Maximum Likelihood Estimation for Allpass Time Series Models

Transcription

1 Maximum Likelihood Esimaio or Allass Time Series Models Richard A. Davis Dearme o Saisics Colorado Sae Uiversiy h:// Joi work wih Jay Breid, Colorado Sae Uiversiy Alex Tridade, Uiversiy o Florida Beh Adrews, Colorado Sae Uiversiy

2 Iroducio roeries o iacial ime series moivaig examle all-ass models ad heir roeries Esimaio likelihood aroximaio MLE ad LAD asymoic resuls order selecio Emirical resuls simulaio NZ/USA exchage raes Noiverible MA rocesses relimiaries a wo-se esimaio rocedure Microso radig volume Summary

3 Fiacial Time Series Log reurs, X *l P - l P -, o iacial asses oe exhibi: heavy-ailed margial disribuios PX > x ~ C x α, < α < 4. lack o serial correlaio ˆ h ear or all lags h > MGD sequece ρ X X ad X have slowly decayig auocorrelaios ρ X h ad ρˆ coverge o slowly as h ˆ h X rocess exhibis sochasic volailiy Noliear models X Z, {Z } ~ IID, ARCH ad is varias Egle `8; Bollerslev, Chou, ad Kroer 99 Sochasic volailiy Clark 973; Taylor 986 3

4 3 4 5 lag h lag h lag h ACFX^ ACFX X ACF Moivaig examle 5-daily log-reurs o NZ/US exchage rae 4

5 All-ass model o order 3 oise ACF : allass A C F : alla ss ACF X ACF model samle 3 4 Lag 3 4 Lag 5

6 All-ass Models Causal AR olyomial: Deie MA olyomial: m, or. m θ / - / or MA olyomial is o-iverible. Model or daa {X } : BX θb Z, {Z } ~ IID o-gaussia Examles: B k X X -k All-ass: X X - Z Z -, <. All-ass: X X - X - Z + / Z - / Z - 6

7 7 Proeries: causal, o-iverible ARMA wih MA rereseaio ucorrelaed la secrum ero mea daa are deede i oise is o-gaussia e.g. Breid & Davis 99. squares ad absolue values are correlaed. X is heavy-ailed i oise is heavy-ailed. π π ω ω ω ω i i i X e e e j j j ψ Z Z B B B X

8 Esimaio or All-Pass Models Secod-order mome echiques do o work leas squares Gaussia likelihood Higher-order cumula mehods Giaakis ad Swami 99 Chi ad Kug 995 No-Gaussia likelihood mehods likelihood aroximaio quasi-likelihood leas absolue deviaios miimum disersio 8

9 Aroximaig he likelihood Daa: X,..., X Model: X where r is he las o-ero coeicie amog he j s. Noise: where Z / r. X Z Z Z / X X X X r, More geerally deie,, + + B X i + +, i,..., +. Noe: is a close aroximaio o iiialiaio error,..., +, 9

10 Assume ha Z has desiy ucio ad cosider he vecor Joi desiy o : ad hece he joi desiy o he daa ca be aroximaed by where qmax{ j : j }. ',...,,...,,,...,,,..., OOO O N OOO LO M OOOO O N OOO LO M X X + ideede ieces,,...,,...,,,..., q q h X X h h + q q h x

11 Log-likelihood: L, l / + q where /. Leas absolue deviaios: choose Lalace desiy ex ad log-likelihood becomes cosa l κ Coceraed Lalacia likelihood l l cosa l Maximiig l is equivale o miimiig he absolue deviaios m. / κ, q κ / q

12 Assumios Assume {Z } iid wih a scale arameer mea, variace For kow, use maximum likelihood urher smoohess assumios iegrabiliy, symmery, ec. o Fisher iormaio: For ukow, use quasi-likelihood Leas absolue deviaios assume has media ~ I ' / d assume coiuous i eighborhood o ac as i Lalace o ge crierio ucio

13 Resuls Le γh ACVF o AR model wih AR oly. ad Γ [ γj- k] j,k Maximum likelihood: ˆ MLE D N, ~ I Γ Leas absolue deviaios: ˆ LAD D N, Var 4 Z Γ 3

14 4 Furher commes o MLE Le α,...,, /, β,..., β q, where β,..., β q are he arameers o d. Se Fisher Iormaio { } d d d d T β β β β β β β β β β β β β β α α + + ; ; ; I ; ; ; ' L ; / ; ' Kˆ ; / ; ' Î,,

15 5 Uder smoohess codiios o wr β,..., β q we have where Noe: is asymoically ideede o ad,, ˆ MLE N D α α Γ ' ˆ ˆ ' ˆ ' ˆ ' ˆ ' ˆ ˆ L LK I LK L LK I L LK I L K L L I K I ˆ MLE ˆ MLE +, α ˆ MLE β

16 Ideiiabiliy i LAD case? Miimier may o be uique. Gaussia case: {Z } iid, so Z Z E E E E Cosider {c j }wih a leas wo o-ero elemes ad Jia ad Pawia 998 show j c j < ad c E c j Z j > E Z j j - N, N, j holds or Lalace, Sude s, coamiaed ormal, ec. - No-Gaussia case: E B B E Z E > B B 6

17 7 Ceral Limi Theorem LAD case Thik o u / as a eleme o R Deie The S u Su i disribuio o CR, where Hece, + + m - S -/ -/ u u u, Var, ~, ' ' r r Z N S Γ + Γ N N u u u u Var, ~ arg mi ˆ arg mi 4 4 / Γ Γ r D LAD Z N S S N u u

18 Asymoic Resuls LAD case: Theorem. Le {Y } be he liear rocess Y j c j j, j where c, <, { }~IID,, media, c j g> g desiy o. The S Var Y -/ - Y - g + N where N ~ N, γ * + γ * h ad γ*h is he covariace ucio or Y sg h 8

19 9 Key idea: { } sg } { } { / / -/ / / g Y Var N Y Y - Y - S Y Y + + < < < <

20 Theorem. O CR, where ad Γ is he covariace marix o a causal AR., -/ u u u S - S +,, ~, ' ' r r Z Var N S Γ + Γ N N u u u u

21 Limi heory or LAD esimae. Noe ha ˆ LAD + uˆ / so ha ˆ LAD u ˆ uˆ arg mi S u. arg mi S u Miimiig S, we id ha he miimier or limi radom variable is r Γ uˆ ˆ LAD N r Γ Var Z N ~ N, Γ 4

22 Asymoic Covariace Marix For LS esimaors o AR: ˆ LS D N, Γ For LAD esimaors o AR: ˆ LAD D N, 4 Γ For LAD esimaors o AP: ˆ LAD D N, Var Z 4 Γ For MLE esimaors o AP: ˆ MLE D N, Iˆ Γ

23 Lalace: LADMLE Sudes ν, ν >: Var Z 4 Iˆ LAD: MLE: Sude s 3 : LAD:.7337 MLE:.5 Var Z 4 Iˆ ARE:.7337/ Γ ν / ν π Γ ν + / ν ν + 3 ν ν 3

24 Order Selecio: Parial ACF From he revious resul, i rue model is o order r ad ied model is o order > r, he / ˆ, LAD Var N, 4 where ˆ is he h eleme o ˆ.,LAD Z Procedure:. Fi high order P-h order, obai residuals ad esimae scalar, θ Var Z 4 by emirical momes o residuals ad desiy esimaes., LAD 4

25 . Fi AP models o order,,..., P via LAD ad obai -h ˆ, coeicie or each. 3. Choose model order r as he smalles order beyod which he esimaed coeicies are saisically isigiica. Noe: Ca relace wih i usig MLE. I his case or > r / ˆ ˆ,, MLE ˆ, MLE N,. Iˆ 5

26 AIC: or o? A aroximaely ubiased esimae o he Kullback-Leiber idex o ied o rue model: AIC : L ˆ, κˆ + Pealy erm or Lalace case: Var Z E Z X E Var Z / Z / / Esimaed ealy erm: var ˆ ave{ ˆ } ˆ Var Z E Z P ˆ 6

27 Samle realiaio o all-ass o order a Daa From Allass Model b ACF o Allass Daa X -4 - ACF c ACF o Squares 3 4 Lag d ACF o Absolue Values ACF ACF Lag Lag 7

28 Esimaes: ˆ ˆ.97.38, Sadard errors comued as ˆ ˆ θ sqr{ / 5} where θ ˆ.99 Order selecio: cu-o value or PACF is.96*.98/sqr5.796 AIC : LX ˆ, κˆ hi_ AIC

29 Simulaio resuls: relicaes o all-ass models model order arameer value.5.3,.4 oise disribuio is wih 3 d.. samle sies 5, 5 esimaio mehod is LAD 9

30 3 To guard agais beig raed i local miima, we adoed he ollowig sraegy. 5 radom sarig values were chose a radom. For model o order, k-h sarig value was comued recursively as ollows:. Draw iid uiorm -,.. For j,,, comue Selec o based o miimum ucio evaluaio. Ru Hooke ad Jeeves wih each o he sarig values ad choose bes oimied value.,...,, k k k,,,,, k j k j j k jj k j j k j k j j k j

31 Asymoic Emirical N mea sd dev mea sd dev %coverage rel e* Asymoic Emirical N mea sd dev mea sd dev %coverage *Eiciecy relaive o maximum absolue residual kurosis: 4 3, v / v 3

32 MLE Simulaios Resuls usig -disr3.5 Asymoic Emirical N mea sd dev mea sd dev %coverage ν ν Asymoic Emirical N mea sd dev mea sd dev %coverage ν ν

33 Miimum Disersio Esimaor: Miimie he objecive c S + where { } are he ordered { }. Emirical Emirical LAD N mea sd dev mea sd dev

34 3 4 5 lag h lag h lag h ACFX^ ACFX X ACF Alicaio o iacial daa 5-daily log-reurs o NZ/US exchage rae 34

35 All-ass model ied o NZ-USA exchage raes usig LAD: Order 6, -.367, -.75, , 4.88, , AIC had local miima a 6 ad ac ACF: residuals ACF: residuals ac Lag Lag 35

36 Noiverible MA models wih heavy ailed oise... X Z +θ Z θ q Z -q, a. {Z } ~ IIDα wih Pareo ails... b. θ + θ + + θ q q No eros iside he ui circle Some eros iside he ui circle iverible oiverible 36

37 Realiaios o a iverible ad oiverible MA rocesses Model: X θ B Z, {Z } ~ IIDα, where θ i B +/B + /3B ad θ i B + B + 3B ACF ACF Lag Lag 37

38 Alicaio o all-ass o oiverible MA model iig Suose {X } ollows he oiverible MA model X θ i B θ i B Z, {Z } ~ IID. Se : Le {U } be he residuals obaied by iig a urely iverible MA model, i.e., X θˆbu ~ ~ θ B θ BU, θ is he iverible versio o So U i θ ~ B i θ B i i Z i θ i. Se : Fi a urely causal AP model o {U } ~ θ BU θ BZ. i i 38

39 Volumes o Microso MSFT sock raded over 755 rasacio days 6/3/96 o 5/8/99 X *^5 6*^5 ^

40 Aalysis o MSFT: Se : Logvolume ollows MA4. X +.53B+.77B +.7B 3 +.B 4 U iverible MA4 Se : All-ass model o order 4 ied o {U } usig MLE -dis: +.84B +.3B +.833B.65B.48B.34B.586B 3.8B Z. Coclude ha {X } ollows a oiverible MA4 which aer reiig has he orm: X +.34B+.374B +.54B B 4 Z, {Z }~IID U 4 4

41 a ACF o Squares o U b ACF o Absolue Values o U ACF..4.8 ACF Lag c ACF o Squares o Z 3 4 Lag d ACF o Absolue Values o Z ACF..4.8 ACF Lag Lag 4

42 Summary: Microso Tradig Volume Two-se i o oiverible MA4: iverible MA4: residuals o iid causal AP4; residuals iid Direc i o urely oiverible MA4: +.34B+.374B +.54B B 4 For MCHP, iverible MA4 is. 4

43 Summary All-ass models ad heir roeries liear ime series wih oliear behavior Esimaio likelihood aroximaio MLE ad LAD order selecio Emrirical resuls simulaio sudy AP6 or NZ/USA exchage raes Noiverible movig average rocesses wo-se esimaio rocedure usig all-ass oiverible MA4 or Microso radig volume 43

44 Furher Work Leas absolue deviaios urher simulaios order selecio heavy-ailed case oher smooh objecive ucios e.g., mi disersio Maximum likelihood Gaussia mixures simulaio sudies alicaios Noiverible movig average modelig iiial esimaes rom wo-se all-ass rocedure adaive rocedures 44