Dynamic Conditional Eigenvalue GARCH

Abstract

In this paper we consider a multivariate generalized autoregressive conditional heteroskedastic (GARCH) class of models where the eigenvalues of the conditional covariance matrix are time-varying. The proposed dynamics of the eigenvalues is based on applying the general theory of dynamic conditional score models as proposed by Creal, Koopman and Lucas (2013) and Harvey (2013). We denote the obtained GARCH model with dynamic conditional eigenvalues (and constant conditional eigenvectors) as the-GARCH model. We provide new results on asymptotic theory for the Gaussian QMLE, and for testing of reduced rank of the (G)ARCH loading matrices of the time-varying eigenvalues. The theory is applied to US data, where we …nd that the eigenvalue structure can be reduced similar to testing for the number in factors in volatility models.

Spectral Targeting Estimation of $λ$-GARCH models

“…As in Hetland, Pedersen, and Rahbek (2019), we focus on the class of O-GARCH models originally introduced by Alexander and Chibumba (1997). The presented model has more general dynamics than the O-GARCH, allowing for eigenvalue-spillovers, and we denote this version of the model the Eigenvalue GARCH, or λ−GARCH for short.…”

“…While theory for classical joint QMLE of O-GARCH type models have been considered in Hetland, Pedersen, and Rahbek (2019), we consider spectral targeting estimation (STE). The stepwise estimation procedure examined in this paper makes estimation and inference for the λ−GARCH feasible, even in large systems, as long as the time series dimension dominates the cross-sectional dimension (Ledoit andWolf (2004, 2012)).…”

“…The asymptotic theory for this estimator can be derived with relative ease, see e.g. Hetland, Pedersen, and Rahbek (2019) who parameterize the joint QMLE of the λ−GARCH in a similar fashion. One should however, keep in mind that the rotation parameters in φ are not uniquely identified unless we impose restrictions on the parameter space.…”

“…Both of these moment conditions for consistency and asymptotic normality stem from the first step estimator of the unconditional eigenvalues and eigenvectors, which uses the sample covariance estimator. This is in contrast to the joint QML estimator of the λ−GARCH, which requires fractional moments for consistency and finite 2+δ moments, for δ > 0, for asymptotic normality (Hetland, Pedersen, and Rahbek, 2019).…”

“…In the context of orthogonal GARCH models, such as the conditional eigenvalue GARCH (λ−GARCH) model of Hetland, Pedersen, and Rahbek (2019), we can combine the idea behind the two methods in what we denote the spectral targeting estimator (STE): By estimating the unconditional eigenvalues and -vectors using a sample moment estimator, the remainder of the parameters of the GARCH model may be estimated univariately in a stepwise manner, in which we target the unconditional eigenvalues and -vectors. This estimation procedure dramatically reduces the computational complexity of the optimization problem and speeds up numerical estimation compared to the QML estimator.…”