How to solve dynamic stochastic models computing expectations just once: How to solve dynamic stochastic models

We introduce a computational technique—precomputation of integrals—that makes it possible to construct conditional expectation functions in dynamic stochastic models in the initial stage of a solution procedure. This technique is very general: it works for a broad class of approximating functions, including piecewise polynomials; it can be applied to both Bellman and Euler equations; and it is compatible with both continuous-state and discrete-state shocks. In the case of normally distributed shocks, the integrals can be constructed in a closed form. After the integrals are precomputed, we can solve stochastic models as if they were deterministic. We illustrate this technique using oneand multi-agent growth models with continuous-state shocks (and up to 60 state variables), as well as Aiyagari’s (1994) model with discrete-state shocks. Precomputation of integrals saves programming efforts, reduces computational burden, and increases the accuracy of solutions. It is of special value in computationally intense applications. MATLAB codes are provided.

[1]  Serguei Maliar,et al.  Envelope Condition Method versus Endogenous Grid Method for Solving Dynamic Programming Problems , 2012 .

[2]  Viktor Tsyrennikov,et al.  Envelope Condition Method with an Application to Default Risk Models , 2014 .

[3]  Paul L. Fackler,et al.  Applied Computational Economics and Finance , 2002 .

[4]  Mark Huggett,et al.  The risk-free rate in heterogeneous-agent incomplete-insurance economies , 1993 .

[5]  Sean R Eddy,et al.  What is dynamic programming? , 2004, Nature Biotechnology.

[6]  Jesús Fernández-Villaverde,et al.  A Generalization of the Endogenous Grid Method , 2007 .

[7]  K. Judd,et al.  including © notice, is given to the source. A Cluster-Grid Projection Method: Solving Problems with High Dimensionality , 2010 .

[8]  W. D. Haan.  Comparison of Solutions to the Incomplete Markets Model with Aggregate Uncertainty , 2008 .

[9]  Kenneth L. Judd,et al.  Projection methods for solving aggregate growth models , 1992 .

[10]  John Rust.  Numerical dynamic programming in economics , 1996 .

[11]  George Tauchen,et al.  Finite state markov-chain approximations to univariate and vector autoregressions , 1986 .

[12]  Lawrence J. Christiano,et al.  Algorithms for solving dynamic models with occasionally binding constraints , 1997 .

[13]  Lilia Maliar,et al.  Solving the Multi-Country Real Business Cycle Model Using Ergodic Set Methods , 2010 .

[14]  Manuel S. Santos.  Numerical solution of dynamic economic models , 1999 .

[15]  Lilia Maliar,et al.  Numerical Methods for Large-Scale Dynamic Economic Models , 2014 .

[16]  Lilia Maliar,et al.  Merging Simulation and Projection Approaches to Solve High-Dimensional Problems , 2012 .

[17]  John Stachurski,et al.  Economic Dynamics: Theory and Computation , 2009 .

[18]  Serguei Maliar,et al.  The Neoclassical Growth Model with Heterogeneous Quasi-Geometric Consumers , 2006 .

[19]  Felix Kubler,et al.  Computing equilibrium in OLG models with stochastic production , 2004 .

[20]  Harald Uhlig,et al.  Solving Nonlinear Stochastic Growth Models: a Comparison of Alternative Solution Methods , 1989 .

[21]  Lilia Maliar,et al.  Smolyak Method for Solving Dynamic Economic Models: Lagrange Interpolation, Anisotropic Grid and Adaptive Domain , 2013 .

[22]  José-Víctor Ríos-Rull,et al.  Computation of equilibria in heterogeneous agent models , 1997 .

[23]  C. Carroll,et al.  The Method of Endogenous Gridpoints for Solving Dynamic Stochastic Optimization Problems , 2005 .

[24]  Serguei Maliar,et al.  Should Central Banks Worry About Nonlinearities of their Large-Scale Macroeconomic Models? , 2017 .

[25]  Albert Marcet,et al.  Solving the Stochastic Growth Model by Parameterizing Expectations , 1990 .

[26]  Lilia Maliar,et al.  Parameterized Expectations Algorithm: How to Solve for Labor Easily , 2005 .

[27]  Quasi-Geometric Discounting: A Closed-Form Solution Under the Exponential Utility Function , 2004 .

[28]  T. Bewley.  The Permanent Income Hypothesis: A Theoretical Formulation. , 1977 .

[29]  Eric R. Young.  Solving the incomplete markets model with aggregate uncertainty using the Krusell-Smith algorithm and non-stochastic simulations , 2010 .

[30]  Andrew Scott,et al.  Computational Methods for the Study of Dynamic Economies , 1998 .

[31]  Lilia Maliar,et al.  Matlab code for "Numerically stable and accurate stochastic simulation approaches for solving dynamic economic models" , 2011 .

[32]  A. Deaton.  Saving and Liquidity Constraints , 1989 .

[33]  R. Jagannathan,et al.  Uninsured Idiosyncratic Risk and Aggregate Saving , 1994 .

[34]  K. Judd,et al.  Lower Bounds on Approximation Errors to Numerical Solutions of Dynamic Economic Models , 2017 .

[35]  Lilia Maliar,et al.  Matlab code for "Solving Nonlinear Dynamic Stochastic Models: An Algorithm Computing Value Function by Simulations" , 2005 .

[36]  K. Judd.  Numerical methods in economics , 1998 .

[37]  Jesús Fernández-Villaverde,et al.  Comparing Solution Methods for Dynamic Equilibrium Economies , 2003 .

[38]  C. Carroll.  The Buffer-Stock Theory of Saving: Some Macroeconomic Evidence , 1992 .

[39]  George Tauchen,et al.  Quadrature-Based Methods for Obtaining Approximate Solutions to Nonlinear Asset Pricing Models , 1991 .

[40]  Jesús Fernández-Villaverde,et al.  Comparing Solution Methods for Dynamic Equilibrium Economies , 2003 .

[41]  Benjamin A. Malin,et al.  Comparison of solutions to the multi-country Real Business Cycle model , 2011 .

[42]  Kenneth L. Judd,et al.  Solving Large Scale Rational Expectations Models , 1997 .

[43]  John Geweke,et al.  Monte carlo simulation and numerical integration , 1995 .

[44]  Manuel S. Santos.  Chapter 5 Numerical solution of dynamic economic models , 1999 .