M-estimators for augmented GARCH(1,1) processes
|
|
- Catherine Walton
- 5 years ago
- Views:
Transcription
1 M-estimators for augmented GARCH(1,1) processes Freiburg, DAGStat 2013 Fabian Tinkl Chair of Statistics and Econometrics FAU Erlangen-Nuremberg
2 Outline Introduction The augmented GARCH(1,1) model Consistency and asymptotic normality of M-estimators in augmented GARCH models Consistency of the QMLE and the LADE Asymptotic normality of the QMLE and the LADE Application to the LGARCH and TGARCH model Finite sample properties Conclusion References 2
3 Introduction
4 Related works 1. Consistency and asymptotic normality of the Gaussian maximum likelihood estimator (QMLE) have been established for the linear GARCH (LGARCH) model by [Lee and Hansen, 1994] [Lumsdaine, 1996], [Berkes et al., 2003] and [Francq and Zakoïan, 2004]. 2. Consistency and asymptotic normality in asymmetric GARCH (AGARCH) models have been shown by [Straumann and Mikosch, 2006], [Pan et al., 2008] and [Hamadeh and Zakoïan, 2011]. 3. M-estimators in the linear GARCH model have been studied by [Muler and Yohai, 2008] and [Mukherjee, 2008]. 4. Least absolute deviation estimators (LADE) have been proposed by [Peng and Yao, 2003] for the LGARCH and [Pan et al., 2008] for the threshold GARCH (TGARCH) 4
5 What is new in our work? 1. We establish consistency and asymptotic normality for the QMLE and the LADE for the class of augmented GARCH models that contains as special cases the LGARCH and TGARCH and many other models. 2. We apply this general framework to derive consistency and asymptotic normality of the QMLE and LADE in the LGARCH and the TGARCH model. 3. We investigate the finite sample properties of these estimators when the process is not weakly stationary. 5
6 Definition of M-estimators 1. The basic idea is to describe estimators as functionals of the empirical distribution function. 2. Therefore, we write ˆθ n = T (F n ) for any estimator and let K R d be some compact set. 3. The M-estimator ˆθ n = T (X 1,..., X n ) is defined as the solution of the following maximization problem ˆθ n := arg max θ K 1 n M 1 n(θ) = arg max θ K n 4. The true parameter θ 0 is the solution of θ 0 := arg max θ K n ρ(x i, θ). (1) i=1 E θ0 [ρ(x i, θ)]. (2) 6
7 The augmented GARCH(1,1) model
8 Definition and properties of augmented GARCH(1,1) processes We follow [Hörmann, 2008] and define: Definition Let ɛ t be an iid sequence and let g(x) and c(x) be real-valued and measurable functions. Assume that the stochastic recurrence equation h t := h(σ 2 t ) = c(ɛ t 1 )h t 1 + g(ɛ t 1 ) (3) has a strictly stationary solution and assume h t : R + R + is an invertible function. Then, an augmented GARCH(1,1) process (X t ) t Z is defined by the equation X t = σ t ɛ t = h 1 t. (4) σ 2 t 8
9 Some Examples Model Specification LGARCH(1,1) of [Bollerslev, 1986] σt 2 = ω + αxt βσ2 t 1 AGARCH(1,1) of [Ding et al., 1993] σt 2 = ω + α( X t 1 γx t 1 ) 2 + βσt 1 2 NGARCH(1,1) of [Engle and Ng, 1993] σ 2 t = ω + α(ɛ t 1 c) 2 σ 2 t 1 + βσ2 t 1 VGARCH(1,1) of [Engle and Ng, 1993] σ t = ω + α(ɛ t 1 c) 2 + βσ 2 t 1 TSGARCH(1,1) of [Schwert, 1989] σ t = ω + α X t 1 + βσ t 1 Table: GARCH specification 9
10 Consistency and asymptotic normality of M- estimators in augmented GARCH models
11 Consistency of M-estimators: Assumptions and Notation I Assume that the following assumptions hold: (P1) ɛ t is an iid sequence, (P2) E[log + g θ (ɛ 0 ) ] <, E[log + c θ (ɛ 0 ) ] <, (P3) < E[log c θ (ɛ 0 ) ] < Then, we can express the conditional volatility as σ 2 t (θ) = h 1 g θ (ɛ t 1 i ) c θ (ɛ t i ). i=1 1 j<i 2. Since we do not observe the whole process but only for t 1, we work with the approximation: t 1 σ 2 t (θ) = h 1 g θ (ɛ t 1 i ) c θ (ɛ t i ). i=1 1 j<i 11
12 Consistency of M-estimators: Assumptions and Notation II 3. The QMLE is defined as ˆθ n = arg max θ K 1 n M n(θ) = arg max θ K 1 2n n t=1 log σ 2 t (θ) + X t 2 σt 2(θ). 4. The QMLE based on the approximation σ t is defined as ˆθ n = arg max 1 θ K n M n (θ) = arg max 1 n log σ 2 t (θ) + X t 2 θ K 2n σ t 2(θ). The following assumptions are needed to ensure the a.s. convergence of the approximation σ t to σ t holds. t=1 12
13 Consistency of M-estimators: Assumptions and Notation (F1) It holds that a.s. c θ (ɛ 0 ) 0 and g θ (ɛ 0 ) 0 for all θ K. (F2) Additionally, we have E[ c θ (ɛ 0 ) µ ] < 1 and E[ g θ (ɛ 0 ) µ ] < for some µ > 0 and for all θ K. (F3) The function h(σ 2 t (θ)) = σ 2δ t (θ) is continuous in θ K for all t. (F4) For every θ K we have the following identifier condition: σ 2 0 (θ) = σ2 0 (θ 0) if and only if θ = θ 0. 13
14 Consistency of the QMLE for augmented GARCH models I Theorem (Consistency) Let 0 < δ and let θ 0 K for some compact set K R d. Assume that (P1)-(P3) and (F1)-(F4) hold and that E[ɛ 2 0 ] = 1. Then, it follows that a.s. ˆθ n θ The idea of the proof is to show in a first step that a.s. ˆθ n θ In a second step we may demonstrate that a.s. ˆθ n ˆθ n. 14
15 Consistency of the LADE for augmented GARCH models I 1. The LADE for the parameters of an augmented GARCH model can be defined by 1 n ˆθ n = arg max θ K n M n(θ) = arg max n 1 log X 2 t log σ 2 t (θ). (5) θ K 2. ˆθ n is obtained replacing σ t 2 by σ 2 t. Theorem Let 0 < δ and θ 0 K for some compact set K R d. Assume (P1)-(P3) and (F1)-(F4) hold and let the iid sequence v t = log ɛ 2 t has unique median at 0. Then, it follows that a.s. ˆθ n θ 0. t=1 15
16 Asymptotic normality of the QMLE I (N1) The true parameter θ 0 lies in the interior of K, denoted with K. (N2) The distribution of ɛ t is such that E[ɛ 2 0 ] = 1 and E[ɛ4 0 ] <. (N3) There exists a convex set K K, containing θ0, such that σ 2 t (θ) is three times continuously differentiable in θ with measurable derivatives such that (i) E θ0 [log + θ σ 2 t (θ) K ] <, (ii) E θ0 [log + 2 θ σ2 t (θ) K ] <, (iii) E θ0 [log + 3 θ σ2 t (θ) K ] < and In addition, the following moment conditions hold: (iv) E θ0 [ [ θ ψ 0 (θ) K ] <. ] 1 (v) E θ0 σ θσ 2 0 2(θ 0) 0 (θ 0) <, [ ] 1 (vi) E θ0 σ0 2(θ 0) 2 θ σ2 0 (θ 0) <, [ ] 1 (vii) E θ0 σ θσ 2 0 4(θ 0) 0 (θ 0) θ σ2 0 (θ 0) <, (N4) The components of θ σ 2 t (θ) are linearly independent random variables. 16
17 Asymptotic normality of the QMLE II Theorem Let δ > 0 and let θ 0 K for some compact set K R d. Assume that (P1)-(P3), (F1)-(F4) and (N1)-(N4) hold. Then, it follows that n(ˆθn θ 0 ) d N(0, Σ), where [ ] 1 Σ = (E[ɛ 4 1 0] 1)E θ0 σ0 4(θ 0) θσ0(θ 2 0 ) θσ0(θ 2 0 ). 17
18 Asymptotic normality of the LADE I For the LADE we replace assumption (N2) by (L2) v 0 = log ɛ 2 0 has a continuous distribution G(v) with G(0) = 1/2 and density g(0) > 0 such that sup v g(v) B 1 <. Theorem Let δ > 0 and v t = log ɛ 2 t have a unique median at 0 with density function g(v) such that g(0) > 0. Assume that (P1)-(P3), (F1)-(F4) and (N1),(L2),(N3)-(N4) hold. Then, it follows that n(ˆθn θ 0 ) d N(0, Σ), (6) where [ ] 1 1 Σ = E θ0 σ0 4(θ 0) θσ0(θ 2 0 ) θσ0(θ 2 0 ) /(4(g(0)) 2 ). 18
19 Application to the LGARCH and TGARCH model
20 Consistency and asymptotic normality of the QMLE in LGARCH(1,1) model I 1. Let g(ɛ t ) = ω, c(ɛ t ) = αɛ 2 t + β h(x) = x, we get the LGARCH(1,1) model of [Bollerslev, 1986]: σ 2 t = ω + αx 2 t 1 + βσ 2 t The parameter space is given by K c = [c, 1/c] {α R + 0, β [0, 1 c]} 3. Consistency and asymptotic normality of the QMLE and LADE follows by application of the previous theorems. 4. The assumptions are the same as in [Francq and Zakoïan, 2004]. 5. The assumptions for the LADE are weaker than those in [Peng and Yao, 2003]. 20
21 Consistency and asymptotic normality of the QMLE in TGARCH(1,1) model I 1. Let g(ɛ t ) = ω, c(ɛ t ) = α + ɛ t + α max(0, ɛ t ) + β h(x) = x 1/2, we get the TGARCH(1,1) model of [Zakoïan, 1994]: 2. The parameter space is given by σ t = ω + α + X t 1 + α X t 1 + βσ t 1. K d = [d, 1/d] {α + [d, 1 d], α [d, 1 d], β [0, 1 d]} 3. Consistency and asymptotic normality of the QMLE and LADE follows by application of the previous theorems. 4. The assumptions for the QMLE are the same as in [Hamadeh and Zakoïan, 2011]. 5. A similar result for a different type of LADE have been established by [Pan et al., 2008]. 21
22 Finite sample properties
23 Simulation study 1. We simulate different GARCH models with different error distributions, namely ɛ t N(0, 1), t(5), t(3). 2. The length of the time series are T = 200, 500, 1000, We repeat the estimation of the parameters 1000 times for each time-length and error distribution and averaged the results i=1 (ˆθ i,j θ 0,j ) 2 for the jth 4. We also report the RMSE j = parameter. 5. The following boxplots report the absolute average error when an integrated LGARCH model is considered with θ 0 = (ω 0, α 0, β 0 ) = (0.1, 0.6, 0.4). 23
24 Results for the LGARCH(1,1) Average absolute error for N=200 Average absolute error for N= QMLE_NV LADE_NV QMLE_t5 LADE_t5 QMLE_t3 LADE_t3 QMLE_NV LADE_NV QMLE_t5 LADE_t5 QMLE_t3 LADE_t3 24
25 Results for the LGARCH(1,1) Average absolute error for N=1000 Average absolute error for N= QMLE_NV LADE_NV QMLE_t5 LADE_t5 QMLE_t3 LADE_t3 QMLE_NV LADE_NV QMLE_t5 LADE_t5 QMLE_t3 LADE_t3 25
26 Conclusion
27 Summary of the results 1. We discuss the properties of the QMLE and the LADE for augmented GARCH models. 2. We derive conditions for consistency and asymptotic normality of the QMLE and the LADE that may easily checked whenever a specific model is considered. 3. Since we also propose a correction routine the results of the LADE are directly comparable with the QMLE. 4. The heavier the tails of the error distribution the more preferable is the LADE but the QMLE may still be useful for instance to obtain starting values for the LADE optimization routine. 27
28 References
29 References I Berkes, I., Horváth, L., and Kokoska, P. (2003). GARCH processes: Structure and estimation. Bernoulli, 9: Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31: Ding, Z., Granger, C. W., and Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1: Engle, R. F. and Ng, V. K. (1993). Measuring and testing the impact of news on volatility. The Journal of Finance, 48: Francq, C. and Zakoïan, J. (2004). Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli, 10: Hamadeh, T. and Zakoïan, J.-M. (2011). Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH processes. Journal of Statistical Planning and Inference, 141: Hörmann, S. (2008). Augmented GARCH sequences: Dependence structure and asymptotics. Bernoulli, 14:
30 References II Lee, S.-W. and Hansen, B. E. (1994). Asymptotic theory for the GARCH (1,1) quasi-maximum likelihood estimator. Econometric Theory, 10: Lumsdaine, R. L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica, 64: Mukherjee, K. (2008). M-estimation in GARCH models. Econometric Theory, 24: Muler, N. and Yohai, V. (2008). Robust estimates for GARCH models. Journal of Statistical Planning and Inference, 138: Pan, J., Wang, H., and Tong, H. (2008). Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics, 142: Peng, L. and Yao, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika, 90: Schwert, G. W. (1989). Why does stock market volatility change over time? Journal of Finance, 44:
31 References III Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Annals of Statistics, 34: Zakoïan, J. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18:
Location Multiplicative Error Model. Asymptotic Inference and Empirical Analysis
: Asymptotic Inference and Empirical Analysis Qian Li Department of Mathematics and Statistics University of Missouri-Kansas City ql35d@mail.umkc.edu October 29, 2015 Outline of Topics Introduction GARCH
More informationHausman tests for the error distribution in conditionally heteroskedastic models
MPRA Munich Personal RePEc Archive Hausman tests for the error distribution in conditionally heteroskedastic models Ke Zhu Institute of Applied Mathematics, Chinese Academy of Sciences 30. September 205
More informationEstimating GARCH models: when to use what?
Econometrics Journal (2008), volume, pp. 27 38. doi: 0./j.368-423X.2008.00229.x Estimating GARCH models: when to use what? DA HUANG, HANSHENG WANG AND QIWEI YAO, Guanghua School of Management, Peking University,
More informationAnalytical derivates of the APARCH model
Analytical derivates of the APARCH model Sébastien Laurent Forthcoming in Computational Economics October 24, 2003 Abstract his paper derives analytical expressions for the score of the APARCH model of
More informationConsistency of Quasi-Maximum Likelihood Estimators for the Regime-Switching GARCH Models
Consistency of Quasi-Maximum Likelihood Estimators for the Regime-Switching GARCH Models Yingfu Xie Research Report Centre of Biostochastics Swedish University of Report 2005:3 Agricultural Sciences ISSN
More informationGARCH processes probabilistic properties (Part 1)
GARCH processes probabilistic properties (Part 1) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationGARCH Models. Eduardo Rossi University of Pavia. December Rossi GARCH Financial Econometrics / 50
GARCH Models Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics - 013 1 / 50 Outline 1 Stylized Facts ARCH model: definition 3 GARCH model 4 EGARCH 5 Asymmetric Models 6
More informationJuly 31, 2009 / Ben Kedem Symposium
ing The s ing The Department of Statistics North Carolina State University July 31, 2009 / Ben Kedem Symposium Outline ing The s 1 2 s 3 4 5 Ben Kedem ing The s Ben has made many contributions to time
More informationParameter Estimation for ARCH(1) Models Based on Kalman Filter
Applied Mathematical Sciences, Vol. 8, 2014, no. 56, 2783-2791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43164 Parameter Estimation for ARCH(1) Models Based on Kalman Filter Jelloul
More informationAsymptotic inference for a nonstationary double ar(1) model
Asymptotic inference for a nonstationary double ar() model By SHIQING LING and DONG LI Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong maling@ust.hk malidong@ust.hk
More informationGARCH Models Estimation and Inference
GARCH Models Estimation and Inference Eduardo Rossi University of Pavia December 013 Rossi GARCH Financial Econometrics - 013 1 / 1 Likelihood function The procedure most often used in estimating θ 0 in
More informationASYMPTOTIC NORMALITY OF THE QMLE ESTIMATOR OF ARCH IN THE NONSTATIONARY CASE
Econometrica, Vol. 7, No. March, 004), 4 4 ASYMPOIC NORMALIY OF HE QMLE ESIMAOR OF ARCH IN HE NONSAIONARY CASE BY SØREN OLVER JENSEN AND ANDERS RAHBEK We establish consistency and asymptotic normality
More informationDiagnostic Test for GARCH Models Based on Absolute Residual Autocorrelations
Diagnostic Test for GARCH Models Based on Absolute Residual Autocorrelations Farhat Iqbal Department of Statistics, University of Balochistan Quetta-Pakistan farhatiqb@gmail.com Abstract In this paper
More informationAsymptotic Theory for GARCH-in-mean Models
Western University Scholarship@Western Electronic Thesis and Dissertation Repository January 2014 Asymptotic Theory for GARCH-in-mean Models Weiwei Liu The University of Western Ontario Supervisor Reg
More informationInterval Estimation for AR(1) and GARCH(1,1) Models
for AR(1) and GARCH(1,1) Models School of Mathematics Georgia Institute of Technology 2010 This talk is based on the following papers: Ngai Hang Chan, Deyuan Li and (2010). Toward a Unified of Autoregressions.
More informationQUASI-MAXIMUM-LIKELIHOOD ESTIMATION IN CONDITIONALLY HETEROSCEDASTIC TIME SERIES: A STOCHASTIC RECURRENCE EQUATIONS APPROACH 1
The Annals of Statistics 2006, Vol. 34, No. 5, 2449 2495 DOI: 10.1214/009053606000000803 Institute of Mathematical Statistics, 2006 QUASI-MAXIMUM-LIKELIHOOD ESTIMATION IN CONDITIONALLY HETEROSCEDASTIC
More informationGARCH Models Estimation and Inference. Eduardo Rossi University of Pavia
GARCH Models Estimation and Inference Eduardo Rossi University of Pavia Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function
More information13. Estimation and Extensions in the ARCH model. MA6622, Ernesto Mordecki, CityU, HK, References for this Lecture:
13. Estimation and Extensions in the ARCH model MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: Robert F. Engle. GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics,
More informationECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications
ECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications Yongmiao Hong Department of Economics & Department of Statistical Sciences Cornell University Spring 2019 Time and uncertainty
More informationA Stochastic Recurrence Equation Approach to Stationarity and phi-mixing of a Class of Nonlinear ARCH Models
TI 2017-072/III Tinbergen Institute Discussion Paper A Stochastic Recurrence Equation Approach to Stationarity and phi-mixing of a Class of Nonlinear ARCH Models Francisco F.) Blasques Marc Nientker2 1
More informationarxiv: v1 [math.st] 21 Nov 2018
Portmanteau test for the asymmetric power GARCH model when the power is unknown Y. Boubacar Maïnassara a, O. Kadmiri a, B. Saussereau a arxiv:1811.08769v1 [math.st] 21 Nov 2018 Abstract a Université Bourgogne
More informationContinuous invertibility and stable QML estimation of the EGARCH(1,1) model
Continuous invertibility and stable QML estimation of the EGARCH(1,1) model Olivier Wintenberger To cite this version: Olivier Wintenberger. Continuous invertibility and stable QML estimation of the EGARCH(1,1)
More informationGoodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach
Goodness-of-Fit Tests for Time Series Models: A Score-Marked Empirical Process Approach By Shiqing Ling Department of Mathematics Hong Kong University of Science and Technology Let {y t : t = 0, ±1, ±2,
More informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
More informationOn the threshold hyperbolic GARCH models. Citation Statistics And Its Interface, 2011, v. 4 n. 2, p
Title On the threshold hyperbolic GARCH models Authors) Kwan, W; Li, WK; Li, G Citation Statistics And Its Interface, 20, v. 4 n. 2, p. 59-66 Issued Date 20 URL http://hdl.hle.net/0722/35494 Rights Statistics
More informationGARCH Models Estimation and Inference
Università di Pavia GARCH Models Estimation and Inference Eduardo Rossi Likelihood function The procedure most often used in estimating θ 0 in ARCH models involves the maximization of a likelihood function
More informationECON3327: Financial Econometrics, Spring 2016
ECON3327: Financial Econometrics, Spring 2016 Wooldridge, Introductory Econometrics (5th ed, 2012) Chapter 11: OLS with time series data Stationary and weakly dependent time series The notion of a stationary
More informationNonlinear Parameter Estimation for State-Space ARCH Models with Missing Observations
Nonlinear Parameter Estimation for State-Space ARCH Models with Missing Observations SEBASTIÁN OSSANDÓN Pontificia Universidad Católica de Valparaíso Instituto de Matemáticas Blanco Viel 596, Cerro Barón,
More informationGeneralized Autoregressive Score Models
Generalized Autoregressive Score Models by: Drew Creal, Siem Jan Koopman, André Lucas To capture the dynamic behavior of univariate and multivariate time series processes, we can allow parameters to be
More informationGARCH Model Estimation Using Estimated Quadratic Variation
GARCH Model Estimation Using Estimated Quadratic Variation John W. Galbraith, Victoria Zinde-Walsh and Jingmei Zhu Department of Economics, McGill University Abstract We consider estimates of the parameters
More informationStrong Approximation for the Sums of Squares of Augmented GARCH Sequences
Strong Approximation for the Sums of Squares of Augmented GARCH Sequences Alexander Aue 1 István Berkes Lajos Horváth 3 Abstract: We study so called augmented GARCH sequences, which include many submodels
More informationAsymptotic theory for the QMLE in GARCH-X models with stationary and non-stationary covariates
Asymptotic theory for the QMLE in GARCH-X models with stationary and non-stationary covariates Heejoon Han Dennis Kristensen The Institute for Fiscal Studies Department of Economics, UCL cemmap working
More informationLes Cahiers de la Chaire / N 13
Les Cahiers de la Chaire / N 3 GARCH(,) models with exogenously-driven volatility : structure and estimation Nazim Regnard, Jean-Michel Zakoian GARCH(,) models with exogenously-driven volatility : structure
More informationStationarity, Memory and Parameter Estimation of FIGARCH Models
WORKING PAPER n.03.09 July 2003 Stationarity, Memory and Parameter Estimation of FIGARCH Models M. Caporin 1 1 GRETA, Venice Stationarity, Memory and Parameter Estimation of FIGARCH Models Massimiliano
More informationQML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS. By Christian Francq. CREST (CNRS) and University Lille 3 (EQUIPPE) and
QML ESTIMATION OF A CLASS OF MULTIVARIATE ASYMMETRIC GARCH MODELS By Christian Francq CREST CNRS) and University Lille 3 EQUIPPE) and By Jean-Michel Zakoïan CREST and University Lille 3 EQUIPPE) We establish
More informationDYNAMIC CONDITIONAL CORRELATIONS FOR ASYMMETRIC PROCESSES
J. Japan Statist. Soc. Vol. 41 No. 2 2011 143 157 DYNAMIC CONDITIONAL CORRELATIONS FOR ASYMMETRIC PROCESSES Manabu Asai* and Michael McAleer** *** **** ***** The paper develops a new Dynamic Conditional
More informationRank-Based Estimation for GARCH Processes
Rank-Based Estimation for GARCH Processes Beth Andrews Northwestern University September 7, 2011 Abstract We consider a rank-based technique for estimating GARCH model parameters, some of which are scale
More informationLeast squares estimation for GARCH (1,1) model with heavy tailed errors
MPRA Munich Personal RePEc Archive Least squares estimation for GARCH (1,1) model with heavy tailed errors Arie Preminger and Giuseppe Storti Ben Gurion University of the Negev, Beer-Sheva, Israel., University
More informationA new hyperbolic GARCH model
A new hyperbolic GARCH model Muyi Li, Wai Keung Li and Guodong Li Xiamen University and University of Hong Kong April 16, 2014 Abstract There are two commonly used hyperbolic GARCH processes, the FIGARCH
More informationUnconditional skewness from asymmetry in the conditional mean and variance
Unconditional skewness from asymmetry in the conditional mean and variance Annastiina Silvennoinen, Timo Teräsvirta, and Changli He Department of Economic Statistics, Stockholm School of Economics, P.
More informationUnified Discrete-Time and Continuous-Time Models. and High-Frequency Financial Data
Unified Discrete-Time and Continuous-Time Models and Statistical Inferences for Merged Low-Frequency and High-Frequency Financial Data Donggyu Kim and Yazhen Wang University of Wisconsin-Madison December
More informationBayesian Semiparametric GARCH Models
Bayesian Semiparametric GARCH Models Xibin (Bill) Zhang and Maxwell L. King Department of Econometrics and Business Statistics Faculty of Business and Economics xibin.zhang@monash.edu Quantitative Methods
More informationThomas J. Fisher. Research Statement. Preliminary Results
Thomas J. Fisher Research Statement Preliminary Results Many applications of modern statistics involve a large number of measurements and can be considered in a linear algebra framework. In many of these
More informationBayesian Semiparametric GARCH Models
Bayesian Semiparametric GARCH Models Xibin (Bill) Zhang and Maxwell L. King Department of Econometrics and Business Statistics Faculty of Business and Economics xibin.zhang@monash.edu Quantitative Methods
More informationEmpirical likelihood and self-weighting approach for hypothesis testing of infinite variance processes and its applications
Empirical likelihood and self-weighting approach for hypothesis testing of infinite variance processes and its applications Fumiya Akashi Research Associate Department of Applied Mathematics Waseda University
More informationIntroduction to Estimation Methods for Time Series models Lecture 2
Introduction to Estimation Methods for Time Series models Lecture 2 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 2 SNS Pisa 1 / 21 Estimators:
More informationInference in VARs with Conditional Heteroskedasticity of Unknown Form
Inference in VARs with Conditional Heteroskedasticity of Unknown Form Ralf Brüggemann a Carsten Jentsch b Carsten Trenkler c University of Konstanz University of Mannheim University of Mannheim IAB Nuremberg
More informationRecursive estimation of GARCH processes
Proceedings of the 9th International Symposium on Mathematical Theory of Networks and Systems MTNS 5 9 July, Budapest, Hungary Recursive estimation of GARCH processes László Gerencsér, Zsanett Orlovits
More informationThe Nonlinear Iterative Least Squares (NL-ILS) Estimator: An Application to Volatility Models
he Nonlinear Iterative Least Squares (NL-ILS Estimator: An Application to Volatility Models Gustavo Fruet Dias Queen Mary University of London his version: November 21, 2012 Job Market Paper Abstract he
More informationPeter M. Robinson and Paolo Zafaroni London School of Economics and Banca d Italia
Pseudo-Maximum Likelihood Estimation of ARCH(1) Models Peter M. Robinson and Paolo Zafaroni London School of Economics and Banca d Italia Contents: 1 Introduction Assumptions And Main Results 3 Fractional
More informationModel estimation through matrix equations in financial econometrics
Model estimation through matrix equations in financial econometrics Federico Poloni 1 Joint work with Giacomo Sbrana 2 1 Technische Universität Berlin (A. Von Humboldt postdoctoral fellow) 2 Rouen Business
More informationThomas Mikosch and Daniel Straumann: Stable Limits of Martingale Transforms with Application to the Estimation of Garch Parameters
MaPhySto The Danish National Research Foundation: Network in Mathematical Physics and Stochastics Research Report no. 11 March 2004 Thomas Mikosch and Daniel Straumann: Stable Limits of Martingale Transforms
More informationRobust M-Estimation of Multivariate GARCH Models
Robust M-Estimation of Multivariate GARCH Models Kris Boudt Christophe Croux Faculty of Business and Economics, K.U.Leuven, Belgium Abstract In empirical work on multivariate financial time series, it
More informationDynamicAsymmetricGARCH
DynamicAsymmetricGARCH Massimiliano Caporin Dipartimento di Scienze Economiche Università Ca Foscari di Venezia Michael McAleer School of Economics and Commerce University of Western Australia Revised:
More informationISSN Article. Selection Criteria in Regime Switching Conditional Volatility Models
Econometrics 2015, 3, 289-316; doi:10.3390/econometrics3020289 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Article Selection Criteria in Regime Switching Conditional Volatility
More informationContinuous Time Approximations to GARCH(1, 1)-Family Models and Their Limiting Properties
Communications for Statistical Applications and Methods 214, Vol. 21, No. 4, 327 334 DOI: http://dx.doi.org/1.5351/csam.214.21.4.327 Print ISSN 2287-7843 / Online ISSN 2383-4757 Continuous Time Approximations
More informationStochastic Volatility Models with Long Memory
Stochastic Volatility Models with Long Memory Clifford M. Hurvich Philippe Soulier 1 Introduction In this contribution we consider models for long memory in volatility. There are a variety of ways to construct
More informationThe properties of L p -GMM estimators
The properties of L p -GMM estimators Robert de Jong and Chirok Han Michigan State University February 2000 Abstract This paper considers Generalized Method of Moment-type estimators for which a criterion
More informationResearch Article The Laplace Likelihood Ratio Test for Heteroscedasticity
International Mathematics and Mathematical Sciences Volume 2011, Article ID 249564, 7 pages doi:10.1155/2011/249564 Research Article The Laplace Likelihood Ratio Test for Heteroscedasticity J. Martin van
More informationA Functional Central Limit Theorem for an ARMA(p, q) Process with Markov Switching
Communications for Statistical Applications and Methods 2013, Vol 20, No 4, 339 345 DOI: http://dxdoiorg/105351/csam2013204339 A Functional Central Limit Theorem for an ARMAp, q) Process with Markov Switching
More informationConfidence Intervals for the Autocorrelations of the Squares of GARCH Sequences
Confidence Intervals for the Autocorrelations of the Squares of GARCH Sequences Piotr Kokoszka 1, Gilles Teyssière 2, and Aonan Zhang 3 1 Mathematics and Statistics, Utah State University, 3900 Old Main
More informationSCORE BASED GOODNESS-OF-FIT TESTS FOR TIME SERIES
Statistica Sinica 21 (2011), 1807-1829 doi:http://dx.doi.org/10.5705/ss.2009.090 SCORE BASED GOODNESS-OF-FIT TESTS FOR TIME SERIES Shiqing Ling and Howell Tong Hong Kong University of Science & Technology
More informationGraduate Econometrics I: Maximum Likelihood I
Graduate Econometrics I: Maximum Likelihood I Yves Dominicy Université libre de Bruxelles Solvay Brussels School of Economics and Management ECARES Yves Dominicy Graduate Econometrics I: Maximum Likelihood
More informationOn Fréchet autoregressive conditional duration models
Title On Fréchet autoregressive conditional duration models Authors) Zheng, Y; Li, Y; Li, G Citation Journal of Statistical Planning and Inference, 206, v. 75, p. 5-66 Issued Date 206 URL http://hdl.handle.net/0722/2354
More informationEcon 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationLEAST SQUARES ESTIMATION OF ARCH MODELS WITH MISSING OBSERVATIONS
LEAST SQUARES ESTIMATION OF ARCH MODELS WITH MISSING OBSERVATIONS Pascal Bondon CNRS UMR 8506, Université Paris XI, France. and Natalia Bahamonde Instituto de Estadística, Pontificia Universidad Católica
More informationMisspecification and domain issues in fitting Garch(1,1) models: a Monte Carlo investigation
Misspecification and domain issues in fitting Garch(,) models: a Monte Carlo investigation Leonardo Bottolo To cite this version: Leonardo Bottolo. Misspecification and domain issues in fitting Garch(,)
More informationThe autocorrelation and autocovariance functions - helpful tools in the modelling problem
The autocorrelation and autocovariance functions - helpful tools in the modelling problem J. Nowicka-Zagrajek A. Wy lomańska Institute of Mathematics and Computer Science Wroc law University of Technology,
More informationIntroduction to ARMA and GARCH processes
Introduction to ARMA and GARCH processes Fulvio Corsi SNS Pisa 3 March 2010 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March 2010 1 / 24 Stationarity Strict stationarity: (X 1,
More informationStudies in Nonlinear Dynamics & Econometrics
Studies in Nonlinear Dynamics & Econometrics Volume 9, Issue 2 2005 Article 4 A Note on the Hiemstra-Jones Test for Granger Non-causality Cees Diks Valentyn Panchenko University of Amsterdam, C.G.H.Diks@uva.nl
More informationGARCH processes continuous counterparts (Part 2)
GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationDocuments de Travail du Centre d Economie de la Sorbonne
Documents de Travail du Centre d Economie de la Sorbonne A note on self-similarity for discrete time series Dominique GUEGAN, Zhiping LU 2007.55 Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital,
More informationON CONDITIONALLY HETEROSCEDASTIC AR MODELS WITH THRESHOLDS
Statistica Sinica 24 (2014), 625-652 doi:http://dx.doi.org/10.5705/ss.2012.185 ON CONDITIONALLY HETEROSCEDASTIC AR MODELS WITH THRESHOLDS Kung-Sik Chan, Dong Li, Shiqing Ling and Howell Tong University
More informationGeneralized Autoregressive Score Smoothers
Generalized Autoregressive Score Smoothers Giuseppe Buccheri 1, Giacomo Bormetti 2, Fulvio Corsi 3,4, and Fabrizio Lillo 2 1 Scuola Normale Superiore, Italy 2 University of Bologna, Italy 3 University
More informationVILNIUS UNIVERSITY ANDRIUS ŠKARNULIS QUADRATIC ARCH MODELS WITH LONG MEMORY AND QML ESTIMATION
VILNIUS UNIVERSITY ANDRIUS ŠKARNULIS QUADRATIC ARCH MODELS WITH LONG MEMORY AND QML ESTIMATION Doctoral Dissertation Physical Sciences, Mathematics (01 P) Vilnius, 2017 The dissertation was written in
More informationThe Size and Power of Four Tests for Detecting Autoregressive Conditional Heteroskedasticity in the Presence of Serial Correlation
The Size and Power of Four s for Detecting Conditional Heteroskedasticity in the Presence of Serial Correlation A. Stan Hurn Department of Economics Unversity of Melbourne Australia and A. David McDonald
More informationFinite Sample and Optimal Inference in Possibly Nonstationary ARCH Models with Gaussian and Heavy-Tailed Errors
Finite Sample and Optimal Inference in Possibly Nonstationary ARCH Models with Gaussian and Heavy-Tailed Errors J and E M. I Université de Montréal University of Alicante First version: April 27th, 2004
More informationCONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS
EVA IV, CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS Jose Olmo Department of Economics City University, London (joint work with Jesús Gonzalo, Universidad Carlos III de Madrid) 4th Conference
More informationABSTRACT. Robust GARCH methods and analysis of partial least squares regression. Joseph Egbulefu
ABSTRACT Robust GARCH methods and analysis of partial least squares regression by Joseph Egbulefu New approaches to modeling volatility are evaluated and properties of partial least squares (PLS) regression
More informationGARCH( 1, 1) processes are near epoch dependent
Economics Letters 36 (1991) 181-186 North-Holland 181 GARCH( 1, 1) processes are near epoch dependent Bruce E. Hansen 1Jnrrwsity of Rochester, Rochester, NY 14627, USA Keceived 22 October 1990 Accepted
More informationSUPPLEMENT TO TESTING FOR REGIME SWITCHING: A COMMENT (Econometrica, Vol. 80, No. 4, July 2012, )
Econometrica Supplementary Material SUPPLEMENT TO TESTING FOR REGIME SWITCHING: A COMMENT Econometrica, Vol. 80, No. 4, July 01, 1809 181 BY ANDREW V. CARTER ANDDOUGLAS G. STEIGERWALD We present proof
More informationJOURNAL OF MANAGEMENT SCIENCES IN CHINA. R t. t = 0 + R t - 1, R t - 2, ARCH. j 0, j = 1,2,, p
6 2 2003 4 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 6 No 2 Apr 2003 GARCH ( 300072) : ARCH GARCH GARCH GARCH : ; GARCH ; ; :F830 :A :1007-9807 (2003) 02-0068 - 06 0 2 t = 0 + 2 i t - i = 0 +( L) 2 t
More informationA note on adaptation in garch models Gloria González-Rivera a a
This article was downloaded by: [CDL Journals Account] On: 3 February 2011 Access details: Access Details: [subscription number 922973516] Publisher Taylor & Francis Informa Ltd Registered in England and
More informationThe Slow Convergence of OLS Estimators of α, β and Portfolio. β and Portfolio Weights under Long Memory Stochastic Volatility
The Slow Convergence of OLS Estimators of α, β and Portfolio Weights under Long Memory Stochastic Volatility New York University Stern School of Business June 21, 2018 Introduction Bivariate long memory
More informationModified Quasi-Likelihood Ratio Test for Regime Switching
Modified Quasi-Likelihood Ratio Test for Regime Switching Hiroyuki Kasahara Vancouver School of Economics University of British Columbia hkasahar@mail.ubc.ca Tatsuyoshi Okimoto Graduate School of International
More informationMODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A FLEXIBLE COEFFICIENT GARCH MODEL
MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A FLEXIBLE COEFFICIENT GARCH MODEL MARCELO C. MEDEIROS AND ALVARO VEIGA ABSTRACT. In this paper a flexible multiple regime GARCH-type model is developed
More informationTesting for non-stationarity
20 November, 2009 Overview The tests for investigating the non-stationary of a time series falls into four types: 1 Check the null that there is a unit root against stationarity. Within these, there are
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More informationNonlinear Time Series Modeling
Nonlinear Time Series Modeling Part II: Time Series Models in Finance Richard A. Davis Colorado State University (http://www.stat.colostate.edu/~rdavis/lectures) MaPhySto Workshop Copenhagen September
More informationCointegration Lecture I: Introduction
1 Cointegration Lecture I: Introduction Julia Giese Nuffield College julia.giese@economics.ox.ac.uk Hilary Term 2008 2 Outline Introduction Estimation of unrestricted VAR Non-stationarity Deterministic
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationBootstrapping Long Memory Tests: Some Monte Carlo Results
Bootstrapping Long Memory Tests: Some Monte Carlo Results Anthony Murphy and Marwan Izzeldin Nu eld College, Oxford and Lancaster University. December 2005 - Preliminary Abstract We investigate the bootstrapped
More informationBootstrapping Long Memory Tests: Some Monte Carlo Results
Bootstrapping Long Memory Tests: Some Monte Carlo Results Anthony Murphy and Marwan Izzeldin University College Dublin and Cass Business School. July 2004 - Preliminary Abstract We investigate the bootstrapped
More informationA Course on Advanced Econometrics
A Course on Advanced Econometrics Yongmiao Hong The Ernest S. Liu Professor of Economics & International Studies Cornell University Course Introduction: Modern economies are full of uncertainties and risk.
More informationFinite Sample Performance of the MLE in GARCH(1,1): When the Parameter on the Lagged Squared Residual Is Close to Zero
Finite Sample Performance of the MLE in GARCH1,1): When the Parameter on the Lagged Squared Residual Is Close to Zero Suduk Kim Department of Economics Hoseo University Asan Si, ChungNam, Korea, 336-795
More informationFinancial Econometrics
Financial Econometrics Nonlinear time series analysis Gerald P. Dwyer Trinity College, Dublin January 2016 Outline 1 Nonlinearity Does nonlinearity matter? Nonlinear models Tests for nonlinearity Forecasting
More informationNonlinear minimization estimators in the presence of cointegrating relations
Nonlinear minimization estimators in the presence of cointegrating relations Robert M. de Jong January 31, 2000 Abstract In this paper, we consider estimation of a long-run and a short-run parameter jointly
More informationSwitzerland, July 2007
GARCH A Case Study presented at the Meielisalp Workshop on Computational Finance and Financial Engineering www.rmetrics.org itp.phys.ethz.ch SP500 Yohan Chalabi, EPFL Lausanne, Diethelm Würtz, ITP ETH
More informationIWQW. Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung. Diskussionspapier Discussion Papers. No. 05/2012
IWQW Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung Diskussionspapier Discussion Papers No. 05/2012 Detecting Outliers in Time Series Vlad Ardelean University of Erlangen-Nuremberg
More information