is, loug e deerminisic rend nd ˆ ˆ,, deerminisic sesonl effecs remin in e forecs profile, s e forecs orizon goes o infiniy, e cyclicl pr evenully vnises in is effec. T mkes sense in e respec e rend nd sesonl effecs re ssumed o be deerminisic i.e. fixed roug ime wile e cycle is modeled s being socsic nd men revering. Te sndrd error of e -sep-ed forecs esimed of e squre roo of ˆ, denoed by se ˆ, is e y y y D, D,. 5 In e Appendix we sow. 6 Tis vrince cn en be consisenly esimed by ˆ Vr ˆ ˆ 7 ˆ were ˆ cn be obined from 4 nd ˆ ˆ / T 8 wi being e Ordinry Les Squres residuls obined from 3. I follows e sndrd error of e -sep-ed forecs of y is ˆ se yˆ ˆ ˆ Vr. 9 ˆ Noice is sndrd error pproces e following limi s ˆ se yˆ 0 ˆ wic, in urn, pproces e sndrd deviion of, nmely 9

10 sd / s e smple size of e ime series goes o infiniy. T e Deerminisic Trend / Deerminisic Sesonl model's represenion of e unceriny ssocied wi fuure forecs errors pproces finie limi / s e orizon of e forecs goes o infiniy is seen by mny forecsers s mor limiion of e model. Inuiively, is resul rises from e fc e DTDS model is deerminisic model nd men revering in nure. In conrs, we will see in e Box-Jenkins models s well s e UC nd exponenil smooing models e rend in e d is reed s being socsic nd us e sndrd errors of e Box-Jenkins forecss pproc infiniy i.e. re unbounded s e forecs orizon pproces infiniy. As consequence, mny forecsers believe in prcicl forecsing problems in business nd economics e confidence inervls produced by e DTDS model invrible underse e forecs unceriny cully presened by e d. Of course, one cn cully observe e ou-of-smple predicion coverge of e compeing forecs confidence inervls nd cn udge for oneself wic of e compeing forecs confidence inervls s e more ccure coverge re. Puing ese disgreemens side, e % confidence inervl for -sep-ed forecs produced by e Deerminisic Trend/ Sesonl model is expressed s Pr yˆ ˆ ˆ ˆ se y Z / < y < y se y Z / were is e cosen level of confidence usully 0.0, 0.05, or 0.0 nd nd Z / is vlue of e sndrd norml cumulive disribuion suc Pr Z / /. For exmple, wen 0.05 nd we re ineresed in consrucing 95% predicion Z confidence inervl for y we ve Z nd e 95% predicion confidence inervl for n -sep-ed forecs of y is yˆ se yˆ.96, yˆ se yˆ.96]. 3 [ Tus, ˆ is e poin forecs nd e 3 is e predicion inervl forecs. y 0

11 Using e SAS progrm Plno_Forecs.ss we genere 3 forecss for December 005 roug December 006 following e recursive use of 4 nd clculing 95% predicion confidence inervls bsed on formul similr o 9. Te forecss were genered using n AR,3 model insed of e simple AR model we used o explin e logic of sndrd errors of forecss. Forecss of Plno Sles Tx Revenues for December 005 roug December 006 Produced by Deerminisic Trend / Deerminisic Seson model wi AR,3 errors p_r3 poin forecss, l_r3 lower 95% cl, u_r3 upper 95% cl Obs p_r3 l_r3 u_r

12 VI. Te Coice of e Order r in e Auoregressive Pr of e DTDS Model In Proc Auoreg in SAS one s e opion of specifying mximum order of of e uoregression, sy, r m x, nd en leing e progrm sequenilly elimine ll of e uoregressive erms re no significn prespecified level. equenll Consider e les squre esime of e r mx -order uoregression: ˆ ˆ ˆ r r mx â Ten e progrm sequenlly es e following ypoeses, dropping ose coefficiens re no sisiclly significn bu reining ose re. H H 0, 0, : : r mx 0 vs 0 vs H, H : r mx : 0 r mx, r mx 0 ec. 4 If ll uoregressive coefficiens re found o be sisiclly insignificn, en Proc Auoreg simply repors e Ordinry Les Squres esimes nd e ssocied es sisics. VII. Tess for e Presence of Trend nd Sesonliy in e DTDS Model Te DTDS offers very convenien frmework of esing for e presence or bsence of rend or sesonliy or bo in ime series d pproxime men revering bevior. ven if e d is no men revering in bevior bu insed s socsic rend, e ess o be discussed could nevereless be useful. To llow for e grees generliy le us consider e following DTDS model were we ve dded qudric erm o e ime rend pr of e model o llow for d mig ve some curvure in is rend. Consider nd y D 3D 3 D, 3A

13 r r. 4 A. Tesing for e Presence or Absence of Sesonliy If we wn o es for e presence or bsence of sesonliy in our d en we re ineresed in e ypoeses H 0 : 3 0 versus 5 H : A les one of e ' s is no equl o zero. Te null ypoesis ses e slopes of e rends of e non-jnury mons ll equl e slope of e rend for Jnury. As resul, ll of e rends of e mons excly coincide nd us ere is no sesonl vriion in e d. Te lernive ypoesis signifies e presence of sesonliy in e d since one or more of e slopes of e rends of e non-jnury mons do no coincide wi e Jnury rend. Te pproprie F-sisic o es is ypoesis is derived from e Generlized Les equion 3 or e ARr generlizion of i. Le one obins from e Generlized Les Squres equion nd SSRu be e sum of squred residuls SSR R be e sum of squred residuls one obins from e resriced Generlized Les Squres equion were e resricions 0 ve been imposed in e esimion process. 3 Te pproprie F-sisic is of e form SSR F SSR u R SSR U / / T r 4 6 wic, under e ssumed ru of e null ypoesis, H 0, s n F-disribuion of numeror degrees of freedom nd T-r-4 denominor degrees of freedom in repeed smples. Te numeror degrees of freedom corresponds o e number of resricions being imposed by e null ypoesis wile e denominor degrees of freedom corresponds o e number of observions minus e number of prmeers in e unresriced full model. Tere re 4 coefficiens o be esimed in 3A nd r 3

14 coefficiens o be equion 4. Of course, if e uoregression 4 is simplified somew by bckwrd secion s deiled in Secion VI bove en r in 6 becomes e mximum order is specificed for e uoregressive coefficiens in e model. Te inuiion beind 6 is, if sesonliy is no presen in e d i.e. H 0 is rue, e fi offered by e resriced model will be lmos s good s e fi offered by e unresriced full model nd loug SSR R mus be greer n equl o SSR U by necessiy, i will no be oo muc greer nd, s resul, e F-sisic will be smll implying p-vlue Pr F Fobserved is greer n e cosen size of e es usully 0.0, 0.05, or 0.0. In conrs, if sesonliy is presen in e d en e resriced model will no provide very good fi of e d nd SSR R will subsnilly exceed SSR U. Ten e F-sisic 6 will be lrge nd will imply p-vlue less n e cosen size of e es. In summry, F-sisic derived from 6 s smll vlue nd lrge p-vlue suppors e supposiion of no sesonliy in e d wile lrge F-vlue nd smll p-vlue suppoe e supposiion of sesonliy in e d. As n exmple of is es, consider gin e Plno Sles Tx Revenue d. Te SAS progrm Plno_Tes_Sesonliy.ss provides e following resuls concerning e es sisic 6 : Tes Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor As expeced, e numeror degrees of freedom equls. Te denominor degrees of freedom is observions sesonl dummy coefficiens e mximl order of e uoregression on e errors, 4. Te numeror men squre error represens e clculed vlue for e numeror in 6 wile e denominor men squre error represens e clculed vlue for e denominor in 6. In e Generlized Les 4

15 Squres esimion process i urned ou e qudric erm ws no needed -.59 wi wo-side p-vlue 0.38 wile e OLS residuls indiced e need o use Generlized Les squres for conducing sisicl inference ess. See e below Durbin-Wson ble were e DW sisics of orders 4 re ll igly significn. Durbin-Wson Sisics Order DW Pr < DW Pr > DW.3805 < < < Noe: Pr<DW is e p-vlue for esing posiive uocorrelion, nd Pr>DW is e p-vlue for esing negive uocorrelion. As i urns ou Proc Auoreg reined e firs nd ird order uoregressive coefficiens in 4 s reveled in e following Proc Auoreg bles: Bckwrd liminion of Auoregressive Terms Lg sime Vlue Pr > simes of Auoregressive Prmeers Lg Coefficien Sndrd rror Vlue In ddiion, e fied model ppers o ve les ner wie noise residuls s indiced by e following Durbin-Wson ble for e Generlized Les Squres residuls nd e corresponding uocorrelion funcion ACF of e sme residuls. 5

16 Durbin-Wson Sisics Order DW Pr < DW Pr > DW Noe: Pr<DW is e p-vlue for esing posiive uocorrelion, nd Pr>DW is e p-vlue for esing negive uocorrelion. B. Tesing for Combinions of Trend nd Sesonliy Oer ess sugges emselves s well in e DTDS model. Consider e possibiliy of esing for no rend in e d bu in e presence of sesonliy. Te null nd lernive ypoeses for is es re H 0 : 0 versus 7 H : ier 0 or 0 or bo. In is cse e pproprie F-sisic is SSR F SSR u R SSR U / / T r 4 wic under e ssumed ru of e null ypoesis is disribued in repeed smples s n F-disribuion wi wo numeror degrees of freedom nd e sme denominor degrees of freedom implied by e unresriced model 3A nd 4. 8 For esing e d is bsen bo rend nd sesonliy we ve e following null nd lernive ypoeses: H 0 : versus 9 0 H : A les one of e bove coefficiens is no equl o zero. 6

17 In is cse e pproprie F-sisic is SSR F SSR u R SSR U /3 / T r Tis ssumes of course we ve bo e liner nd qudric erms in e unresriced model. Tis es is equivlen o e F-es for e overll significnce of e DTDS regression nd is vilble in e sndrd ANOVA ble mos sisicl regression pckges produce including SAS. C. Some D Ses Demonsring e Vrious Types of Tess Consider e following Mone Crlo d ses genered nd esed by e SAS progrm Compreensive Trend_Sesonl Tes.ss ll by e meod of Generlized Les Squres. D wi No Trend or Sesonl ffecs Figure XTi me YY Ser i es

18 D wi No Trend bu Sesonl ffecs Figure XTi me YY Ser i es D wi Trend bu No Sesonl ffecs Figure 3 XTi me YY Ser i es

19 D wi Trend nd Sesonl ffecs Figure 4 XTi me YY Ser i es For Figure e following es resuls re produced by e SAS progrm Compreensive Trend_Sesonl Tes.ss: Tes Trend Source DF Men Squre F Vlue Pr > F Numeror Denominor Tes Seson Source DF Men Squre F Vlue Pr > F Numeror Denominor

20 Tes Trend_Seson Source DF Men Squre F Vlue Pr > F Numeror Denominor You cn see none of ese ess re sisiclly significn since ll of e p-vlues of e ess re greer n, sy, 0.05 us implying e bsence of bo rend nd sesonl effecs s rougly indiced by visul inspecion of e d. For Figure e es resuls re: Tes Trend Source DF Men Squre F Vlue Pr > F Numeror Denominor Tes Seson Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor

21 Tes Trend_Seson Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor Here, e sesonl es indices significn sesonl effecs p<.000 wile e rend es indices no rend in e d. Te oin es of e bsence of bo rend nd sesonl effecs is nurlly reeced. For Figure 3 e es resuls re: Tes Trend Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor Tes Seson Source DF Men Squre F Vlue Pr > F Numeror Denominor Tes Trend_Seson Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor

22 Here, e rend es indices significn rend wile e sesonl effecs re insignificn, s expeced. Te oin es of e bsence of bo rend nd sesonl effecs is nurlly reeced. For Figure 4 e es resuls re: Tes Trend Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor Tes Seson Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor Tes Trend_Seson Source DF Men Squre F Vlue Pr > F Numeror <.000 Denominor Here, e rend es indices significn rend wile e sesonl es indices significn sesonl effecs s well. Te oin es is lso igly significn s expeced.

23 VIII. Conclusion Te Deerminisic Trend / Deerminisic Sesonl model is one of e oldes models used for forecsing ime series. Ineresingly i is specil cse of e UC model pr from us ssuming iger order uoregressive error process. However, mny yers go ere did no exis e sofwre or compuing power o execue e UCM us e pplicbiliy of e DTDS model isoriclly. Te DTDS does provide useful esing frmework for deecing rend, sesonl effecs, nd combinion ereof. Te bigges drwbck, owever, my be e fc e infinie orizon predicion confidence inervls re bounded wic is no very relisic given mos economic nd business d seem o ve socsic rends in em wi lile men reversion. Despie is drwbck, e DTDS model is quie inerpreble, nd in e presence of fixed sesonl effecs is useful for idenifying mons wic re sesonlly srong s compred o ose re sesonlly wek nd for mking comprisons of ese srengs nd weknesses. 3

24 4 Appendix Properies of e AR rror Term: Now we prove e resuls sed in 7. By bckwrd subsiuion in 4, i is esy o sow 4 I follows e men of e is Te vrince of e is 0 4 +ddiionl erms involving were s Te covrince beween nd is

25 In similr mnner i cn be sown Ten e uocorrelion funcion for e AR model 4 is Vr Vr cov Corr,,... Derivion of e Cocrne-Orcu equion 3 for use in obining e esimes of e prmeers in e DTDS model wi AR errors D D y, 3 Lgging 3 one period provides,, D D y. 3 Muliplying 3 by provides,, D D y. 3 Subrcing 3 from 3 provides,,, D D D D y y

26 6 D D y *, *, * * * 3 s required. Derivion of e -sep-ed sndrd error of y ˆ : Given e AR error process, by recursive subsiuion we cn sow I en follows. Terefore Bu ime is no longer rndom nd, s resul, we ve ] [ ] [ ] [ 4 4 s desired.