Lower Bounds on Approximation Errors to Numerical Solutions of Dynamic Economic Models

Abstract

We propose a novel methodology for evaluating the accuracy of numerical solutions to dynamic economic models. It consists in constructing a lower bound on the size of approximation errors. A small lower bound on errors is a necessary condition for accuracy: If a lower error bound is unacceptably large, then the actual approximation errors are even larger, and hence, the approximation is inaccurate. Our lower‐bound error analysis is complementary to the conventional upper‐error (worst‐case) bound analysis, which provides a sufficient condition for accuracy. As an illustration of our methodology, we assess approximation in the first‐ and second‐order perturbation solutions for two stylized models: a neoclassical growth model and a new Keynesian model. The errors are small for the former model but unacceptably large for the latter model under some empirically relevant parameterizations.

Quantitative Macroeconomics: Lessons Learned from Fourteen Replications

“…The issue of the use of linear and log-linear, and first-and second-order perturbation in welfare evaluations has already been described in the Introduction. The results of Judd et al (2017) showing that the minimum error bounds on linear, log-linear, and first-order approximations are large enough to be problematic for most Economic models should dissuade Economists from using them in any application. This is especially true thanks to the implementation of second-order and higher methods in many available codebases (including Dynare).…”

“…This finding is not entirely novel, but it's importance is widely underappreciated. Kim and Kim (2007) show that 1st-order approximation methods deliver incorrect welfare results if even when using the correct (to 1st-order) optimal policies (although these can be largely avoided by putting the 1st-order solution into the unapproximated welfare function), while Judd et al (2017) show further that 1st-order solution methods are simply incorrect for many Macroeconomic models, deriving minimum error bounds that are large enough to be troubling. I conjecture that this problem, inaccurate welfare results, is likely widespread in early Quantitative Macroeconomics papers and recommend that any welfare result from pre-2000 should be treated as quantitatively suspect until replicated.…”

Near-Rational Equilibria in Heterogeneous-Agent Models: A Verification Method

“…One can view our method as a form of forward error analysis with provable guarantees of approximation quality, where the distance between the approximate and exact solution is measured in economic terms -in terms of individual welfare loss under approximate policies. Judd, Maliar, and Maliar (2017) provide a complementary view of the solution quality. They establish a lower bound on the (forward) error of an approximate solution to an equilibrium model.…”

Reliably Computing Nonlinear Dynamic Stochastic Model Solutions: An Algorithm with Error Formulas

“…This work contrasts with the analysis provided in(Judd et al, 2017a;Peralta-Alva and Santos, 2014;Santos and Peralta-alva, 2005;Santos, 2000). Their approaches identify Euler Equation Errors, but use statistical techniques to estimate the impact of these errors on solution accuracy.…”

“…The series representation presented here should also prove useful for applications based on value function iteration but I defer treating this topic for future work.2 Others have also studied approximation error for dynamic stochastic models(Judd et al, 2017a;Santos and Peralta-alva, 2005;Santos, 2000).…”