scholarly journals Properties and estimation of GARCH(1,1) model

2005 ◽  
Vol 2 (2) ◽  
Author(s):  
Petra Posedel
Keyword(s):  
Fourth Moment ◽  
Asymptotic Covariance Matrix ◽  
Likelihood Estimator ◽  
Consistent Estimation ◽  
Strong Stationarity ◽  
Asymptotic Covariance ◽  
Sampling Behavior ◽  
Garch Processes ◽  
Heavy Tailed ◽  
Quasi Maximum Likelihood

We study in depth the properties of the GARCH(1,1) model and the assumptions on the parameter space under which the process is stationary. In particular, we prove ergodicity and strong stationarity for the conditional variance (squared volatility) of the process. We show under which conditions higher order moments of the GARCH(1,1) process exist and conclude that GARCH processes are heavy-tailed. We investigate the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1,1) model. A bounded conditional fourth moment of the rescaled variable (the ratio of the disturbance to the conditional standard deviation) is sufficient for the result. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator

1994 ◽  
Vol 10 (1) ◽  
pp. 29-52 ◽  
Author(s):  
Sang-Won Lee ◽  
Bruce E. Hansen
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Asymptotic Theory ◽  
Fourth Moment ◽  
Asymptotic Covariance Matrix ◽  
Likelihood Estimator ◽  
Consistent Estimation ◽  
Asymptotic Covariance ◽  
Sampling Behavior ◽  
Quasi Maximum Likelihood

This paper investigates the sampling behavior of the quasi-maximum likelihood estimator of the Gaussian GARCH(1,1) model. The rescaled variable (the ratio of the disturbance to the conditional standard deviation) is not required to be Gaussian nor independent over time, in contrast to the current literature. The GARCH process may be integrated (α + β = 1), or even mildly explosive (α + β > 1). A bounded conditional fourth moment of the rescaled variable is sufficient for the results. Consistent estimation and asymptotic normality are demonstrated, as well as consistent estimation of the asymptotic covariance matrix.

scholarly journals R-estimators in GARCH models; asymptotics and applications

2021 ◽  
Author(s):  
Hang Liu ◽  
Kanchan Mukherjee
Keyword(s):  
Likelihood Estimation ◽  
Real Data ◽  
Superior Performance ◽  
Garch Models ◽  
Fourth Moment ◽  
Asymptotically Normal ◽  
Asymmetric Distributions ◽  
Class Of Estimators ◽  
Heavy Tailed ◽  
Quasi Maximum Likelihood

Abstract The quasi-maximum likelihood estimation is a commonly-used method for estimating the GARCH parameters. However, such estimators are sensitive to outliers and their asymptotic normality is proved under the finite fourth moment assumption on the underlying error distribution. In this paper, we propose a novel class of estimators of the GARCH parameters based on ranks of the residuals, called R-estimators, with the property that they are asymptotically normal under the existence of a finite 2 + δ moment of the errors and are highly efficient. We propose fast algorithm for computing the R-estimators. Both real data analysis and simulations show the superior performance of the proposed estimators under the heavy-tailed and asymmetric distributions.

scholarly journals Block Empirical Likelihood for Longitudinal Single-Index Varying-Coefficient Model

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yunquan Song ◽  
Ling Jian ◽  
Lu Lin
Keyword(s):  
Longitudinal Data ◽  
Empirical Likelihood ◽  
Coverage Probability ◽  
Confidence Region ◽  
Asymptotic Covariance Matrix ◽  
Varying Coefficient Model ◽  
Varying Coefficient ◽  
Single Index ◽  
Asymptotic Covariance ◽  
Normal Approximations

In this paper, we consider a single-index varying-coefficient model with application to longitudinal data. In order to accommodate the within-group correlation, we apply the block empirical likelihood procedure to longitudinal single-index varying-coefficient model, and prove a nonparametric version of Wilks’ theorem which can be used to construct the block empirical likelihood confidence region with asymptotically correct coverage probability for the parametric component. In comparison with normal approximations, the proposed method does not require a consistent estimator for the asymptotic covariance matrix, making it easier to conduct inference for the model's parametric component. Simulations demonstrate how the proposed method works.

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