Lecture 15 Forecasting

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1 RS EC - Lcur 5 Lcur 5 Forcasng Forcasng A shock s ofn usd o dscrb an unxpcd chang n a varabl or n h valu of h rror rs a a parcular prod. A shock s dfnd as h dffrnc bwn xpcd (a forcas) and wha acually happnd. On of h os poran objcvs n srs analyss s o forcas s fuur valus. I s h prary objcv of ARIMA odlng: wo yps of forcass. - In sapl (prdcon): h xpcd valu of h RV (n-sapl), gvn h sas of h parars. - Ou of sapl (forcasng): h valu of a fuur RV ha s no obsrvd by h sapl.

2 RS EC - Lcur 5 Forcasng Basc Concps Any forcass nds an nforaon s, I. hs ncluds daa, odls and/or assupons avalabl a. h forcass wll b condonal on I. h varabl o forcas +l s a RV. I can b fully characrzd by a pdf. In gnral, s dffcul o g h pdf for h forcas. In pracc, w g a pon sa (h forcas) and a C.I. Noaon: - Forcas for +l ad a : - +l forcas rror: - Man squard rror (MSE):,, ( ) MSE( ) E[ ] E[( ) ] Forcasng Basc Concps o g a pon sa,, w nd a cos funcon o judg varous alrnavs. hs cos funcon s call loss funcon. Snc w ar workng wh forcas, w work wh a xpcd loss funcon. A popular loss funcons s h MSE, whch s quadrac and syrc. W can us asyrc funcons, for xapl, funcons ha pnalz posv rrors or han ngav rrors. If w us h MSE as h cos funcon, w look for a nzs. ha s n E[ ] E[( ) ] E[ hn, foc ply E[ ] [ ] E. ] ha

3 RS EC - Lcur 5 Forcasng Basc Concps h opal pon forcas undr MSE s h (condonal) an: E I Dffrn loss funcons lad o dffrn opal forcas. For xapl, for h MAE, h opal pon forcas s h dan. h copuaon of E[ +l I ] dpnds on h dsrbuon of {ε }. If {ε } ~ WN, hn E[ε +l I ] =, whch graly splfs copuaons, spcally n h lnar odl. hn, for ARMA(p,q) saonary procss (wh a Wold rprsnaon), h nu MSE lnar forcas (bs lnar prdcor) of +l, condonng on I s: Forcasng Sps for ARMA Modls Procss: - ARIMA odl - Esaon (Evaluaon n-sapl) - Forcas (Evaluaon ou-of-sapl) (Esaof ) (Prdcon) (Forcas) a 3

4 RS EC - Lcur 5 Forcasng Fro ARMA Modls W obsrv h srs: I ={,,, }. -A, w wan o forcas: +, +,, +l. -: h forcas orgn. - l: Forcas horzon - ( ) : l-sp ahad forcas = Forcasd valu +l Us h condonal xpcaon of +l, gvn h obsrvd sapl. E,,, Exapl: On-sp ahad forcas: E,,, Forcas accuracy o b asurd by MSE condonal xpcaon, bs forcas. 7 Forcasng Fro ARMA Modls h saonary ARMA odl for s or B p Assu ha w hav daa,,...,. W wan o forcas +l. hn, Consdrng h Wold rprsnaon: q B B B q B a p p p p p q q q q 4

5 RS EC - Lcur 5 5 akng h xpcaon of +l, w hav whr hn, w dfn h forcas rror: h xpcaon of h forcas rror: Forcasng Fro ARMA Modls,,, E,,,, j j E j j E h xpcaon of h forcas rror: => h forcas n unbasd. h varanc of h forcas rror: Exapl : On-sp ahad forcas (l=). Forcasng Fro ARMA Modls E Var Var Var

6 RS EC - Lcur 5 Forcasng Fro ARMA Modls Exapl : On-sp ahad forcas (l=). No: Var l l Var As w forcas no h fuur, h forcass ar no vry nrsng (uncondonal forcass!). ha s why ARMA (or ARIMA) forcasng s usful only for shor-r forcasng. Forcasng Fro ARMA Modl: C.I. A (- )% prdcon nrval for +l (l-sps ahad) s z z / / Exapl: 95% C.I. for h sp-ahad forcas Whn copung prdcon nrvals fro daa, w subsu sas for parars, gvng approxa prdcon nrvals. No: Snc s ar RV, MSE[ε +l ]=MSE[ +l ]= Var.96 6

7 RS EC - Lcur 5 7 Suppos w hav obsrvaons a =. W hav a good ARMA odl for. W oban h forcas for +, +, c. A =+, w obsrv +. Now, w wan o upda our forcass usng h orgnal valu of + and h forcasd valu of. h forcas rror s: W can also wr hs as Forcasng Fro ARMA Modl: Updang hn, Exapl: l=, =. Forcasng Fro ARMA Modl: Updang

8 RS EC - Lcur 5 Forcasng Fro ARMA Modl: ransforaons If w us varanc sablzng ransforaon, afr h forcasng, w nd o convr h forcass for h orgnal srs. For xapl, f w us log-ransforaon, hn, E, xp E ln ln,, ln, If X ~ N(, ), hn, E xp X xp h MSE forcas for h orgnal srs s: xp Z n Var. whr Z ln n n EZ Z,, VarZn Z,, Z n n Z n n Forcasng Fro ARMA Modl: Rarks In gnral, w nd a larg. Br sas and s possbl o chck for odl sably and chck forcasng ably of odl by whholdng daa. Sasonal parns also nd larg. Usually, you nd 4 o 5 sasons o g rasonabl sas. Parsonous odls ar vry poran. Easr o copu and nrpr odls and forcass. Forcass ar lss snsv o dvaons bwn parars and sas. 8

9 RS EC - Lcur 5 Forcasng Fro Spl Modls: ES Indusral copans, wh a lo of npus and oupus, wan quck and nxpnsv forcass. Easy o fully auoa. Exponnal Soohng Modls (ES) fulfll hs rqurns. In gnral, hs odls ar ld and no opal, spcally copard wh Box-Jnkns hods. Goal of hs odls: Supprss h shor-run flucuaon by soohng h srs. For hs purpos, a wghd avrag of all prvous valus works wll. hr ar any ES odls. W wll go ovr h Spl Exponnal Soohng (SES) and Hol-Wnr s Exponnal Soohng (HW ES). Forcasng Fro Spl Modls: ES Obsrvd srs:,,, h quaon for h odl s whr - : h soohng parar, - : h valu of h obsrvaon a - S : h valu of h soohd obsrvaon a. S S h quaon can also b wrn as S S S S forcas rror hn, h forcas s: S ha s, a spl updang quaon. S S S 9

10 RS EC - Lcur 5 Forcasng Fro Spl Modls: ES Q: Why Exponnal? For h obsrvd srs,,, n, n+ can b xprssd as a wghd su of prvous obsrvaons. whr c s ar h wghs. c c Gvng or wghs o h rcn obsrvaons, w can us h gorc wghs (dcrasng by a consan rao for vry un ncras n lag): c c ;,,...;. hn, S S 9 Forcasng Fro Spl Modls: Slcng Choos bwn and. -If =, bcos a nav odl; f s clos o, or wghs ar pu on rcn valus. h odl fully ulzs forcas rrors. -If s clos o, dsan valus ar gvn wghs coparabl o rcn valus. S whn hr ar bg rando varaons n. - s ofn slcd as o nz h MSE. In prcal work,.5.3 ar usd. ar usd rarly Nurcal Mnzaon Procss: - ak dffrn valus rangng bwn and. - Calcula -sp-ahad forcas rrors for ach. - Calcula MSE for ach cas. n Choos whch has h n MSE: S n

11 RS EC - Lcur 5 Forcasng Fro Spl Modls: Exapl S + (=.) ( S ) (.)5+(.9)5= (.)7+(.9)5= (.)6+(.9)5.= (.)3+(.9)5.8=5.5.7 OAL.945 MSE SSE n.74 Calcula hs for =.,.3,,.9, and copar h MSEs. Choos wh nu MSE Forcasng Fro Spl Modls: Rarks So copur progras auoacally chooss h opal usng h sarch hod or non-lnar opzaon chnqus. Inal Valu Probl - S S o s on hod of nalzaon. - Also, ak h avrag of, say frs 4 or 5 obsrvaons. Us hs avrag as an nal valu. hs odl gnors rnds or sasonals. No vry ralsc. Bu, drnsc coponns, D, can b asly ncorporad. h odl ha ncorporas boh faurs s calld Hol-Wnr s ES.

12 RS EC - Lcur 5 Forcasng Fro Spl Modls: HW ES Now, w nroduc rnd ( ) and sasonaly (I ) facors. Boh can can b ncludd as addvly or ulplcavly facors. Dals - W us ulplcav sasonals.., /I - and addv rnd. - h forcas, S, s adjusd by h drnsc rnd: S +. - h rnd,, s a wghd avrag of - and h chang n S. - h sasonaly s also a wghd avrag of I -S and h /S hn, h odl has hr quaons: S I I s S S S S I s 3 Forcasng Fro Spl Modls: HW ES W hnk of ( /S ) as capurng sasonal ffcs. s = # of prods n h sasonal cycls (s = 4, for quarrly daa) W hav only hr parars : = soohng parar = rnd coffcn = sasonaly coffcn Q: How do w drn hs 3 parars? - Ad-hoc hod: α, and can b chosn as valu bwn.<,, <. 4 - Mnzaon of h MSE, as n SES.

13 RS EC - Lcur 5 Forcasng Fro Spl Modls: HW ES s h-sp ahad forcas h S h I h No: Sasonal facor s ulpld n h h-sp ahad forcas Inal valus for algorh - W nd a las on copl sason of daa o drn h nal sas of I -s. - Inal valus:. S. or / s s s s s s s s s / s / s / s s ss s s 5 Forcasng Fro Spl Modls: HW ES Algorh o copu nal valus for sasonal coponn I s. Assu w hav obsrvaon and quarrly sasonaly (s=4): () Copu h avrags of ach of yars. 4 A, / 4,,,,6 (yarly avrags) () Dvd h obsrvaons by h appropra yarly an:, /A. (3) I s s ford by copung h avrag, /A pr yar: I s, s / A 4 s,,3,4 6 3

14 RS EC - Lcur 5 Forcasng Fro Spl Modls: HW ES Rarks - No ha, f a copur progra slcs = =, hs dos no a lack of rnd or sasonaly. I pls a consan (drnsc) coponn. - In hs cas, an ARIMA odl wh drnsc rnd ay b a or appropra odl. - For HW ES, a sasonal wgh nar on pls ha a non-sasonal odl ay b or appropra. - W odld sasonals as ulplcav: => Mulplcav sasonaly: Forcas = S * I -s. - Bu, sasonal coponns can also b addv. For xapl, durng h onh of Dcbr sals a a parcular sor ncras by $X vry yar. In hs cas, w jus $X o h Dcbr forcas. => Addv sasonaly: Forcas = S + I -s. 7 ES Modls Dffrn yps. No rnd and addv sasonal varably (,). Addv sasonal varably wh an addv rnd (,) 3. Mulplcav sasonal varably wh an addv rnd (,) 4. Mulplcav sasonal varably wh a ulplcav rnd (,) 4

15 RS EC - Lcur 5 5 Slc h yp of odl o f basd on h prsnc of - rnd addv or ulplcav, dapnd or no - Sasonal varably addv or ulplcav 5. Dapnd rnd wh addv sasonal varably (,) 6. Mulplcav sasonal varably and dapnd rnd (,) ES Modls Dffrn yps Evaluaon of forcass Suary of loss funcons of ou-of-sapl forcas accuracy: y y ) ( y y y y ) ( Man Error = Man Absolu Error (MAE) = Roo Man Squar Error (RMSE)= y y ) ( Man Squard Error (MSE) = y U hl s U-sa =

16 RS EC - Lcur 5 Evaluaon of forcass DM s o drn f on odl prdcs br han anohr, w dfn h loss dffrnal bwn wo forcass: d = g( M ) - g( M ). whr g(.) s h forcasng loss funcon. M and M ar wo copng ss of forcass could b fro odls or sohng ls. W only nd { M } & { M }, no h srucur of M or M. In hs sns, hs approach s odl-fr. ypcal (syrc) loss funcons: g( ) = & g( ) =. Bu ohr g(.) s can b usd: g( ) =xp(λ )- λ (λ>). Evaluaon of forcass DM s hn, w s h null hypohss of qual prdcv accuracy: H : E[d ]= vs. H : E[d ]=μ. - Dbold and Marano (995) assu { M } & { M } s covaranc saonary and ohr rgulary condons (fn Var[d ], ndpndnc of forcass afr l prods) ndd o apply CL. hn, d Var [ d ] / d N (,), d d hn, h DM s s a spl z-s: DM d Var [ d ] / d N (,) 6

17 RS EC - Lcur 5 Evaluaon of forcass DM s whr Var [ d ] s a conssn saor of h varanc, usually basd on sapl auocovarancs of d : V ar[ d ] () ( j) l j k Assu h l-sp ahad forcas rrors hav zro auocorrlaons a ordr l. Harvy al. (998) propos a sall-sapl odfcaon of h DM s: DM* = DM/{[+-l+l (l-)/]/}/ ~ -. If ARCH s suspcd, rplac l wh [.5 ()]+l. ([.]=ngr par). No: If { M } & { M } ar prfcly corrlad, h nuraor and dnonaor of h DM s ar boh convrgng o as. Avod DM s whn hs suaon s suspcd (say, wo nsd odls.) hough, n sall sapls, s OK. Evaluaon of forcass DM s Exapl: Cod n R d.s <- funcon (,, h =, powr = ) { d <- c(abs())^powr - c(abs())^powr d.cov <- acf(d, na.acon = na.o, lag.ax = h -, yp = "covaranc", plo = FALSE)$acf[,, ] d.var <- su(c(d.cov[], * d.cov[-]))/lngh(d) dv <- d.var#ax(-8,d.var) f(dv > ) SAISIC <- an(d, na.r = RUE) / sqr(dv) ls f(h==) sop("varanc of DM sasc s zro") ls { warnng("varanc s ngav, usng horzon h=") rurn(d.s(,,alrnav,h=,powr)) } n <- lngh(d) k <- ((n + - *h + (h/n) * (h-))/n)^(/) SAISIC <- SAISIC * k nas(saisic) <- "DM" } 7

18 RS EC - Lcur 5 Evaluaon of forcass DM s: Rarks h DM ss s rounly usd. Is odl-fr approach has appal. hr ar odl-dpndn ss, s Ws (996), Clark and McCrackn (), and, or rcn, Clark and McCrackn (), wh or coplcad asypoc dsrbuons. h loss funcon dos no nd o b syrc (lk MSE). h DM s s basd on h noon of uncondonal.., on avrag ovr h whol sapl- xpcd loss. Followng Morgan, Grangr and Nwbold (977), h DM sasc can b calculad by rgrsson of d, on an nrcp, usng NW SE. Bu, w can also condon on varabls ha ay xplan d.w ov fro an uncondonal o a condonal xpcd loss prspcv. Evaluaon of forcass Condonal s Gacon and Wh (6) prsn a gnral frawork for ou-ofsapl prdcv ably sng, characrsd by h forulaon of ss (such as ss for qualy of forcass) basd on condonal xpcd loss. Now, E[ F - ] = => E[h - ] =, whr h - s a F - asurabl funcon of dnson q. No: G&W (6) also dffrs fro h sandard approach o sng for prdcv ably n ha copars forcasng hods (saon + odl) rahr han forcasng odls. h s bcos a Wald s, wh an asypoc χ (q) dsrbuon. 8

19 RS EC - Lcur 5 Cobnaon of Forcass Ida fro Bas & Grangr (Opraons Rsarch Quarrly, 969): - W hav dffrn forcass fro R odls: M M MR,,... Q: Why no cobn h? M M MR... Cob M M MR Vry coon pracc n conocs, fnanc and polcs, rpord by h prss as consnsus forcas. Usually, as a spl avrag. Q: Advanag? Lowr forcas varanc. Dvrsfcaon argun. Inuon: Indvdual forcass ar ach basd on paral nforaon ss (say, prva nforaon) or odls. 37 Cobnaon of Forcass Opal Wghs h varanc of h forcass s: R Mj Mj M ] ( ) Var[ ] Co var[ Var[ ] Cob j Mj No: Idally, w would lk o hav ngavly corrlad forcass. R j R Mj j M Assung unbasd forcass and uncorrlad rrors, Var R [ Cob Mj j j ] ( ) Exapl: Spl avrag: ω j =/R. hn, ] /. [ Cob Var R W can drvd opal wghs,., ω j s ha nz h varanc of h forcas. Undr h uncorrlad assupon: Mj * j R j j R j j 9

20 RS EC - Lcur 5 Cobnaon of Forcass Opal Wghs Undr h uncorrlad assupon: Mj * j h ω j * s ar nvrsly proporonal o hr varancs. R j j In gnral, forcass ar basd and corrlad. h corrlaons wll appar n h abov forula for h opal wghs. For h wo forcass cas: Mj * ( ) ( ) ( ) ( ) W do no obsrv h forcas varancs and covarancs, nor h bass. W nd a hsory of forcass o sa h opal wghs. Cobnaon of Forcass: Rgrsson Wghs In gnral, forcass ar basd and corrlad. h corrlaons wll appar n h abov forula for h opal wghs. Idally, w would lk o hav ngavly corrlad forcass. Grangr and Raanahan(984) usd a rgrsson hod o cobn forcass. - Rgrss h acual valu on h forcass. h sad coffcns ar h wghs. y M MR R... M Should us a consrand rgrsson O h consan Enforc non-ngav coffcns. Consran coffcns o su o on 4

21 RS EC - Lcur 5 Cobnaon of Forcass: Rgrsson Wghs Rarks: - o g wghs, w do no nclud a consan. Hr, w ar assung unbasd forcass. If h forcass ar basd, w nclud a consan. - o accoun for ponal corrlaon of rrors, Coulson and Robbns (993) suggss allowng for ARMA rsduals or nclud y +l- n h rgrsson. - varyng wghs ar also possbl. Should wghs ar? wo vws: - Spl avrags ouprfor or coplcad cobnaon chnqus --Sock and Wason (999) and Flds and Ord (). - Saplng varably ay affc wgh sas o h xn ha h cobnaon has a largr MSFE --Harvy and Nwbold (5). - Baysan chnqus, usng prors, ay hlp n h lar suaon. 4 Cobnaon of Forcass: Baysan Wghs In our dscusson of odl slcon, w nond ha h BIC s conssn. ha ans, h probably ha a odl s ru, gvn h daa s proporonal o BIC: P(M j daa) α xp(-bic j /). Basd on hs, w us h BIC of dffrn odls o drv wghs. hs s a splfd for of Baysan odl avragng (BMA). Easy calculaon of wghs. L BIC* b h salls BIC aong h R odls consdrd. Dfn ΔBIC Mj =BIC Mj BIC*. hn, * xp( Mj BIC Mj Mj Mj * / R j Mj / ) * 4

22 RS EC - Lcur 5 Cobnaon of Forcass: Baysan Wghs Sps: () Copu BIC for h R dffrn odls. () Fnd bs-fng BIC*. (3) Copu ΔBIC & xp( ΔBIC/). (4) Add up all valus and r-noralz. BMA pus h os wgh on h odl wh h salls BIC. So auhors hav suggsd rplacng BIC wh AIC n h wgh forula.., ω j αxp(-aic j /). - hr s no clar hory for hs forula. I s spl and works wll n pracc. - hs hod s calld wghd AIC (WAIC). 43 Cobnaon of Forcass: Baysan Wghs Q: Dos ak a dffrnc h crra usd? wo suaons: () h slcon crron (AIC, BIC) ar clos for copng odls. hn, s dffcul o slc on ovr h ohr. - WAIC and BMA wll produc slar wghs. () h slcon crron ar dffrn. - WAIC and BMA wll produc dffrn wghs. - hy wll gv zro wgh f h dffrnc s larg, say, abov. Q: Whch on o us? - No clar. WAIC works wll n pracc. Gnral fndng: Spl avragng works wll, bu s no opal. A cobnaon bas h lows crra usd. 44

23 RS EC - Lcur 5 Forcasng: Fnal Cons Snc Bas and Grangr (969) and Grangr and Raanahan (984), cobnaon wghs hav gnrally bn chosn o nz a syrc, squard-rror loss funcon. Bu, asyrc loss funcons can also b usd. Ello and rann (4) allow for gnral loss funcons (and dsrbuons). hy fnd ha h opal wghs dpnd on hghr ordr ons, such a skwnss. I s also possbl o forcas quanls and cobn h. sng of quanl forcass can b basd on h gnral approach of G&W (6). Gacon and Kounjr (5) prsn an applcaon. 45 3

innovations shocks white noise

innovations shocks white noise Innovaons Tm-srs modls ar consrucd as lnar funcons of fundamnal forcasng rrors, also calld nnovaons or shocks Ths basc buldng blocks sasf var σ Srall uncorrlad Ths rrors ar calld wh nos In gnral, f ou