THEORETICAL AUTOCORRELATIONS Cov( y, y ) E( y E( y))( y E( y)) ρ = = Var( y) E( y E( y)) =,, L ρ = and Cov( y, y ) s ofen denoed by whle Var( y ) f ofen denoed by γ. Noe ha γ = γ and ρ = ρ and because of hs syery he heorecal auocorrelaon funcon and he saple auocorrelaon funcon (below) only need be exaned over he posve lags =, L. SAMPLE AUTOCORRELATIONS γ T ( y y)( y y) = + = = T ( y y) = r,,, L. The r are conssen esaors of he heorecal auocorrelaon coeffcens ρ. Under he assupon ha y follows a whe nose process he sandard errors of hese r are approxaely equal o T. Thus, under he null hypohess ha y follows a whe nose process, roughly 95% of he r should fall whn he range of ±.96 / T. If ore han 5% of he r fall ousde of hs range, hen os lkely y does no follow a whe nose process. THEORETICAL PARTIAL AUTOCORRELATIONS Cov( y, y y, L, y + ) φ = Var( y y, L, y ) + E( y E( y y, L, y ))( y E( y y, L, y = + + E( y E( y y, L, y + )) =,, L = he correlaon beween y and y afer neng ou he effecs he nervenng values y, L, have on boh of he y + ))
SAMPLE PARTIAL AUTOCORRELATIONS ˆ φ are calculaed usng he forulas for he heorecal auocorrelaons for a gven ARMA(p,q) odel (see y ACF_PACF_Table.doc Word docuen for he forulas) bu replacng all of he heorecal auocorrelaons ( ρ ) wh he above saple auocorrelaons ( r ) and all of he unknown Box-Jenkns coeffcens ( φ, θ ) wh her correspondng esaes ( ˆ φ, ˆ θ ) obaned by he ehod of oens or soe oher ehod. The ˆ φ are conssen esaors of he heorecal paral auocorrelaons, φ. Under he assupon ha follows a whe nose process he sandard errors of hese ˆ y φ are approxaely equal o T. Thus, under he null hypohess ha y follows a whe nose process, roughly 95% of he φ should fall whn he range of ±.96 / T. If ore han 5% of he ˆ φ fall ousde of hs range, hen os lkely whe nose process. ˆ y does no follow a GOODNESS-OF-FIT MEASURES. AIC (Akake Inforaon Creron) AIC = L aˆ + K ( ) where K = p + q +, L( aˆ ) = he log of he lkelhood funcon of he Box- Jenkns ARMA(p,q) odel, a = he resdual a e for he Box-Jenkns odel ˆ and he log lkelhood funcon, L( a ˆ ), s a onooncally decreasng funcon of he su of squared resduals, a ˆ. In oher words, he saller a ˆ s, he larger L( aˆ ) s and vce versa.. SBC (Schwarz Bayesan Creron) SBC = L aˆ + K n ( ) ln( ) SBC = L aˆ + K n ( ) ln( ) where n s he nuber of resduals copued for he odel. In ers of choosng a Box-Jenkns odel, he saller hese goodness-of-f easures, he beer. Tha s, we prefer he Box-Jenkns odel ha has he salles AIC and SBC
easures. Noce ha, as you add coeffcens o he Box-Jenkns odel, ( φ, θ ), he f of he odel, as easured by he su of squared resduals, herefore, addng coeffcens always ncreases he log lkelhood, aˆ, always decreases and, L ˆ ), of he Box- ( a Jenkns odel. To offse he endency for addng coeffcens o a odel us o prove s f, he above goodness-of-f (nforaon) crera each nclude a "penaly" er. (For he AIC creron he penaly er s +K whle for he SBC easure he penaly er s +Kln(T). Thus, wh hese crera, as one adds coeffcens o he Box-Jenkns odel, he proveen n f cong fro reducon n he su of squared resduals wll evenually be offse by he penaly er ovng n he oppose drecon. The goodness-of-f crera are hen nended o keep us fro buldng large order Box- Jenkns odels us o prove he f us o fnd ha such large order odels don' forecas very well. Shbaa (976) has shown ha, for a fne-order AR process, he AIC creron asypocally overesaes he order wh posve probably. Thus, an esaor of he AR order (p) based on AIC wll no be conssen. (By conssen we ean ha, as he saple sze goes o nfny, he correc order of an AR(p) Box-Jenkns odel wll be correcly chosen wh probably one.) In conras, he SBC creron s conssen n choosng he correc order of an AR(p) odel. Ofen hese wo crera choose he sae Box-Jenkns odel as beng he bes odel. However, when here s a dfference n choce, he AIC easure nvarably ples a Box-Jenkns odel of bgger order (K = p + q + ) han he order of he odel pled by he SBC creron. In oher words, he SBC creron ends o pck he ore parsonous odel when here s a "spl" decson arsng fro usng hese crera. Personally, I prefer o rely on he SBC creron n he case of "spl" decsons. A TEST FOR WHITE NOISE RESIDUALS (and hus he Box-Jenkns odel's "copleeness") H : Resduals of Esaed Box-Jenkns odel are whe nose (.e. uncorrelaed a all lags). Oher hngs held consan, he esaed Box-Jenkns odel s adequae. H : Resduals of Esaed Box-Jenkns odel are no whe nose. In hs case, a beer odel can be found by addng ore paraeers o he odel. The ch-square es used o es for whe nose resduals s calculaed as where χ = nn ( + ) r ( aˆ ) = n k = aa ˆˆ = n aˆ = + k r ( aˆ ) ( n ),
n = nuber of resduals, and aˆ s he e resdual of he Box-Jenkns odel. Ths sasc was suggesed by Lung and Box (978) and s called he Lung-Box ch-square sasc for esng for whe nose resduals. The null hypohess above s acceped f he observed ch-square sasc s sall (.e. has a probably value greaer han.5) and s reeced f he ch-square sasc s "large" (.e. has a probably value less han.5). As far as he choce of he nuber of lags,, o use, I would sugges = for quarerly daa and = 4 for onhly daa o ncrease he power of he es gven he frequency wh whch he daa s observed. CONSTRUCTION OF THE P-Q BOX In hs class we wll be consrucng a "P-Q Box" of he for P Q...... where. represens he followng nubers n each cell: AIC, SBC, χ, and he p-value of he Lung-Box ch-square sasc, χ. These cells represen he os prevalen Box- Jenkns odels ha apply o non-seasonal econoc e seres daa, naely, he ARMA(,), AR(), AR(), MA(), MA(), and ARMA(,) odels. Usng he saple ACF and saple PACF of he daa one can ofen narrow down he choce beween hese cell (odels) bu no always wh cerany. Thus, he P-Q Box can ofen help confr whch Box-Jenkns odel s bes for he daa. The odel wh he lowes AIC and SBC easures and havng whe nose resduals s he odel ha he P-Q Box sascs sugges. Hopefully, afer lookng a he saple ACF and saple PACF and he P-Q Box resuls one can coe o a enave choce for he p and q orders of he Box-Jenkns odel. OVERFITTING EXERCISE To confr he choce of odel suggesed by he saple ACF, saple PACF, and he P-Q Box, one should conduc an overfng exercse. Tha s, you should f wo addonal Box-Jenkns odels, one havng one ore auoregressve coeffcen and one havng one ore ovng average coeffcen and hen exanng (ndvdually) he sascal sgnfcance of he exra coeffcen n each odel. For exaple, f your enave choce s p = and q = (an AR() odel), you should exane he AR coeffcen n an AR() odel and deerne wheher hs "overfng" coeffcen s
sascally sgnfcan or no. If s no sascally sgnfcan (.e. he p-value s >.5), you can "fall" back o your orgnal choce. The oher overfng odel for he AR() odel s he ARMA(,) odel. So when you f, he overfng paraeer s he MA paraeer. If s no sascally sgnfcan, hen you can "fall" back o your orgnal "alos fnal" choce agan and ake your "fnal" choce for forecasng purposes. Of course, f eher of he overfng paraeers s sascally sgnfcan, you need o connue he odel buldng process.