Fast Multigrid Solvers for Calibration and Estimation of Dynamic Structural Models

Abstract

A new methodology for calibrating parameters when working with intractable, dynamic structural models is developed. A straight-forward extension also allows for formal estimation and hypothesis testing in a Generalized Method of Moment framework. The method is based on multigrid techniques used in many state of the art solvers from engineering applications. These techniques are adapted to solve Bellman and Euler-type equations and to handle subtleties arising from the interaction of statistical and numerical errors. The method works on a joint mode of analysis incorporating both statistical and numerical errors in the spirit of "forward-backward" error analysis of Kubler and Schmedders (2005). Numerical results from example problems - and experience from thirty years of multigrid literature - support the papers main finding: a fully-identified model that is smooth in parameters can be calibrated and solved with only about three to five times the work required to solve the model and compute associated "moments" for a single set of parameters. As with other multigrid methods, the solvers can be efficiently and naturally implemented on parallel processors. This work also shows that the size of numerical error can be less important than the qualitative type of error when parameters are fitted to a numerically solved model subject to discretization error. An example is presented in which a popular, consistent discretization of a portfolio problem with endogenous retirement produces an ill-posed, unstable calibration problem. This example and subsequent analysis show greater care must be taken when discretizing a model for a calibration or estimation problem: To avoid corrupting sensitivity analysis, identification, and inference, it is important to perform a joint error analysis that includes both discretization and statistical errors.