Principal Component Models for Generating Large GARCH Covariance Matrices

Abstract

The implementation of multivariate GARCH models in more than a few dimensions is extremely difficult: because the model has many parameters, the likelihood function becomes very flat, and consequently the optimization of the likelihood becomes practicably impossible. There is simply no way that full multivariate GARCH models can be used to estimate directly the very large covariance matrices that are required to net all the risks in a large trading book. This paper begins by describing the principal component GARCH or ‘orthogonal GARCH’ (O‐GARCH) model for generating large GARCH covariance matrices that was first introduced in Alexander and Chibumba (1996) and subsequently developed in Alexander (2000, 2001b). The O‐GARCH model is an accurate and efficient method for generating large covariance matrices that only requires the estimation of univariate GARCH models. Hence, it has many practical advantages, for example in value–at–risk models. It works best in highly correlated systems, such as term structures. The purpose of this paper is to show that, if sufficient care is taken with the initial calibration of the model, equities and foreign exchange rates can also be included in one large covariance matrix. Simple conditions for the final covariance matrix to be positive semi‐definite are derived.(J.E.L.: C32, C53, G19, G21, G28).