Forecasting Realized (Co)Variances with a Block Structure Wishart Autoregressive Model

Abstract

The increased availability of high-frequency data provides new tools for forecasting of variances and covariances between assets. However, recent realized (co)variance models may suffer from a 'curse of dimensionality' problem similar to that of multivariate GARCH specifications. As a result, they need strong parameter restrictions, in order to avoid non-interpretability of model coefficients, as in the matrix and log exponential representations. Among the proposed models, the Wishart autoregressive model introduced by Gourieroux et al. (2005) analyzes the realized covariance matrices without any restriction on the parameters while maintaining coefficient interpretability. Indeed, the model, under mild stationarity conditions, provides positive definite forecasts for the realized covariance matrices. Unfortunately, it is still not feasible for large asset cross-section dimensions. In this paper we propose a restricted parametrization of the Wishart Autoregressive model which is feasible even with a large cross-section of assets. In particular, we assume that the asset variances-covariances have no or limited spillover and that their dynamic is sector-specific. In addition, we propose a Wishart-based generalization of the HAR model of Corsi (2004). We present an empirical application based on variance forecasting and risk evaluation of a portfolio of two US treasury bills and two exchange rates. We compare our restricted specifications with the traditional WAR parameterizations. Our results show that the restrictions may be supported by the data and that the risk evaluations of the models are extremely close. This confirms that our model can be safely used in a large cross-sectional dimension given that it provides results similar to fully parametrized specifications.

Keywords: realized volatility, forecasting, Value-at-Risk, Wishart