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1 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

2 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

3 Movng Avrag Procsss Dbold, Chapr 7 Ths modls ar lnar funcons of sochasc rrors

4 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

5 Man of MA Th uncondonal man of s

6 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

7 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

8 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

9 Auocovaranc of MA Th frs auocovaranc s σ γ

10 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

11 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

12 Frs Auocorrlaon Th frs auocorrlaon has h sam sgn as ρ As rangs from - o, ρ rangs from - ½ o ½ Θ< : ngav auocorraon

13 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.

14 Invrson of an MA W can wr an MA n rms of laggd Rwr as Thn lag hs quaon on prod Thn combn

15 Invrson, Connud Do hs agan Rpa o nfn Thn

16 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

17 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 L L L L

18 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.

19 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

20 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

21 Auocorrlaons Th frs q auocorrlaons of a MAq ar non-zro, h auocorrlaons abov q ar zro

22 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

23 Lnar Procss L Normalzaon: Squar summabl <

24 Inrpraon of Wold s Thorm Thr s a bs lnar approxmaon for n rms of s pas valus MAq ma b a usful approxmaon

25 Man and Varanc Uncondonal man Uncondonal varanc var var σ

26 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

27 Quarrl Consumpon Growh Consumpon Growh Ral Prsonal Consumpon xpndurs - 95q 96q 97q 98q 99q q q q m

28 MA Modl W wll sma a MA Saa command arma, arma,,

29 MA smaon arma s a nonlnar opmzr, so h algorhm ras unl convrgnc. arma pc, arma,, sng opmzaon o BHHH Iraon : l og lklhood Iraon : log lklhood Iraon : log lklhood Iraon 3: log lklhood Iraon 4: log lklhood swchng opmzaon o BFGS Iraon 5: log lklhood Iraon 6: log lklhood Iraon 7: log lklhood Iraon 8: log lklhood

30 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 Prob > ch. OPG pc Cof. Sd. rr. z P> z [95% Conf. Inrval] pc ARMA _cons ma L L /sgma

31 Rsuls MA modl for consumpon growh

32 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

33 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

34 Imporanc of β < If β hn dos no convrg, so h sum s no dfnd.

35 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

36 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

37 β < If β hn s nfn var σ β

38 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.

39 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

40 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

41 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

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