DETERMINISTIC TREND / DETERMINISTIC SEASON MODEL

Transcription

1 DTRMINISTIC TRND / DTRMINISTIC SASON MODL Professor Toms B. Fomby Deprmen of conomics Souern Meodis Universiy Dlls, TX June 008 Appendix B on NP Trend Tess Added Sep. 00 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 jus 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 j is sesonl dummy vrible kes e vlue of one if e - observion is observed in e j- 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 jus 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 j, j, 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 j coefficien is posiive, en e j 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 j ' s is posiive en, by deful, Jnury's sesonl effec is e wekes. Te verge sesonl effec is, of course, [ / [ j ]/ j /. 5 Ten e srong mons in erms of sesonl effec re ose wose rend inercep for Jnury nd j for e oer mons re greer n wile e wek mons ve or j inerceps re less n. If one wns o sndrdize e sesonl effecs, one could do so by forming e coefficiens *, * *,., j j. 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 j iii Cov, j 7 iv Corr, j j For exmple, in e specil AR model 0.8 e uocorrelion funcion of 5

6 e errors is represened grpiclly by e following grp. j , , , , , , Given is uocorrelion funcion, we cn see djcen errors, ve n uocorrelion coefficien of 0.8 bu errors nd j re j-periods pr ve smller correlion of beween em j 0.8. 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 j 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 mjor 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 judge 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 join es of e bsence of bo rend nd sesonl effecs is nurlly rejeced. 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

22 Numeror <.000 Denominor Here, e rend es indices significn rend wile e sesonl effecs re insignificn, s expeced. Te join es of e bsence of bo rend nd sesonl effecs is nurlly rejeced. 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

23 Here, e rend es indices significn rend wile e sesonl es indices significn sesonl effecs s well. Te join es is lso igly significn s expeced. 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 A 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 j j Ten e uocorrelion funcion for e AR model 4 is j j j j j j Vr Vr cov Corr j,,... 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

26 6,,, D D D D y y 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.

27 Appendix B Non-Prmeric Tess for Trend I. Inroducion Recll we ve lredy discussed ow we cn seprely or simulneously es for rend nd sesonliy in ime series d by using nd F-ess of e join significnce of coefficiens in e Deerminisic Trend/Deerminisic Seson DTDS model. Te DTDS model is given by y D 3D3 D, r r. Te es from rend en consiss of esing versus H 0 : β 0 no rend 3 H : β 0 rend. 4 Tese ypoeses cn be esed srigforwrdly using e -sisic for e coefficien esime of β obined from pplying Generlized Les Squres o model. Seprely, one cn es for e presence sesonliy by esing e following ypoeses: versus H 0 : γ γ 3 γ 0 5 H : les one of e γ i s is no e0qul o zero. 6 Tese ypoeses cn be esed by using join F-es of 5 obined from pplying Generlized Les Squres o model. Finlly, bo rend nd sesonliy cn be esed joinly by using join F-es of 3 nd 5 simulneously. Tis mouns o es of e overll significnce of e regression model. Of course, ll of ese ess re prmeric in nure s ey re bsed on e prmeric model. As n lernive o e bove ess, one cn es for rend using bery of non-prmeric ess. We ve lredy discussed Friedmn s non-prmeric es for sesonliy. We will discuss ree ess in succession: e Mnn-Kendll es, e Mnn-Kendll Sesonl es, nd e Mnn-Kendll Sesonl/Auocorrelion es. 7

28 II. Kendll Fmily of Tess for Trend. Mnn-Kendll Tes for Trend In 945 pper in conomeric Henry Mnn suggesed esing for rend by using Kendll s non-prmeric u sisic o es for significn correlion beween e ime series in quesion,y, nd e ime index vrible ssocied wi e ime series,,,, T, T being e ol number of observions on e ime series. Te so-clled Mnn-Kendll es for rend cn be sed s follows: versus H 0 : Prob y i > y j 0.5, were i > j no monoone rend 7 H : Prob y i > y j 0.5 -sided es presence of monoone rend. 8 Of course, if one s srong convicion e rend is eier posiive or negive, e lernive ypoesis cn be convered ino one-il es if desired. So we re esing for monoonic rend weer i be liner rend or non-liner rend by exmining o w exen e ime series ends o move monooniclly wi ime. A crucil ssumpion in is es is e observions y occur independenly over ime wic, by e wy, is rre occurrence in mos economic nd business ime series. Aloug is independence ssumpion doesn ofen old in ime series d we discuss e Mnn-Kendll es becuse i serves s e bsis for e wo below ess for rend do djus for lck of independence. Le e sign funcion, sgn, wen pplied o vrible, sy w, be defined by sgnw +, w > 0 sgnw 0, w 0 9 sgnw -, w < 0. Ten, under e null ypoesis of no monoone rend, e Mnn-Kendll sisic S sgny j y i i<j 0 is disribued sympoiclly norml wi men 0 nd vrince σ T T T + 5/8. S is en e sum of e sgn vlues for ll of e TT-/ possible ordered differences y j y i, j > i. Anoer wy of viewing S is S P M were P denoes e number of imes e y pirs indiced n increse nd M denoes e number of imes e y pirs indiced decrese. Obviously, rre vlues of e S sisic ving p-vlues less n e cosen level of significnce would indice e d s monoone rend. 8

29 Oerwise e d re rndomly ordered wi respec o ime nd disply no significn rend. To mke for beer sympoic pproximion, i is cusomry o use e following sisic in lieu of S: Z s S σ S if S > 0 0 if S 0 S+ σ S if S < 0 were σ S T T T+5 8 disribued s sndrd norml rndom vrible. b. Sesonl Mnn-Kendll Tes for Trend. Under e ssumed ru of e null ypoesis Z S is To ccommode for e cse were e d is sesonl nd us no independen yer-by-yer by mon, Hirsc, Slck, nd Smi 98 exended e Mnn-Kendll es. Consider e following ordering of e ime series d by yer: Y y y,. y n y n, Here we ve mde e noion simplifying ssumpion e d consiss of n complee yers of d resuling in e number of observions being T n. Furermore, le e following mrix conin e mrix of rnks by mon rnking by columns s in e Friedmn Sesonliy es of e originl d. R R R,. 3 R n, R n, Specificlly, n R jg n + + i sgny jg y ig /, j,,, n; g,,,. 4 W we re going o be doing ere is simulneously esing for significn monoone rend in ec of e mon s rends, i.e. e rend mde up of ll of e Jnury observions, e rend mde up of ll of e Februry observions,, e rend mde up of ll of e December observions. Recll e Buys-Bllo plo of sesonl ime series wi sepre ime rend plos for ec mon.. Tere will be monoone rend in e d 9

30 if one or more of e rends ssocied wi e mons ve monoone rends in em. Te Mnn-Kendll es sisic for ec mon is S g i<j sgn y jg y ig, g,,,. 5 Te Sesonl Mnn-Kendll es sisic for rend is given by S g S g. 6 Under e ssumed ru of e null ypoesis ere is no monoone rend in e d nd us no monly rends conin monoone rend, S is sympoiclly normlly disribued wi men 0 nd vrince given by Vr S g σ g + g, σ g, were g 7 nd σ g vrs g nd σ g covs g, S. Hirsc e. l. 98 ssumes e d re independen pr from e sesonliy nd erefore ssume e covrince erms σ g re ll equl o zero. In erms of esing, wen e probbiliy vlue ssocied wi e es sisic 6 is less n e sed level of significnce, we ccep e lernive ypoesis e d d rend in i. Oerwise, we ssume monoone rend does no exis in e d. c. Mnn-Kendll Tes for Trend Adjused for Bo for Sesonliy nd Auocorrelion Te Hirsc, Slck, nd Smi 98 es for rend s been exended by Hirsc nd Slck 984 o ccommode, no only sesonliy in e d bu uocorrelion s well. In order o do, ey d o come up wi consisen esime of e covrince second erm in 7 bove. Tey did so in prescribing were n σ g [K g + 4 i R ig R i nn + ]/3 8 K g sgn[ y jg y ig y j y i ] i<j. 9 Wi is djusmen, e Mnn-Kendll es for rend s been fully djused for e possible presence of sesonliy nd uocorrelion in e rend. III. A DOS-bsed progrm o compue e Mnn-Kendll Fmily of Tess for Trend In eir pper Compuer Progrm for e Kendll Fmily of Trend Tess Scienific Invesigions Repor of e U.S. Geologicl Survey by Dennis R. 30

31 Helsel, Dvid K. Mueller, nd Jmes R. Slck p://pubs.usgs.gov/sir/005/575, e uors repor compuer code developed e USGS o perform e Mnn-Kendll bery of ess described bove. Tis compuer code cn be downloded from e bove URL s well s pdf file describes e synx nd oupu genered by e code. Te code is wrien in DOS nd requires e coding of n inpu file including e d o be esed nd nming n oupu file for reporing e es resuls. Te DOS execuion file is clled Kendll.exe. Consider e pplicion of e Mnn-Kendll bery of rend ess o e Plno Sles Tx Revenue d. Here is e inpu file processed in pr by Kendll.exe. 0 Sesonl Kendll es - on Plno Rev D, inpu ype In e firs column of e firs line bove, e snds for e ess o be repored nd e inpu form used for e d. Te second iem, 0, indices non-prmeric fi of e rend in e d is no o be repored. Te res of e firs line is jus ile for e job. For more informion on e synx of e inpu file see e bove menioned scienific repor. 3

32 Te oupu produced by e Kendll.exe progrm s pplied o e Plno Sles Tx Revenue d is s follows: Sesonl Kendll Tes for Trend US Geologicl Survey, 005 D se: Sesonl Kendll ess - on Plno Rev D, inpu ype Te record is 7 complee wer yers wi sesons per yer beginning in wer yer 990. Te u correlion coefficien is 0.90 S 70. z p p djused for correlion mong sesons suc s seril dependence Te djused p-vlue sould be used only for d wi more n 0 nnul vlues per seson. Te esimed rend my be described by e equion: Y * Time were Time Yer s deciml beginning of firs wer yer In e bove oupu, e reference o wer yers is jus e convenion of e USGS s mny of e ime series e gency nlyzes concerns wer mesures. Above wo Mnn-Kendll ess for rend re repored, nmely e Sesonl Mnn-Kendll es sisic nd is ccompnying wo-sided p-vlue Z 6.735, p nd e djused wo-sided p-vlue is produced wen djusing e Mnn-Kendll Sesonl Trend es for uocorrelion in e d p We feel comforble wi is uocorrelion djusmen of e p-vlue s e Plno d consiss of 7 > 0 complee yers of d. Bo of ese ess srongly suppor e supposiion ere is rend in e Plno Sles Tx Revenue d s would be suspeced from looking e plo of e d. Bu remember no ll ime series re s long s e Plno Revenue d series nd us deecing rend my no lwys be s esy of sk s ere. Also noe, if one wises o specify one-sided lernive given e invesigors s reson o expec eier posiive or negive rend, e repored wo-sided p-vlues sould be divided by o ge e desired one-sided p-vlues. 3

33 References: Mnn, H. B. 945, Nonprmeric Tess Agins Trend, conomeric, Vol. 3, No. 3, pp Hirsc, R. M., J. R. Slck, nd R. A. Smi 98, Tecniques of Trend Anlysis for Monly Wer Quliy D, Wer Resources Reserc, Vol. 8, No., pp Hirsc, R. M. nd J. R. Slck 984, A Nonprmeric Trend Tes for Sesonl D wi Seril Dependence, Wer Resources Reserc, Vol. 0, No. 6, pp Helsel, D.R., D.K. Mueller, nd J.R. Slck 005, Compuer Progrm for e Kendll Fmily of Trend Tess, Scienific Invesigions Repor , U.S. Deprmen of Inerior, U.S. Geologicl Survey. Avilble p://pubs.usgs.gov/sir/005/