Data-driven methods for predictive control rely on input-output data to give a Hankel matrix representation of the space of trajectories. They are poorly suited to situations where both process noise and measurement noise dominate the behaviour whereas Kalman filters optimally estimate system states in this scenario. We derive a data-driven Kalman filter formulation based on the dynamic evolution of Hankel matrix output predictions. This leads to an extended state space model that describes the evolution of both the future inputs and outputs. By applying measurement feedback one arrives at a Kalman filter for the system. The Kalman filter design is performed purely on the basis of the input and output signals and without the need for a specific state-space representation. A benchmark simulation illustrates that the resulting prediction-based control significantly out-performs predictive controllers based on current data-driven methods.