Another Look at Estimation for MA(1) Processes With a Unit Root

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1 Aother Look at Etimatio for MA Procee With a Uit Root F. Jay Breidt Richard A. Davi Na-Jug Hu Murray Roeblatt Colorado State Uiverity Natioal Tig-Hua Uiverity U. of Califoria, Sa Diego

2 Program Model: Y t = Z t θ Z t-, {Z t } ~ IID, σ 2 Itroductio The MA uit root problem Why tudy o-ivertible MA? over-differecig radom walk + oie Gauia Likelihood Etimatio Idetifiability Limit reult Exteio o-zero mea heavy tail Laplace Likelihood/LAD etimatio Joit ad exact likelihood Limit reult Limit ditributio/imulatio compario Pile-up probabilitie joit likelihood exact likelihood 2

3 MA uit root problem MA: world implet time erie model! Y t = Z t θ Z t-, {Z t } ~ IID, σ 2 Propertie: θ < θ > θ = j θ j= j= Z t = Y t j - j Z t = θ Y t Z P t + j ivertible o-ivertible p{ Yt, Yt,...,} ad Zt p{ Yt +, Yt + 2,...,} p{ Y, } Y = Y perfect iterpolatio θ < ˆ i AN, 2 MLE θ θ θ / MLE = maximum Gauia likelihood, = ample ize What if θ =? 3

4 Why tudy MA with a uit root? a differecig to remove o-tatioarity liear tred model: X t = a + bt + Z t. Y t = X t X t- = b + Z t Z t- ~ MA with θ =. eaoal model: X t = t + Z t, t eaoal compoet w/ period 2. Y t = X t X t-2 = Z t Z t-2 ~ MA2 with θ =. b radom walk + oie X t = X t- + U t Y t = X t + V t radom walk igal radom walk igal + oie The Y t Y t- = U t + V t V t- ~MA with θ= if ad oly if VarU t =. 4

5 Idetifiability ad the Gauia likelihood Idetifiability Y t = Z t θ Z t-, {Z t } ~ IID, σ 2 θ > Y t = ε t θ ε t-, where {ε t } ~ WN,θ 2 σ 2. {ε t } i IID if ad oly if {Z t } i Gauia Breidt ad Davi `9 {ε t } i a pecial cae of a All-Pa Model Breidt, Davi, Tridade `, Adrew et al. `5a, `5b Gauia Likelihood L G θ, σ 2 = L G /θ, θ 2 σ 2 θ i oly idetifiably for θ. Note: i thi implie L G θ = L G /θ for the profile likelihood ad θ = i a critical poit, L ' G =. ii a pile-up effect eue, i.e., eve if θ <. P θ ˆ = > 5

6 Gauia likelihood, o-gauia data obervatio from Y t = Z t θ Z t-, {Z t } ~ IID, Laplace pdf θ =.8 θ =. θ =.25 Gauia likelihood theta 6

7 Gauia MLE for ear-uit root Idea: build parameter ormalizatio ito the likelihood fuctio. Model: Y t = Z t --/ Z t-, t =,,. = -θ, θ = - /, θ = - γ/ Gauia Likelihood: L = l - / - l, l = profile log-like. Theorem Davi ad Dumuir `96: Uder θ = -γ /, L d Z γ o C[,. Reult: θˆ ˆ = argmax Ζ γ θˆ ˆ L L = arglocalmax Ζ γ P θˆ = P ˆ = =.658 if γ =. L mle mle L 7

8 Exteio of MLE Gauia likelihood i o-zero mea Che ad Davi `: ame type of limit, except pile-up i more exceive. Pˆ =.955 θ mle Thi make hypothei tetig eay! Reject H : θ = if ˆ < ize of tet i.45 θ mle ii heavy tail Davi ad Mikoch `98: {Z t } ymmetric alpha table SαS. The the max Gauia likelihood etimator ha the ame ormalizig rate, i.e., θˆ L d P θˆ P ˆ L = L = The pile-up decreae with icreaig tail heavie. ˆ L 8

9 Laplace likelihood/lad etimatio If oie ditributio i o-gauia, the MA parameter θ i idetifiable for all real value. Q. For MLE o-gauia doe oe have / or / /2 aymptotic? Q2. I there a pile-up effect? Look at thi problem with o-gauia likelihood Specifically, coider Laplace likelihood / Leat Abolute Deviatio for uit root oly ot ear-uit root Some reult are prelimiary oly! 2

10 No-Gauia likelihood Joit ad Exact Model. Y t = Z t θ Z t-, {Z t } ~ IID with media ad EZ 4 <. Iitial variable. Z iit = Z, Z t = Y t, if θ, otherwie. Joit deity: Let Y =Y,...,Y, the iit f y, z = f z, z,..., z + θ, { θ } { θ > } where the z t are olved forward by: z t = Y t + θ z t-, t =,, for θ with z = z iit backward by: z t- = θ z t Y t, t =,, for θ > with z = z iit + Y Y Note: itegrate out z iit to get Exact likelihood. f y = f y, z iit dz iit 3

11 Laplace likelihood example obervatio from Y t = Z t θ Z t-, {Z t } ~ IID Laplace pdf 4 Z θ =.8 θ = theta Z z.iit theta z.iit - -2

12 Laplace likelihood, Laplace oie obervatio from Y t = Z t θ Z t-, {Z t } ~ IID Laplace pdf θ =.8 θ =. θ =.25 Exact likelihood Joit likelihood at z max θ Laplace likelihood Laplace likelihood theta theta 5

13 Laplace likelihood-lad etimatio Joit Laplace log-likelihood. σ = E Z i a cale parameter L θ, z iit, σ = Maximizig wrt σ, we obtai + log 2σ σ σ ˆ = t= t= o that maximizig L i equivalet to miimizig l θ, z iit = z t / + t= t= z z t t z t log, if θ, θ, otherwie. θ { θ > } 7

14 8 Joit Laplace likelihood limit reult Reult. Uder the parameterizatio, θ = + / ad z iit = Z + ασ/ /2, we have where for, ad for >.,,,, α θ σ = α U Z l z l U d iit d e t ds e f dw e t ds e U t t 2, + α + + α = α d e t ds e f dw e t ds e U t t 2, + α + + α = α +

15 Joit Laplace likelihood limit reult The limit cotai correlated Browia Motio St ad Wt, obtaied a the limit of the partial um procee S [ t ] [ t ] t = Z i d S t, W t = ig Z i d σ i= σ i= Wt. From the limit, U, α U, α, d it ugget from the cotiuou mappig theorem? that limitoptimumcriterio = optimumlimitcriterio. So for the optimizer of the Joit likelihood where ˆ, ˆ ˆ, ˆ iit θ σ z Z α ˆ J, αˆ J J J = arglocal mi U, α. d J J 9

16 Coitet etimatio of oie? Note that the previou reult imply that σ zˆiit = Z + ˆ α = Z + O p / 2 o that a uoberved radom oie ca be coitetly etimated. Doe thi make ay ee? Recall that i the uit root cae, Z p{ Y, Y 2, K, Y, K } o that i fact, coitet etimatio i poible. 2

17 Exact Laplace likelihood limit reult Exact Laplace Likelihood: L θ, σ = f y, z iit dz iit Reult 2. For the local optimizer of the Exact likelihood, θˆ ˆ E d E, where ˆ arg mi * E = U, ad U * i a tochatic proce defied i term of St ad Wt. 22

18 Simulatig from the limit proce Step. Simulate two idep equece W,..., W m ad V,..., V m of iid N, radom variable with m=. Step 2. Form Wt ad Vt by the partial um procee, W t = [ t ] j= W j / ad V t = [ t ] j= V j /. Step 3. Set St = Wt + c Vt, where 2 c = Var Z t / E Z. Limit proce deped oly o c ad f. Step 4. Compute U,α ad U * from the defiitio. Step 5. Determie the repective Local ad Global miimizer of Joit limit U,α ad Exact limit U * umerically. 23

19 Simulated realizatio of the limit procee Simulate Joit ad Exact limit procee, U,α, U*. t5 pdf Simulate realizatio of each limit proce, joit ad exact Compute local ad global optima Repeat Build up limit ditributio fuctio U ad Utar beta 24

20 Limit cdf red graph = Laplace pdf for Z t blue graph = Gauia pdf for Z t cdf Joit Lap Like cdf Exact Lap Like beta.mi beta.mi 27

21 Simulatio reult: Global Exact ad Global Joit Exact = MLE Joit = maximize over θ ad z iit Laplace oie θ =, σ = rep = 2 bia rme Exact Joit θˆ ˆ E θ J Note: Joit domiate Exact rme i half the ize = 5 = bia rme bia rme = 2 bia rme

22 Aalyi of pile-up probabilitie Look back at realizatio of the limit procee, U,α, U*. Whe i there a local optimum at θ =? Check derivative Negative derivative from the left Poitive derivative from the right Local optimum at θ = U ad Utar t5 pdf -2-2 beta 32

23 Pile-up probabilitie Joit Reult 3. Joit Laplace likelihood where Y = P θˆ J = P- < Y <, W S dw -W S d + 2 W f d-w / 2 Idea: look at derivative ˆ P θ = = Plim U, αˆ < ad lim, αˆ U Plim U, αˆ < ad lim U, αˆ Now, lim U, αˆ = Y + lim U, αˆ = Y ad the reult follow. J > > 33

24 Pile-up probabilitie Exact Reult 4. Exact Laplace likelihood ˆ P θ E = P < Y < - = 2 2 The pile-up probability i alway zero for the Exact, ad alway poitive for the Joit ee Reult 3. Remark. Laplace pile-up - z /σ If Z t ha a Laplace deity f z = e, the 2σ Y = [ W W ] dv +. 2 where W ad V are idepedet tadard Browia motio. 34

25 Laplace pile-up probabilitie cot It follow that the Joit etimator ha pile-up probability P θˆ J = P < Y < = P-.5 [ W W ] < dv <.5 = E P-.5 < / 2 = E 2Φ.5 [ ] W W 2 d - [ W W ] dv <.5 W t, t [,].82 But o pile-up probability for Local Exact: Remark: if Local doe ot pile up, Global doe ot pile up if Local doe pile up, Global probably doe a well 35

26 Simulatio reult pile-up probabilitie Pile-up probabilitie for Joit: P θˆ J = Gau Lap Uif t No pile-up probabilitie for Exact. 36

27 Summary ad Future Work Reviewed MA uit root ad ear-uit root with Gauia likelihood / aymptotic, pile-up eve if θ< New reult for MA uit root with Leat Abolute Deviatio / aymptotic for Joit or Exact Joit beat Exact; Joit ha pile-up ad Exact doe ot Further work: Nail dow prelimiary reult, coduct further imulatio Other o-gauia criterio fuctio MLE? No-zero mea? Near-uit root? -γ/ Performace of Joit with Gauia likelihood? 37

Pile-up Probabilities for the Laplace Likelihood Estimator of a Non-invertible First Order Moving Average

Pile-up Probabilities for the Laplace Likelihood Estimator of a Non-invertible First Order Moving Average IMS Lecture Note Moograph Serie c Ititute of Mathematical Statitic, Pile-up Probabilitie for the Laplace Likelihood Etimator of a No-ivertible Firt Order Movig Average F. Jay Breidt 1,, Richard A. Davi,3,