The accuracy of the Kalman filter in state estimation depends on the knowledge of the process and measurement noise covariances. These are usually treated as tuning parameters and adjusted in a heuristic manner to fine-tune the state predictions. While several methods to identify the noise covariance from data exist, some require the use of optimization algorithms, or inversion of large matrices, which is numerically inefficient. In this work we explore a direct approach to estimate the covariance of possibly correlated process and measurement noises, which is based on subspace identification. It is shown that the subspace-based method outperforms the established autocovariance least-squares scheme and provides a good initial guess on the noise covariance in case the system is subjected to model errors. We validate the proposed scheme on a laboratory experiment, where it is shown that the predictions of the system outputs at sensor locations that are not used as observations in the identification procedure match well with the actual measurements.