Conditional-sum-of-squares estimation of models for stationary time series with long memory

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1 IMS Lecture Notes Moograph Series Time Series ad Related Topics Vol. 52 (2006) c Istitute of Mathematical Statistics, 2006 DOI: / Coditioal-sum-of-squares estimatio of models for statioary time series with log memory P. M. Robiso 1, Lodo School of Ecoomics Abstract: Employig recet results of Robiso (2005) we cosider the asymptotic properties of coditioal-sum-of-squares (CSS) estimates of parametric models for statioary time series with log memory. CSS estimatio has bee cosidered as a rival to Gaussia maximum likelihood ad Whittle estimatio of time series models. The latter kids of estimate have bee rigorously show to be asymptotically ormally distributed i case of log memory. However, CSS estimates, which should have the same asymptotic distributioal properties uder similar coditios, have ot received comparable treatmet: the trucatio of the ifiite autoregressive represetatio iheret i CSS estimatio has bee essetially igored i proofs of asymptotic ormality. Ulike i short memory models it is ot straightforward to show the trucatio has egligible effect. 1. Itroductio Cosider a real-valued, strictly ad covariace statioary time series x t, t Z. It is believed that x t has a parametric autoregressive (AR) represetatio (1.1) α j (θ 0 )x t j = ε t, t Z. Here ε t is a sequece of zero-mea, ucorrelated ad homoscedastic radom variables, with variace σ0,theα 2 j (θ) are give fuctios with p 1 vector argumet θ, θ 0 is a ukow p 1vector,adα 0 (θ) 1 for all θ. The rage of structures {α j (θ)} covered by (1.1) is very broad, but of iterest to us are oes which allow x t to have log memory. Usually, these are fractioal, it is assumed that the fuctio (1.2) α(s; θ) = α j (θ)s j, with complex-valued argumet s o the uit circle, is of form (1.3) α(s; θ) =(1 s) δ(θ) α (s; θ), δ(θ) is a scalar fuctio of θ such that (1.4) 0 <δ(θ 0 ) < Lodo School of Ecoomics, p.m.robiso@lse.ac.uk Research supported by ESRC Grat R AMS 2000 subject classificatios: 62M10. Keywords ad phrases: log memory, coditioal-sum-of-squares estimatio, cetral limit theorem, almost sure covergece. 130

2 Estimatio of models for statioary time series with log memory 131 ad (1.5) 0 < α (s; θ 0 ) <, s =1. It follows that x t has spectral desity of form σ0 2 (1.6) f(λ) = α (e iλ ; θ 0 ) 2 = 1 e iλ 2δ(θ0) σ2 0 α (e iλ ; θ 0 ) 2. The leadig choice of α is a ratioal fuctio of s, iwhichcasex t is said to be a fractioal autoregressive itegrated movig average (FARIMA) model; δ(θ 0 )is caled the memory parameter. Leadig methods of estimatio of θ 0, give observatios x 1,...,x, are Gaussia maximum likelihood (ML), ad approximatios thereto. They are approximatios i the sese that uder similar coditios they have the same asymptotic ormal distributio as ML, ad are thus asymptotically efficiet uder Gaussiaity. At the same time, uder may departures from Gaussiaity, though the efficiecy is lost the limit ormal distributio of all these estimates is uaffected. Assumig Gaussiaity, asymptotic ormality of oe form of approximatio, a Whittle estimate ivolvig itegratio over frequecy, was first established by Fox ad Taqqu [4], ad the by Dahlhaus [3] i case of ML estimatio. Giraitis ad Surgailis [5] established asymptotic ormality for the estimate cosidered by Fox ad Taqqu [4] whe ε t eed ot be Gaussia but is idepedet ad idetically distributed with fiite fourth momet. Due to the pole i the spectral desity at λ = 0 (see (1.6)), the asymptotic ormality proofs are cosiderably more challegig tha those of Haa [6] for short memory time series models, icisive though these were for such models. A alterative estimate that has bee cosidered i the literature is coditioalsum-of-squares (CSS) estimatio, which was previously employed by Box ad Jekis [1] for short memory time series models. Defie (1.7) (1.8) ad estimate θ 0 by t 1 e t (θ) = α j (θ)x t j, s (θ) = 1 e 2 t (θ), (1.9) ˆθ =argmi s (θ), θ Θ Θ R p is a compact set. Oe ca motivate ˆθ bythehopethats (θ 0 ) is a good approximatio to 1 ε2 t, which is itself proportioal to the expoet i the desity fuctio of idepedet idetically distributed zero-mea ormal variates. Thus oe hopes that (after ceterig at θ 0 ad 1 2 ormig) ˆθ has the same limit distributioal properties as the Gaussia ML ad Whittle estimates metioed previously. Give a iitial cosistecy proof of ˆθ, a stadard approach to provig asymptotic ormality etails applyig the mea value theorem to r (ˆθ )aboutθ 0, (1.10) r (θ) = s (θ) = 2 h t (θ)e t (θ), θ

3 132 P. M. Robiso for (1.11) h t (θ) = e t(θ). θ The mai part of the proof the ivolves establishig that 1 2 r (θ 0 ) coverges i distributio to a zero-mea ormal vector. If the ε t are assumed to be coditioally homoscedastic martigale differeces, ad coditios esurig that h t (θ) has fiite variace are imposed, such covergece is easily see to hold (see e.g. [2]) for (1.12) r (θ 0 )= 2 h t ε t, h t = h t (θ 0 ). However this is oly useful if also (1.13) r (θ 0 ) r (θ 0 )=o p ( 1 2 i other words, if the effect of replacig e t = e t (θ 0 )byε t is sufficietly small. Ulike the h t ε t,theh t e t ad h t (e t ε t ) are ot zero-mea, orthogoal radom variables. We ca employ the Schwarz iequality: (1.14) E r (θ 0 ) r (θ 0 ) 2 1 ), [ E(e t ε t ) 2 E h t (θ t (θ 0 ) 2] 1 2. The if, say, it were true that E(e t ε t ) 2 = O p (t 1 η )forsomeη>0, the right had side of (1.14) would be O p ( 1 2 η 2 ), ad (1.13) established. For short memory models E(e t ε t ) 2 typically decays fast eough, ideed eve expoetially. But uder quite geeral coditios permittig log memory (see [8]), (1.15) E(e t ε t ) 2 Kt 1 oly, K is a arbitrarily large geeric costat, which is isufficiet to establish (1.13) usig (1.14). A more delicate proof of (1.13) is required, ad this was give by Robiso [8]. As discussed there, this delicacy relates to that see i the proofs of Fox ad Taqqu [4] ad others for alterative estimates of θ 0. Ideed, give that these estimates ad CSS should have the same limit distributioal properties, it would be extraordiary if the proof for CSS were very much easier tha for the other estimates. A cetral limit theorem for ˆθ is give i Sectio 3. Prior to that, i the followig sectio, we provide the almost sure covergece of ˆθ (uder somewhat more geeral coditios). Haa [6] proved this for various estimates, assumig strict statioarity ad ergodicity, which is cosistet with log memory. However, he did ot cover CSS estimatio. 2. Almost sure covergece I the preset sectio we do ot require that x t ecessarily has spectral desity of form (1.6), with (1.5) holdig, but simply that it is a zero-mea strictly statioary, ergodic process with AR represetatio (1.1), with the setece after (1.1) holdig, ad also θ 0 Θ, for all θ Θ\{θ 0 } (2.1) α(s; θ) α(s; θ 0 )

4 Estimatio of models for statioary time series with log memory 133 o a subset of s = 1 of positive measure, α(s; θ) is cotiuous i θ for all s : s = 1, ad (2.2) sup α j (θ) <. θ Θ Coditio (2.1) is a stadard idetifiability coditio, ad (2.2) is reasoable i that log memory models (e.g. (1.6), such as FARIMAs) typically have AR represetatios with summable coefficiets. Note that this setup allows the spectral desity to have poles at o-zero frequecies (as i certai cyclic ad seasoal models), as (1.6) does ot, i view of (1.5). Theorem 1. Uder the above coditios (2.3) lim ˆθ = θ 0, a.s. Proof. Theorem 1 of Haa [6] ad Theorem 1 of Fox ad Taqqu [4] cover the estimate (2.4) θ =argmi Θ s (θ), s (θ) is the objective fuctio for the itegral form of Whittle estimate, i.e. σ 2 N (θ) of Haa [6] or σ2 N (θ) of Fox ad Taqqu [4]. We ca write 1 (2.5) s (θ) =c (0)ξ 0 (θ)+2 c (j)ξ j (θ), (2.6) (2.7) c (j) = 1 j x t x t+j, 0 j 1, ξ j (θ) = α k (θ)α k+j (θ). k=0 From Theorem 1 of Haa [6], ad its proof, it is clear that it suffices to show that (2.8) lim s (θ) s (θ) =0, a.s. Now sup Θ (2.9) s (θ) s (θ) = 1 x 2 t k= t j x t x t+j 4 = a i (θ), i=1 α 2 k(θ) k= t j+1 α k (θ)α k+j (θ)

5 134 P. M. Robiso (2.10) (2.11) (2.12) (2.13) 1 1 a 1 (θ) =γ(0) jα 2 j(θ)+ αj(θ) 2, a 2 (θ) = 1 j= { x 2 t γ(0) } αk(θ), 2 k= t+1 a 3 (θ) = 2 1 j γ(j) α k (θ)α k+j (θ), k= t j+1 1 j 1 a 4 (θ) =2 (x t x t+j γ(j)) α k (θ)α k+j (θ), k= t j+1 (2.14) γ(j) =cov(x 0,x j ). It remais to prove (2.15) lim sup θ Θ a i (θ) = 0 a.s., i =1, 2, 3, 4. As the proofs for i =1, 2 are similar to but simpler tha those for i =3, 4, we give oly the latter. We have 2 (2.16) sup a 3 (θ) 2 1 γ(j) sup α j (θ). Θ θ Θ The quatity i braces is fiite ad sice, by the Riema-Lebesgue theorem, existece of a spectral desity implies lim j γ(j) = 0, it follows from the Toeplitz lemma that (2.16) 0 as. Next, by summatio-by-parts (2.17) 1 a 4 (θ) = 2 1 j 1 t {c t(j) γ(j)} α t j+1 (θ)α t+1 (θ) j 1 +2 {x t x t+j γ(j)} α k (θ)α k+j (θ). The modulus of the first term o the right has supremum, over Θ, bouded by (2.18) K sup 1 j k=1 c t (j) γ(j) sup α t+1 (θ) Θ usig (2.2). Usig (2.2) agai, ad Theorem 1 of Haa [7] ad the Toeplitz lemma, it follows that (2.18) is o(1) a.s. The secod term i (2.17) ca be similarly hadled. 3. Asymptotic ormality We assume ow i additio that x t has spectral desity (1.6), with (1.4), (1.5) satisfied, that θ 0 is a iterior poit of Θ, that the ε t i (1.1) are idepedet with

6 Estimatio of models for statioary time series with log memory 135 zero mea, variace σ 2 0 ad uiformly bouded fourth momet, that α(s; θ) is twice cotiuously differetiable i θ, ad that the matrix (3.1) Ω = 1 π [ 2π π is positive defiite. log 1 e iλ 2 2 θ log α ( e iλ ; θ 0 ) ][ log 1 e iλ 2 2 θ log α ( e iλ ; θ 0 ) ] dλ Theorem 2. Uder the above coditios, as 1 2 (ˆθ θ 0 ) coverges i distributio to a p-variate ormal vector with zero mea ad covariace matrix Ω 1. Proof. As discussed i Sectio 1, we have (3.2) 0 = r (ˆθ )=r (θ 0 )+ T (ˆθ θ 0 ), T is the matrix formed by evaluatig, for i =1,...,p,thei-th row of the matrix T (θ) =( 2 / θ θ )s (θ) atθ = θ i, θ i θ 0 ˆθ θ 0, deotig Euclidea orm. Defie (3.3) ζ j = θ α j(λ; θ), so that t 1 (3.4) h t = ζ j x t j, ad defie also (3.5) (3.6) ρ t = ζ j x t j, r = 1 ρ t ε t. Write r (θ 0 ) r = r 1 + r 2 + r 3, (3.7) (3.8) (3.9) r 1 =2 1 (h t ρ t )ε t, r 2 =2 1 ρ t (e t ε t ), r 3 =2 1 (h t ρ t )(e t ε t ). We show that r i = o p ( 1 2 ), i =1, 2, 3. To deal with r 1, we may write (3.10) h t ρ t = ζ j x t j = χ jt ε j, j=t

7 136 P. M. Robiso (3.11) χ jt = j ζ k+j β j k. k=0 Sice (3.12) E h t ρ t 2 = σ0 2 χ jt 2 (log t)2 K t as oted o p of [8], ad ε t is idepedet of h t ρ t, it follows that (3.13) E r 1 2 K 2 t 1 K log 2. Next, we ca write (3.14) e t ε t = λ jt ε j, (3.15) λ jt = Thus, from Lemma 16 of Robiso [8], j α k+j β t k. k=0 (3.16) E r 2 2 (log )3 K 2. Fially, (3.17) E r 3 1 K K ( E h t ρ t 2 E (e t ε t ) 2) 1 2 log t t (log )2, usig (3.12) ad also Lemma 14 of Robiso [8]. This completes the proof that r i = o p ( 1 2 ), i =1, 2, 3. The remaider of the proof is easier, ad more stadard, ad is omitted. Ackowledgmet I thak a referee for a careful readig of the paper.

8 Estimatio of models for statioary time series with log memory 137 Refereces [1] Box, G. E. P. ad Jekis, G. M. (1970). Time Series Aalysis, Forecastig ad Cotrol. Holde-Day, Sa Fracisco. [2] Brow, B. M. (1971). Martigale cetral limit theorems. A. Math. Statist [3] Dahlhaus, R. (1989). Efficiet parameter estimatio for self-similar processes. A. Statist (Correctio A. Statist., forthcomig.) [4] Fox, R. ad Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strogly depedet statioary Gaussia time series. A. Statist [5] Giraitis, L. ad Surgailis, D. (1990). A cetral limit theorem for quadratic forms i strogly depedet liear variables ad its applicatio to asymptotic ormality of Whittle s estimate. Probab. Theory Related Fields [6] Haa, E. J. (1973). The asymptotic theory of liear time series models. J. Appl. Probability (Correctio: J. Appl. Probability ) [7] Haa, E. J. (1974). The uiform covergece of autocovariaces. A. Statist [8] Robiso, P. M. (2005). Efficiecy improvemets i iferece o statioary ad ostatioary fractioal time series. A. Statist