Dynamic Principal Components: A New Class of Multivariate GARCH Models

Abstract

The OGARCH specification is the leading model for a class of multivariate GARCH (MGARCH) models that are based on linear combinations of univariate GARCH specifications. Most MGARCH models in this class adopt a spectral decomposition of the covariance matrix, allowing for heteroskedasticity on at least some of the principal components, while the loading matrix, which maps the conditional principal components to the asset returns, is constant over time. This paper extends the OGARCH model class to allow for time-varying loadings. Our approach closely parallels the DCC modelling approach, introduced as an extension of the CCC model, to allow for dynamic correlations. After introducing an auxiliary process that captures the relevant features of the unobservable loading dynamics, we compute the time-varying loading matrix from the auxiliary process, subject to the necessary orthonormality constraints. The resulting model (the Dynamic Principal Components, or DPC, model) preserves the OGARCH models ease of interpretation and feasibility. In particular, we show that the eigenvectors of the sample covariance matrix can consistently estimate the time-varying loadings intercept term. This property extends to the dynamic framework the well-known analogous property of the OGARCH model. Empirical examples demonstrate the benefits to the loading matrix of introducing time-varying properties.