A finite-state model for sequential minimum-mean-square-error estimation of a random variable in additive noise is analyzed to determine the dependence of optimum performance and structure on the memory size of the estimator. Necessary conditions for determining the structure of the optimum finite-state estimator are derived for arbitrary statistics. Numerical results are presented for Gaussian statistics. The performance of several different estimators is used to show the trade-off one may obtain between memory size, observation quality, and number of observations.