A Penalised OLS Framework for High-Dimensional Multivariate Stochastic Volatility Models

Although multivariate stochastic volatility (MSV) models usually produce more accurate forecasts compared to multivariate GARCH models, their estimation techniques such as Monte Carlo likelihood or Bayesian Markov Chain Monte Carlo are computationally demanding and thus suffer from the so-called gcurse of dimensionality": using such methods, the applications are typically restricted to low-dimensional vectors. In this paper, we propose a fast estimation approach for MSV models based on a penalised ordinary least squares framework. Specifying the MSV model as a multivariate state-space model, we propose a two-step penalised procedure for estimating the latter using a broad range of potentially non-convex penalty functions. In the first step, we approximate an EGARCH type dynamic using a penalised AR process with a sufficiently large number of lags, providing a sparse estimator. Conditionally on this first step estimator, we estimate the state vector based on a AR type dynamic. This two-step procedure relies on OLS based loss functions and thus easily accommodates high-dimensional vectors. We provide the large sample properties of the two-step estimator together with the so- called support recovery of the first step estimator. The empirical performances of our method are illustrated through in-sample simulations and out-of-sample variance-covariance matrix forecasts, where we consider as competitors commonly used MGARCH models.