TESTING FOR THE BUFFERED AUTOREGRESSIVE PROCESSES SUPPLEMENTARY MATERIAL) By Ke Zhu, Philip L.H. Yu ad Wai Keug Li Chiese Academy of Scieces ad Uiversity of Hog Kog APPENDIX: PROOFS I this appedix, we first give the proofs of Lemmas 2.-2.2. Deote C as a geeric costat which may vary from place to place i the rest of this paper. The proofs of Lemmas 2.-2.2 rely o the followig three basic lemmas: Lemma A.. Suppose that y t is strictly statioary, ergodic ad absolutely regular with mixig coefficiets βm) = Om A ) for some A > v/v ) ad r > v > ; ad there exists a A 0 > such that 2A 0 rv/r v) < A. The, for ay γ = r L, r U ) Γ, we have r v)/2a j 0 rv E Ir L < y t i r U ) <. j= i= Proof. First, deote ξ i = Ir L < y t i r U ). The, ξ i is strictly statioary, ergodic ad α-mixig with mixig coefficiets αm) = Om A ). Next, take ι [2A0 rv/r v) + ]/A + ), ), ad let p = j ι ad s = j/j ι, where x is the largest iteger ot greater tha x. Whe j j 0 is large eough, we ca always fid {ξ kp+ } s k=0, a subsequece of {ξ i} j i=. Furthermore, let Fm = σξ i, m i ). The, ξ kp+ F kp+2 kp+. Note that E [ξ kp+ ] < P a y t b) ρ 0, ). Hece, by Propositio 2.6 i Fa ad Yao 2003, p.72), we have that for j j 0, j E ξ i E i= { = [ s E ] ξ kp+ k=0 [ s k=0 ξ kp+ ] 6s )αp) + ρ s s k=0 C j/j ι j ι A + ρ j/jι. E [ξ kp+ ] } + s k=0 E [ξ kp+ ]
2 Therefore, sice r v)/2a 0 rv > 0, by usig the iequality x + y) k Cx k + y k ) for ay x, y, k > 0, it follows that A.) r v)/2a j 0 rv r v)/2a j 0 rv E ξ i j 0 ) + E ξ i j= i= j=j 0 i= [ j 0 ) + C j/j ι j ι A] r v)/2a 0 rv j=j 0 + C ρ j/jι r v)/2a 0rv. j=j 0 Sice ι > [2A 0 rv/r v) + ]/A + ), we have ιa + ι )r v)/2a 0 rv >, ad hece j= j ιa+ι )r v)/2a 0rv <, which implies that A.2) [ j/j ι ] r v)/2a0 rv j=j 0 j ι A [ ] r v)/2a0 rv j j= j ι j ι ) A <. O the other had, sice ρ j/jι r v)/2a 0 rv ) /j <, by Cauchy s root test, we have A.3) ρ j/jι r v)/2a 0rv < ρ j/jι r v)/2a 0 rv <. j=j 0 j= Now, the coclusio follows directly from A.)-A.3). This completes the proof. Lemma A.2. Suppose that the coditios i Lemma A. hold, ad y t has a bouded ad cotiuous desity fuctio. The, there exists a B 0 > such that for ay γ, γ 2 Γ, we have R t γ ) R t γ 2 ) 2rv/r v) C γ γ 2 r v)/2b0rv. Proof. Let γ = r L, r U ) ad γ 2 = r 2L, r 2U ). Sice R t γ) = Iy t d r L ) + R t γ)ir L < y t d r U ), we have R t γ ) R t γ 2 ) = t γ, γ 2 ) + Ir L < y t d r U ) [R t γ ) R t γ 2 )], where t γ, γ 2 ) = Ir 2L < y t d r L ) + R t γ 2 ) [Ir L < y t d r U ) Ir 2L < y t d r 2U )].
Thus, by iteratio we ca show that R t γ ) R t γ 2 ) TESTING FOR BAR PROCESSES 3 A.4) j = t γ, γ 2 ) + t j γ, γ 2 ) Ir L < y t i d r U ). j= i= Next, for brevity, we assume that r 2L r L r 2U r U, because the proofs for other cases are similar. Note that for ay j 0, R t j γ 2 ) ad Ir L < y t j d r U ) Ir 2L < y t j d r 2U ) = Ir 2U < y t j d r U ) Ir 2L < y t j d r L ). Let fx) be the desity fuctio of y t. Sice x fx) < ad t j γ, γ 2 ) 2, by Hölder s iequality ad Taylor s expasio, it follows that for ay s, A.5) E t j γ, γ 2 ) s 2 s E t j γ, γ 2 ) 2 s [2 x C γ γ 2. fx) r L r 2L + fx) r U r 2U x ] Let A 0 > be specified i Lemma A., ad choose B 0 such that /A 0 + /B 0 =. By Hölder s iequality ad A.5), we ca show that j 2rv/r v) E t jγ, γ 2 ) Ir L < y t i d r U ) i= { E[ t j γ, γ 2 )] } 2B /B 0rv/r v) 0 j E Ir L < y t i d r U ) i= /A 0 A.6) 2 [2B0rv/r v)] {E t j γ, γ 2 ) } /B 0 j E Ir L < y t i d r U ) i= /A 0 j C γ γ 2 /B 0 E Ir L < y t i d r U ) i= /A 0. By A.4)-A.6), Mikowski s iequality, Lemma A. ad the compactess of Γ, we
4 have R t γ ) R t γ 2 ) 2rv/r v) C γ γ 2 r v)/2rv + C γ γ 2 r v)/2b 0rv j E Ir L < y t i d r U ) j= This completes the proof. i= C γ γ 2 r v)/2b 0rv. r v)/2a 0 rv Lemma A.3. Suppose that the coditios i Lemma A.2 hold ad Ey 4 t <. The, X γx Σ γ 0 a.s. as. Proof. For brevity, we oly prove the uiform covergece for t= φ t γ), the last compoet of X γx, where φ t γ) = yt pr 2 t γ). First, for fix ε > 0, we partitio Γ by {B,, B Kε }, where B k = {r L, r U ); ω k < r L ω k+, ν k < r U ν k+ } Γ. Here, {ω k } ad {ν k } are chose such that A.7) ω k+ ω k ) r v)/2b 0rv < C ε ad ν k+ ν k ) r v)/2b 0rv < C ε, where B 0 > is specified as i Lemma A.2, ad C > 0 will be selected later. Next, we set f u t ε) = y 2 t pr t ω k+, ν k+ ) ad f l tε) = y 2 t pr t ω k, ν k ). By costructio, sice R t γ) is a odecreasig fuctio with respect to r L ad r U, for ay γ Γ, there is some k such that γ B k ad f l tε) φ t γ) f u t ε). Furthermore, sice rv/2rv r + v) <, we have yt p 2 < y 2 2rv/2rv r+v) t p <. 2
TESTING FOR BAR PROCESSES 5 Thus, by Hölder s iequality, Lemma A.2 ad A.7), we have E [ ft u ε) ftε) ] l yt p 2 R t ω k+, ν k+ ) R t ω k, ν k ) 2rv/2rv r+v) 2rv/r v) C [ ω k+ ω k ) r v)/2b0rv + ν k+ ν k ) ] r v)/2b 0rv 2CC ε. By settig C = 2C), we have E [ f u t ε) f l tε) ] ε. Thus, the coclusio holds accordig to Theorem 2 i Pollard 984, p.8). This completes the proof. Proof of Lemma 2.. First, sice K γγ is positive defiite by Assumptio 2., we kow that both Σ ad Σ γ are positive defiite. By usig the same argumet as for Lemma 2.iv) i Cha 990), it is ot hard to show that for every γ Γ, Σ γ Σ γ Σ Σ γ is positive defiite. Secod, by the ergodic theorem, it is easy to see that A.8) Third, by Lemma A.3 we have A.9) X γx X X Σ γ 0 ad as. Note that if H 0 holds, we have Σ a.s. as. X γx γ T γ = { X γ X γxx X) X } ε = X γ X)X X), I ) Z γε. Σ γ 0 a.s. The, i) ad ii) follow readily from A.8)-A.9). This completes the proof. Proof of Lemma 2.2. Deote G γ) Z γε = N x t γ)ε t. It is straightforward to show that the fiite dimesioal distributio of {G γ)} coverges to that of {σg γ }. By Pollard 990, Sec.0), we oly eed to verify the stochastic equicotiuity of {G γ)}. To establish it, we use Theorem, Applicatio
6 4 i Doukha, Massart, ad Rio 995, p.405); see also Adrews 993) ad Hase 996). First, the evelop fuctio is γ x t γ)ε t = x t ε t, where x t = γ x t γ). By Hölder s iequality ad Assumptio 2., we kow that the evelop fuctio is L 2v bouded. Next, for ay γ, γ 2 Γ, by Assumptios 2.-2.3, Lemma A.2 ad Hölder s iequality, we have x t γ )ε t x t γ 2 )ε t 2v = h t γ )ε t h t γ 2 )ε t 2v x t ε t 2r R t γ ) R t γ 2 ) 2rv/r v) C x t 4r ε t 4r γ γ 2 r v)/2b 0rv C γ γ 2 r v)/2b 0rv for some B 0 >, where the last iequality holds sice x t 4r ε t 4r <. Now, followig the argumet i Hase 996, p.426), we kow that G γ) is stochastically equicotiuous. This completes the proof. Next, we give Lemmas A.4-A.6, i which Lemma A.4 is crucial for provig Lemma A.5, ad Lemmas A.5 ad A.6 are eeded to prove Corollary 2. ad Theorem 3., respectively. Lemma A.4. Suppose that y t is strictly statioary ad ergodic. The, i) 0 = O p ); ii) furthermore, if E y t 2 < ad E ε t 2 <, for ay a = o), we have a 0 x t x A.0) tr t γ) = O p) ad A.) a 0 h t γ)ε t = O p). Proof. First, by the ergodic theory, we have that M M Ia y t d b) = P a y t d b) κ > 0 as M. Thus, η > 0, there exists a iteger Mη) > 0 such that P M Ia y t d b) < κ < η. M 2 a.s.
By the defiitio of 0, it follows that A.2) TESTING FOR BAR PROCESSES 7 M P 0 > M) = P Ia y t d b) = 0 = P M Ia y t d b) = 0 M P M Ia y t d b) < κ M 2 < η, i.e., i) holds. Furthermore, by takig M = M 2, from A.2) ad Markov s iequality, it follows that η > 0, A.3) 0 P a x t x tr t γ) > M 0 = P a x t x tr t γ) > M, 0 M P max k a x t x tr t γ) > M p k M M k P a x t 2 > M k=p a M = O k k=p a M 2 ) M E x t 2 M = O a ) < η as is large eough. Thus, we kow that equatio A.0) holds. Next, by Hölder s iequality ad a similar argumet as for A.3), it is ot hard to show that η > 0, P a 0 h t γ)ε t > M O a ) < η as is large eough, i.e., A.) holds. This completes the proof.
8 Lemma A.5. If Assumptios 2.-2.3 hold, the it follows that uder H 0 or H, i) Xγ X ) γ X = o p ), ii) X γx γ X X ) γ γ = o p ), iii) T γ T γ = op ), where X γ ad T γ are defied i the same way as X γ ad T γ, respectively, with R t γ) beig replaced by R t γ). Proof. i) Note that Xγ X ) γ X = 0 x t x tr t γ). Hece, we kow that i) holds by takig a = /2 i equatio A.0). ii) By a similar argumet as for i), we ca show that ii) holds. iii) Note that whe λ 0 = φ 0, h / ), we have T γ T γ = Xγ X γ ) ε Xγ X γ ) XX X) X ε Xγ X γ ) XX X) X X γ0 h X γxx X) X X γ0 X γ0 ) h + X γ X γ0 X γ X ) γ0 h I γ) I 2 γ) I 3 γ) I 4 γ) + I 5 γ) say. First, sice I γ) = 0 h /4 /4 t γ)ε t, it follows that γ I γ) = o p ) by takig a = /4 i equatio A.). Next, sice I 2 γ) = 0 x t x tr t γ) X X ) X ε, we have that γ I 2 γ) = o p ) from i). Similarly, we ca show that γ I i γ) = o p ) for i = 3, 4, 5. Hece, uder H 0 i.e., h 0) or H, we kow that iii) holds. This completes the proof.
TESTING FOR BAR PROCESSES 9 Lemma A.6. If Assumptios 2.-2.3 hold, the it follows that uder H 0 or H, λ γ) λ 0 = O p ). Proof. First, for ay γ Γ, by Taylor s expasio we have [ ε 2 t λ γ), γ) ε 2 t λ 0, γ) ] A.4) = λ γ) λ 0 ) + λ γ) λ 0 ) 2ε t λ 0, γ)x t γ) x t γ)x t γ) λ γ) λ 0 ). Next, whe λ 0 = φ 0, h / ), we ca show that A.5) N ε t λ 0, γ)x t γ) = Z γε + N x t γ)[x t γ 0 ) x t γ)] λ 0 = Z γε + G γ). x t x t[r t γ 0 ) R t γ)] x t x t[r t γ)r t γ 0 ) R t γ)] h Let λ mi γ) > 0 be the miimum eigevalue of K γγ. The, by equatios A.4)- A.5), η > 0, there exists a Mη) > 0 such that ) P λ γ) λ 0 > M = P λ γ) λ 0 > M, for some γ Γ) [ ε 2 t λ γ), γ) ε 2 t λ 0, γ) ] 0 P λ γ) λ 0 > M, 2 λ γ) λ 0 G γ) + λ γ) λ 0 2 [λ mi γ) + o p )] 0 for some γ Γ ) P M < λ γ) λ 0 2[λ mi γ) + o p )] G γ) for some γ Γ) P G γ) > M[λ mi γ) + o p )]/2 for some γ Γ) η,
0 where the last iequality holds because G γ) = O p ) by Lemma 2.2 ad Lemma A.3. Hece, uder H 0 i.e., h 0) or H, our coclusio holds. This completes the proof. Proof of Corollary 2.. The coclusio follows directly from Theorems 2.- 2.2 ad Lemma A.5. Proof of Theorem 3.. We use the method i the proof of Theorem 2 i Hase 996). Let W deote the set of samples ω for which A.6) A.7) lim lim x t γ) ε 2 t < a.s., x t γ)x t δ) ε 2 t σ 2 K γδ γ,δ Γ 0 a.s. Sice x t γ) 2 x t ad E x t ε 2 t ergodic theorem we have < due to Assumptio 2., by the lim 2 x t γ) ε 2 t lim x t ε 2 t < a.s., i.e., A.6) holds. Furthermore, by Assumptios 2.-2.3 ad a similar argumet as for Lemma A.3, it is ot hard to see that lim x t γ)x t δ) ε 2 t σ 2 K γδ 0 a.s., γ,δ Γ i.e., A.7) holds. Thus, P W ) =. Take ay ω W. For the remaider of the proof, all operatios are coditioally o ω, ad hece all of the radomess appears i the i.i.d. N0, ) variables {v t }. Defie Zγ) = N x t γ)ε t v t. By usig the same argumet as i Hase 996, p.426-427), we have A.8) Z γ) σg γ a.s. as. Note that Ẑγ) Z γ) x t γ)x t γ) v t λ γ) λ 0 ).
TESTING FOR BAR PROCESSES Usig the same argumet as for A.8) see, e.g., Hase 996, p.427)), we have A.9) x t γ)x t γ) v t 0 a.s. as. Now, by Lemma A.6 ad A.9), it follows that uder H 0 or H, A.20) Ẑ γ) Z γ) 0 i probability as. Thus, by A.8) ad A.20), we kow that uder H 0 or H, A.2) Ẑ γ) σg γ i probability as. Next, we cosider the fuctioal L : x ) D 2p+2 Γ) σ xγ) Ω 2 γ xγ), where D 2p+2 Γ) deotes the fuctio spaces of all fuctios, mappig R 2 Γ) ito R 2p+2, that are right cotiuous ad have right-had limits. Clearly, L ) is a cotiuous fuctioal; see e.g., Cha 990, p.89). By the cotiuous mappig theory ad A.2), it follows that uder H 0 or H, A.22) LẐγ)) LσG γ ) i probability as. Furthermore, sice σ 2 σ 2 a.s. ad X γ), I) [X 2 γ)] X γ), I) Ω γ uiformly i γ by Lemma A.3, we have that A.23) LR ˆ γ) = LẐγ)) + o p ). Fially, the coclusio follows from A.22)-A.23). This completes the proof. Proof of Corollary 3.. Coditioal o the sample {y 0,, y N }, let ˆF,J ad ˆF be the coditioal empirical c.d.f. ad c.d.f. of LR ˆ, respectively. The, P ) LR c J,α = E [ P )] LR c J,α y 0,, y N = E [ P ˆF,J LR ) α y 0,, y N )].
2 By the Gliveko-Catelli Theorem ad Theorem 3., it follows that uder H 0 or H, A.24) lim lim P ) LR c J,α J = lim E [ P )] ˆF LR ) α y 0,, y N = lim E [P F 0 LR ) α y 0,, y N )] = lim P F 0 LR ) α), where F 0 is the c.d.f. of G γω γ G γ. Thus, by A.24) ad Theorem 2., uder H 0 we have lim lim P ) ) LR c J,α = P G γω γ G γ F0 α) J = α, i.e., i) holds. Furthermore, by A.24) ad Theorem 2.2, uder H we have lim lim lim P ) LR c J h,α = lim P B h F0 α) ) =, J h where B h { G γ Ω γ G γ + h µ γγ0 h } ad the last equatio holds sice B h i probability as h. Thus, ii) holds. This completes the proof. REFERENCES [] Adrews, D.W.K. 993) A itroductio to ecoometric applicatios of fuctioal limit theory for depedet radom variables. Ecoometric Reviews 2, 83-26. [2] Cha, K.S. 990) Testig for threshold autoregressio. Aals of Statistics 8, 886-894. [3] Doukha, P., Massart, P., ad Rio, E. 995) Ivariace priciples for absolutely regular empirical processes. Aales de I Istitut H. Poicare 3 393-427. [4] Fa, J. ad Yao, Q. 2003) Noliear time series: Noparametric ad parametric methods. Spriger, New York. [5] Hase, B.E. 996) Iferece whe a uisace parameter is ot idetified uder the ull hypothesis. Ecoometrica 64, 43-430. [6] Pollard, D. 984) Covergece of Stochastic Processes. New York: Spriger. [7] Pollard, D. 990) Empirical processes: Theory ad applicatios. NSF-CMBS Regioal Coferece Series i Probability ad Statistics, Vol 2. Hayward, CA: Istitute of Mathematical Statistics. Chiese Academy of Scieces Istitute of Applied Mathematics HaiDia District, Zhogguacu Bei Jig, Chia E-mail: zkxaa@ust.hk Departmet of Statistics ad Actuarial Sciece Uiversity of Hog Kog Pokfulam Road, Hog Kog E-mail: plhyu@hku.hk hrtlwk@hku.hk