On the Numerical Accuracy of First-Order Approximate Solutions to Dsge Models

Abstract

Many algorithms that provide approximate solutions for dynamic stochastic general equilibrium (DSGE) models employ the QZ factorization because it allows a flexible formulation of the model and exempts the researcher from identifying equations that give raise to infinite eigenvalues. We show, by means of an example, that the policy functions obtained by this approach may differ from both the solution of a properly reduced system and the solution obtained from solving the system of nonlinear equations that arises from applying the implicit function theorem to the model's equilibrium conditions. As a consequence, simulation results may depend on the specific algorithm used and on the numerical values of parameters that are theoretically irrelevant. The sources of this inaccuracy are ill-conditioned matrices as they emerge, e.g., in models with strong habits. Researchers should be aware of those strange effects, and we propose several ways to handle them.

On the uniqueness of solutions to rational expectations models

“…In practice, i.e., taking into account finite precision computer arithmetic, the particular algorithm that factorsà andB comes into play. In a companion paper, Heiberger et al (2014), we show by means of a model from the asset pricing literature that there can be noticeable differences in the matrices L w w and L y w depending on the factorization employed.…”

Higher-order dynamic effects of uncertainty risk under thick-tailed stochastic volatility

“…However, most studies only focused on the impact of specific uncertainties and did not systematically account for the various characteristics of uncertainty risk, significantly reducing a model’s ability to portray disaster risks. Moreover, most previous studies that analyzed uncertainty often adopted a first-order perturbation method to approximate the DSGE model (Heiberger et al 2017 ; Niu et al 2018 ; Heiberger 2020 ). However, uncertainty risk exists in the third-order term of the DSGE model, which is related to the third-order disturbance approximation (Andreasen 2012 ).…”