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## HIGH MOMENT PARTIAL SUM PROCESSES OF RESIDUALS IN GARCH MODELS AND THEIR APPLICATIONS 1 (2006)

Citations: | 14 - 0 self |

### Citations

5193 | Probability and Measure - Billingsley - 1995 |

3800 |
Density Estimation for Statistics and Data Analysis
- Silverman
- 1986
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Citation Context ...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 ... |

2400 | Generalized Autoregressive Conditional Heteroskedasticity
- Bollerslev
- 1986
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Citation Context ...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... |

2088 |
The Econometrics of Financial Markets
- Campbell, Lo
- 1996
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Citation Context ...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... |

1909 |
Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50
- Engle
- 1982
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Citation Context ...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... |

1050 | New Introduction to Multiple Time Series Analysis - Lütkepohl - 2005 |

754 |
Theoretical statistics
- Cox, Hinckley
- 1974
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Citation Context ...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... |

521 |
Techniques for testing constancy of regression relationships over time" (with discussion
- Brown, Durbin, et al.
- 1975
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Citation Context ...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... |

255 | Efficient Tests for Normality, Homoscedasticity and Serial Independence of Regression Residuals, - Jarque, Bera - 1980 |

253 |
Almost Sure Convergence
- Stout
- 1974
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Citation Context ...| + 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... |

249 | A Test for Normality of Observations and Regression Residuals.
- Jarque, Bera
- 1987
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Citation Context ...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... |

219 | Convergence of probability measures, 2nd ed - Billingsley - 1999 |

161 |
Stationarity of GARCH processes and of some non-negative time series.
- Bougerol, Picard
- 1992
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Citation Context ...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... |

131 | Time Series: Theory and Methods. Second edition, - Brockwell, Davis - 1991 |

103 |
Strict stationarity of generalized autoregressive processes.
- Bougerol, Picard
- 1992
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Citation Context ...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... |

66 |
GARCH processes: structure and estimation.
- Berkes, Horvath, et al.
- 2003
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Citation Context ... 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... |

58 |
Omnibus test contours for departures from normality based
- Bowman, Shenton
- 1975
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Citation Context ...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) + ... |

55 |
Inference in ARCH and GARCH models with heavy-tailed errors
- Hall, Yao
- 2003
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Citation Context ...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 ... |

45 |
Change-point estimation in ARCH models,
- Kokoszka, Leipus
- 2000
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Citation Context ...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... |

40 | Weak convergence of the sequential empirical processes of residuals in ARMA models. - Bai - 1994 |

35 | Properties of Sequences of Partial Sums of Polynomial Regression Residuals With Application to Tests for Change of Regression at Unknown Times,” The Annals of - MacNeill - 1978 |

21 | Estimation of the distribution of noise in an autoregression scheme. Theory Probab. - Boldin - 1982 |

10 | A weak convergence result useful in robust autoregression - Koul - 1991 |

10 | Estimation of the distribution function of noise in stationary processes - Kreiss - 1991 |

9 |
The rate of consistency of the quasi-maximum likelihood estimator
- Berkes, Horvath
- 2003
(Show Context)
Citation Context ...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... |

8 |
Modeling Stock Market Volatility: Bridging the Gap to Continuous Time
- Rossi
- 1996
(Show Context)
Citation Context ...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... |

7 |
On the partial sums of residuals in autoregressive and moving average models.
- Bai
- 1993
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Citation Context ... 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... |

6 |
On the CUSUM test for parameter changes in GARCH(1, 1) models
- Kim, Cho, et al.
(Show Context)
Citation Context ...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... |

6 | On Testing Hypotheses in Sliding Average Scheme by the Kolmogorov-Smirnov and ! Tests," Theory of Probability and its Applications - Boldin - 1989 |

4 |
Goodness-of-fit tests for correlated data
- Gasser
- 1975
(Show Context)
Citation Context ...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... |

3 | On the residuals of autoregressive processes and polynomial regression - Kulperger - 1985 |

3 | Limit processes for sequences of partial sums of regression residuals, - MacNeill - 1978 |

2 |
Residual based tests for normality in autoregressions: asymptotic theory and simulations.
- Kilian, Demiroglu
- 2000
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Citation Context ...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 ... |

1 |
The efficiency of the estimatiors of the parameters in GARCH processes
- Berkes, Horváth
- 2004
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Citation Context ...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.... |

1 |
Identification of distribution for ARCH innovation
- Lu
- 2001
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Citation Context ...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]... |

1 |
Analyzing residual processes of (G)ARCH time series models
- Yu
- 2004
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Citation Context ...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... |

1 | Weak convergence of the sequential empirical processes of residuals of arma models with unknown mean - Yu - 2003 |