Stochastic simulation of forward{looking models

Recently, there has been an increasing interest in analyzing the the quantitative properties of dynamic macroeconomic models. The existence of constraints, heterogeneity, non{di erentiability that economists introduce in these models forbid any analytical solution. They thus develop a set of tools that allows them to solve these models numerically. These methods rely heavily on dynamic programming (value iteration, policy iteration ...) and more recently on smooth approximation methods such as (parameterized expectations algorithm, neural networks, projection methods...) In this note we will focus on a particular method, introduced in economics by Judd [1992], that approximate decision rules, specifying a rule{of{thumb, whose parameters are identi ed using orthogonality conditions. The rst section presents the neo{classical growth model as a simple benchmark. In this model, there is an explicit solution for the distribution of the variables which is used to evaluate the proposed approximations. Section