Long and Short run dynamics in Realized Covariance Matrices: a Robust MIDAS Approach

Forecasting time-varying conditional (co)variances is an interesting research topic, due to the importance of asset returns correlation for financial applications: hedging, asset allocation, pricing, risk management, and so on. Early multivariate volatility models (e.g. the BEKK of Engle and Kroner, 1995) were based on daily cross product returns and assumed a constant average (or long-run) level of (co)variances, though empirical evidence suggests that it is time-varying (see, for example, the results in Gallo and Otranto, 2015, for the S&P500 volatility). In the last decade, a great deal of effort was put into the development of models based on Realized Covariance (see, for example, the Conditional Autoregressive Wishart (CAW) model of Golosnoy et al., 2012), modeling directly a nonparametric estimation of the covariance matrices, based on intra-daily returns. A recent stream of literature is devoted to detect long and short-run components that characterize, with different dynamics, the realized covariance series (see, for example, Bauwens et al., 2016). By decomposing the covariance matrix into a short-run and a long-run component, it is possible to capture, in a parsimonious way, the long memory behavior of (co)variances. The short-run component is aimed to capture daily fluctuations and transitory effects; conversely, the long-run component represents the average level that varies over time according to economic conditions. However, dynamic component models are based on the Cholesky decomposition, which makes the short-run component potentially sensible to asset order. We propose a new additive component model belonging to the MIDAS family, in the spirit of Colacito et al. (2011), with features that help overcome some drawbacks: • it does not depend on the Cholesky decomposition to the covariance matrix, so that the order of the series is not relevant in the estimation of the model parameters; • the multiplicative decomposition of the covariance matrix, adopted in other models, requires the calculation at each time of the inverse of the Cholesky factor, thus slowing down the optimization algorithm. Our additive specification does not require this step, with a clear computational gain; • multivariate volatility models, to overcome the curse of dimensionality problem, usually assume a scalar specification of the conditional (co)variances, imposing the same dynamics for each series. This hypothesis is very strong and not supported by empirical evidence. The model we propose adopts the Hadamard exponential function proposed by Bauwens and Otranto (2022), which allows asset-pair-specific and time-varying parameters. This specification offers a more flexible dynamics with only one more parameter than the baseline specification, thus preserving the parsimony of the model. In the empirical analysis we fit our set of models to the Realized Covariance series of 9 assets belonging to the Dow Jones Industrial Average (DJIA) index. Then, we compare the in-sample tting of the estimated models through some information criteria and statistical loss functions. Finally, we verify the superior forecasting performance of our model with respect to the competitive ones in terms of statistical and economic loss functions.