Abstract

This chapter considers estimation of autoregressive conditional heteroscedasticity (ARCH) and the generalized autoregressive conditional heteroscedasticity (GARCH) models using quasi-likelihood (QL) and asymptotic quasi-likelihood (AQL) approaches. The QL and AQL estimation methods for the estimation of unknown parameters in ARCH and GARCH models are developed. Distribution assumptions are not required of ARCH and GARCH processes by QL method. Nevertheless, the QL technique assumes knowing the first two moments of the process. However, the AQL estimation procedure is suggested when the conditional variance of process is unknown. The AQL estimation substitutes the variance and covariance by kernel estimation in QL. Reports of simulation outcomes, numerical cases, and applications of the methods to daily exchange rate series and weekly prices’ changes of crude oil are presented.

Highlights

  • The generalized autoregressive conditional heteroscedasticity (GARCH(p,q)) process yt is defined by yt 1⁄4 μ þ ξt, t 1⁄4 1, 2, 3, ⋯, T: (3)

  • Given ^ξ0 1⁄4 0, θ0 1⁄4 Àμ0, α0, α1, ⋯, αqÁ, Σðt,0nÞ 1⁄4 I2, and ^ξ2tÀ1 1⁄4 ÀytÀ1 À μ0Á2, the asymptotic quasi-likelihood (AQL) estimation of σ2t is the solution of GðtÞÀσ2t Á 1⁄4 0, that is, σ^2t 1⁄4 α0 þ α1^ξ2tÀ1 þ ⋯ þ αq^ξ2tÀq, t 1⁄4 1, 2, 3⋯, T: (15)

  • The estimations of unknown parameters are considered without any distribution assumptions concerning the processes involved, and the estimation is based on different scenarios in which the conditional covariance of the error’s terms are assumed to be known or unknown

Read more

Summary

Introduction

The autoregressive conditional heteroscedasticity (ARCH(q)) process is defined by yt 1⁄4 μ þ ξt, t 1⁄4 1, 2, 3, ⋯, T:. For estimation and applications of ARCH models, see [1–19]. ARCH models have become the standard textbook material in econometrics and finance as exemplified by, for example, [20–23]. GARCH models have become the standard textbook material in econometrics and finance as exemplified by, for example, [20–23]. This chapter considers estimation of ARCH and GARCH models using quasilikelihood (QL) and asymptotic quasi-likelihood (AQL) approaches. The AQL estimation procedure is suggested when the conditional variance of process is unknown.

The QLE and AQL methods
The QL method
The AQL method
Parameter estimation of ARCH(q) model using the QL method
Parameter estimation of ARCH(q) model using the AQL method
Simulation studies for the ARCH(1) model
Parameter estimation of ARCH(1) model using the QL method
Parameter estimation of ARCH(1) model using the AQL method
Empirical applications
Parameter estimation of GARCH(p,q) model using the QL method
Parameter estimation of GARCH(p,q) model using the AQL method
Parameter estimation of GARCH(1,1) model using the QL method
Parameter estimation of GARCH(1,1) model using the AQL method
Conclusions

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.