Modelling and Forecasting Covariance Matrices: A Parsimonious Approach

This paper proposes a simple approach for forecasting large dimensional covariance
matrices by modelling the dynamics of realized covariance matrices with the
help of a latent factor structure with stochastic volatility components. The factor
structure together with the conditional Wishart distribution (1) assures automatically
the positive-definiteness and symmetry of the forecasts, (2) captures the commonality
in the dynamics and (3) the long-persistence of the autocorrelation of the realized
(co)variances within a unified and very parsimonious framework with no parameter
constraints. The Factor Autoregressive model we propose profits from what alternatives suffer, namely the curse of dimensionality: increasing the dimension of the
realized covariance matrix leads to only a linear increase in the number of parameters
as well as in an improvement in the efficiency of the parameter estimates and
forecasts. The model has a non-Gaussian non-linear state-space representation that
we estimate by maximum likelihood and numerical integration using a non-Gaussian
filtering technique. Monte Carlo simulations provide evidence for the accuracy of the
estimates and the comprehensive empirical application to DJIA components proves
the usefulness of the model to accurately forecast large dimensional covariance matrices.