State estimation and prediction problems are discussed for stationary and nonstationary processes represented by state space forms, which include unknown parameters. First we calculate the variance of the difference between the state variable and its estimate constructed by the Kalman filter in which unknown parameters are replaced by any fixed values. Second we consider a special case when the process is purely explosive and show that the prediction error, which is obtained by substituting least squares estimates of parameters into unknown parameters, has a limit distribution with a bounded variance, though the variance of the process diverges to infinity.