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 s an rror, should b nrprd as wh nos. W wll wr s WN, σ
Unforcasabl Innovaons Wh nos procsss ar lnarl unforcasabl A srongr condon s unforcasabl. Th nnovaons ar unforcasabl f Ω - Ths mans h bs forcas s zro For som purposs, w wll assum h rrors ar unforcasabl
Movng Avrag Procsss Dbold, Chapr 7 Ths modls ar lnar funcons of sochasc rrors
MA Procss Th frs-ordr movng avrag procss, or MA procss, s whr s WN, σ Th MA coffcn conrols h dgr of sral corrlaon. I ma b posv or ngav. Th nnovaons mpac ovr wo prods An conmporanous sam prod mpac A on-prod dlad mpac
Man of MA Th uncondonal man of s
Varanc of MA Th uncondonal varanc of s Ths s a funcon of boh h nnovaon varanc σ and h MA coffcn., cov var var var var σ σ σ
Condonal Man of MA If h rror s unforcasabl Ω - hn h condonal man of s Ths s h bs forcas of. Th opmal forcas rror s Ω Ω Ω Ω Ω
Condonal Varanc of MA Th condonal varanc of s Th condonal varanc, h forcas varanc, and h nnovaon varanc ar all h sam hng var var var σ Ω Ω Ω Ω
Auocovaranc of MA Th frs auocovaranc s σ γ
Auocovaranc of MA Th auocovaranc for k> ar Thus h auocovaranc funcon s zro for k> k k k k k k k k γ
Auocorrlaons of MA Snc hn γ γ k var γ σ ρ k, k σ ρ σ, k Th auocorrlaon funcon of an MA s zro afr h frs lag. σ
Frs Auocorrlaon Th frs auocorrlaon has h sam sgn as ρ As rangs from - o, ρ rangs from - ½ o ½ Θ< : ngav auocorraon
Lag Opraor Noaon Rmmbr h lag opraor L L W can wr h MA as or L L L whr LL s a funcon of h lag opraor.
Invrson of an MA W can wr an MA n rms of laggd Rwr as Thn lag hs quaon on prod Thn combn
Invrson, Connud Do hs agan Rpa o nfn Thn 3 3 3 3 3 3 3 3
xsnc of Invrs Ths srs convrgs and h nvrson xss f <. Rcall h lag opraor xprsson W can wr hs as L Ths nvrson s vald f < L
Invrson of Lag Polnomal Wha dos hs man? B akng a powr srs xpanson from calculus Ths xpanson convrgs f < Applng hs xprsson as ndd L 3 3 L L L L 3 3 3 3 L L L L
Opmal Forcas In h MA modl h opmal forcas s - bu h rror s no drcl obsrvd. On approach s o us h auorgrssv rprsnaon Ω Bu hs s cumbrsom.
Rcursv Forcas for MA Anohr approach s o us h quaon and ralz ha hs gvs a rcursv formula o numrcall compu h rror Gvn, and gvn h nal condon Ths gvs a rcursv formula o compu all h rrors. Th ou-of-sampl forcas s T T T
MAq Procss Th movng avrag procss of ordr q, or MAq, s whr s WN, σ W can wr h quaon as whr L s a q h ordr polnomal n L q q q q L L L L
Auocorrlaons Th frs q auocorrlaons of a MAq ar non-zro, h auocorrlaons abov q ar zro
Wold s Thorm If s a zro-man covaranc saonar procss, hn can b wrn as an nfn ordr movng avrag, also known as a gnral lnar procss L whr s WN, σ
Lnar Procss L Normalzaon: Squar summabl <
Inrpraon of Wold s Thorm Thr s a bs lnar approxmaon for n rms of s pas valus MAq ma b a usful approxmaon
Man and Varanc Uncondonal man Uncondonal varanc var var σ
Rlvanc of MAq Modls MAq modls hlp o buld our undrsandng and nuon for sral dpndnc and auocorrlaon Bu, no commonl usd for forcasng To sma n STATA, us command arma, arma,,q
Quarrl Consumpon Growh Consumpon Growh Ral Prsonal Consumpon xpndurs - 95q 96q 97q 98q 99q q q q m
MA Modl W wll sma a MA Saa command arma, arma,,
MA smaon arma s a nonlnar opmzr, so h algorhm ras unl convrgnc. arma pc, arma,, sng opmzaon o BHHH Iraon : l og lklhood -733.4995 Iraon : log lklhood -78.7674 Iraon : log lklhood -75.9669 Iraon 3: log lklhood -75.44537 Iraon 4: log lklhood -75.333 swchng opmzaon o BFGS Iraon 5: log lklhood -75.785 Iraon 6: log lklhood -75.389 Iraon 7: log lklhood -75.373 Iraon 8: log lklhood -75.373
MA smas, con. Th smad MA coffcns ar shown as L and L. No h MA cof s small, h MA cof s largr ARIMA rgrsson Sampl: 947q - 6q4 Numbr of obs 79 Wald ch 9.5 Log lklhood -75.373 Prob > ch. OPG pc Cof. Sd. rr. z P> z [95% Conf. Inrval] pc ARMA _cons 3.36877.7568.45..837974 3.89858 ma L..3889.37795.8.4 -.48595.5373 L..36567.4443 8.8..839999.4464535 /sgma 3.39894.734445 4.75..995945 3.83843
Rsuls MA modl for consumpon growh.3.36.4.4
Auorgrssv Procsss Th frs-ordr auorgrssv procss, AR s β whr s WN, σ Usng h lag opraor, w can wr βl If β>, - and ar posvl corrlad If β<, - and ar ngavl corrlad
Invrson B back-subsuon a gnral lnar procss wh gomrcall dclnng coffcns Ths nvrson rqurs ha β < β < s rqurd for saonar β β β β β β
Imporanc of β < If β hn dos no convrg, so h sum s no dfnd.
Man and Varanc B h formula for h uncondonal man and varanc of a gnral lnar procss var var β σ σ β β β
Anohr Varanc Calculaon Tak varanc of boh sds of β Thus var If s varanc saonar, w solv and fnd var var β var β var β var σ var σ β
β < If β hn s nfn var σ β
β W calculad ha var β var σ Whn β, hn var var σ > var so h varanc s ncrasng wh β s nconssn wh varanc saonar. β < s ncssar for saonar.
Random Walk An AR wh β s known as a random walk or un roo procss B back-subsuon Th pas nvr dsappars. Shocks hav prmann ffcs
Un Roo Th random walk s calld a un roo procss bcaus h lag opraor -L has a roo nrscon wh h x-axs a L I s calld a random walk bcaus nds o wandr whou man-rvrson. If s an AR wh a un roo β hn s frs dffrnc Δ - s wh nos
Assgnmns Rad Dbold hrough Chapr 7. Problm S # 5 Du Tusda / Rad Chapr 4 from Th Sgnal and h Nos Radng Rflcon Du Thursda /6