Approximation Methods in Continuous Time Economic Models

Abstract

As the number of state variables increases, solving heterogeneous agent problems become infeasible. Hence, this dissertation studies various methods of increasing the feasibility frontier. The following three chapters present \emph{(i)} a method of general function approximations, \emph{(ii)} a model reduction method for Hamilton-Jacobi-Bellman (HJB) equations in continuous time, and \emph{(iii)} a model reduction method for solving heterogeneous agent models with aggregate shocks.

Chapter 1 examines approximations of general functions on a linear subspace. It presents how many different approximations used by economists can be considered as an approximation on a linear subspace, and presents concise and efficient implementations that would allow researchers of heterogeneous agents models to build custom approximations instead of relying on a pre-existing black box codes written by others.

Chapter 2 studies how to solve continuous time dynamical programming problems more efficiently. Even with efficiency gains from the continuous time framework over the discrete time counterpart, the curse of dimensionality makes solving HJB equations impossible for models with a moderate number of state variables. In this chapter, I show how continuous time HJB equations can be approximated on a linear subspace and implement the Smolyak method as an example. The benefits of modeling in continuous time are preserved under the reduction, and hence, the Smolyak method can be applied without nonlinear solvers nor numerical quadratures in comparison to discrete time models.

Chapter 3, co-authored with Greg Kaplan, Benjamin Moll, Thomas Winberry, and Christian Wolf, studies how to solve heterogeneous agent models with aggregate shocks. With aggregate shocks, the distribution of agents become state variables. As the distribution is infinite-dimensional, the literature has resorted to problem-specific reduction methods like \citet{krusell-smith} that rely on the approximate aggregation. Using the model reduction literature from engineering, this chapter introduces a model-free method to solve heterogeneous agent models with aggregate shocks. This new method gives around 1000-fold speed gain over the next best algorithm in the literature while achieving the best approximation error.