Modern quantitative macroeconomics necessarily involves the use of numerical methods in order to compute the equilibrium behavior of a model economy. As initially introduced by Magill (1977) and later used by Kydland and Prescott (1982) in their seminal work on business cycle models, linearization methods have been the preferred solution approach. Such methods are easy to implement and, as shown by Christiano (1990), do not introduce significant approximation errors for many settings studied by macroeconomists.
But, as discussed in Judd (2002), linearization methods are not trouble free. Quoting from that paper: “For example, Tesar (1995) uses the Kydland-Prescott method and found an example where completing asset markets will make agents worse off. This result violates general equilibrium theory and can only be attributed to the numerical method used.” (p.2) More recently, Kim and Kim (2003) have shown that this error in welfare analysis is symptomatic of linearized models and argue in favor of second-order approximation methods; variations on this theme have been developed by Sims (2000) and Schmitt-Grohe and Uribe (2004).
In order to study more complicated settings, non-linear methods have also been proposed that employ either projection techniques (Judd (1992) and McGrattan (1999)) or perturbation techniques (Judd and Guu (1997)). In a recent contribution to understanding these approaches, Aruoba, Fernandez-Villaverde, and Rubio-Ramirez (2003) examine the accuracy of these methods (along with traditional linearization and log-linearization methods) within the context of a prototypical real business cycle model. Their results replicate Christiano’s earlier analysis in that, for economies characterized by low risk aversion (i.e. little curvature in the utility function) and shocks that do not push the economy far from the steady-state (i.e. small variance of technology shocks), linearization methods do quite well. (footnotelog linearization is poor.) However, linearization methods, not surprisingly, deteriorate quickly in the presence of large shocks and high risk aversion.
This paper complements and extends the analysis by Aruoba, et al. (2003) in two dimensions. First, we examine discrete state settings so that heteroskedasticity in the technology shock can be introduced. In particular, we examine a crash state scenario and demonstrate that linearization methods perform poorly in this environment. We show that, even though the magnitude of the unconditional variance of the technology shock would lead one to conjecture reasonably small approximation errors (as suggested by Christiano and Aruoba et al.) for linear methods, the volatility of the conditional variances undermines this conjecture. Recent papers by Barro (2005) and Bloom (2005) have argued forcefully for the presence of large shocks to uncertainty in the economy and, hence, our analysis motivates the use of more sophisticated solution methods in such settings.
The second and major contribution of this paper is the introduction of a new algorithm to solve stochastic dynamic economies; for our analysis we use Hansen’s (1985) real business cycle model (again, with the shocks following a discrete-state Markov process). Our approach involves two parts: first, a one-pass continuous modification of the Upwind Gauss-Seidel Algorithm (Judd (1998)) is used to solve for the non-stochastic problem, and, second, an implicit iterative scheme is employed to account for the stochastic effects. In the latter iterative approach, the small numerical magnitudes of the stochastic terms (e.g. cross-state transition probabilities in the case of discrete-state Markovian processes or variances for continuous AR processes) produces relatively fast convergence. We will refer below to our method as to the RUGS (Recursive Upwind Gauss-Seidel) method. The algorithm has two strengths: (1) It is computationally fast; and (2) It has high global (i.e. non-local) accuracy. We test the performance and accuracy of our algorithm in comparison with other popular nonlinear methods using the analysis by Aruoba et al. as a template.
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A New Algorithm for Solving Dynamic Stochastic Macroeconomic Models
