Comparison of solutions to the multi-country Real Business Cycle model

Abstract

a b s t r a c tWe compare the performance of perturbation, projection, and stochastic simulation algorithms for solving the multi-country RBC model described in Den Haan et al.(this issue). The main challenge of solving this model comes from its large number of continuous-valued state variables, ranging between four and 20 in the specifications we consider. The algorithms differ substantially in terms of speed and accuracy, and a clear trade-off exists between the two. Perturbation methods are very fast but invoke large approximation errors except at points close to the steady state; the projection methods considered are accurate on a large area of the state space but are very slow for specifications with many state variables; stochastic simulation methods have lower accuracy than projection methods, but their computational cost increases only moderately with the state-space dimension. Simulated series generated by different methods can differ noticeably, but only small differences are found in unconditional moments of simulated variables. On the basis of our comparison, we identify the factors that account for differences in accuracy and speed across methods, and we suggest directions for further improvement of some approaches.

Solving the multi-country real business cycle model using a Smolyak-collocation method

“…Since u c , f k and the probability density function for the exogenous state variables are not polynomials, we need to approximate the integral numerically. It is well known that for integration in relatively low dimensions (say around [10][11][12][13][14][15], if the integrand is su¢ ciently smooth, routines based on interpolatory cubature rules turn out to deliver much more accurate results than Monte Carlo or quasi Monte Carlo methods (see Cools (2002) or Schürer (2003)). Since Judd's (1998) textbook contains an excellent description of these various rules, we do not discuss them in detail here.…”

“…We therefore only repeat it here insofar as is needed for the description of the algorithm. Furthermore, Juillard and Villemont (2010) describe the accuracy tests with which our and competing solution methods are evaluated, while the paper by Kollmann, Maliar, Malin and Pichler (2010) compares the performance of our algorithm relative to these competing methods. 2 We therefore defer the detailed discussion of the performance of the algorithm to the latter paper.…”

Scalable high-dimensional dynamic stochastic economic modeling

“…Therefore, it is common practice in economics (see, e.g., [45]) to compute (unit-free) errors in the M + 1 equilibrium conditions. As in our model, these conditions often mainly consist of Euler equations 13 , the respective errors are therefore called Euler errors.…”

“…We overcome these difficulties by minimizing both the number of points to be evaluated and the time needed for each evaluation. For the first purpose (i) we use adaptive sparse grids (see, e.g., [8,9]), while the second task (ii) is accomplished using a hybrid parallelization scheme that minimizes interprocess 45 communication by using Intel Threading Building Blocks (TBB) [10] and partially offloads the function evaluations to accelerators using CUDA/Thrust [11]. This scheme enables us to make efficient use of modern hybrid high-performance computing facilities, whose performance nowadays reaches multiple petaflops [12].…”

“…To describe this challenge more formally, we first describe the general structure common to many (infinite-horizon) dynamic economic models. In a second step, we describe one concrete example, the 130 IRBC model (see, e.g., [44,45]). …”