. Definiions Saionay Time Seies- A ime seies is saionay if he popeies of he pocess such as he mean and vaiance ae consan houghou ime. i. If he auocoelaion dies ou quickly he seies should be consideed saionay ii. If he auocoelaion dies ou slowly his indicaes ha he pocess is non-saionay Nonsaionaiy- A ime seies is nonsaionay if he popeies of he pocess ae no consan houghou ime i. Uni Roo Nonsaionaiyii. Random Walk wih Dif- Whie Noise- A ime seies is called a whie noise if a sequence of independen and idenically disibued andom vaiables wih finie mean and vaiance, usually WN(, σ ). Whie noise has covaiance Backwad shif opeao a sho hand fo shif backwad in he ime seies. βy Y - β p Y Y -p. Auocoelaion Measues he linea dependence o he coelaion beween and -p. (summaizes seial dependence) ρ l Cov(, l ) Cov(, l ) Va( ) ( ) Va( ) Va l whee Va( ) Va( - ) fo weakly saionay pocess A way o check andomness in he daa Lag of he auocoelaion is by definiion i. If he auocoelaion dies ou slowly his indicaes ha he pocess is non-saionay. ii. If all he ACFs ae close o zeo, hen he seies should be consideed whie noise. No Memoy Seies i. Auocoelaion funcion is zeo Sho Memoy Seies i. Auocoelaion funcion decays exponenially as a funcion of lag Long Memoy Seies i. Auocoelaion funcion decays a polynomial ae ii. The diffeencing exponen is beween -½ and ½. 3. Paial Auocoelaion Coelaion beween obsevaions X and X +h afe emoving he linea elaionship of all obsevaions in ha fall beween X and X +h.
,,,3 + φ + φ + φ, M Each φ is he lag-p PACF ˆ p, p,,,3 + e + φ + φ,,3 + e, + φ + e 3, 3 The PACF shows he added conibuion of -p o pedicing. 4. Diagnosics and Model Selecion Residual Diagnosics i. The esiduals should be saionay whie noise ii. The ACF and PACF should all be zeo a. If hee is a long memoy in he esiduals hen he assumpions ae violaed nonsaionaiy of esiduals 3, AIC (Akaike s Infomaion Cieion) i. A measue of fi plus a penaly em fo he numbe of paamees ii. Coeced AIC- songe penaly em ~ makes a diffeence wih smalle sample sizes iii. Choose he model ha minimizes his adjused measue fi iv. AIC k log(mle esimae of he noise vaiance) + k/t, whee T is he sample size and k is he numbe of paamees in he model Pomaneau Tes i. Tess whehe he fis m coelaions ae zeo vs. he alenaive ha a leas one diffes fom zeo. ii. The sum of he fis m squaed coelaion coefficiens H : ρ... ρ m iii. whee ρ i is he auocoelaion H a : ρi iv. Box and Piece * Q ( m) T m pˆ l l Q*(m) is asympoically a chi-squaed andom vaiable wih m degees of feedom v. Ljung and Box m ˆ ρl Q( m) T ( T + ) l T l Modified Box & Piece saisic o incease powe Uni Roo Tes i. Deived in 979 by Dickey and Fulle o es he pesence of a uni oo vs. a saionay pocess ii. φ ρ + e ρ φ + φ + e ρ ρ
If φ hen he seies is said o have uni oo and is no saionay. The uni oo es deemines if φ is significanly close o. H : φ H A : φ < iii. The behavio of he es saisics diffes if i is a andom walk wih dif o if i is a andom walk wihou dif. 5. Uni Roo Nonsaionay Pocess Random Walk i. The equaion fo a andom walk is ρ ρ + a, whee ρ denoes he saing values and a is whie noise. ii. A andom walk is no pedicable and his can no be foecased. iii. All foecass of a andom-walk model ae simply he value of he seies a he foes oigin. iv. The seies has a song memoy Random Walk wih Dif i. ρ µ + ρ + a, whee µ E( ρ ρ ) ρ µ + ρ + a 6. Diffeencing ρ µ + ρ + a M µ + ρ + a ρ µ + ρ + a + a +... + a A posiive µ implies ha he seies evenually goes o infiniy. Reasons why he Diffeence is aken i. To ansfom non-saionay daa ino a saionay ime seies ii. To emove seasonal ends a. ake 4 h diffeence fo qualy daa b. ake h diffeence fo monhly daa Fis Diffeence- The fis diffeence of a ime seies is z y y i. A way o handle song seial coelaion of ACF is o ake he fis diffeence Second Diffeence- The second diffeence is z y y ) ( y y ) 7. Log Tansfomaion + a Reasons o ake log ansfomaion i. Used o handle exponenial gowh of a seies ii. Used o sabilize he vaiabiliy Values mus all be posiive befoe he log is aken (
8. Auoegessive Model i. If no all values ae posiive a posiive consan can be added o evey daa poin A egession model in which is pediced using pas values, -, -, i. AR() : + φ + a, whee a is a whie noise seies wih zeo mean and consan vaiance ii. AR(p): φ + φ +... + φ + a p p Weak saionay is he sufficien and necessay condiion of an AR model i. Fo an AR model o be saionay all of is chaaceisic oos mus be less han in modulus ACF fo Auoegessive Model i. The ACF decays exponenially o zeo ) Fo φ >, he plo of ACF fo AR() should decay exponenially ) Fo φ <, he plo should consis of wo alenaing exponenial decays wih ae φ. ii. The ACF fo AR() ρ l ρ l, because ρ hen ρ l. So he ACF fo he AR() should decay o exponenially wih ae φ saing a ρ PACF fo Auoegessive Model ii. The PACF is zeo afe he lag of he AR pocess iii. ˆ φ conveges o zeo fo all l > p. Thus fo AR(p) he PACF cus l, l off a lag p. 9. Moving Aveage Model A linea egession of he cuen value of he seies agains he whie noise o andom shocks of one o moe pio values of he seies. i. X µ + a θa, whee µ is he mean of he seies, a -i ae whie noise, and θ is a model paamee. The MA model is always saionay as i is he linea funcion of uncoelaed o independen andom vaiables. The fis wo momens ae ime-invaian MA model can be viewed as a infinie ode AR model ACF fo Moving Aveage Model ii. The ACF is zeo afe he lages lag of he pocess PACF fo Moving Aveage Model i. The PACF decays o zeo. ARMA [p,q]
The seies is a funcion of pas values plus cuen and pas values of he noise. Combines an AR(p) model wih a MA(q) model The equaion fo a ARMA(,) is + a + θa ACF fo ARMA i. The ACF begins o decay exponenially o zeo afe he lages lag of he MA componen.. ARIMA is an ARIMA model if he fis diffeence of is an ARMA model. In an ARMA model, if he AR polynomial has as he chaaceisic oo, hen he model is a ARIMA Uni-oo nonsaionay because i s AR has uni oo. ARIMA has song memoy. ARFIMA A pocess is a facional ARMA (ARFIMA) pocess if he facional diffeenced seies follows an ARMA(p,q) pocess. Thus if a seies ( B) d x follows ARMA(p,q) model, hen he seies is an ARFIMA(p,d,q). 3. Foecasing The mulisep foecas conveges o he mean of he seies and he vaiances of foecas eos convege o he vaiance of he seies. Fo AR Model i. The -sep ahead foecas is he condiional expecaion ˆ () E( h h+ h, h,...) + ii. Fo mulisep ahead foecas: p i φ i h+ i ˆ ( l) + h p φ i h+ l i i ˆ h ( ) h + h () ah+ iii. The foecas eo fo sep ahead: e iv. Mean eveing. Fo a saionay AR(p) model, long em poin foecass appoach hen uncondiional mean. Also, he vaiance of he foecas appoaches he uncondiional vaiance of. Fo MA Model i. Because he model has finie memoy, is poin foecass go o he mean of he seies quickly. ii. The -sep ahead foecas fo MA() is he condiional expecaion ˆ h () E( h + h, h,...) co θah The -sep ahead foecas fo MA() ˆh () E( h + h, h,...) co
iii. Fo a MA(q) model, he mulisep ahead foecass go o he mean afe he fis q seps. 4. Specal Densiy A way of epesening a ime seies in ems of hamonic componens a vaious fequencies. Tells he dominan cycles o peiods in he seies Specal Densiy is only appopiae fo saionay ime seies daa. A Peiodogam a a paicula fequency ω is popoional o he squaed ampliude of he coesponding cosine wave, α cos( ω) + β sin( ω), fied o he daa using leas squaes. Fo a Covaiance saionay ime seies(csts) wih auocovaiance funcion γ (v), v, ±, ± he specal densiy is given by f ( v) γ ( h) n / / γ ( h) e 5. VaR Value a Risk e πivh πivh f ( v) dv whee v [-/, /] Esimaes he amoun which an insiuion s posiion in a isk caegoy could decline due o geneal make movemens duing a given holding peiod. Concened wih make isk In ealiy, used o assess isk o se magin equiemens i. Ensues ha financial insiuions can sill be in business afe a caasophic even Deemined via foecasing If mulivaiae: i. VaR VaR + VaR + ρvarvar 6. VAR Veco Auoegessive Model A veco model used fo mulivaiae ime seies + Φ + a i. VAR(): - ; whee φ is a k-dim veco, Φ is a k x k maix, and {a } is a sequence of seially uncoelaed andom vecos wih mean zeo and covaiance maix Σ. Σ posiive definie. ii. VAR(p): + Φ - +... + Φ p-p + a Can also model VMA and VARMA models i. One issue, VARMA has an idenifiabiliy poblem (i.e. may no be uniquely defined ii. When VARMA models ae used, you should only eneain lowe ode models.
7. Volailiy Models ARCH i. Only an AR em a σ ε ii. ARCH(m): σ α + αa +... + α ma m iii. Weaknesses: Assume +ve & -ve shocks have same effecs on volailiy (i.e. use squae of pevious shocks o deemine ode) use leveage o accoun fo he fac ha ve shocks (i.e. bad news ) have lage impac on volailiy han +ve shocks (i.e. good news ). Model is esicive (see p.86, 3.3.()) Only descibes he behavio of he condiional vaiance. Does no explain he souce of he vaiaions. Likely o ove-pedic he volailiy since he espond slowly o lage isolaed shocks o he eun seies. GARCH genealized ARCH i. Mean sucue can be descibed by an ARMA model a σ ε ii. GARCH(m,s): σ α + m i α a i + i s j β σ iii. Same weaknesses as he ARCH iv. If he AR componen has a uni oo, hen we have an IGARCH model (i.e. Inegaed GARCH; a.k.a. uni-oo GARCH model) v. EGARCH (i.e. Exponenial GARCH) allows fo asymmeic effecs beween +ve & -ve asse euns. Models he log(cond. vaiance) as an ARMA. PRO: vaiances ae guaaneed o be posiive. GARCH-M - GARCH in mean i. Used when he eun of a secuiy depends on is volailiy a σ ε ii. GARCH(,)-M: µ + cσ + a σ α + αa + βσ indicaes ha he eun is posiively elaed o is pas volailiy. iii. Coss-Coelaion: seies coelaed agains seies ; used o deemine whehe hee exiss volailiy in he mean sucue. Alenaive GARCH models ) CHARMA Condiional heeoscedasic ARMA uses andom coefficiens o poduce condiional heeoscedasiciy. ) RCA Random Coefficien Auoegessive model accouns fo vaiabiliy among diffeen subjecs unde sudy. Bee suied fo modeling he condiional mean as i allows fo he paamees o evolve ove ime. 3) SV Sochasic Volailiy model is simila o an EGARCH bu incopoaes an innovaion o he condiional vaiance equaion. j j ; whee µ, c consan. A +ve c
4) LMSV Long-Memoy SV model allows fo long memoy in he volailiy. NOTE: Diffeencing ONLY effecs mean sucue, Log Tansfomaion effecs volailiy sucue. 8. MCMC Mehods (Makov Chain Mone Calo) Makov chain simulaion ceaes a Makov pocess on Θ, which conveges o a saionay ansiion disibuion, P(θ, X). GIBBS SAMPLING (p.397) o Likelihood unknown, condiional dis n s known. o Need saing values o Sampling fom cond. dis n s conveges o sampling fom he join dis n. o PRO: Compaed o MCMC, Gibbs can decompose a high-dim esimaion poblem ino seveal lowe-dim ones. o CON: When paamees ae highly coelaed, you should daw hem joinly. o In pacice, epea seveal imes wih diffeen saing values o ensue he algoihm has conveged. BAYESIAN INFERENCE (p. 4) o Combines pio belief wih daa o obain poseio dis n s on which saisical infeence is based.