New Integration Methods for Rational Expectations Models

Ken Judd
Hoover Institution
Judd@hoover.stanford.edu

Abstract

A key feature of every rational expectations problems is computing expected utility, a task usually requiring numerical integration. Another task is computing the coefficients of an unknown equilibrium behavior rule (such as consumption functions), a problem usually solved by using nonlinear equation solvers which often need gradient information. I also exploit automatic differentiation methods. This paper develops integration methods which also produce information used to compute gradients, via automatic differentiation, a triple combination which greatly improves the overall performance of the algorithm.

In this paper I first develop multidimensional integration rules which use both level and derivative information about the integrand. These formulas are generalizations of multidimensional monomial formulas and interpolatory integration rules.

Standard theory says that optimal integration rules do not use derivative information. However, that theory assumes that the cost of evaluating a derivative equals the cost of evaluating the function. Automatic differentiation shows that derivatives are far cheaper to compute. We exploit that to develop efficient integration rules.

We then go on to incorporate these ideas into solution methods for rational expectations methods, where the derivatives computed for integration purposes can also be used to make Jacobian computation particularly cheap.

We find that large rational expectations models can be computed in far less time than previous methods.