"Regularized Wishart Autoregressive Stochastic Volatility"

(with Guilherme Valle Moura and Julian Spies)

We develop extensions of Uhlig's (1994) Wishart stochastic volatility (WSV) approach to model high-dimensional financial return series with time varying volatility and correlations. The WSV is a state-space model in which the multiplicative autoregressive evolution of the latent precision matrix is driven by a multivariate beta matrix variate. The resulting predictions of the covariance matrix are weighted moving averages with the classical exponential forgetting scheme that discounts information in past returns uniformly over time and across the space of the return vector. Our proposed extension of the WSV regularizes this forgetting scheme by shrinking the covariance matrix predictions toward a pre-specified prior target matrix. This regularization restricts the variation of the latent covariance matrix of the WSV and keep it away from zero, thereby ensuring the stationarity of returns and stabilizing the eigenvalues of the predictions for the covariance matrix. Furthermore, we combine this regularized forgetting with a time varying and directional forgetting scheme to achieve robustness to sudden changes in the correlation structure triggered by transitions from one market regime to another. Our proposed regularized WSV approach maintains closed-form sequential updating formulas for filtering, prediction and likelihood evaluation facilitating practical implementation for very high-dimensional empirical applications. The performance of our approach is evaluated using a historical dataset of daily returns on up to 1000 US stocks. We consider performance along several dimensions: the ability to predict the conditional covariance matrix and the probability density function of future returns, and to forecast the weights of the global minimum variance portfolio. The results show that our regularized WSV approach performs well compared to several existing alternative models.