A Goodness-of-fit test for GARCH innovation density. Hira L. Koul 1 and Nao Mimoto Michigan State University. Abstract

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1 A Goodness-of-fit test for GARCH innovation density Hira L. Koul and Nao Mimoto Micigan State University Abstract We prove asymptotic normality of a suitably standardized integrated square difference between a kernel type error density estimator based on residuals and te expected value of te error density estimator based on innovations in GARCH models. Tis result is similar to tat of Bickel-Rosenblatt under i.i.d. set up. Consequently te goodness-of-fit test for te innovation density of te GARCH processes based on tis statistic is asymptotically distribution free, unlike te tests based on te residual empirical process. A simulation study comparing te finite sample beavior of tis test wit Kolmogorov-Smirnov test and te test based on integrated square difference between te kernel density estimate and null density sows some superiority of te proposed test. Introduction Te problem of fitting a given density or distribution function to a random sample, oterwise known as te goodness-of-fit testing problem, is classical in statistics. Often omnibus tests are based on a discrepancy measure between a density estimate and te null density being fitted or between empirical and null distribution functions (d.f. s. Test based on empirical d.f. s are easy to implement as long as tere are no nuisance parameters under te null ypotesis. For example, wen fitting a known continuous d.f. to te given data, Kolmogorov test is known to be distribution free for all sample sizes and ence easy to implement. But it looses tis property wen fitting an error distribution in te one sample location-scale model. In comparison, as noted by Bickel and Rosenblatt (973, some goodness-of-fit tests based on density estimates do not suffer from tis draw back. One suc statistic is te integrated square difference. More precisely, let ε,, ε n be i.i.d. observations from a density f. Let f 0 be a given density and consider te problem of testing te ypotesis H 0 : f = f 0, vs. H : f f 0. Supported in part by te NSF Grant DMS AMS 00 subject classifications. Primary 6M0; secondary 6G0 Key words and prases. L difference of densities.

2 Let K be a density kernel, n be a sequence of positive numbers, define te density estimator f n (x = n ( x εk K. Bickel and Rosenblatt (973 proposed to use te statistic (fn T n = (x E 0 f n (x dx for testing H 0, vs. H. Here E 0 is expectation under H 0. Tey proved, under H 0 and under some conditions tat included second order differentiability of f 0, tat as n, n ( T n (. K (tdt D N (0, τ, τ = f0 (xdx (K K(x dx, n were g g (x := g (x tg (tdt, for any two integrable functions g, g. Bacmann and Dette (005 weakened teir conditions required for (., and also establised te following asymptotic normality result for T n under te alternatives H : f f 0, (f(x f 0 (x dx > 0. Assuming only f, f 0 to be continuous and square integrable, tey proved ( {K n T n (f f 0 (x } (. (xdx D N(0, 4ω, n, were ω = Var[f(ε 0 f 0 (ε 0 ], and K ( = (/K( /. Bacmann and Dette (005 mention tey assume f, f 0 to be twice continuously differentiable wit bounded second derivatives, but a close inspection of teir proofs of (. and (. sows tat all tey need is f, f 0 to be continuous and square integrable. Now consider te problem of fitting a zero mean density f 0 to te error density of a stationary linear autoregressive model of a known order. Lee and Na (005 and Bacmann and Dette (005 sowed tat (. and (. continue to old for an analog of T n based on autoregressive residuals, under H 0 and H, respectively. In oter words, not knowing nuisance autoregressive parameters as no effect on asymptotic level of te test based on tis analog for fitting f 0 to te error density in tis model. In tis paper we consider te problem of fitting density f 0 to te error density of a generalized autoregressive conditionally eteroscedastic (GARCH(p, q, were p and q are known positive inegers. We provide some sufficient conditions under wic (. continues to old for T n, an analog of T n based on GARCH residuals, defined at (.4 below. In addition, we establis a first order expansion of T n under H. Tis expansion sows tat unlike in linear autoregressive models, te estimation of te model parameters affects te asymptotic distribution of T n under H.

3 Mimoto (008 sowed tat te goodness-of-fit test for te error density in GARCH models based on a suitably standardized sup-norm statistic f n f 0 as te same asymptotic null distribution as in te i.i.d. set up. Ceng (008 contains asymptotic distribution of similar statistics in te case of ARCH models. Tis paper is organized as follows. In te next section we describe te model, assumptions and recall some preliminaries from Berkes, Horvát and Kokoszka (003. Asymptotic normality of T n under H 0 and a first order approximation of T n under H are given in section 3 wit proofs appearing in section 5. Section 4 contains a simulation comparing T n test wit te Kolmogorov-Smirnov test and te one based on ( f n f 0. Te proofs given below use several results from Berkes et al. (003 and Horvát and Zitikis (006 about some properties of GARCH models. Many details of te proofs are different from tose appearing in tese papers and Mimoto (008. Model, some preliminaries and assumptions In tis section, we describe te model, review some known results about te model, and state te needed assumptions. Let p, q be known positive integers, y k, k Z := {0, ±,, } be te GARCH(p, q process satisfying (. y k = σ k ε k, σ k = ω + i p α i y k i + j q β j σ k j, were θ = (ω, α,, α p, β,, β q T is te parameter vector of te process, wit ω > 0, α i 0, i p, β i 0, j q. Te innovations ε k, k Z, are assumed to be i.i.d. random variables wit a density function f aving zero mean and finite variance. Necessary and sufficient conditions for te existence of a unique stationary solution of (. ave been specified by Nelson (990 for p =, q =, and by Bougerol and Picard (99a, 99b for p, q. In particular, for GARCH(, model, E log(β + α ε 0 < 0 implies stationarity of te process. Because e x + x, for all x, we ave E log(β + α ε 0 β + α E(ε. Tus, if error density is standardized to ave zero mean and unit variance, ten β + α < implies te stationarity of te GARCH(, process. A major caracteristic of te GARCH process is tat te past dependency of te observations y k is only troug te unobservable conditional variance σk. Berkes, et al. (003 sow tat under suitable conditions, σk admits a unique representation as te infinite sum of yk, allowing σ k to be estimated from te observations. One of tem is to assume tat te polynomials (. α x + + α p x p and β x β q x q are coprimes on te set of polynomials wit real coefficients. 3

4 Tis is to ensure tat te equations (. are true only wit θ and tere is no oter parameter tat satisfies te equation. Let u = (r, s,, s p, t,, t q denote a generic element of te parameter space U := { u : t + + t q ρ 0, u < min(r, s,, s p, t,, t q max(r, s,, s p, t,, t q u }, were 0 < u < u, 0 < ρ 0 <, qu < ρ 0. Wit coefficients c i (u, 0 i < as in Berkes et al. (003, let w k (u = c 0 (u + i< c i (uy k i. Assuming tat E ε 0 δ < for some δ > 0 and θ is an interior point of U wit none of te coordinates equal to zero, Berkes et al. sowed tat σ k = w k(θ, for all k Z, a.s. Assuming furter tat te distribution of ε 0 is non-degenerate, tis representation is almost surely unique. Wit given observations y k, k n, tis representation allows one to estimate σ k by te truncated version k (.3 σ k = ŵ k (θ n = c 0 (θ n + c i (θ n yk i, were θ n is an estimator of θ based on y k, k n. Tis leads to te GARCH residuals { ε k = y k / σ k } and to te GARCH error density estimate f n (x = n Te proposed test for H 0 is to be based on (.4 i= ( x εk K. T n = ( fn (x E 0 f n (x dx. We sall now state te needed assumptions for obtaining asymptotic distributions of T n under H 0 and H. About θ n assume (.5 n(θn θ = O p (. Lee and Hansen (994, Lumsdaine (996 and Berkes et al. (003 discuss some sufficient conditions tat imply (.5 for te sequence of quasi-maximum likeliood estimators. About te kernel K, assume (.6 K is a bounded symmetric density on [, ], vanising off (,, and twice differentiable wit bounded derivative K and ( K (z dz <, 4

5 were g and g denote, respectively, te first and second derivatives of a smoot function g. In order to use results from Berkes et al. (003, we need to assume (.7 (.8 E ε 0 +δ <, for some δ > 0. P {ε 0 t} = o(t δ, as t 0. We say density f satisfies condition C(f if te following olds. (.9 f is bounded, absolutely continuous, x f(xdx <, xf(xdx = 0, ( and its a.e. derivative f satisfies f(x dx <, x { f(x z f(x } (f(x dxk(zdz 0, + x f(x dx <. Note tat te last integral above is te Fiser information in te one sample scale model. Many smoot densities can be sown to satisfy C(f. We sall now sow tat double exponential density f(x = e x / also satisfies tis condition. Clearly, a.e. derivative of tis f(x is f(x = -sign(xf(x. Hence, because f is a density bounded by /, x { f(x z f(x } dxk(zdz 4 x I( z x z f (xdxk(zdz z K(zdz 0. Te rest of te conditions in (.9 are easy to verify in tis case. For te bandwidt, we assume (.0 0, n 5, as n. Trougout te rest of te paper, for any square integrable function g on R, its L norm is denoted by g := ( g (xdx /, and all limits are taken as n, unless stated oterwise. 3 Main Results In te first teorem below we give some preliminary results about f n. To state tis result we need to introduce g n (x = (θ n θ T n w k (θ w k (θ [ E ( x ε 0 K ε0 were w k (θ is te column vector of lengt p + q +, consisting of te first derivatives of w k (θ w.r.t. θ, and were T denotes te transpose. We are now ready to state 5 ],

6 Teorem 3. Suppose te GARCH(p, q model (. is stationary wit te true error density f satisfying C(f. In addition, assume (., and (.5-(.8 old. Ten, (3. and (3. f n f n g n = O p ( n n gn = O p (. 5/, Consequently, if also (.0 olds, ten (3.3 n fn f n = O p (. Teorem 3. is useful in establising an analog of te result (. for T n as follows. We sall first approximate T n by T n. Assume H 0 olds and recall E 0 denotes te expectation under H 0. Direct calculations sow tat n ( (3.4 Tn T n = n ( dx ( dx fn (x E 0 f n (x n f n (x E 0 f n (x = n ( dx ( fn (x f n (x + n fn (x f n (x ( f n (x E 0 f n (x dx. By (3.3, te first term is o p (. Te following proposition sows te same olds for te second term. Proposition 3. Under te conditions of Teorem 3. and (.0, n ( fn (x f n (x ( f n (x E 0 f n (x dx p 0. Tis proposition togeter wit (3.4 yields te following corollary. Corollary 3. Suppose te GARCH(p, q model (. is stationary and (.5 - (.8, C(f 0, and (.0 old. Ten, under H 0, n ( Tn T n = o p (, and ence, (3.5 n ( Tn n K (tdt D N (0, τ, τ = f0 (xdx (K K (xdx. 6

7 Remark 3. An alternative test of H 0 using density estimates could be based on T n := ( fn (x f 0 (x dx. Upon taking v = in (3.5 of Horvát and Zitikis (006 one obtains tat under (.0 and E ε 0 3+δ <, and under some conditions on K and f tat are stronger tan tose given above, n ( Tn K /n D N (0, τ. It is important to point out tat (3.5 does not follow directly from tis asymptotic normality result about T n, for n / T n T n = O p (n 5/ + O p (n / p, under (.0. Te next teorem gives te first order limiting beavior of T n under te alternative H. Teorem 3. Suppose te GARCH(p, q model (. is stationary, and (.5 - (.8, C(f, C(f 0, and (.0 old. Ten, under H, ( { dx n Tn K (f f 0 (x} = ( { } dx n T n K (f f 0 (x n(ˆθ n θ T E [ w 0 (θ ] ( f(x + x f(x (f(x f 0 (xdx + o p (. w 0 (θ Tis result is unlike te result (. in AR models because of te presence of te second term in te rigt and side above. A primary reason tat analog of tis term is absent in te AR model is tat te analog of w 0(θ/w 0 (θ in te AR(p model is (y p, y p,, y 0 T wose expected value is zero wen fitting an error density wit zero mean. But ere tese entities come from a scale factor and ence teir expectation can not be zero. 4 Simulation Study Tis section contains results of a simulation study illustrating a finite sample performance of te goodness-of-fit tests based on T n of Corollary 3. and T n of Remark 3. against te Kolmogolov-Smirnov (KS goodness-of-fit test based on n / sup x R F n (x F 0 (x, were F n (t = n I( ɛ i t. i= In te study, GARCH(, process wit ω = 0.5, α =.4, β =. so tat θ = (.5,.4,. T of varying lengt n were simulated, eac iterated 0,000 times. Eac time first 500 observations were not included. Note tat α + β =.6 <. Estimator θ n was obtained by te quasimaximum likeliood metod. For te underlying density f, we used 7 densities: standard normal density (N, Student-t densities wit degrees of freedom 40, 0, 0, and 5 (T 40, T 0, T 0, T 5, double exponential density (D, and logistic density (L. All of tese densities were standardized to ave mean 7

8 zero and variance. For f 0, we cose normal, double exponential, and logistic densities, all standardized. bandwidt = (3 5/4n /4.9. For te kernel function, we used K(u = (3/4( u I( u and Test based on empirical (asymptotic critical values of T n is denoted by T n,e ( T n,a. Define T n,e and T n,a similarly. Tables -3 contain empirical sizes and powers of tese tests and of te KS test using empirical critical values only. Te first row entries in all te tables and sub-tables are empirical sizes and sould be close to te nominal level.05. In Table, f 0 is te standard normal. Test T n,e clearly outperforms all te oter tests for all cosen sample sizes. Te empirical power is especially significantly iger tan tat of oter tests against alternative densities T 0 and L. Table : Empirical sizes and powers of te tests wen f 0 = N(0,. n f\ Tests Tn,e Tn,e KS Tn,a Tn,a f 0 =N T T T T D L f 0 =N T T T T D L f 0 =N T T T T D L In Table, f 0 = D, te double exponential density. Empirical powers of all te tests quickly becomes large for n = 500, 000. For all sample sizes cosen, KS test as worse empirical power compared to tat of T n,e and T n,e tests wit T n,e test dominating T n,e for 8

9 n = 00. For all te values of n considered, Tn,a test suffers from large bias of te kernel density estimation around its peak. Table : Empirical sizes and powers of te tests wen f 0 = D. n f\ Tests Tn,e Tn,e KS Tn,a Tn,a f 0 =D N T T T T L f 0 =D N T T T T L f 0 =D N T T T T L In Table 3, f 0 = L, te logistic density. Against N, T40, T0 and T0 alternatives, T n,e performs te best. On te oter and, against te alternatives T5 and D, T n,e test performs te best for all n considered, and KS test as iger empirical power tan T n,e test for n = 00 and 500. Monte Carlo distributions of statistics T n and T n do not appear to approximate teir asymptotic distribution well even for n = 000. Tis results in misspecified size and lack of power for te asymptotic tests T n,a and T n,a for all cases considered. Empirical level of te T n,a test is especially worse wen fitting double exponential density. 9

10 5 Proofs Table 3: Empirical sizes and powers of te tests wen f 0 = L. n f\ Tests Tn,e Tn,e KS Tn,a Tn,a f 0 =L N T T T T D f 0 =L N T T T T D f 0 =L N T T T T D Tis section contains te proofs of some of te above claims. Te following Caucy-Swarz inequality is used repeatedly in te proofs. For any real sequences a k, b k, k n, (5. ( N ( ( a k b k a k a k b k. We also need to recall te following facts from Berkes, et al. (003. Facts below are Lemmas.,.3, 5., 5.6, and (5.35 in Berkes, et al. (003, respectively. Fact 5. Let log + x = log x if x >, and 0 oterwise. If {ζ k, 0 k < } is a sequence of identically distributed random variables satisfying E log + ζ 0 <, ten 0 k< ζ kz k converges a.s., for all z <. Under te assumptions of Teorem 3. te following facts old. 0

11 Fact 5. Tere exists a δ > 0 depending on δ of (.7, suc tat E y 0 δ + E σ 0 δ <. Fact 5.3 E [ supu U σ k /w k(u ] ν <, for any 0 < ν < δ, were δ is as in (.7. Fact 5.4 E sup u U w 0(u w 0 (u ν < and E sup u U w 0(u w 0 (u ν <, ν > 0, were denotes te maximum norm of vectors and matrices. Tis implies tat E sup w 0(u/w 0 (u ν <, ν > 0, E u U m were, for any m r matrix A = ((a i,j, A E := r i= j= a ij. Fact 5.5 For all u U, w k (u ŵ k (u ŵ k (u C ρ k/q 0 C 0 j< ρ j/q 0 y j were ρ 0 is from te definition of U, and 0 < C, C <. We would like to point out tat Fact 5.4 is te corrected version of Lemma 3.4 of Mimoto (008 were w0(u appears in te denominators. We now proceed wit te proof of Teorem 3.. Trougout te proof below, let (5. r k (u := w k (u w k (u, R k(u := w k (u w k (u, r k := w k (θ w k (θ, S n := r k, S n := n S n. Because te underlying process is stationary, by te Ergodic Teorem, in view of Fact 5.4, (5.3 S n Er 0 = E w 0(θ w 0 (θ, a.s. Tis notation and fact will be often used below. Proof of Teorem 3.. Define σ kn = w k (θ n = c 0 (θ n + i< c i (θ n y k i. Let { ε k = y k /σ kn } be te non-truncated version of te residuals and let f n (x := n ( x εk K.

12 Now write f n f n g n = f n f n + f n f n g n, so tat (5.4 f n f n g n f n f n + f n f n g n. (5.5 We claim fn f n = O ( n 3/, Use te Mean-Value Teorem and te triangle inequality to obtain f n (x f n (x = n n a.s. [ ( x εk ( x εk ] K K ε k ε k K ( x η k, were η k = ε k + c ( ε k ε k, for some 0 < c <. Hence, (5. and a routine argument yields f n (x f n (x dx = = ( ( ( x ε n 3 k ε k ε k ε k K ηk dx ( ( ε n 3 k ε k ( K ε n 3 k ε k (z dz. ε k ε k K (z dx (5.6 We sall sow ε k ε k = O(, a.s. Tis fact and (.6 ten completes te proof of (5.5. To prove (5.6, observe tat ε k ε k = y k wk (θ n ŵk (θ n y k w = k (θ n ŵ k (θ n ( wk wk (θ n ŵk (θ n (θ n + ŵ k (θ n sup u U sup u U y k wk (u y k wk (u w k(u ŵ k (u ŵ k (u ( C ρ k/q 0 C j=0 ρ j/q 0 y j, k,

13 were 0 < C, C <, and ρ 0 is from te definition of U. We obtain te last but one upper bound above by te fact tat ŵ k (u w k (u, for all k and u U, and te last upper bound by Fact 5.5. Terefore, ε k ε k C ( C j=0 ( ρ j/q 0 y j sup u U y k wk (u ρk/q Note tat supu U y k / w k (u is a stationary sequence, and so is y j. By Fact 5., (.7 implies E y0 δ <, for some δ > 0, and also independence of ε k and w k (u and te Fact 5.3 yield [ E sup u U y k ] ] [ = E [ε 0 E sup wk (u u U σ0 ] / <. w 0 (u Since log + moments of bot supu U y k / w k (u and y j are finite, and ρ 0 <, Fact 5. implies 0 j< y jρ j/q 0 < and tereby completing te proof of (5.6. sup u U k< 0. y k wk (u ρk/q 0 <, a.s., Next, we sall analyze f n f n g n, te second term of te upper bound in (5.4. By te Taylor expansion up to te second order, (5.7 f n (x f n (x g n (x = n = n { ( x εk ( x εk } K K g n (x ( x ( ε k ε k K εk g n (x + ( x ( ε n 3 k ε k K ξk were ξ k = ε k + c ( ε k ε k, for some 0 < c <. (5.8 We claim [ n 3 Let n = θ n θ, and write ( ε k ε k K ( x ξk ] dx = Op ( n 5. ( ( ε k ε k = y k wk (θ n ( = y k wk (θ wk ( n + θ wk (θ, 3

14 Recall (5.. Te first and second order Taylor expansions of / w k ( n + θ around θ yield te following two equations, respectively. (5.9 (5.0 ( ε k ε k = ε k T n r k (θ = ε k T n r k + 3ε k 8 ( T nr k (θ ε k 4 T nr k (θ n were θ = θ n + c n, and θ = θ n + c n, for some 0 < c, c <. (5. By (5.9, te left and side of (5.8 is equal to [ ( ε n 4 6 k n T nr k (θ ( K x ξk ] dx [ ( ε n 5 k n T n nr k (θ ] dz, (K (z ( were te last inequality is obtained by applying (5. wit a k = ε k n T nr k (θ, bk = K ( (x ξ k /, and by a cange of variable in te integration. Te above bound is O p (/n 5, by (.5, (.6, (.7 and Fact 5.4, tereby proving (5.8. Upon combining (5.7 wit (5.8, we obtain fn f n g n (5. [ ( x = ( ε n k ε k K εk By (5.0, (5.3 n We sall now sow [ (5.4 n (5.5 [ n ( x ( ε k ε k K εk = ( n ( T x n 3/ nr k ε k K εk + n n ( T nr k (θ T nr k (θ n K ε k ( T nr k (θ 3ε k ( x εk ] dx ( g n (x + Op. n 5 ( x εk 8 K ε ( k x εk 4 K. ] dx = Op ( n 3. ( x ε k T nr k (θ n K εk ] dx ( = Op. n 3 4

15 To prove (5.4, use (5. in a similar fasion as for (5. and a cange of variable formula to obtain tat te left and side of (5.4 is bounded above by n 3 [ n ε k ( n T nr k (θ ] dz. (K (z Tis bound in turn is O p (/n 3, by (.5, (.6, (.7, and Fact 5.4. Tis proves (5.4. Te proof of (5.5 is exactly similar. Tus, upon combining (5. to (5.5, we obtain (5.6 = f n f n g n [ ( n T x n 3/ nr k ε k K εk ] dx ( g n (x + Op. n 5 Let Z n denote te first term in te rigt and side above. To obtain its rate of convergenc, introduce ( x G k (x = ε k K εk [ E ε 0 K ( x ε0 Also, write n = ( n,,..., n,p+q+, and r k = (r k,, r k,,, r k,p+q+. Ten, ( n T x nr k ε k K εk g n (x = n T n and by te Caucy-Scwarz inequality, ]. r k G k (x, Z n = = [ n 3/ n T n n 3 4 n 3 4 [ p+q+ p+q+ j= j= dx r k G k (x] n n,j dx r k,j G k (x] p+q+ ( n n,j j= [ dx. r k,j G k (x] Ten, Fix a j p + q +. Let F l denote te σ algebra generated by te r.v. s {ε k, k l}. E ( r k,j G k (x Fk = E ( rk,j EG0 (x = 0, k, x. Hence, [ E r k,j G k (x] dx = [ ] dx E r k,j G k (x = ner 0,j E [ G 0(x ] dx. 5

16 But, since G 0 is centered, [ ( x EG 0(xdx E ε 0 K ε0 ] dx = (x z ( K (z f(x zdzdx K y f(ydy = O(, because K is bounded, as compact support, and because x f(xdx <. Tis togeter wit Fact 5.4 proves tat Z n = O p (/n 3, wic in turns, togeter wit (5.6, (5.5 and (5.4 completes te proof of (3.. Next, we prove (3.. Wit S n defined at (5., observe tat [ n g n ] [ = n T { ( x ns n E ε 0 K ε0 }] dx. Let ψ(x := f(x + xf(x. We sall sortly prove [ { ( x E ε 0 K ε0 } dx (5.7 ψ(x] 0. Consequently, in view of (5.3, (5.8 n g n = [ ] n T ner 0 ψ (xdx + o(, Tis result togeter wit (.5 completes te proof of (3.. To prove (5.7, recall K is a density on [, ], vanising at te end points. Use te cange of variable formula, integration by parts and f being a.c. to write { ( x E ε 0K ε0 } = (x zk (zf(x zdz = (x zf(x zdk(z = K(z { f(x z + (x z( f(x z } dz = K(z { f(x z + (x z f(x z } dz. a.s. Hence, [ { ( x E ε 0K ε0 } dx ψ(x] [ {f(x } { = z f(x K(zdz + x f(x z f(x } K(zdz dx zf(x zk(zdz] 4 ( B + B + B 3, 6

17 were B := B := [ {f(x z f(x } K(zdz ] dx, [ x { f(x z f(x } dx, [ K(zdz] B3 = z f(x zk(zdz] dx. By te Caucy-Scwarz inequality and Fubini Teorem, and because K is a density, {f(x } { x dxk(zdz dxk(zdz B z f(x = f(sds} x z x ( z f(s dsdxk(zdz ( z K(zdz f(s ds = O(, x z by assumption (.9. Similarly, te same assumption implies x { f(x z f(x } dxk(zdz 0. B Finally, B 3 z ( f(x z dxk(zdz = z K(zdz ( f(s ds = O(. Tis completes te proof of (5.7. Claim (3.3 follows from (3. and (3., tereby completing te proof of Teorem 3.. Proof of Proposition 3.. To begin wit note tat by te Caucy-Swarz inequality, (., (3. and (.0, n ( f n (x f n (x g n (x(f n (x Ef n (x dx n fn f n g n n fn Ef n = O p (n / 5/ O p ( = o p (. Hence, (5.9 n ( ( fn f n fn Ef n dx = n g n ( fn Ef n dx + op (. Moreover, (5.0 (5. We claim n ( g n fn (x Ef n (x dx = n T n S n n [ ε0 ( x ε0 ]( E K f n (x Ef n (x dx. ( x n [ε E 0 K ε0 ]( ψ(x f n (x Ef n (x dx p 0. 7

18 Tis follows from (5.7 and te fact E ( n f n Ef n = O(, because te square of te left and side of (5. is bounded above by [ { ( x E ε 0 K ε0 } ] dx ψ(x n fn Ef n. Terefore, in view of (5.9 and (5.0, n ( ( fn f n fn Ef n dx = n T n S n n ψ(x[f n (x Ef n (x]dx + o p (. Next, we sall sow tat ( (5. V := Var n ψ(x[f n (x Ef n (x]dx = o(. We ave V = n { ψ(xψ(ye f n (x Ef n (x ( } f n (y Ef n (y dxdy. But { (fn E (x Ef n (x ( f n (y Ef n (y } { [ ( x ε0 ( y ε0 ] ( y ε0 ( x ε0 } = E K K EK EK n = ( x y K + w K(wf(y wzdw n K(tf(x tdt K(sf(y sds n =: A n, (x, y B n, (x, y, say. Hence, one can write V = V V, were V := ψ(xψ(ya n, (x, ydxdy = ( x y ψ(xψ(yk + w K(wf(y wdwdxdy = ψ(t sψ(tk(s + wk(wf(t wdwdsdt f (K K(s ψ(t sψ(tdtds 0, by C(f. Similarly, V := ψ(xψ(yb n, (x, ydxdy = ψ(xψ(y K(tf(x tk(sf(y sdtdsdxdy f ψ(xψ(y K(tK(sdtdsdxdy = f ( ψ(x dx 0. 8

19 Terefore, in view of (5.3, (5., (5. and (.5, n ( fn f n ( fn Ef n dx = op (. Tis concludes te proof of Proposition 3.. Proof of Teorem 3.. Recall := f f 0. We ave ( ( (xdx n Tn K = n ( fn (x f n (x + f n (x E 0 f n (x dx n ( K (xdx = n ( fn (x f n (x dx + ( ( dx ( (xdx n f n (x E 0 f n (x K + n ( fn (x f n (x ( f n (x Ef n (x dx + n ( fn (x f n (x ( Ef n (x E 0 f n (x dx Te first term is o p ( by Teorem 3.. Te second term is exactly te left and side of (.. Te tird term is o p ( by Proposition 3.. Next, observe tat ( ( n fn f n Efn E 0 f n dx n g n (Ef n E 0 f n dx n f Efn n f n g n E 0 f n = o p (, by Teorem 3., and C(f and C(f 0. Terefore, ( n fn (x f n (x ( Ef n (x E 0 f n (x dx = (5.3 n g n (Ef n E 0 f n dx + o p (. Note tat n g n (Ef n (x E 0 f n (x dx = n T n S n n [ ε0 ( x ε0 ]( E K Ef n (x E 0 f n (x dx. Recall ψ(x = f(x + x f(x. Because of C(f and C(f 0, and (5.7, one readily sees E [ ( x ɛ 0 K ɛ0 ]( Ef n (x E 0 f n (x dx ψ(x ( Ef n (x E 0 f n (x dx = o p (. 9

20 But, since under C(f, C(f 0, f and f 0 are bounded and f(x+x f(x is square integrable, te dominated convergence teorem implies, ψ(x[ef n (x E 0 f n (x]dx = ψ(x K(z[f(x z f 0 (x z]dzdx Tis, in view of (.5 and (5.3, completes te proof of Teorem 3.. ψ(x[f(x f 0 (x]dx. References Bacmann D. and Dette H. (005. A note on te Bickel-Rosenblatt test in autoregressive time series. Statist. Probab. Lett., 74, -34. Berkes I., Horvát L., Kokoszka P. (003. GARCH process: structure and estimation. Bernoulli, Bickel, P. and M. Rosenblatt (973. On some global measures of te deviations of density function estimates. Annals of Statistics, Bougerol, P. and Picard, N. (99a. Strict stationarity of generalized autoregressive process. Ann. Probaab, Bougerol, P. and Picard, N. (99b. Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics, Ceng, F. (008. Asymptotic properties in ARCH(p-time series. Journal of Nonparametric Statistics 0, Horvát L. and Zitikis, R. (006. Testing goodness of fit based on densities of GARCH innovations. Econometric Teory, Lee S. and Na S. (00. On te BickelRosenblatt test for first-order autoregressive models. Statist. Probab. Lett Lee, S.W. and B.E. Hansen. (994. Asymptotic teory for te GARCH(, quasimaximum likeliood estimator. Econometric Teory, Lumsdaine, R.L. (996. Consistency and asymptotic normality of te quasi-maximum likeliood estimator in IGARCH(, and covariance stationary GARCH(, models. Econometrica, Nelson, D.B. (990. Stationarity and persistence in te GARCH(, model. Econometric Teory, 6, Mimoto, N. (008. Convergence in distribution for te sup-norm of a kernel density estimator for GARCH innovations Statist. Probab. Lett., 78,