A two-stage state and parameter estimation algorithm for linear systems has been developed. Stage 1 uses a stochastic approximation method for state estimation, while stage 2 considers parameter estimation through a linear Kalman filter. These two stages are coupled in a bootstrap manner. The algorithm is computationally much simpler than the usual extended Kalman filter. A fourth-order numerical example has been solved, and results have been compared with those obtained using an extended Kalman filter.