UNIFORM INTERVAL ESTIMATION FOR AN AR(1) PROCESS WITH AR ERRORS

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1 Statistica Siica 6 (06), 9-36 doi: UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS Joatha Hill, Deyua Li ad Liag Peg Uiversity of North Carolia, Fuda Uiversity ad Georgia State Uiversity Abstract: A empirical likelihood method was proposed i Hill ad Peg (04) to costruct a uified iterval estimatio for the coefficiet i a AR() model, regardless of whether the sequece was statioary or ear itegrated. The error term, however, was assumed idepedet, ad this method fails whe the errors are depedet. Testig for a uit root i a AR() model has bee studied i the literature for depedet errors, but existig methods caot be used to test for a ear uit root. I this paper, assumig the errors are govered by a AR(p) process, we exploit the efficiet empirical likelihood method to give a uified iterval for the coefficiet by takig the structure of errors ito accout. Furthermore, a jackkife empirical likelihood method is proposed to reduce the computatio of the empirical likelihood method whe the order i the AR errors is ot small. A simulatio study is coducted to examie the fiite sample behavior of the proposed methods. Key words ad phrases: AR model, empirical likelihood, jackkife empirical likelihood method, weighted score.. Itroductio Cosider the autoregressive time series model t = β 0 t ϵ t for t =,...,, (.) with the sample size, ad the ϵ ts idetically distributed with zero mea ad fiite variace. A commoly estimator for the true parameter value β 0 is the least squares estimator ˆβ LS = t t / t which miimizes t β t. As is well kow, the asymptotic distributio of ˆβ LS is ormal for the statioary case, β 0 <, ad o-ormal for the early itegrated case, β 0 = δ/ for some δ R, cf., Phillips (987), makig iterval estimatio quite challegig sice iferece by least squares is ot uified. I a more geeral eviromet, whe ϵ t may be a martigale differece sequece, Cha, Li, ad Peg (0) applied the empirical likelihood method to the weighted score equatio t β t t = 0, (.) t

2 0 JONATHAN HILL, DEYUAN LI AND LIANG PENG so that a uified iterval for β 0 ca be obtaied, regardless of whether t is statioary or early itegrated. Other weighted fuctios ca be employed i order to achieve weighted score equatios similar to those i Lig (007) ad Pa, Wag, ad Yao (007). The empirical likelihood method based o estimatig equatios was studied by Qi ad Lawless (994) amogst may others, ad a overview of the method ca be foud i Owe (00). Refereces for more computatioally-ivolved methods for costructig uified itervals for β 0 ca be foud i Cha, Li, ad Peg (0). I practice, assumig idepedet, or eve martigale differece, errors is restrictive sice this requires (.) to be the true data geeratig process. I may cases, however, we simply wat to kow if t is differece statioary or ot, i which case aturally ϵ t may be depedet (Phillips (987), Phillips ad Perro (988)). Some recet refereces o processes with depedet errors iclude Wu (005), Wu ad Mi (005), ad Che, Mi, ad Che (03). By assumig that ϵ t is a α-mixig sequece, Phillips ad Perro (988) proposed statistics for testig a uit root uder model (.) with a itercept. However, these statistics caot be employed to costruct a cofidece iterval for β 0 i the case of a ear uit root. Therefore, it remais ukow how to costruct a uified iterval for β 0 uder the assumptio that (.) holds with possibly depedet ϵ t. The procedure i Cha, Li, ad Peg (0) does ot work sice it does ot accout for geeral weak depedece, but the blockwise empirical likelihood method has the ability to deal with mixig data (see Kitamura (997)). Oe ca attempt to apply the blockwise empirical likelihood method to some weighted score equatios for obtaiig a uified iterval. But there are serious issues i doig so. We ivestigate ways to costruct a uified iterval for β 0 whe ϵ t i (.) is govered by a AR(p) process. The least squares estimator ˆβ LS is clearly iefficiet sice it does ot take the structure of errors ito accout, ad the empirical likelihood method based o (.) does ot lead to a chi-squared limit due to the depedet errors. We propose a uified empirical likelihood method based o some differet weighted score equatios that take the structure of ϵ t ito accout. We the propose a jackkife empirical likelihood method to reduce the computatio of the proposed empirical likelihood method whe the order i the errors is ot small. Our uified itervals ca be immediately applied to a uit root test with a geeral alterative. To keep argumets short, we assume the AR(p) error has a iid term e t. The theory developed here ca be exteded to the case where e t is a martigale differece, coverig coditioal volatility of ukow form. A simulatio study i Sectio 3 cofirms this argumet. We orgaize the remaider of the paper as follows. Sectio presets the methodologies for a AR() model without or with a time tred, whe the errors follow a AR process. Simulatio results are preseted i Sectio 3. Proofs are i Sectio 4.

3 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS. Methodologies.. Profile empirical likelihood method Cosider model (.) with ϵ t followig from a AR(p) process: e t = ϵ t γ 0,j ϵ t j for t =,...,, (.) j= where e t is iid with zero mea ad fiite variace σ > 0, ad γ 0 = (γ 0,,..., γ 0,p ) T is such that ϵ t is a statioary process. Oe ca efficietly estimate β 0 ad γ 0 simultaeously by miimizig t β t, γ j ( t j β t j ) hece the proposed estimators of (β 0, γ 0 ) T solve the score equatios t β t t β t j= γ k ( t k β t k ) ( t j β t j ) = 0 for j =,..., p, (.) γ j ( t j β t j ) t γ j t j = 0. (.3) j= Ituitively this ew estimator for β 0 should be more efficiet tha the least squares estimator ˆβ LS = t t / t sice it takes the error structure ito accout. I order to costruct a cofidece iterval for β 0 without estimatig the asymptotic variace, oe ca apply the profile empirical likelihood method to (.) ad (.3). However, whe t is early itegrated, Wilks theorem fails. As i Cha, Li, ad Peg (0), we propose to apply the profile empirical likelihood method to (.), ad to a weighted versio of (.3), t t j t β t γ j ( t j β t j ) γ j =0. j= t j= t j (.4) I particular, write Z t (β, γ) (Z t, (β, γ),..., Z t,p (β, γ)) T R p, where Z t,j (β, γ) = t β t j= γ k ( t k β t k ) ( t j β t j ) for j =,..., p,

4 JONATHAN HILL, DEYUAN LI AND LIANG PENG Z t,p (β, γ) = t β t t t γ k ( t k β t k ) γ k t k, t k ad defie the empirical likelihood fuctio for (β 0, γ 0 ) as L(β, γ) = sup (p t ) : p,..., p 0, p t =, p t Z t (β, γ) = 0. Sice we are oly iterested i β 0, we cosider the profile empirical likelihood fuctio for β 0 : L P (β) = max L(β, γ). γ Rp Theorem. Suppose (.) holds with (.), ad E e t d < for some d > 0. The log L P (β 0 ) d χ () as wheever β 0 <, idepedet of, or β 0 = δ/ for some δ R. Cosider the same model with a liear time tred t = α 0, α 0, t β 0 t ϵ t for t =,...,, (.5) where ϵ t follows the AR(p) process i (.). I this case, take Z t (β, α, α, γ) [ Z t, (β, α, α, γ),..., Z t,p3 (β, α, α, γ)] T R p3, where Z t,j (β, α, α, γ) = t α α t β t γ k ( t k α α (t k) β t k ) ( t j α α (t j) β t j ) for j =,..., p, ad Z t,p (β, α, α, γ) = t α α t β t γ k ( t k α α (t k) β t k ), Z t,p (β, α, α, γ) = t α α t β t γ k ( t k α α (t k) β t k ) t γ k (t k),

5 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 3 Z t,p3 (β, α, α, γ) = t α α t β t γ k ( t k α α (t k) β t k ) t (t )q t j γ j ( j= t j )q W t for some q >0. The W ts are simulated iid radom variables, distributed as N(0, σ ), ad σ > 0 is chose to be smaller tha E(e t ). Here E(e t ) ca be estimated by first estimatig the errors ϵ ts based o the least squares estimate for α, α, β, ad the usig these estimated errors to obtai the least squares estimate for γ, ad fially estimatig e ts by these least squares estimates. I order to avoid the effect of a radom seed i geeratig W ts, we use W t = /, 000,000 W t,i i our simulatio study, where the W t,i s are iid N(0, σ ) draws for t =,..., ad i =,...,,000. p Whe σ = 0, q = /, ad t, the joit limit of / Z t,p3 (β, α, α, γ) ad / Z t,p (β, α, α, γ) is o loger ormal, which makes the applicatio of the empirical likelihood method fail. This is why we eed to add the pseudo sample W t to achieve uiform iferece. Based o argumets i Li, Cha, ad Peg (04, Sec. ), ad Hill ad Peg (04, p.88), i the ostatioary case a choice of q > / makes Z t,p3 (β 0, α 0,, α 0,, γ 0 ) asymptotically equivalet to W t, while small q allows Z t,p3 (β, α 0,, α 0,, γ 0 ) to better detect departures from β 0. We therefore eforce q (/, ] to balace power ad size, ad i practice simply use q = The empirical likelihood fuctio for (β, α, α, γ) based o Z t (β, α, α, is defied as γ) L(β, α, α, γ) = sup (p t ) : p,..., p 0, p t =, p t Zt (β, α, α, γ) = 0 ad, as before, we oly cosider the profile empirical likelihood fuctio L P (β) = max L(β, α, α, γ). (α,α,γ T ) T R p Theorem. Suppose (.5) holds with E e t d < for some costat d > 0. The log L P (β 0 ) d χ () as wheever β 0 <, idepedet of, or β 0 = δ/ for some δ R. Remark. Whe we cosider (.) with a costat tred, α 0, = 0 i (.5) is kow, Theorem still holds if the term Z t,p is removed ad α is replaced by zero.

6 4 JONATHAN HILL, DEYUAN LI AND LIANG PENG.. Jackkife empirical likelihood method The profile empirical likelihood methods become computatioally itesive whe p is large. I order to reduce computatio time, oe ca estimate γ 0 first by solvig (.), which results i a explicit fuctio of β, ad the apply the empirical likelihood method to (.4) with γ replaced by this estimator. However, this does ot lead to a chi-squared limit due to the plug-i estimator, but rather a weighted sum of idepedet chi-squared variables, see Che ad Va Keilegom (009). Recetly a jackkife empirical likelihood method was proposed by Jig, Yua, ad Zhou (009) to deal with o-liear fuctioals, ad Li, Peg, ad Qi (0) employed this idea to reduce the computatio of the empirical likelihood method based o estimatig equatios. Here, we employ the jackkife empirical likelihood method. Cosider model (.) with the error ϵ t followig the autoregressive process (.). Let ˆγ(β) = (ˆγ (β),..., ˆγ p (β)) T be, for arbitrary β, the solutio to (.), ad for each i =,..., let ˆγ (i) (β) = (ˆγ (i) (β),..., ˆγ(i) p (β)) T be the solutio to t β t γ k ( t k β t k ) ( t j β t j ) = 0,t i Defie the pseudo sample as Z,j(β) = t β t t t t β t,t j t t for j =,..., p. (.6) ˆγ k (β)( t k β t k ) ˆγ k (β) ˆγ (j) t k t k ˆγ (j) k (β)( t k β t k ) k (β) t k t k for j =,...,. Based o this pseudo sample, the jackkife empirical likelihood fuctio for β is L (β) = sup (p i ) : p 0,..., p 0, p i =, p i Z,i(β) = 0. Theorem 3. Suppose (.) holds with (.) ad E e t 4d < for some d > 0. The log L (β 0 ) d χ () as wheever β 0 <, idepedet of, or β 0 = δ/ for some δ R.

7 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 5 Remark. The jackkife empirical likelihood methods i Theorem 3 require a higher momet for e t tha the profile empirical likelihood methods i Theorems ad. The reaso is that we eed to derive a accurate approximate distace betwee ˆγ(β 0 ) ad ˆγ (i) (β 0 ), ad this ultimately requires a cetral limit theory for cross products of the e ts. Now cosider this model with a liear time tred, model (.5). Defie the parameter subset θ = [α, α, β] T. Let γ(θ) = ( γ (θ),..., γ p (θ)) T deote, for arbitrary θ, the solutio to Z t,j (θ, γ) = 0 for j =,..., p, (.7) ad let γ (i) (θ) = ( γ (i) (θ),..., γ(i) p (θ)) T for i =,..., deote the solutio to,t i Next, defie the pseudo sample as Z,i,k (θ) = Z t,pk (θ, γ) Z t,j (θ, γ) = 0 for j =,..., p. (.8),t i Z t,pk (θ, γ (i) ), where i =,..., ad k =,, 3. Take Z,i (θ) = ( Z,i, (θ), Z,i, (θ), Z,i,3 (θ)), ad defie the jackkife empirical likelihood fuctio for θ = (α, α, β) T as L (β, α, α ) = L (θ) = sup (p i ) : p 0,..., p 0, p i =, p i Z,i (θ) = 0. Sice we are oly iterested i β, we cosider the profile jackkife empirical likelihood fuctio L P (β) = max L (β, α, α ). (α,α ) T R Theorem 4. Suppose (.5) holds with E e t 4d < for some costat d > 0. The log L P (β 0 ) d χ () as wheever β 0 <, idepedet of, or β 0 = δ/ for some δ R. Remark 3. I Theorem 4 we do ot estimate α, α, γ simultaeously i the first step sice this does t lead to explicit formulas. If oe wats to estimate them simultaeously, the the approximate jackkife empirical likelihood method i Peg (0) ca be employed, where explicit formulas for estimatig uisace parameters are ot required.

8 6 JONATHAN HILL, DEYUAN LI AND LIANG PENG Remark 4. Whe the process is explosive, β 0 > ad idepedet of, the proposed methods are still valid as i Cha, Li, ad Peg (0). The ear uit root case icludes the slightly explosive case whe δ < 0. Remark 5. Whe ϵ t = j=0 c je t j, where e t is a sequece of idepedet ad idetically distributed radom variables, similar jackkife empirical likelihood methods may be developed by otig C (B) t = β 0 C (B) t e t with C(B) = j=0 c jb j, B deotig the usual backward operator, ad usig the trucatio techiques i iao et al. (003). Detailed ivestigatio is a future project. 3. Simulatio Study I this sectio we examie the fiite sample behavior of the proposed methods for models (.), (.5), ad (.5) with kow α 0, = 0. We drew 0,000 radom samples with size = 50 ad 00 from either model (.) or (.5), with β 0 0.9, 0.99,, the error AR order is p = or 5 with γ 0 = ( ( p ) 0.,..., ( p p) 0. p ), ad e t is a sequece of idepedet stadard ormals. Uder (.5) we use α 0, = 0.5 ad α 0, 0, 0.. The added pseudo sample W t is computed usig W t = /, 000,000 W t,i, where W t,i s are iid N(0,.5). We employ the R package emplik to compute the empirical likelihood fuctio ad the use the R package lm to calculate the profile empirical likelihood fuctio. Iitial values are computed by least squares estimates. For example, i order to compute coverage probabilities for (.5), we miimize t α α t β t with respect to α ad α to obtai fuctios ˆα i (β), ad the miimize i ˆα (β) ˆα (β) t β t γ k ( t k ˆα (β) ˆα (β) (t k) β t k ) with respect to γ = (γ,..., γ p ) T to achieve ˆγ(β). Coverage probabilities are computed usig simply ˆα i (β 0 ) ad ˆγ(β 0 ). We cosider (.) with e t = e t ξ t, where ξ t is a sequece of idepedet stadard ormals. Coverage probabilities are reported i Tables 4. We observe from these tables that (i) whe the model has o tred, the jackkife empirical likelihood method performs better tha the empirical likelihood method for = 50, but they are comparable for = 00; (ii) whe the model has a liear tred, the jackkife empirical likelihood method is worse tha the empirical likelihood method

9 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 7 Table. Model (): o tred. Coverage probabilities based o Theorem (ELM for level 0.9 ad ELM for level 0.95) ad Theorem 3 (JELM for level 0.9 ad JELM for level 0.95) are reported for model (.). (β 0, ) ELM JELM ELM JELM ELM JELM ELM JELM p = p = p = p = p = 5 p = 5 p = 5 p = 5 (0.9, 50) (0.99, 50) (, 50) (0.9, 00) (0.99, 00) (, 00) Table. Model (7): a liear time tred. Coverage probabilities based o Theorem (ELM for level 0.9 ad ELM for level 0.95) ad Theorem 4 (JELM for level 0.9 ad JELM for level 0.95) are reported for model (.5). (β 0, ) ELM JELM ELM JELM ELM JELM ELM JELM p = p = p = p = p = 5 p = 5 p = 5 p = 5 (0.9, 50) (0.99, 50) (, 50) (0.9, 00) (0.99, 00) (, 00) Table 3. Model (7): costat tred. Coverage probabilities based o Theorem (ELM for level 0.9 ad ELM for level 0.95) ad Theorem 4 (JELM for level 0.9 ad JELM for level 0.95) are reported for model (.5) with α 0, = 0 kow. (β 0, ) ELM JELM ELM JELM ELM JELM ELM JELM p = p = p = p = p = 5 p = 5 p = 5 p = 5 (0.9, 50) (0.99, 50) (, 50) (0.9, 00) (0.99, 00) (, 00) for = 50 ad p = 5, but they are comparable for = 00; (iii) whe the model has a costat tred, the methods are comparable; (iv) the methods work whe e t is a martigale differece. The proposed methods perform quite well. The jackkife empirical likelihood method is preferred whe the model has o tred or a costat tred, or the

10 8 JONATHAN HILL, DEYUAN LI AND LIANG PENG Table 4. Model (): o tred, but heteroscedastic e t. Coverage probabilities based o Theorem (ELM for level 0.9 ad ELM for level 0.95) ad Theorem 3 (JELM for level 0.9 ad JELM for level 0.95) are reported for model (.) whe e t = e t ξ t, ξ ts are iid with N(0, ). (β 0, ) ELM JELM ELM JELM (0.9, 50) (0.99, 50) (, 50) (0.9, 00) (0.99, 00) (, 00) sample size is ot too small, e.g., = 00. Whe the model has a liear tred, the jackkife empirical likelihood is worse tha the empirical likelihood method for a large order p ad a small, although its computatio is less itesive. 4. Proofs The proof of Theorem exploits the followig results. Lemma. Let (.) hold with (.) ad E e t d < for some d > 0. Let either β 0 < idepedet of, or β 0 = δ/ for some δ R. The / Z d t(β 0, γ 0 ) N(0, Σ) as, where Σ = (σ i,j ) i,j p with σ i,j = E(e t )E(ϵ t i ϵ t j ) for i, j =,..., p, ( E(e t )E ϵ t i t ) p σ i,p = j= γ 0,j t j whe β 0 <, t t j 0 whe β 0 = δ for i =,..., p, ad E(e t )E t γ 0,j t j whe β0 <, t j= t j σ p,p = ( ) E(e t ) γ 0,j whe β 0 = δ. j= Proof. Note that Z t,j (β 0, γ 0 ) = e t ϵ t j for j =,..., p, Z t,p (β 0, γ 0 ) = e t t γ 0,j t j= t j t j. (4.)

11 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 9 I the local to uity case β 0 = δ/, t ( t t j= γ 0,j ) t j p ( t j p as t, hece ) γ 0,j as t. (4.) The result follows from (4.), (4.), ad the cetral limit theorem for martigale differeces. See McLeish (974) ad Hall ad Heyde (980), ad see argumets i the appedices of Cha, Li, ad Peg (0) ad Hill ad Peg (04). Lemma. Uder the coditios of Lemma, ZT t (β 0, γ 0 )Z t (β 0, γ 0 ) p Σ as. Proof. The claim follows from (4.) (4.) ad the weak law of large umbers for martigale differeces (see Hall ad Heyde (980)). Lemma 3. Uder the coditios of Lemma, as, with probability oe L(β 0, γ) attais its maximum or local maximum value at some poit γ i the iterior of the ball γ γ 0 d/3, ad γ ad λ satisfy Q ( γ, λ) = 0 ad Q ( γ, λ) = 0, where ad Q (γ, λ) Q (γ, λ) Z i (β 0, γ) λ T Z i (β 0, γ) λ T Z i (β 0, γ) j= ( ) Zi (β 0, γ) T λ. γ Proof. The proof is similar to the proof of Lemma i Qi ad Lawless (994) by usig Lemmas ad. Proof of Theorem. Apply Lemmas 3 ad argumets i Qi ad Lawless (994). Proof of Theorem. The proof is similar to the proof of Theorem. Before provig Theorems 3 ad 4, we eed some otatios ad lemmas. Put, for i =,...,, e t ϵ t e t ϵ t,t i A =, A (i) =, e t ϵ t p e t ϵ t p,t i

12 30 JONATHAN HILL, DEYUAN LI AND LIANG PENG B = B (i) = ϵ t ϵ t p ϵ t, ϵ t p ϵ t p ϵ t,t i,t i ϵ t ϵ t p ϵ t ϵ t p ϵ t, ϵ t p,t i,t i Σ = E(B), D = ( Σ B) Σ A, D (i) = ( Σ B (i) ) Σ A (i). Lemma 4. Uder the coditios of Theorem 3, ˆγ(β 0 ) γ 0 Σ A = O p ( ), (4.3) max i ˆγ(i) (β 0 ) γ 0 Σ A (i) = O p ( ), (4.4) max i B(ˆγ(i) (β 0 ) ˆγ(β 0 )) B Σ (A (i) A) D (i) D = o p ( 3/ ). (4.5) Proof. Equatio (4.3) follows from 0 = A B(ˆγ(β 0 ) γ 0 ) = D B(ˆγ(β 0 ) γ 0 Σ A), (4.6) A = O p ( / ), ad Σ B = O p ( / ). Write where B (i) = Bi = B ϵ i ϵ i p ϵ i B i, A(i) = A ϵ i p ϵ i, A i = ϵ i p A i, (4.7) e i ϵ i e i ϵ i p.

13 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 3 Sice max i, j p ϵ i ϵ i j = o p ( / ), it follows from (4.7) that B (i) = O p () ad B (i) B = o p ( / ) uiformly i i =,...,. (4.8) Similarly, Σ B (i) = O p ( / ), A (i) = O p ( / ) ad A (i) A = o p ( / ) uiformly i i =,...,. (4.9) Therefore, (4.4) follows from (4.9) ad 0 = D (i) B (i) (ˆγ (i) (β 0 ) γ 0 Σ A (i) ). By writig 0 = D (i) (B (i) B)(ˆγ (i) (β 0 ) γ 0 Σ A (i) )B(ˆγ (i) (β 0 ) γ 0 Σ A (i) ), (4.5) follows from (4.4), (4.6), ad (4.8). Lemma 5. Uder the coditios of Theorem 3, we have ad Z,t(β 0 ) d = e t ẽ t o p () d N(0, E(e t ẽ t )) Z,t(β 0 ) p E(e t ẽ t ) as, where t ẽ t = t γ 0,k t k t k T ϵ t E(ϵ E p) E(ϵ p ϵ ) p ϵ t p E(ϵ p ϵ ) E(ϵ ) ( E ϵ p p Proof. By (4.7) (4.9) ad ϵ p ( p γ 0,k γ 0,k p k p k p k p k ). ) (A (i) A) = 0 ad (B (i) B) = 0, (4.0) we have D (i) D = (B B (i) ) Σ (A (i) A)

14 3 JONATHAN HILL, DEYUAN LI AND LIANG PENG = ( B B(i) ) Σ (A (i) = ( ) Bi Σ A i O p ( ) A) ( ) B Σ A = O p ( ), (4.) D (i) D = A (i) A (B (i) B) Σ A (i) B Σ (A (i) A) = o p ( / ) uiformly i i =,...,. (4.) Usig (4.5), (4.0), (4.), ad (4.), we ca show that, for ay p p matrix, o p ( 3/ ) = = (ˆγ(β 0 ) ˆγ (i) (β 0 )) Σ (A (i) A) B (D (i) D) (ˆγ(β 0 ) ˆγ (i) (β 0 )) o p ( / ), (A (i) A) T Σ Σ (A (i) A) = = (A (i) A)T Σ Σ (A (i) A) O p( ) ( ) A T i Σ Σ A i O p ( ) = O p ( ), (A (i) A) T Σ B (D (i) D) = = (A (i) A)T Σ B (D (i) D) O p ( ) A T i Σ B (D (i) D) O p ( ) = O p ( / )o p ( / ) O p ( ) = O p ( ), (D (i) D) T B B (D (i) D) = A (i) A T B B (D (i) D)

15 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 33 (B (i) B) Σ A (i) T B B (D (i) D) B Σ (A (i) A) T B B (D (i) D) = O p ( ) o p ( / )O p ( / )o p ( / ) O p ( ) = o p ( / ), (A (i) A) T Σ (ˆγ(β 0 ) ˆγ (i) (β 0 )) = (A (i) A) T Σ ˆγ(β 0 ) ˆγ (i) (β 0 ) Σ (A (i) A) B (D (i) D) O p ( ) = o p ( / )o p ( 3/ ) O p ( ) = O p ( ), (D (i) D) T B (ˆγ(β 0 ) ˆγ (i) (β 0 )) = O p ( ), o p ( 3/ )o p ( 3/ ) = ˆγ(β 0 ) ˆγ (i) (β 0 ) Σ (A (i) A) B (D (i) D) T ˆγ(β 0 ) ˆγ (i) (β 0 ) Σ (A (i) A) B (D (i) D) = (ˆγ(β 0 ) ˆγ (i) (β 0 )) T (ˆγ(β 0 ) ˆγ (i) (β 0 ))o p ( / ), which imply that (ˆγ(β 0 ) ˆγ (i) (β 0 )) = o p ( / ), (ˆγ(β 0 ) ˆγ (i) (β 0 )) T (ˆγ(β 0 ) ˆγ (i) (β 0 )) = o p ( / ). (4.3) For j =,...,, put W j = l= (ˆγ l (β 0 ) ˆγ (j) l (β 0 )) l= ( t ϵ t l t (ˆγ l (β 0 ) ˆγ (j) l (β 0 )) e t t l t l γ 0,k ) t k t k

16 34 JONATHAN HILL, DEYUAN LI AND LIANG PENG W j = ( l= ϵ j, ϵ j p (ˆγ k (β 0 ) ˆγ (j) k j j j j (β 0))(ˆγ l (β 0 ) ˆγ (j) l (β 0 )) γ 0,k j k j k γ 0,k j k j k ϵ t k t l, t l j e j, j The it follows from Lemma 4, (4.3) ad Taylor expasios that Z,i(β 0 ) = = W i (ˆγ (i) (β 0 ) γ 0 ) T W i ( i e i i (A (i) ) T Σ W i ( i e i i = ( A) T Σ = W i ( e i i i e i ẽ i o p (). γ 0,k γ 0,k γ 0,k ) T j p e j. j p ) i k o p () i k ) i k o p () i k ) i k o p () i k Now apply a martigale cetral limit theorem argumet as i Lemma to (/ ) e iẽ i to achieve (/ ) Z,i (β 0) d N(0, E(e ẽ )). Similarly, we ca show that Z,i(β 0 ) = e i ẽ i o p () p E(e ẽ ). This completes the proof. Proof of Theorem 3. The claim ca be prove by usig Lemma 5, ad argumets i Qi ad Lawless (994).

17 UNIFORM INTERVAL ESTIMATION FOR AN AR() PROCESS WITH AR ERRORS 35 Proof of Theorem 4. The argumet is similar to the proof of Theorem 3. Ackowledgemet We thak two reviewers for their helpful commets. Li s research was partly supported by the NSFC Grat Peg s research was partly supported by the Simos Foudatio. Refereces Cha, N. H., Li, D. ad Peg, L. (0). Toward a uified iterval estimatio of autoregressios. Ecoom. Theory 8, Che, S., Mi, W. ad Che, R. (03). Model idetificatio for time series with depedet iovatios. Statist. Siica 3, Che, S. ad Va Keilegom, I. (009). A review o empirical likelihood methods for regressio. Test 8, Hall, P. ad Heyde, C. (980). Martigale Limit Theory ad Its Applicatios. Academic Press. Hill, J. B. ad Peg, L. (04). Uified iterval estimatio for radom coefficiet autoregressive models. J. Time Ser. Aal. 35, Jig, B. Y., Yua, J. ad Zhou, W. (009). Jackkife empirical likelihood. J. Amer. Statist. Assoc. 04, 4-3. Kitamura, Y. (997). Empirical likelihood methods with weakly depedet processes. A. Statist. 5, Li, D., Cha, N. H. ad Peg, L. (04). Empirical likelihood test for causality for bivariate AR() processes. Ecoom. Theory 30, Li, M., Peg, L. ad Qi, Y. (0). Reduce computatio i profile empirical likelihood method. Caad. J. Statist. 39, Lig, S. (007). Self-weighted ad local quasi-maximum likelihood estimators for ARMA- GARCH/IGARCH models. J. Ecoom. 40, McLeish, D. (974). Depedet cetral limit theorems ad ivariace priciples. A. Probab., Owe, A. (00). Empirical Likelihood. Chapma ad Hall, New York. Pa, J., Wag, H. ad Yao, Q. (007). Weighted least absolute deviatios estimatio for ARMA models with ifiite variace. Ecoom. Theory 3, Peg, L. (0). Approximate jackkife empirical likelihood method for estimatig equatios. Caad. J. Statist. 40, 0-3. Phillips, P. C. B. (987). Time series regressio with a uit root. Ecoometrica 55, Phillips, P. C. B. ad Perro, P. (988). Testig for a uit root i time series regressio. Biometrika 75, Qi, J. ad Lawless, J. (994). Empirical likelihood ad geeral estimatig fuctios. A. Statist., Wu, W. B. (005). Uit root testig for fuctioals of liear processes. Ecoom. Theory, -4. Wu, W. B. ad Mi, W. L. (005). O liear processes with depedet iovatios. Stochastic Process. Appl. 5,

18 36 JONATHAN HILL, DEYUAN LI AND LIANG PENG iao, Z., Lito, O. B., Carroll, R. J. ad Mamme, E. (003). More efficiet local polyomial estimatio i oparametric regressio with autocorrelated errors. J. Amer. Statist. Assoc. 98, Departmet of Ecoomics, Uiversity of North Carolia, Chapel Hill, NC , USA. jbhill@ .uc.edu School of Maagemet, Fuda Uiversity, Shaghai 00433, P.R. Chia. deyuali@fuda.edu.c Departmet of Risk Maagemet ad Isurace, Georgia State Uiversity, Atlata, GA 30303, USA. lpeg@gsu.edu (Received July 04; accepted March 05)