Chapter 4 Mechanics of forming and estimating dynamic linear economies This paper describes the recent advances for rapidly and accurately solving matrix Riccati and Sylvester equations and applies them to devise efficient computational methods for solving and estimating dynamic linear economies. The chapter explores the most promising solution methods available and compares their speed and accuracy for some particular economic examples. Except for the simplest dynamic linear models it is necessary to compute solutions numerically. In estimation contexts computation speed is important because climbing a likelihood function can require that a model be solved many times. Methods that are faster than direct iterations on the Riccati equation and are more reliable than solutions based on eigenvalueeigenvector decompositions of the statecostate evolution equation are discussed in the chapter. Two generalizations are presented in the chapter: The first generalization introduces forcing sequences or uncontrollable states into the deterministic regulator problem while the second generalization introduces among other things discounting and uncertainty into the augmented regulator problem.