Convergence of Computed Dynamic Models with Unbounded Shock

TL;DR

This paper establishes theoretical conditions under which numerical solutions of dynamic models with unbounded shocks converge to the true solutions, extending prior results limited to bounded shocks.

Contribution

It provides the first theoretical proof of convergence for approximate invariant measures in models with unbounded shocks, broadening applicability.

Findings

Convergence of approximate invariant measures is proven under specific conditions.

Extends convergence results from bounded to unbounded shock scenarios.

Supports the validity of numerical methods in more realistic economic models.

Abstract

This paper studies the asymptotic convergence of computed dynamic models when the shock is unbounded. Most dynamic economic models lack a closed-form solution. As such, approximate solutions by numerical methods are utilized. Since the researcher cannot directly evaluate the exact policy function and the associated exact likelihood, it is imperative that the approximate likelihood asymptotically converges -- as well as to know the conditions of convergence -- to the exact likelihood, in order to justify and validate its usage. In this regard, Fernandez-Villaverde, Rubio-Ramirez, and Santos (2006) show convergence of the likelihood, when the shock has compact support. However, compact support implies that the shock is bounded, which is not an assumption met in most dynamic economic models, e.g., with normally distributed shocks. This paper provides theoretical justification for most dynamic models used in the literature by showing the conditions for convergence of the approximate invariant measure obtained from numerical simulations to the exact invariant measure, thus providing the conditions for convergence of the likelihood.