Estimation and tests for power-transformed and threshold GARCH models

J. Pan, H. Wang, H. Tong

Research output: Contribution to journalArticle

33 Citations (Scopus)

Abstract

Consider a class of power transformed and threshold GARCH(p,q) (PTTGRACH(p,q)) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa (2004) and includes the standard GARCH model and many other models as special cases. We ¯rst establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has ¯nite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH(p,q) model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH(p,q) models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy tailed errors. Furthermore the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the ¯nancial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH(p,q), we give in the appendix a necessary and su±cient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.
LanguageEnglish
Pages352-378
Number of pages27
JournalJournal of Econometrics
Volume142
Issue number1
DOIs
Publication statusPublished - Jan 2008

Fingerprint

Threshold Model
GARCH Model
Least Absolute Deviation
Generalized Autoregressive Conditional Heteroscedasticity
Quasi-maximum Likelihood
Asymptotic Normality
Maximum Likelihood Estimator
Maximum likelihood
Strictly
Model
Order Selection
Moment
Wald Test
Tail Behavior
Moment Conditions
Statistical Inference
Stationary Solutions
Asymmetry
GARCH model
Standard Model

Keywords

  • threshold garch
  • power transformation
  • asymptotic normality
  • quasi-maximum likelihood
  • estimator
  • least absolute deviations estimation
  • wald test
  • order selection
  • pttgarch structure

Cite this

Pan, J. ; Wang, H. ; Tong, H. / Estimation and tests for power-transformed and threshold GARCH models. In: Journal of Econometrics. 2008 ; Vol. 142, No. 1. pp. 352-378.
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Estimation and tests for power-transformed and threshold GARCH models. / Pan, J.; Wang, H.; Tong, H.

In: Journal of Econometrics, Vol. 142, No. 1, 01.2008, p. 352-378.

Research output: Contribution to journalArticle

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AB - Consider a class of power transformed and threshold GARCH(p,q) (PTTGRACH(p,q)) model, which is a natural generalization of power-transformed and threshold GARCH(1,1) model in Hwang and Basawa (2004) and includes the standard GARCH model and many other models as special cases. We ¯rst establish the asymptotic normality for quasi-maximum likelihood estimators (QMLE) of the parameters under the condition that the error distribution has ¯nite fourth moment. For the case of heavy-tailed errors, we propose a least absolute deviations estimation (LADE) for PTTGARCH(p,q) model, and prove that the LADE is asymptotically normally distributed under very weak moment conditions. This paves the way for a statistical inference based on asymptotic normality for heavy-tailed PTTGARCH(p,q) models. As a consequence, we can construct the Wald test for GARCH structure and discuss the order selection problem in heavy-tailed cases. Numerical results show that LADE is more accurate than QMLE for heavy tailed errors. Furthermore the theory is applied to the daily returns of the Hong Kong Hang Seng Index, which suggests that asymmetry and nonlinearity could be present in the ¯nancial time series and the PTTGARCH model is capable of capturing these characteristics. As for the probabilistic structure of PTTGARCH(p,q), we give in the appendix a necessary and su±cient condition for the existence of a strictly stationary solution of the model, the existence of the moments and the tail behavior of the strictly stationary solution.

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