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date: 08 December 2022

Sparse Grids for Dynamic Economic Modelslocked

Sparse Grids for Dynamic Economic Modelslocked

  • Johannes Brumm, Johannes BrummDepartment of Economics and Management, Karlsruhe Institute of Technology
  • Christopher Krause, Christopher KrauseDepartment of Economics and Management, Karlsruhe Institute of Technology
  • Andreas SchaabAndreas SchaabColumbia Business School, Columbia University
  •  and Simon ScheideggerSimon ScheideggerFaculty of Business and Economics, University of Lausanne

Summary

Solving dynamic economic models that capture salient real-world heterogeneity and nonlinearity requires the approximation of high-dimensional functions. As their dimensionality increases, compute time and storage requirements grow exponentially. Sparse grids alleviate this curse of dimensionality by substantially reducing the number of interpolation nodes, that is, grid points needed to achieve a desired level of accuracy. The construction principle of sparse grids is to extend univariate interpolation formulae to the multivariate case by choosing linear combinations of tensor products in a way that reduces the number of grid points by orders of magnitude relative to a full tensor-product grid and doing so without substantially increasing interpolation errors. The most popular versions of sparse grids used in economics are (dimension-adaptive) Smolyak sparse grids that use global polynomial basis functions, and (spatially adaptive) sparse grids with local basis functions. The former can economize on the number of interpolation nodes for sufficiently smooth functions, while the latter can also handle non-smooth functions with locally distinct behavior such as kinks. In economics, sparse grids are particularly useful for interpolating the policy and value functions of dynamic models with state spaces between two and several dozen dimensions, depending on the application. In discrete-time models, sparse grid interpolation can be embedded in standard time iteration or value function iteration algorithms. In continuous-time models, sparse grids can be embedded in finite-difference methods for solving partial differential equations like Hamilton-Jacobi-Bellman equations. In both cases, local adaptivity, as well as spatial adaptivity, can add a second layer of sparsity to the fundamental sparse-grid construction. Beyond these salient use-cases in economics, sparse grids can also accelerate other computational tasks that arise in high-dimensional settings, including regression, classification, density estimation, quadrature, and uncertainty quantification.

Subjects

  • Economic Theory and Mathematical Models
  • Financial Economics
  • International Economics
  • Macroeconomics and Monetary Economics

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