Normalité asymptotique de l'estimateur du pseudo-maximum de vraisemblance d'un modèle GARCH

Boussama, Farid

doi:10.1016/s0764-4442(00)01593-7

Cited by 32 publications

(14 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From (A.1), we obtain h t (θ ) ≥ ω + (ω + αy 2 t−1−i )β i for all i ≥ 1. Using the fact that x/(1 + x) < x p/r for all x ≥ 0 and any p ∈ (0, 1), r ≥ 1 -this idea of exploiting this inequality is due to Boussama (2000) -we obtain ∂h t ∂β…”

Section: Lemma A2mentioning

confidence: 99%

Least‐squares estimation of GARCH(1,1) models with heavy‐tailed errors

Preminger

Storti

2017

The Econometrics Journal

View full text Add to dashboard Cite

GARCH(1,1) models are widely used for modelling processes with time-varying volatility. These include financial time series, which can be particularly heavy tailed. In this paper, we propose a novel log-transform-based least-squares approach to the estimation of GARCH(1,1) models. Within this approach, the scale of the estimated volatility is dependent on an unknown tuning constant. By means of a backtesting exercise on both real and simulated data, we show that knowledge of the tuning constant is not crucial for Value at Risk prediction. However, this does not apply to many other applications where correct identification of the volatility scale is required. In order to overcome this difficulty, we propose two alternative two-stage least-squares estimators and we derive their asymptotic properties under very mild moment conditions for the errors. In particular, we establish the consistency and asymptotic normality at the standard convergence rate of √ n for our estimators. Their finite sample properties are assessed by means of an extensive simulation study.

show abstract

Section: Lemma A2mentioning

confidence: 99%

Least‐squares estimation of GARCH(1,1) models with heavy‐tailed errors

Preminger

Storti

2017

The Econometrics Journal

View full text Add to dashboard Cite

show abstract

“…A possible approximation of σ 2 t is obtained by mimicking the recursion (1.2), for example, with an initialization X −p+1 = · · · = X 0 = 0 and σ 2 −q+1 = · · · = σ 2 0 = 0. Recently, Berkes, Horváth and Kokoszka [2] showed under minimal assumptions that the resulting estimator, which is called the (Gaussian) quasi-maximumlikelihood estimator (QMLE), is consistent and asymptotically normal, thereby generalizing work by Lumsdaine [24], Lee and Hansen [22] and Boussama [7,8]. Related to [2] is Francq and Zakoïan [15].…”

Section: Introductionmentioning

confidence: 97%

“…To the best of our knowledge, the theoretical properties of the QMLE in EGARCH have not been studied in the literature. Work of Boussama [7,8] and Comte and Lieberman [12] on multivariate GARCH indicates that the contraction technique cannot be employed in the analysis of the QMLE in multivariate conditionally heteroscedastic time series models.…”

Section: Introductionmentioning

confidence: 99%

Estimation in Conditionally Heteroscedastic Time Series Models

Lund¹

2005

Lecture Notes in Statistics

View full text Add to dashboard Cite

This paper studies the quasi-maximum-likelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form X t = σ t Z t , where the unobservable volatility σ t is a parametric function of (X t−1 , . . . , X t−p , σ t−1 , . . . , σ t−q ) for some p, q ≥ 0, and (Z t ) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (X t ) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models. Introduction. Gaussian quasi-maximum-likelihood estimation, that is, likelihood estimation under the hypothesis of Gaussian innovations, is a popular method which is widely used for inference in time series models. However, it is often a nontrivial task to establish the consistency and asymptotic normality of the quasi-maximum-likelihood estimator (QMLE) applied to specific models and, therefore, an in-depth analysis of the probabilistic structure generated by the model is called for. A classical example of this kind is the seminal paper by Hannan [18] on estimation in linear ARMA time series.In this paper we study the QMLE for a general class of conditionally heteroscedastic time series models, which includes GARCH, asymmetric GARCH and exponential GARCH. Recall that a GARCH(p, q) [generalized autoregressive conditionally heteroscedastic of order (p, q)] process [4] is defined by

show abstract

“…Berkes et al (2003) proved the consistency and asymptotic normality if the QMLE of the parameters of the GARCH(p,q) model under second-and fourth-order moment conditions, respectively. Boussama (2000), McAleer et al (2007), and Francq and Zakoïan (2004) also considered the properties of the QMLE under different specifications of the symmetric and asymmetric GARCH(p,q) model.…”

Section: Introductionmentioning

confidence: 99%

Structure and asymptotic theory for nonlinear models with GARCH errors

2015

View full text Add to dashboard Cite

Nonlinear time series models, especially those with regime-switching and/or conditionally heteroskedastic errors, have become increasingly popular in the economics and finance literature. However, much of the research has concentrated on the empirical applications of various models, with little theoretical or statistical analysis associated with the structure of the processes or the associated asymptotic theory. In this paper, we derive sufficient conditions for strict stationarity and ergodicity of three different specifications of the first-order smooth transition autoregressions with heteroskedastic errors. This is essential, among other reasons, to establish the conditions under which the traditional LM linearity tests based on Taylor expansions are valid. We also provide sufficient conditions for consistency and asymptotic normality of the Quasi-Maximum Likelihood Estimator for a general nonlinear conditional mean model with first-order GARCH errors. ResumoModelos não-lineares com múltiplos regimes e heterpcedasticidade condicional são muito populares em economia e finanças. Neste artigo derivamos condições de estacionaridade para modelos com transição suave e erros heterocedásticos. Além disso derivamos condições suficientes para consistência e normalidade assintótica do estimador de quase-máxima verossimilhança.

show abstract

Normalité asymptotique de l'estimateur du pseudo-maximum de vraisemblance d'un modèle GARCH

Cited by 32 publications

References 0 publications

Least‐squares estimation of GARCH(1,1) models with heavy‐tailed errors

Least‐squares estimation of GARCH(1,1) models with heavy‐tailed errors

Estimation in Conditionally Heteroscedastic Time Series Models

Structure and asymptotic theory for nonlinear models with GARCH errors

Contact Info

Product

Resources

About