An Adaptive Empirical Likelihood Test For Time Series Models 1

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1 A Adaptive Empirical Likeliood Test For Time Series Models 1 By Sog Xi Ce ad Jiti Gao We exted te adaptive ad rate-optimal test of Horowitz ad Spokoiy (001 for specificatio of parametric regressio models to weakly depedet time series regressio models wit a empirical likeliood formulatio of our test statistic. It is foud tat te proposed adaptive empirical likeliood test preserves te rate-optimal property of te test of Horowitz ad Spokoiy (001. KEYWORDS: Empirical likeliood, goodess of fit test, kerel estimatio, rate-optimal test. 1. INTRODUCTION Cosider a time series eteroscedasticity regressio model of te form (1.1 Y t = m(x t + σ(x t e t, t = 1,,..., were bot m( ad σ( are ukow fuctios defied over R d, te data {(X t, Y t } t=1 are weakly depedet statioary time series, ad e t is a error process wit zero mea ad uit variace. Suppose tat {m θ ( θ Θ} is a family of parametric specificatio to te regressio fuctio m(x were θ R q is a ukow parameter belogig to a parameter space Θ. Tis paper cosiders testig te validity of te parametric specificatio of m θ (x agaist a series of local alteratives, tat is to test (1. H 0 : m(x = m θ (x versus H 1 : m(x = m θ (x + C (x for all x S, were C is a o-radom sequece tedig to zero as, (x is a sequece of fuctios i R d ad S is a compact set i R d. Bot C ad (x caracterize te departure of te local alterative family of regressio models from te parametric family {m θ ( θ Θ}. Tere ave bee extesive ivestigatios o employig te kerel smootig metod to form oparametric specificatio tests for a ull ypotesis like H 0 ; see Härdle ad Mamme (1993, Hjellvik, Yao ad Tjøsteim (1998 ad oters. A commo feature amog tese tests is tat te test statistics are formulated based o a sigle kerel smootig badwidt wic coverges to 0 as. Tis leads to a commo cosequece tat C, wic defies te gap betwee H 0 ad H 1, as to be at least of te order of 1/ d/4 i order to ave cosistet tests. I oter words, tese tests are uable to distiguis betwee H 0 ad H 1 for C at a order smaller ta 1/ d/4, wic ca be muc larger ta 1/, te order acieved by some oter oparametric tests for te above H 0 versus H 1 wit (x (x, for istace te coditioal Kolmogorov test cosidered i Adrews (1997. Te sigle badwidt based kerel tests also as o built-i adaptability to te smootess of (. 1 We would like to tak Mr Ceg Yog Tag for valuable computatio assistace wo was supported uder a Natioal Uiversity of Sigapore researc grat R Te secod autor ackowledges support from a Australia Researc Coucil Discovery Grat. 1

2 I a sigificat developmet, Horowitz ad Spokoiy (001 propose a test by combiig a studetized versio of te kerel based test statistic of Härdle ad Mamme (1993 over a set of badwidts. Tey establis i te more restrictive case of (x (x tat te test is cosistet for C = O( 1/ log log( wic almost reaces te order of 1/. Most remarkably, te test is adaptive to te ukow smootess of m i H 1 wic forms a Hölder smootess class of fuctios. I particular, if fuctios i te Hölder class possess a ukow s-order derivatives, te test is cosistet for C = O{( 1 log log( s/(4s+d } for s max (, d 4, wic is te optimal rate of covergece for C i te miimax sese of Spokoiy (1996 ad Igster ad Suslia (003. We cosider i tis paper two extesios of te adaptive test of Horowitz ad Spokoiy. Oe is to iclude weakly depedet observatios; te oter is to use te empirical likeliood (EL of Owe (1988 to formulate te test statistic, wic is desiged to equip te test statistic wit some favorable features of te EL. We sow tat te above metioed optimal or ear optimal rates for C establised by Horowitz ad Spokoiy (001 are maitaied uder tese extesios. Te rest of tis paper is orgaized as follows. Sectio proposes te adaptive empirical likeliood test ad presets te rate-optimal property of te test. Sectio 3 presets simulatio results. All te tecical proofs are provided i te appedix.. ADAPTIVE EMPIRICAL LIKELIHOOD TEST Like existig kerel based goodess-of-fit tests, our test is based o a kerel estimator of te coditioal mea fuctio m(x. Let K be a d-dimesioal bouded fuctio wit a compact support o [ 1, 1] d. Let be a smootig badwidt satisfyig (.1 ad K (u = d K(u/. 0 ad d / log 6 ( as, Te Nadaraya-Watso (NW estimators of m(x is ˆm(x = t=1 K (x X t Y t t=1 K (x X i. Let θ be a cosistet estimator of θ uder H 0. Like Härdle ad Mamme (1993, let m θ(x t=1 = K (x X t m θ(x t t=1 K (x X t be a kerel smoot of te parametric model m θ (x wit te same kerel ad badwidt as i ˆm(x. Tis is to avoid te bias of te kerel estimator i te goodess-of-fit test. Ce, Härdle ad Li (003 propose a test statistic based o te EL as follows. Let Q t (x = K (x X t {Y t m θ(x}. At a arbitrary x S, let p t (x be oegative real Oter kerel based EL tests wit a sigle badwidt are Fa ad Zag (004 ad Tripati ad Kitamura (004.

3 umbers represetig weigts allocated to eac (X t, Y t. Te EL for m(x evaluated at te smooted parametric model m θ(x is (. L{ m θ(x} = max p t (x t=1 subject to t=1 p t(x = 1 ad t=1 p t(xq t (x = 0. As te EL is maximized at p t (x = 1, te log-el ratio is l{ m θ(x} = log[l{ m θ(x} ]. Te EL test statistic at a give badwidt is (.3 l( m θ; = l{ m θ(x}π(xdx, were π( is a o-egative weigt fuctio supported o te compact set S R d satisfyig π(xdx = 1 ad π (xdx <. Let R(K = K (xdx, v(x = R(Kσ (xf 1 (x ad (K C(K, π = R (K ( (x (.4 dx π (ydy, were K ( is te covolutio of K. Ce, Härdle ad Li (003 sow tat as { } d/ l( m θ; 1 d/ v 1/ (x (xπ(xdx d (.5 N(0, C(K, π for te case of π(x = S 1 I(x S were I is te idicator fuctio ad S is te volume of S. A extesio of (.5 to a geeral weigt fuctio is automatic. Ce, Härdle ad Li (003 te proposes a sigle badwidt based EL test based o critical values obtaied by simulatig a Gaussia radom field. Like all oparametric kerel goodess-of-tests based o a sigle badwidt, te test is cosistet oly if C is at te order of 1/ d/4 or larger, idicatig tat C as to coverge to zero more slowly ta 1/. Te latter is te rate establised for oparametric goodessof-fit tests based o te residuals we tere is o smootig ivolved. To reduce te order of C, we employ te adaptive test procedure of Horowitz ad Spokoiy (001 for te EL test as follows. Let H = { } (.6 = max a k : mi, k = 0, 1,,... J be a set of badwidts, were 0 < a < 1, J = log 1/a ( max / mi is te umber of badwidts i H, max = c max (log log( 1 d ad mi = c mi γ for 0 < γ < 1 ad some positive d costats < c mi, c max <. Te coice of max is vital i reducig C to almost 1/ rate i te case of ( (. Te rage of γ allows mi = O{ 1/(4+d }, te optimal 3

4 order i te kerel estimatio of m(x. I view of te fact tat E{l( m θ; } = 1 uder H 0 ad var{l( m θ; } = C(K, π d as give i (.4 te adaptive EL test statistic is proposed as follows: (.7 l( L = max m θ; 1. H C(K, π d Let l α (0 < α < 1 be te 1 α quatile of L were α is te sigificace level of te test. Motivated by te bootstrap procedure of Horowitz ad Spokoiy, we propose te followig bootstrap procedure to approximate l α : 1. For eac t = 1,,...,, let Yt = m θ (X t + σ (X t e t, were σ ( is a cosistet estimator of σ( ad {e t} is sampled radomly from a distributio wit E[e t] = 0, E [e t ] = 1 ad E [ e t 4+δ] < for some δ > 0.. Let ˆθ be te estimate of θ based o te resample {(X t, Yt } t=1. Compute te statistic L by replacig Y t ad θ wit Yt ad ˆθ accordig to ( Estimate l α by l α, te 1 α quatile of te empirical distributio of L, wic ca be obtaied by repeatig steps 1 may times. Te estimator σ ( ca be te followig kerel estimator (.8 σ (x = t=1 K b(x X t {Y t ˆm(x} t=1 K b(x X t wit a badwidt b suc tat mi b d as. Te proposed adaptive EL test rejects H 0 if L > l α. 3. MAIN RESULTS Te followig teorem sows tat te adaptive EL test as a correct size asymptotically. Teorem 3.1. Suppose Assumptios A.1 ad A.(i(ii(iv old. Te uder H 0, lim P (L > lα = α. I te followig, we establis te cosistecy of te adaptive EL test agaist a sequece of fixed, local ad smoot alteratives, respectively. Let te parameter space Θ be a ope subset of R q. Let M = {m θ ( : θ Θ} ad f(x be te margial desity of X i. We ow defie te distace betwee m ad te parametric family M as [ ( 1/ (3.1 ρ(m, M = if [m θ (x m(x] f(xdx]. θ Θ x S Te cosistecy of te test agaist a fixed alterative is establised i Teorem 3. below. 4

5 Teorem 3.. Assume tat Assumptios A.1 ad A.(i(iii(iv old. If tere is a C ρ > 0 suc tat ρ(m, M C ρ for 0 wit some large 0, te lim P (L > l α = 1. We te cosider te cosistecy of te EL test agaist special from of H 1 of te form (3. m(x = m θ (x + C (x were C 0 as, θ Θ ad for positive ad fiite costats D 1, D ad D 3, (3.3 0 < D 1 (xf(xdx D < ad ρ(m, M D 3 C. x S Teorem 3.3. Assume Assumptios A.1 ad A.(i(iii. Let Assumptio A.(iv old wit max = C (log log( 1 d for some fiite costat C. Let m satisfy (3. ad (3.3 wit C C 1/ log log( for some costat C > 0. Te lim P (L > lα = 1. To discuss te cosistecy of te adaptive EL test over alteratives i a Hölder smootess class, we itroduce te followig otatio. Let j = (j 1,..., j d were j 1,..., j d 0 are itegers, j = d k=1 j d ad D j m(x = j m(x weever te derivative exists. Defie te x j 1 1 x j d d Hölder orm m H,s = sup x S j s ( Dj m(x. Te smootess class tat we cosider cosist of fuctios m S(H, s {m : m H,s C H } for some ukow s ad C H <. For s max(, d/4 ad all sufficietly large D m <, defie (3.4 B H, = ( {m S(H, s : ρ(m, M D m 1 } s/(4s+d log log(. Teorem 3.4. Assume tat Assumptios A.1 ad A. old. Let m satisfy (1. uder H 1 ad (3.4. Te for 0 < α < 1 ad B H, defied i (3.4, lim if m BH, P (L > l α = SIMULATION RESULTS We carried out two simulatio studies wic were desiged to evaluate te empirical performace of te proposed adaptive EL test. I te first simulatio study, we coducted simulatio for te followig regressio model used i Horowitz ad Spokoiy (001: (4.1 Y i = β 0 + β 1 X i + (5/τφ(X i /τ + ɛ i, were {ɛ i } are idepedet ad idetically distributed from tree distributios wit zero mea ad costat variace, {X i } are uivariate desig poits ad θ = (β 0, β 1 τ = (1, 1 τ is cose as te true vector of parameters ad φ is te stadard ormal desity fuctio. Te ull ypotesis H 0 : m(x = β 0 + β 1 x specifies a liear regressio correspodig to τ = 0, wereas te alterative ypotesis H 1 : m(x = β 0 + β 1 x + (5/τφ(x/τ for τ = 1.0 5

6 ad 0.5. Readers sould refer to Horowitz ad Spokoiy (001 for details o te desigs X i, te tree distributios of ɛ i ad oter aspects of te simulatio. We used te same umber of simulatio, te bootstrap resamples ad estimatio procedures for θ as i Horowitz ad Spokoiy (001. We also employed te same kerel, te same badwidt set H, ad te same estimator σ ad te distributio for e i i te bootstrap procedure as i Horowitz ad Spokoiy (001. Like Horowitz ad Spokoiy, te omial size of te test was 5%. Table 1 summaries te performace of te adaptive EL test by addig oe colum to Table 1 of Horowitz ad Spokoiy (001. Our results sow tat te proposed adaptive EL test as sligtly better power ta te adaptive test of Horowitz ad Spokoiy (001, wile te sizes are similar to tose of Horowitz ad Spokoiy (001. Tis may ot be surprisig as te two tests are equivalet i te first order. Te differeces betwee te two tests are (i te EL test statistic carries out te studetizig implicitly ad (ii certai iger order features like te skewess ad kurtosis are reflected i te EL statistic. Tese migt be te uderlyig cause for te sligtly better power observed for te EL test. Te secod simulatio study was coducted o a ARCH type time series regressio model of te form: (4. Y i = Y i 1 + C cos(8y i Yi e i, were te iovatio {e i } i=1 was cose to be idepedet ad idetically distributed N(0, 1 radom variables. Te sample sizes cosidered i te simulatio were = 300 ad = 500. Te vector of parameters θ = (α, β, σ was estimated usig te pseudo-maximum likeliood metod, wic is commoly used i te estimatio of ARCH models. I te bootstrap implemetatio, we cose e iid i N(0, 1 ad te estimator σ(x give i (.8. We cose te badwidt set H = {0.3, 0.33, 0.367, 0.407, 0.45} wit a = for = 300 ad H = {0.5, 0.81, 0.316, 0.356, 0.4} wit a = for = 500. Bot te power ad te size of te adaptive test are reported i Table. We foud te test ad good approximatio to te omial sigficiace level of 5%, wic cofirms Teorem 3.1 ad te quality of te bootstrap calibratio to te distributio of te adaptive EL test statistic. As expected we C was icreased, te power of te test was icreased; ad for a fixed level of C, te power icreased we was icreased. Te latter was because te distace betwee H 0 ad H 1 became larger we was icreased altoug C was kept te same. Departmet of Statistics, Iowa State Uiversity, Ames, IA 50010; sogce@iastate.edu; ttp:// ad Scool of Matematics ad Statistics, Te Uiversity of Wester Australia, Crawley WA 6009, Australia; jiti@mats.uwa.edu.au; ttp:// jiti 6

7 APPENDIX Tis appedix gives te assumptios ad proofs of te teorems give i Sectio 3. A.1. Assumptios Assumptio A.1. (i Assume tat te process (X t, Y t is strictly statioary ad α-mixig wit te mixig coefficiet α(t = sup{ P (A B P (AP (B : A Ω s 1, B Ω s+t } for all s, t 1, were Ω j i deotes te σ-field geerated by {(X s, Y s : i s j}. Tere exist costats a > 0 ad ρ [0, 1 suc tat α(t aρ t for t 1. (ii Assume tat for all t 1, P (E[e t Ω t 1 ] = 0 = 1 ad P ( E[e t Ω t 1] = 1 = 1, were Ω t = σ{(x s+1, Y s : 1 s t} is a sequece of σ-fields geerated by {(X s+1, Y s : 1 s t}. (iii Let ɛ t = Y t m(x t. Tere exists a positive costat δ > 0 suc tat E [ ɛ t 4+δ] <. (iv Let µ i (x = E[ɛ i t X = x], S π be a compact subset of R d, S f be te support of f ad S = S π S f be a compact set i R d. Let π be a weigt fuctio supported o S suc tat tat s S π(sds = 1 ad s S π (sds C for some costat C. I additio, te margial desity fuctio, f(x, of X t ad µ i (x for i = or 4 are all Lipscitz cotiuous i S, ad tat all te first two derivatives of f(x, m(x ad µ (x are cotiuous o R d, if x S σ(x C 0 > 0 for some costat C 0, ad o S te desity fuctio f(x is bouded below by C f ad above by C 1 f for some C f > 0. (v Te kerel K is a product kerel as defied by K(x 1,, x d = d i=1 k(x i, were k( is a r-t order uivariate kerel wic is symmetric ad supported o [ 1, 1], ad satisfies k(tdt = 1, t l k(tdt = 0 for l = 1,, r 1 ad t r k(tdt = k r 0 for a positive iteger r > d/. I additio, k(x is Lipscitz cotiuous i [ 1, 1]. Let te parameter set Θ be a ope subset of R q. Let M = {m θ ( : θ Θ}. Defie θ m θ (x = m θ (x θ, θ m θ(x = m θ (x θ θ, ad 3 θ m θ(x = 3 m θ (x θ θ θ weever tese derivatives exist. For ay q q matrix D, defie D = sup Dv v R q v, were v = q i=1 v i for v = (v 1,..., v q τ. Assumptio A.. (i Te parameter set Θ is a ope subset of R q for some q 1. Te parametric family M = {m θ ( : θ Θ} satisfies: For eac x S, m θ (x is twice differetiable almost surely wit respect to θ Θ. Assume tat tere are costats 0 < C 1, C < suc tat [ [ ] E ] sup m θ (X 1 C 1 ad max E θ Θ 1 j 3 were B m = q i=1 q j=1 b ij for B = {b ij} 1 i,j q. sup j θ m θ(x 1 m θ Θ C, For eac θ Θ, m θ (x is cotiuous wit respect to x R d. Assume tat tere is a fiite C I > 0 suc tat for every ε > 0 if [m θ(x m θ (x] f(xdx C I ε. θ,θ Θ: θ θ ε x S 7

8 ( (ii Let H 0 be true. Te θ 0 Θ ad lim P θ θ0 > C L < ε for ay ε > 0 ad all sufficietly large C L. ( (iii Let H 0 be false. Te tere is a θ Θ suc tat lim P θ θ > C L < ε for ay ε > 0 ad all sufficietly large C L. (iv Assume tat te set H as te structure of (.6 wit max > mi γ for some costat γ suc tat 0 < γ < 1 d ad max = C (log log( 1 d for some fiite costat C > 0. Assumptio A.1 is quite stadard i tis kid of problem ad Assumptio A. correspods to Assumptios 1, 4 ad 6 of Horowitz ad Spokoiy (001. A.. Tecical Lemmas From (.6 of Ce, Härdle ad Li (003, oe may sow tat (A.1 l( m θ ; = d Ū1 (x; θv 1 (xπ(xdx + o p ( d/ uiformly i H, were v(x = R(Kf 1 (xσ (x if te issue of boudary is ot cosidered, ad Ū 1 (x; θ ( x = ( d 1 Xt K {Y t m θ (x}. t=1 Let W t (x = 1 K ( x X t d, ast = d x S W s(xw t (xv 1 (xπ(xdx, ad λ t (θ = λ(x t, θ = m(x t m θ (X t. Defie (A. l 0 ( = s,t a st ɛ s ɛ t ad Q (θ = Q (θ; = s,t a st λ s (θλ t (θ. Te te leadig term i l ( m θ; is (A.3 l 1 (, θ d Ū1 (x; θv 1 (xπ(xdx = l 0 ( + Q ( θ + Π ( θ, were Π ( θ = l 1 (; θ l 0 ( Q ( θ is te remaider term. Witout loss of geerality, we assume tat C(K, π = R (K ( K ( (x dx π (ydy = 1. I view of te defiitio of L = max H l( m θ; 1 d/ (A.4 ad (A.3, defie L 0 ( = l 0( 1 d/, L 1 ( = l 1(, θ 1 d/ ad L ( = l 1(, θ 1 d/, were θ = θ 0 we H 0 is true ad θ is as defied i Assumptio A.(iii we H 0 is false. Let L 0 ( ad L 1 ( be te correspodig versios of L 0( ad L 1 ( wit {(X i, Y i } ad θ replaced by {(X i, Yi } ad ˆθ respectively. Te followig two Lemmas are preseted witout proof ere, wic ca be foud from te proofs of Lemmas A.1 ad A.6 of Ce ad Gao (004. 8

9 Lemma A.1. Suppose tat Assumptios A.1 ad A. old. For eac θ Θ ad sufficietly large, we ave tat C 1 d λ(θ τ λ(θ Q (θ C d λ(θ τ λ(θ olds i probability, were λ(θ = (λ 1 (θ,, λ (θ τ ad 0 < C 1 C < are costats. Lemma A.. Suppose tat Assumptios A.1 ad A. old. Te as max L ( H = max L 1 ( + o p (1 = max L ( + o p (1, H H max L 1 ( H = max L 0 ( + o p(1, H ad max H L 1 ( = max H L 0 ( + o p (1 uder H 0. Lemma A.3. Suppose Assumptios A.1 ad A. old. Te te asymptotic distributios of max H L ( ad max H L 0 ( are idetical uder H 0. Proof: I view of Lemma A., i order to prove Lemma A.3, it suffices to sow tat te distributios of max H L 0 ( ad max H L 0 ( are asymptotically te same. Similarly to te proof of Lemma A., we ca sow tat ( ( (A.5 max d/ a ss ɛ s 1 = o p (1 ad max d/ a ss ɛ s 1 = o p (1. H H s=1 Tus, it suffices to sow tat max H s t a stɛ s ɛ t ad max H s t a stɛ sɛ t ave te same asymptotic distributio. For H, let u t = ɛ t or ɛ t ad defie (A.6 s=1 B (u 1,..., u = d/ a st u s u t s t Let B (u 1,..., u be te sequece obtaied by stackig te correspodig B (u 1,..., u ( H. Let G( = G ( be a 3 times cotiuously differetiable fuctio over R J. Defie C (G = sup max 3 G(v v R J i,j,k=1,,...,j v i v j v k. Like Horowitz ad Spokoiy (001, tere are two steps i te proof of Lemma A.3. First, we wat to sow tat ( J E [G(B (ɛ 1,..., ɛ ] E [G(B (ɛ 1,..., ɛ 3 1/ (A.7 ] C 0 C (G for ay 3 times differetiable G(, some fiite costat C 0, ad all sufficietly large. Te i te secod step, (A.7 is used to sow tat B (ɛ 1,..., ɛ ad B (ɛ 1,..., ɛ ave te same asymptotic distributio. Trougout te rest of te proof, we replace a st i (A.6 wit ã st ( = d/ a st. Note tat (A.8 E [G(B (ɛ 1,..., ɛ ] E [G(B (ɛ 1,..., ɛ ] 9

10 E [ G(B (ɛ 1,..., ɛ t, ɛ t+1,, ɛ ] E [G(B (ɛ 1,..., ɛ t 1, ɛ t,..., ɛ ], t=1 were B (ɛ 1,..., ɛ, ɛ +1 = B (ɛ 1,..., ɛ ad B (ɛ 0, ɛ 1,..., ɛ = B (ɛ 1,..., ɛ. We ow derive a upper boud o te last term of te sum o te rigt ad side of (A.8. Similar bouds ca be derived for te oter terms. Let U 1, Λ ad Λ, respectively, deote te vectors tat are obtaied by stackig U, = 1 1 s=1 t=1, s ã st (ɛ s ɛ t, 1 Λ, = ɛ ã s (ɛ s, s=1 1 Λ, = ɛ ã s (ɛ s. s=1 Usig a Taylor expasio to te last term of te sum o te rigt ad side of (A.8 about ɛ = ɛ = 0 gives [ E [G(B (ɛ 1,..., ɛ ] E [G(B (ɛ 1,..., ɛ 1, ɛ ] E G (U 1 (Λ Λ ] + 1 [ ] E Λ τ G (U 1 Λ Λ τ G C (G { (U 1 Λ + E [ Λ 3] [ + E Λ 3]}, 6 were G ad G deote te gradiet ad matrix of secod derivatives of G ad C (G is a positive ad fiite costat. ] [ ] Sice E [ɛ j Ω 1 = E ɛ j Ω 1 for j = 1,, we ave Tis implies (A.9 E [(Λ Λ ] Ω 1 = 0 ad E [(Λ Λ τ Λ ] Λτ Ω 1 = 0. E [G(B (ɛ 1,..., ɛ ] E [G(B (ɛ 1,..., ɛ 1, ɛ ] C (G 6 { E [ Λ 3] [ + E Λ 3]}. To estimate te upper boud of (A.9, we eed te followig result: ( ( 1 x Xs x Xt (A.10 a st = d K K q(xdx = 1 ( K(uK u + X s X t q(x s + udu = 1 ( L Xs X t, X s, were q(x = v 1 (xπ(x ad L (x, y = K(uK(u + xq(y + udu. Usig Assumptios A.1 A. ad (A.10, we ave as (A ã s ( 1ã t ( ɛ s ɛ t ɛ4 E 1 H H s=1 t=1, s 10

11 = = 1 H 1 H [ ( E L Xs X ( ] 1 d, X s L Xt X H 1 d, X t ɛ 1 sɛ tɛ 4 ( ( ( 1 d x z y z L, x L 1 H 1 L (x 1, x 3 + x 1 1 L (x, x 3 + x u v w 4 H 1 H, y u v w 4 f(x, y, z, u, v, wdxdydzdudvdw f(x 3 + x 1 1, x 3 + x, x 3, u, v, wdx 1 dx dx 3 dudvdw C ( J (1 + o(1, were f(x, y, z, u, v, w is te joit desity fuctio of (X s, X t, X, ɛ s, ɛ t, ɛ ad 0 < C < is a costat. Similarly to te proof of Lemma C. of Gao ad Kig (001, we ca sow tat as 1 ( 1 d E ( a s( 1 a s ( a t ( ɛ 3 sɛ t ɛ 4 J (A.1 = o, 1, H 1 s t 1 1 ( 1 d E ( a s( 1 a t ( a u ( ɛ sɛ t ɛ u ɛ 4 J = o, 1, H 1 s t,s u,t u q 1 ( 1 d E ( a s ( 1 a t ( 1 a u ( a v ( ɛ s ɛ t ɛ u ɛ v ɛ 4 J = o 1, H 1 1 s t,s u,s v,t u,t v,u v 1 usig Assumptio A.1(ii tat for every give x, [ ( ] [ ( ] Xt x Xt x (A.13 E L, X t ɛ t = E L, X t E [ɛ t Ω t 1 ] Equatios (A.11 ad (A.1 te imply tat as 1 ( ã s ( 1 ɛ s ã t ( 1 ɛ t ã u ( ɛ u ã v ( ɛ v ɛ 4 J (A.14 C. E 1 H H s,t,u,v=1 Let à s be te vector tat is obtaied by stackig ã s ( ( H. Equatio (A.14 te implies tat as (A.15 E [ Λ 3] 1 = 8E 3 à s ɛ s ɛ 8 E = 8 { s=1 E 1 H H [ 1 s,t,u,v=1 H ( 1 = 0. ã s (ɛ s ɛ s=1 3/4 ã s ( 1 ɛ s ã t ( 1 ɛ t ã u ( ɛ u ã v ( ɛ v ɛ 4 ]} 3/4 C ( J 3/. 11

12 ] A similar result olds for E [ Λ 3. Tus (A.16 E [ Λ 3] [ + E Λ 3] C ( 3/ J. Step : Followig te lies of Horowitz ad Spokoiy (001 by utilizig te above establised boud (A.16 ad usig (A.8, it ca be sow tat as (A.17 ] [max P B (ɛ 1,..., ɛ x P H [ ] max B (ɛ 1,..., ɛ x C H Tis implies (A.7 ad fially completes te proof of Lemma A.3. ( J 3 1/ 0. Lemma A.4. Suppose tat Assumptio A.1 olds. Te for ay x 0, H ad all sufficietly large P (L 0( > x exp ( x. 4 Proof: Te proof is give as tat of Lemma A.8 i Ce ad Gao (004. For 0 < α < 1, defie l α to be te 1 α quatile of max H L 0 (. Lemma A.5. Suppose tat Assumptio A.1 olds. Te for large eoug l α log(j log(α. Proof: Te proof is similar to tat of Lemma 1 of Horowitz ad Spokoiy (001. Lemma A.6. Suppose tat Assumptios A.1 ad A. old. Suppose tat ( (A.18 lim P Q (θ d/ l α = 1 ( for some H, were l α = max l α, log(j + log(j. Te lim P (L > lα = 1. Proof: By (A., (A.3, (A.4 ad Lemma A., L ca be replaced wit max H L (. By Lemmas A. ad A.3, l α ca be replaced by l α. Tus, it suffices to sow tat lim P (max L ( > l α = 1, H wic olds if lim P (L ( > l α = 1 for some H. For ay H, usig (A., (A.3, (A.4 ad Lemma A. agai we ave (A.19 L ( = L 0 ( + d/ Q (θ + d/ Π (θ = L 0( + d/ Q (θ + d/ Π (θ + o p (1 = L 0( + d/ Q (θ (1 + o p (1 + o p (1. 1

13 Coditio (A.18 implies tat as (A.0 P (Q (θ < d/ l α 0. Observe tat P (L ( > l α = P (L ( > l α, Q (θ d/ l α ( + P L ( > l α, Q (θ < d/ l α I 1 + I. Tus, it follows from (A.19 tat as I 1 = P (L 0( + d/ Q (θ + d/ Π (θ > l α Q (θ d/ l α P (Q (θ d/ l α = P (L 0( + d/ Q (θ (1 + o p (1 > l α Q (θ d/ l α P (Q (θ d/ l α P (L 0( > l α l α Q (θ d/ l α P (Q (θ d/ l α 1 because L 0 ( is asymptotically ormal ad terefore bouded i probability ad l α l α as. Because of (A.0, lim I P (Q (θ < d/ l α = 0. Tis fiises te proof. A.3. Proofs of Teorems Proof of Teorem 3.1: By Lemma A., max H L 1 ( = max H L ( + o p (1. By Lemma A.3, uder H 0, max H L ( max H L 0 ( 0 i distributio as. Usig Lemma A. agai implies max H L 1 ( = max H L 0 ( + o p(1. Tis implies tat max H L 1 ( max H L 1 ( 0 i distributio as. Tis, alog wit equatios (A.1 (A.4, fiises te proof. I order to prove Teorems , i view of Lemma A.6, it suffices to verify (A.18. Usig Lemma A.1, it suffices to verify ( (A.1 lim P d λ(θ τ λ(θ 4 l α d/ = 1. Proof of Teorem 3.: I view of te defiitio of l α, equatio (A.1 follows from te fact tat as, 1 (A. λ(θτ λ(θ ρ(m, M 0 olds i probability ad d C 0 l α d/ for some costat 0 < C 0 < ad large eoug. (A.3 Proof of Teorem 3.3: Usig te defiitio of l α, (A., 1 [ (X t E S (X 1 ] = t=1 x S 13 (xf(xdx D 1 > 0 as,

14 ad te fact tat (A.4 1 λ(θτ λ(θ = C (X t D 1 C t=1 olds i probability, oe ca see tat (A.1 olds we = max = (log log( 1 d. Tis fiises te proof of Teorem 3.3. Proof of Teorem 3.4: I order to verify (A.18, we eed to itroduce te followig otatio: ( 1 = 1 l α 4s+d. Tis implies 4s+d 1 = l α. Coose H suc tat 1 < 1. We te ave 4 d l α = 4 d 4s+d 1 4 4s+d + (A.5 d = 4 s+d. (A.6 Tus, i order to verify (A.18, it suffices to sow tat Q (θ 4 s+d olds i probability for te selected H ad θ Θ. Te verificatio of (A.6 ca be doe usig similar teciques detailed i te proof of Lemma A.1 give i Ce ad Gao (004. Alteratively, oe may follow te proof of (A13 of Horowitz ad Spokoiy (001 by otig tat all te derivatios below teir (A13 old i probability for radom X i. REFERENCES Adrews, D. W. K. (1997: A Coditioal Kolmogorov Test, Ecoometrica, 65, Ce, S. X., ad J. Gao (004: A Adaptive Empirical Likeliood Test for Time Series Regressio Models, mauscript. Ce, S. X., W. Härdle, ad M. Li (003: A Empirical Likeliood Goodess of Fit Test for Time Series, Joural of te Royal Statistical Society, Series B., 65, Fa, J., ad J. Zag (004: Sieved Empiricial Likeliood Ratio Tests for Noparametric Fuctios, Te Aals of Statistics, 9, Gao, J. ad M. L. Kig (001: Estimatio ad Model Specificatio Testig i Noparametric ad Semiparametric Regressio, Workig paper available at jiti/jems.pdf. Härdle, W., ad E. Mamme (1993: Comparig Noparametric versus Parametric Regressio Fits, Te Aals of Statistics, 1, Hjellvik, V., Q. Yao, ad D. Tjøsteim (1998: Liearity Testig Usig Local Polyomial Approximatio, Joural of Statistical Plaig ad Iferece, 68, Horowitz, J. L., ad V. G. Spokoiy (001: A Adaptive, Rate Optimal Test of a Parametric Mea Regressio Model Agaist a Noparametric Alterative, Ecoometrica, 69, Igster, Yu. I. ad Suslia, I. A. (003: Noparametric Goodess-of-Fit Testig Uder Guassia Models, Lecture Notes i Statistics, 169, New York: Spriger-Verlag. Owe, A. (1988: Empirical Likeliood Ratio Cofidece Itervals for a Sigle Fuctioal, Biometrika, 75, Spokoiy, V. G. (1996: Adaptive Hypotesis Testig Usig Wavelets, Te Aals of Statistics, 4, Tripati, G., ad Y. Kitamura (004: O Testig Coditioal Momet Restrictios: te Caoical Case, Te Aals of Statistics (to appear. 14

15 TABLE 1 SIMULATION RESULTS ON MODEL (4.1 Probability of Rejectig Null Hypotesis Adrews Härdle-Mamme Horowitz-Spokoiy EL Distributio ɛ τ Test Test Test Test Null Hypotesis Is True Normal Mixture Extreme Value Null Hypotesis Is False Normal Mixture Extreme Value Normal Mixture Extreme Value TABLE SIMULATION RESULTS ON MODEL (4. C = 300 = 500 Null Hypotesis Alterative Hypotesis Alterative Hypotesis