We propose a novel framework for modeling large dynamic covariance matrices via heterogeneous autoregressive volatility and correlation components. Our model provides direct forecasts of monthly covariance matrices and is flexible, parsimonious and simple to estimate using standard least squares methods. We address the problem of parameter estimation risks by employing nonlinear shrinkage methods, making our framework applicable in high dimensions. We perform a comprehensive empirical out-of-sample analysis and find significant statistical and economic improvements over common benchmark models. For minimum variance portfolios with over a thousand stocks, the annualized portfolio standard deviation improves to 8.92% compared to 9.75–10.43% for DCC-type models.