The paper deals with identification of linear state-space models from input-output data in the presence of process and measurement stochastic noise. A three step procedure is proposed to identify a state-space model: the least-squares identification of the corresponding ARX model (input-output system description), computation of system Markov parameters from the identified ARX model, and identification of the state-space system matrices. Next, state estimation of a system is considered in the Kalman filter framework and a three step procedure is suggested for computing the state estimator gain. This includes selection of the deterministic and stochastic part of the system output in the identified state-space model, identification of the MA model describing the stochastic component of the system output (by identifying and inverting the corresponding AR model), calculation of the state estimator gain from the estimated MA model parameters. An experimental example is used to demonstrate the feasibility of the approach.