Giorgio Pauletto

Krylov methods for solving models with forward-looking variables

Abstract The simulation of large macroeconometric models containing forward-looking variables can become impractical when using exact Newton methods. The difficulties generally arise from the use of direct methods for the solution of the linear system in the Newton step. In such cases, nonstationary iterative methods, also called Krylov methods, provide an interesting alternative. In this paper we apply such methods to simulate a real world econometric model. Our numerical experiments confirm the interesting features of these techniques: low computational complexity and storage requirements. We also discuss a block preconditioner suitable for the particular class of models solved.

Rational Expectations Models

Nowadays, many large-scale macroeconometric models explicitly include forward expectation variables that allow for a better accordance with the underlying economic theory and also provide a response to the Lucas critique. These ongoing efforts gave rise to numerous models currently in use in various countries. Among others, we may mention MULTIMOD (Masson et al. [79]) used by the International Monetary Fund and MX-3 (Gagnon [41]) used by the Federal Reserve Board in Washington; model QPM (Armstrong et al. [5]) from the Bank of Canada; model Quest (Brandsma [18]) constructed and maintained by the European Commission; models MSG and NIGEM analyzed by the Macro Modelling Bureau at the University of Warwick.

Sparse direct methods for model simulation

Abstract In this paper, different strategies to exploit the sparse structure in the solution techniques for macroeconometric models with forward-looking variables are discussed. First, the stacked model is decomposed into recursive submodels without destroying its original block pattern. Next, we concentrate on how to efficiently solve the sparse linear system in the Newton algorithm. In this frame, a multiple block diagonal LU factorization and a sparse Gaussian elimination are presented. The algorithms are compared by solving the country model for Japan in MULTIMOD.

Solving finite difference schemes arising in trivariate option pricing

Abstract We investigate computational and implementation issues for the valuation of options on three underlying assets, focusing on the use of the finite difference methods. We demonstrate that implicit methods, which have good convergence and stability properties, can now be implemented efficiently due to the recent development of techniques that allow the efficient solution of large and sparse linear systems. In the trivariate option valuation problem, we use nonstationary iterative methods (also called Krylov methods) for the solution of the large and sparse linear systems arising while using implicit methods. Krylov methods are investigated both in serial and in parallel implementations. Computational results show that the parallel implementation is particularly efficient if a fine spatial grid is needed.

Econometric Model Simulation On Parallel Computers

The solution of large and sparse models presents in many ways a suitable structure for implementation on parallel computers. However, efficient use of these computing devices requires that the code be specifi cally structured to exploit the particular type of parallel computer used. This article discusses the implementa tion of data parallel processing algorithms as well as performance results based on the solution of a macro- econometric model on a Connection Machine 2.

Parallel Krylov Methods for Econometric Model Simulation

This paper investigates parallel solution methods to simulate large-scalemacroeconometric models with forward-looking variables. The method chosen isthe Newton-Krylov algorithm, and we concentrate on a parallel solution to thesparse linear system arising in the Newton algorithm. We empirically analyzethe scalability of the GMRES method, which belongs to the class of so-calledKrylov subspace methods. The results obtained using an implementation of thePETSc 2.0 software library on an IBM SP2 show a near linear scalability forthe problem tested.

Equation reordering for iterative processes–a comment

The ordering of the equations for a nonlinear model plays an important role in the performance of solution algorithms using iterative processes. The paper comments on what is often referred to be an optimal ordering.

Chapter 58 – Preconditioning in Economic Stochastic Growth Models

Publisher Summary This chapter describes the preconditioning in economic stochastic growth models. Preconditioning significantly improves the performance of Krylov subspace methods (KSM) used in policy iteration for solving stochastic growth models. The performance improvement is especially large in models where the discount factor approaches 1. The potential of KSM in solving economic models should be further investigated along the following lines. The role of KSM in solving stochastic growth models with larger values of the risk aversion coefficient and models with more persistent productivity shocks should be evaluated. The effect of drop-off tolerance strategies with ILU on the performance of KSM should be explored, especially for BiCSTAB. The impact of the starting policy iterate on the performance should be evaluated. It is suggested that the performance of KSM with preconditioning should be evaluated on parallel computers. High performance algorithms in such applications should include finely tuned policy improvement search over smaller action sets, multigrid policy interpolation, and potentially multigrid interpolation of preconditioners.

Preconditioning in Economic Stochastic Growth Models

Publisher Summary This chapter describes the preconditioning in economic stochastic growth models. Preconditioning significantly improves the performance of Krylov subspace methods (KSM) used in policy iteration for solving stochastic growth models. The performance improvement is especially large in models where the discount factor approaches 1. The potential of KSM in solving economic models should be further investigated along the following lines. The role of KSM in solving stochastic growth models with larger values of the risk aversion coefficient and models with more persistent productivity shocks should be evaluated. The effect of drop-off tolerance strategies with ILU on the performance of KSM should be explored, especially for BiCSTAB. The impact of the starting policy iterate on the performance should be evaluated. It is suggested that the performance of KSM with preconditioning should be evaluated on parallel computers. High performance algorithms in such applications should include finely tuned policy improvement search over smaller action sets, multigrid policy interpolation, and potentially multigrid interpolation of preconditioners.

A Review of Solution Techniques

This chapter reviews classic and well implemented solution techniques for linear and nonlinear systems. First, we discuss direct and iterative methods for linear systems. Some of these methods are part of the fundamental building blocks for many techniques for solving nonlinear systems presented later. The topic has been extensively studied and many methods have been analyzed in scientific computing literature, see e.g. Golub and Van Loan [56], Gill et al. [47], Barrett et al. [8] and Hageman and Young [60].