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...ε0| < ∞ implies that |K(x) − K(y)| ≤ C|x − y| ∀x,y ∈ R. sup | x∈R ˆ fn(x) − fn(x)| = oP(1). The proof of Theorem 2.2 follows easily from Theorem 1.3. Given the conditions in Theorem 2.2, we have (cf. =-=[21]-=-) Thus, by Theorem 2.2, sup |fn(x) − f(x)| = oP(1). x∈R sup | x∈R ˆ fn(x) − f(x)| = oP(1). Notice in the above result that only the finite first innovation moment and √ n consistency of the parameter ...

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...metric and discrete time financial modeling, Engle’s [12] ARCH model plays a fundamental role; see [10], or the volume edited by Rossi [20]. The ARCH model has been generalized to GARCH by Bollerslev =-=[5]-=-. Received April 2003; revised November 2004. 1 Supported by grants from the Natural Sciences and Engineering Research Council of Canada. AMS 2000 subject classifications. 60F17, 62M99, 62M10. Key wor...

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...tribution is discussed. 1. Introduction and results. In nonlinear time series and in particular econometric and discrete time financial modeling, Engle’s [12] ARCH model plays a fundamental role; see =-=[10]-=-, or the volume edited by Rossi [20]. The ARCH model has been generalized to GARCH by Bollerslev [5]. Received April 2003; revised November 2004. 1 Supported by grants from the Natural Sciences and En...

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...sity function estimation of the innovation distribution is discussed. 1. Introduction and results. In nonlinear time series and in particular econometric and discrete time financial modeling, Engle’s =-=[12]-=- ARCH model plays a fundamental role; see [10], or the volume edited by Rossi [20]. The ARCH model has been generalized to GARCH by Bollerslev [5]. Received April 2003; revised November 2004. 1 Suppor...

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...mong others. They point out that it is asymptotically equivalent to the likelihood ratio test, implying it has the same asymptotic power characteristics and, hence, has maximum local asymptotic power =-=[11]-=-. Therefore, a test based on JB is asymptotically locally most powerful against the Pearson family, and (2.3) shows that JB is asymptotically distributed as χ2 (2). The hypothesis of normality is reje...

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...he CUSUM normalized high moment partial sum process { ˆ S (k) n (u) − u ˆ S (k) n (1),0 ≤ u ≤ 1} defined in Theorem 1.2. It is related to the standard CUSUM test introduced by Brown, Durbin and Evans =-=[9]-=-, which was one of the first tests on structural change with unknown break point.10 R. KULPERGER AND H. YU We first consider a structural change in the conditional mean for GARCH models. We can formu...

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...| + ξE|Y0| ∣n ∣ n 0≤u≤1 t=1 t=1 ξ≤u≤1∣ [nu] 1 ∑ + sup Yt − uEY0 n ∣ t=1 ≤ [nξ] 1 ∑ (|Yt| − E|Y0|) ∣n ∣ t=1 j∑ 1 + 3ξE|Y0| + sup (Yt − EY0) ∣j ∣ . j≥[nξ] Since Yt = h(εt,εt−1,...), by Theorem 3.5.8 of =-=[22]-=- {Yt} is stationary and ergodic. Thus, as n → ∞, t=1 and [nξ] 1 ∑ (|Yt| − E|Y0|) = oP(1) [nξ] t=1 1 sup ∣j j≥[nξ] j∑ (Yt − EY0) = oP(1). ∣ t=1 □ Lemma 3.7. Suppose that (1.6) and (1.14) hold. Then, fo...

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...2 D − 3) −→ χ 2 (2). (2.3) The statistic JB in (2.3) is the Jarque–Bera normality test widely used in econometrics and implemented in standard statistical packages such as S-PLUS, and Jarque and Bera =-=[15]-=- show that JB is a Lagrange multiplier test statistic of normality against alternatives within the Pearson family of distributions, which includes the beta, gamma and Student’s t distributions among o...

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...s. Throughout this paper we assume that (1.1)–(1.4) hold, so that, by definition, µ1 = 0 and µ2 = 1. The existence of a unique strictly stationary solution of (1.1) and (1.2) is well established. See =-=[6, 7]-=- for details. In this paper a minimal set of conditions in [6, 7] for the existence and stationarity of the GARCH(p,q) sequence {Xt, −∞ < t < ∞} is assumed, plus the assumption (1.4). Estimation of th...

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...s. Throughout this paper we assume that (1.1)–(1.4) hold, so that, by definition, µ1 = 0 and µ2 = 1. The existence of a unique strictly stationary solution of (1.1) and (1.2) is well established. See =-=[6, 7]-=- for details. In this paper a minimal set of conditions in [6, 7] for the existence and stationarity of the GARCH(p,q) sequence {Xt, −∞ < t < ∞} is assumed, plus the assumption (1.4). Estimation of th...

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... sample X0,X1,...,Xn, and that it is √ n consistent, that is, √ (1.5) n| θn ˆ − θ| = OP(1), where we use | · | to denote the maximum norm of vectors or matrices. Recently Berkes, Horváth and Kokoszka =-=[3]-=- studied the asymptotic properties of the quasi-maximum likelihood estimator for θ in GARCH(p,q) models under mild conditions. Berkes and Horváth [1] have shown that the quasimaximum likelihood estima...

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...n) The sample skewness and kurtosis of the residuals are ˆγn(1) and ˆκn(1), respectively. Omnibus statistics based on sample skewness and kurtosis have been used to test normality. Bowman and Shenton =-=[8]-=- and Gasser [13] give details of this method. The basic idea is to construct the statistic n where, by (1.16), σ 2 γ (ˆγn(1) − λ3) 2 + n σ2 (ˆκn(1) − λ4) κ 2 , σ 2 γ = E(B (3) (1)) 2 = (λ6 − λ 2 3) + ...

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...for θ in GARCH(p,q) models under mild conditions. Berkes and Horváth [1] have shown that the quasimaximum likelihood estimator cannot be √ n consistent if E|ε0| k = ∞ for some 0 < k < 4. Hall and Yao =-=[14]-=- also studied inference for GARCH models under slightly stronger assumptions on the parameters. To remove such limitations as the finite fourth innovations moment, Berkes and Horváth [1] have used an ...

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...proposed test statistics. They outperform the CUSUM test constructed from the squares of the original data by Kim, Cho and Lee [17]. Some details are given in [23]. Independently, Kokoszka and Leipus =-=[18]-=- also study a change point for an ARCH process, again based on the original observations and not residuals. Here we list empirical sizes and powers of the CUSUM (2) 2 test. The significance level is 5...

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...or matrices. Recently Berkes, Horváth and Kokoszka [3] studied the asymptotic properties of the quasi-maximum likelihood estimator for θ in GARCH(p,q) models under mild conditions. Berkes and Horváth =-=[1]-=- have shown that the quasimaximum likelihood estimator cannot be √ n consistent if E|ε0| k = ∞ for some 0 < k < 4. Hall and Yao [14] also studied inference for GARCH models under slightly stronger ass...

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...ion and results. In nonlinear time series and in particular econometric and discrete time financial modeling, Engle’s [12] ARCH model plays a fundamental role; see [10], or the volume edited by Rossi =-=[20]-=-. The ARCH model has been generalized to GARCH by Bollerslev [5]. Received April 2003; revised November 2004. 1 Supported by grants from the Natural Sciences and Engineering Research Council of Canada...

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... Then the invariance principle for partial sums of iid sequence {kt } (cf. Billingsley (1999)) implies that{ S (k) n (x)− [nx]S(k)n (1)/n νk √ n , 0 ≤ x ≤ 1 } converges weakly in the Skorokhod spaceD=-=[0, 1]-=- to a Brownian bridge {B(x), 0 ≤ x ≤ 1}. Hence the following result follows from Theorem 1.3. Corollary 1.1 If (A1) to (A3) and (A4) or (A4’) hold, then E|0|2k < ∞ for some integer k ≥ 1 implies { Ŝ...

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...rge. Preliminary empirical studies show promising results from the above proposed test statistics. They outperform the CUSUM test constructed from the squares of the original data by Kim, Cho and Lee =-=[17]-=-. Some details are given in [23]. Independently, Kokoszka and Leipus [18] also study a change point for an ARCH process, again based on the original observations and not residuals. Here we list empiri...

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...kewness and kurtosis of the residuals are ˆγn(1) and ˆκn(1), respectively. Omnibus statistics based on sample skewness and kurtosis have been used to test normality. Bowman and Shenton [8] and Gasser =-=[13]-=- give details of this method. The basic idea is to construct the statistic n where, by (1.16), σ 2 γ (ˆγn(1) − λ3) 2 + n σ2 (ˆκn(1) − λ4) κ 2 , σ 2 γ = E(B (3) (1)) 2 = (λ6 − λ 2 3) + 3(3 + 3λ 2 3 − 2...

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...ues of the JB test as JB0.05 = 5.991645 − 15.17n −1/2 + 345.9n −1 − 3110.8n −3/2 , n ≥ 100. We are unaware of any other results studying the Jarque–Bera test for GARCH residuals. Kilian and Demiroglu =-=[16]-=- studied the Jarque–Bera test for autoregressive residuals. 2.3. Kernel density estimation of the innovation distribution. The omnibus type statistic discussed in Section 2.2 provides a means to test ...

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...ρ E i Nci(θ)X 2 t−i 1 + ∑∞ i=1 ci(θ)X2 ) κ∗ < ∞. t−i22 R. KULPERGER AND H. YU Since ρN can be close enough to 1 if N is large enough, the above inequality follows from the same proof of Lemma 3.7 of =-=[2]-=-. For completeness, we give a detailed proof here. By Lemma 3.1 of [3], there are constants C2 and 0 < ρ < 1 such that |ci(u)| ≤ C2ρ i Then for any M ≥ 1, we have ∑ ∞i=1 ρ i N ci(θ)X 2 t−i By Lemma 2....

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...t JB is asymptotically distributed as χ2 (2). The hypothesis of normality is rejected for large sample size, if the computed value of JB is greater than the appropriate critical value of a χ2 (2). Lu =-=[19]-=- has used Monte Carlo simulation to obtain critical values for several different n. Based on this, a finite sample size correction can also be used to improve the choice of the critical value. Lu [19]...

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...es show promising results from the above proposed test statistics. They outperform the CUSUM test constructed from the squares of the original data by Kim, Cho and Lee [17]. Some details are given in =-=[23]-=-. Independently, Kokoszka and Leipus [18] also study a change point for an ARCH process, again based on the original observations and not residuals. Here we list empirical sizes and powers of the CUSU...