Estimation of large financial volatility models is plagued by the curse of dimensionality. As the dimension grows, joint estimation of the parameters becomes unfeasible in practice. This problem is compounded if covariates or conditioning variables ("X") are added to each volatility equation. The problem is especially acute for non-exponential volatility models, e.g., GARCH models, since the variables and parameters in these cases are restricted to be positive. Here, we propose an estimator for a multivariate log-GARCH-X model that avoids these problems. The model allows for feedback among the equations, admits several stationary regressors as conditioning variables in the X-part (including leverage terms), and allows for time-varying conditional covariances of unknown form. Strong consistency and asymptotic normality of an equation-by-equation least squares estimator are proved, and the results can be used to undertake inference both within and across equations. The flexibility and usefulness of the estimator is illustrated in two empirical applications.