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Article

Asymptotic and Finite Sample Properties for Multivariate Rotated GARCH Models

by 1,*,†, 2,3,†, 4,5,6,7,8,† and 6,†
1
Faculty of Economics, Soka University, Tokyo 192-8577, Japan
2
Department of Applied Economics, National Chung Hsing University, Taichung City 402, Taiwan
3
Department of Finance, National Chung Hsing University, Taichung City 402, Taiwan
4
Department of Finance, College of Management, Asia University, Taichung City 41354, Taiwan
5
Department of Bioinformatics and Medical Engineering, College of Information and Electrical Engineering, Asia University, Taichung City 41354, Taiwan
6
Discipline of Business Analytics, University of Sydney Business School, Darlington 2006, Australia
7
Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, 3062 PA Rotterdam, The Netherlands
8
Department of Economic Analysis and ICAE, Complutense University of Madrid, 28040 Madrid, Spain
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Academic Editors: Marc S. Paolella, In Choi and Martin Wagner
Received: 24 December 2020 / Revised: 18 March 2021 / Accepted: 27 April 2021 / Published: 4 May 2021
This paper derives the statistical properties of a two-step approach to estimating multivariate rotated GARCH-BEKK (RBEKK) models. From the definition of RBEKK, the unconditional covariance matrix is estimated in the first step to rotate the observed variables in order to have the identity matrix for its sample covariance matrix. In the second step, the remaining parameters are estimated by maximizing the quasi-log-likelihood function. For this two-step quasi-maximum likelihood (2sQML) estimator, this paper shows consistency and asymptotic normality under weak conditions. While second-order moments are needed for the consistency of the estimated unconditional covariance matrix, the existence of the finite sixth-order moments is required for the convergence of the second-order derivatives of the quasi-log-likelihood function. This paper also shows the relationship between the asymptotic distributions of the 2sQML estimator for the RBEKK model and variance targeting quasi-maximum likelihood estimator for the VT-BEKK model. Monte Carlo experiments show that the bias of the 2sQML estimator is negligible and that the appropriateness of the diagonal specification depends on the closeness to either the diagonal BEKK or the diagonal RBEKK models. An empirical analysis of the returns of stocks listed on the Dow Jones Industrial Average indicates that the choice of the diagonal BEKK or diagonal RBEKK models changes over time, but most of the differences between the two forecasts are negligible. View Full-Text
Keywords: BEKK; rotated BEKK; diagonal BEKK; variance targeting; multivariate GARCH; consistency; asymptotic normality BEKK; rotated BEKK; diagonal BEKK; variance targeting; multivariate GARCH; consistency; asymptotic normality
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MDPI and ACS Style

Asai, M.; Chang, C.-L.; McAleer, M.; Pauwels, L. Asymptotic and Finite Sample Properties for Multivariate Rotated GARCH Models. Econometrics 2021, 9, 21. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics9020021

AMA Style

Asai M, Chang C-L, McAleer M, Pauwels L. Asymptotic and Finite Sample Properties for Multivariate Rotated GARCH Models. Econometrics. 2021; 9(2):21. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics9020021

Chicago/Turabian Style

Asai, Manabu, Chia-Lin Chang, Michael McAleer, and Laurent Pauwels. 2021. "Asymptotic and Finite Sample Properties for Multivariate Rotated GARCH Models" Econometrics 9, no. 2: 21. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics9020021

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