The recursive least-squares (RLS) identification algorithm with a forgetting factor is often used to estimate time-varying parameters in stochastic systems. However, it is hard to say anything about the distribution of the parameter estimates, since they are nonlinear functions of the observations. In this paper we present a way to compute the exact distribution and moments of the weighted least-squares (WLS) estimator in a time-varying Gaussian AR(1) process. The RLS estimate follows as special case of the WLS estimator.