BOX-JENKINS MODEL NOTATION. The Box-Jenkins ARMA(p,q) model is denoted by the equation. pwhile the moving average (MA) part of the model is θ1at

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1 BOX-JENKINS MODEL NOAION he Box-Jenkins ARMA(,q) model is denoed b he equaion + + L+ + a θ a L θ a 0 q q. () he auoregressive (AR) ar of he model is + L+ while he moving average (MA) ar of he model is θa L θqa. he model's inerce is q 0 while a is he "whie noise" error. he arameers (coefficiens),, L,, θ, L, θ of he model are 0 deermined from he daa b he mehod of momens, leas squares, he mehod of maximum likelihood, or some oher mehod ha is consisen. he whie noise errors erms are assumed o have he following roeries: a. Ea ( ) 0, (zero mean assumion). Ea ( ) σ, (consan variance assumion) a 3. Eaa ( ) 0, s (indeendence of errors assumion) s 4. are normall disribued a Assumion 4 is no alwas needed for deriving cerain resuls wih resec o he Box- Jenkins model bu we will assume i here. Equaion () is referred o as he inerce-form of he Box-Jenkins ARMA(,q) model. An algebraicall equivalen form of he model (ofen reored b comuer rograms like SAS) is he so-called deviaion-from-he mean form of he Box-Jenkins model: μ ( μ) + L+ ( μ) + a θ a L θ a. () q q In his form, he mean of, denoed b μ, is relaed o he inerce 0 of equaion () b he formula μ [ 0 /( L ) ]. Probabl he mos comac wa o wrie he Box-Jenkins ARMA(,q) model is b using "backshif" olnomials ( B) B B L B and q θ( B) θb θb L θqb. ( B) is called he auoregressive backshif olnomial and θ ( B) is called he moving average backshif olnomial. he 3 backshif oeraors BB,, B, L siml "shif back" in ime observaions so ha s B s, for examle. Using hese olnomials we can wrie equaion () comacl as

2 ( B)( μ) θ( B) a. (') For examle, he ARMA(,0) model (in brief AR()) can be wrien as ( B)( μ) a. Using he disribuive law and he roeries of he backshif oeraor we have μ B + Bμ a and μ + μ a and finall μ μ) + a. ( his las equaion is he deviaion-from-he-mean form of he AR() Box-Jenkins model. Similarl he ARMA(0,) model (in shor MA()) can be wrien as μ a θ a and in olnomial form μ θ ( Ba ) where he moving average olnomial is defined b θ ( B) θb. he ARMA(,) "mixed" Box-Jenkins model can be wrien as μ ( μ) + a θ a or more comacl using backshif olnomials as in ( B)( μ) θ( B) a where ( B) B and θ ( B) θb. For he Box-Jenkins model o be "esimable" he so-called saionari and inveribili condiions mus hold. If we relace he back shif oeraors B, B, L, B he auoregressive olnomial and he moving average olnomial wih corresonding in

3 owers of z, z, z, L, z, and se hese olnomials o zero we have wha are called he auoregressive olnomial of he ARMA(,q) Box-Jenkins model, namel, ( z ) z z L 0 (3) z and he moving average olnomial of he ARMA(,q) model, namel θ ( z ) θ z θ z L θ z 0. (4) reaing he arameers,, L, and θ, θ, L, θ as known in hese homogenous equaions (evenuall, o be racical, we will have o use esimaes of hese arameers) AR AR AR le z z,, z denoe he roos (zeroes) of he auoregressive olnomial (3) and, L MA MA MA z, z, L, zq denoe he q roos of he moving average olnomial (4). For he Box- Jenkins model () o be saionar i mus be he case ha all of he roos of he auoregressive olnomial (3) mus be greaer han one in magniude (or, if comlex, have modulus greaer han one). For he Box-Jenkins model () o be inverible i mus be he case ha all of he roos of he moving average olnomial mus be greaer han one in magniude (or, if comlex, have modulus greaer han one). SAIONARIY Consider he AR() model μ ( μ) + a. For his model he AR auoregressive olnomial equaion is z 0 and herefore z / is he roo of he auoregressive olnomial. hus, for he AR() model o be saionar i is required ha / > and herefore ha <. Similarl, for an MA() model MA μ a θ a o be inverible i is required ha /θ > and herefore ha z θ <. For he saionari and inveribili condiions for oher oular Box-Jenkins models like he AR(), MA(), and ARMA(,) models, see m ADF and PACF able in he documen ACF_PACF.doc. B definiion, all AR() models are inverible while all MA(q) models are saionar. Now consider he racical imlicaions of saionari and inveriblili in Box- Jenkins models. When a Box-Jenkins model is saionar is observaions saisf he following hree roeries:. E( ). Var( ) μ (i.e. he mean of (i.e. he variance of σ is consan for all ime eriods) is consan for all ime eriods) 3. Cov(, ) γ (i.e. he covariance beween and is consan for all ime eriods and fixed,,, L ) hese hree condiions give rise o wha is called weak saionari (or us saionari for shor). he racical imlicaion of saionari is ha onl one realizaion of he

4 ime series is needed for us o be able o consisenl esimae he mean μ, he variance s σ, he covariance γ, and he auocorrelaion, c, and r. hese saisics are defined as ρ wih he samle saisics, r where denoes he oal number of observaions available on (samle mean) s ( ) (samle variance) c + ( )( ) (samle covariance) r ( )( ) c +,,, L. s ( ) (samle auocorrelaion) On he oher hand, we can see ha if our ime series has, for examle, an ever increasing mean (rend), hen he samle mean would no be aroriae for characerizing he rend in he daa. he samle saisic would likel no be aroriae for series daa ha exhibis, for examle, an increasing or decreasing volaili overime. Furhermore, daa ha exhibis a changing auocorrelaion srucure overime would be incorrecl characerized b he samle saisics c and r. For some examles of various forms of nonsaionari in ime series daa ou should run he SAS rogram MCARLO.sas ha is available on he websie for his course. I will go over some of hese grahs in class and wha daa ransformaions are necessar o change a nonsaionar ime series ino a saionar ime series so ha he ransformed series can be roerl modeled using he Box-Jenkins model. For examle, when a ime series has a linear rend, a ical ransformaion o use o make he daa saionar is o ake he firs difference of he daa, oeraor and Δ, where Δ reresens he firs difference reresens he ransformed daa. Since man economic and business ime series have rend in hem he Box-Jenkins model is generalized o he case where Δ is modeled as an ARMA(,q) rocess. Such a model can be wrien in inerce form as s

5 Δ + Δ + L+ Δ + a θ a L θ a 0 q q (5) or in deviaion-from-he-mean form as Δ μ ( Δ μ ) + L+ ( Δ μ ) + a θ a L θ a (5') Δ Δ Δ q q where μ Δ denoes he mean of he Δ series. In he case of aking he firs difference of he daa o make i saionar (and hus amenable o Box-Jenkins analsis) he models (5) or (5') or denoed b ARIMA(,,q) where he middle number reresens he number of imes he daa has o be differenced in order o make he daa saionar. In general, if he d daa has o be differenced d consuecuive imes o render he daa saionar, as in Δ, hen he d-differenced Box-Jenkins ARMA(,q) model is denoed b ARIMA(,d,q) and is wrien in inerce form as Δ + Δ + + Δ + d d d 0 L a θa L θqa q. (6) I canno be overemhasized how imoran he roer choice of he order of differencing (d) is in roerl modeling economic and business ime series b means of he Box-Jenkins model. If an inaroriae choice of d is made before roceeding o build a Box-Jenkins model for he daa, oor forecasing models will resul. More will be discussed on his oic when we address he issue of "uni roo" esing laer in he course. INVERIBILIY he imlicaion of he inveribili condiion can bes be areciaed b considering he MA() model wih, for simlici, a zero mean for assumed ( 0 0 ). We can rewrie he MA() model b ieraive subsiuion. a θ a θ ( + θ ) a a a θ θ a a θ θ θ θ + L a0. (7) From equaion (7) we can see ha an MA() model can be rewrien as an infinie order (as ) AR model (AR( )). Bu for (7) o be meaningful, we should have θ < so ha curren values of ( ) become less and less deenden on disan as values of as ime roceeds. If sa, θ >, a curren will be infiniel deenden on he as

6 observaions as. his las circumsance doesn' seem aroriae o real world alicaions. herefore, in he MA() case, imosing he inveribili condiion eliminaes models ha u infinie weigh on disan as values of a ime series in deermining curren values of he ime series. Anoher raionale for imosing he inveribili condiions on MA models is o ensure idenificaion of MA models. Consider he following wo MA() models: and a θ a (8) a θ a where θ θ. (9) / he firs MA() model has he same auocorrelaion funcion as he second MA() model! he auocorrelaion funcion of he firs model is ρ θ/( + θ ) and 0 for while he auocorrelaion funcion of he second model is ρ θ /( + θ ) (/ θ ) /( + (/ θ ) ) θ /( + θ ) and 0 for. Since he auocorrelaion funcions of he wo models are exacl he same, he auocorrelaion funcion canno be used o disinguish beween he wo aramerizaions. herefore, a second reason for imosing he inveribili condiions for MA models is o ensure heir uniqueness (as i relaes o he auocorrelaion funcion) for given values of he MA coefficiens. When building Box-Jenkins models i should be recognized ha such models are idenified u o a common facor. ha is, if a common roo exiss beween he auoregressive and moving average olnomials, hen a given ARMA(,q) model can be reduced o a ARMA(-,q-) model. Consider he following ARMA(,) model ( B )( μ) ( θb) a. (0) Obviousl, if θ, hen he ARMA(,) model of (0) can be reduced o he whie noise model (ARMA(0,0)), μ a () b canceling ou he common facors ha exis across he auoregressive and moving average backshif olnomials. he exisence of his common facor is he resul of he auoregressive and moving average olnomials ((3) and (4)) have he common roos AR MA z / z / θ because θ. hus, in he resence of hese common facors (common roos), he daa will no allow us o disinguish beween equaion (0) and () and we sa ha ARMA models are idenified u o a common facor. he lesson o be drawn here is ha when building a Box-Jenkins model for a given ime series we should be careful in comaring wo comeing model, one of which

7 has one more auoregressive arameer and one more moving average arameer han he oher comeing model, ha is when we are comaring a ARMA(-, q-) model wih a ARMA(,q) model. If he bigger model has an (esimaed) auoregressive olnomial wih a roo ha is almos equal o a roo of he (esimaed) moving average olnomial hen we migh susec ha here is in fac a common roo in he oulaion model and hus he simler model is o be referred. So when ou have a saisfacor (in he sense of goodness-of-fi saisics and whie noise residuals) ARMA(-,q-) model bu a somewha beer ARMA(,q) model ou should insec he emirical roos of he auoregressive and moving average olnomials of he ARMA(,q) model o see if here is an almos common roo across he olnomials. If here is, hen ou are robabl beer served (in erms of forecasing accurac) in going wih he simler ARMA(-,q- ) model.