TY - JOUR

T1 - Dynamic Conditional Eigenvalue GARCH

AU - Rahbek, Anders

AU - Pedersen, Rasmus Søndergaard

AU - Hetland, Simon Thinggaard

PY - 2023

Y1 - 2023

N2 - In this paper we introduce a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models with time-varying conditional eigenvalues. The dynamics of the eigenvalues is derived for the cases with nderlying Gaussian and Student’s t-distributed innovations based on the general theory of dynamic conditional score models by Creal, Koopman and Lucas (2013) and Harvey (2013). The resulting time-varying eigenvalue GARCH models – labeled ‘λ-GARCH’ models – differ for the two cases of innovations, similar to, and generalizing, univariate linear Gaussian GARCH and Student’s t-based Beta-t-GARCH models. Asymptotic theory is provided for the Gaussian-based quasi-maximum likelihood estimator (QMLE). In addition, and in order to test for the number of (linear combinations of) the time-varying eigenvalues, we consider testing and inference under the hypothesis of reduced rank of the GARCH loading matrices. The conditional Gaussian and Student’s t distributed λ-GARCH models are applied to US return data, and it is found that the eigenvalue structure for the sample considered indeed satisfies the hypothesis of reduced rank. Specifically, it is possible to disentangle time-varying linear combinations of the eigenvalues, or factors, from time-invariant factors which drive the dynamics of the conditional covariance.

AB - In this paper we introduce a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models with time-varying conditional eigenvalues. The dynamics of the eigenvalues is derived for the cases with nderlying Gaussian and Student’s t-distributed innovations based on the general theory of dynamic conditional score models by Creal, Koopman and Lucas (2013) and Harvey (2013). The resulting time-varying eigenvalue GARCH models – labeled ‘λ-GARCH’ models – differ for the two cases of innovations, similar to, and generalizing, univariate linear Gaussian GARCH and Student’s t-based Beta-t-GARCH models. Asymptotic theory is provided for the Gaussian-based quasi-maximum likelihood estimator (QMLE). In addition, and in order to test for the number of (linear combinations of) the time-varying eigenvalues, we consider testing and inference under the hypothesis of reduced rank of the GARCH loading matrices. The conditional Gaussian and Student’s t distributed λ-GARCH models are applied to US return data, and it is found that the eigenvalue structure for the sample considered indeed satisfies the hypothesis of reduced rank. Specifically, it is possible to disentangle time-varying linear combinations of the eigenvalues, or factors, from time-invariant factors which drive the dynamics of the conditional covariance.

KW - Faculty of Social Sciences

KW - Multivariate GARCH

KW - GO-GARCH

KW - Reduced rank

KW - Asymtotic theory

U2 - 10.1016/j.jeconom.2021.09.003

DO - 10.1016/j.jeconom.2021.09.003

M3 - Journal article

VL - 237

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 2B

M1 - 105175

ER -