Abstract
We study the ability of three different projection methods to solve high-dimensional state space problems: Galerkin, collocation, and least squares projection. The curse of dimensionality can be reduced substantially for both Least Squares and Galerkin projection methods through the use of monomial formulas. Least Squares are shown to require a good initial value in order to give an accurate solution. Alternatively, we suggest a new ad hoc collocation method for complete polynomials that is fast and easy to implement.
Highlights
In this paper, we study projection methods which have become a standard tool in the analysis of business cycle models and, in particular, asset prices
We study the ability of three different projection methods to solve highdimensional state space problems: Galerkin, collocation, and least squares projection
Modern business cycle analyses are based on complex high-dimensional general equilibrium models
Summary
We study projection methods which have become a standard tool in the analysis of business cycle models and, in particular, asset prices. The literature has only paid little attention to the solution of the stochastic growth model or any other dynamic stochastic general equilibrium model with the help of Least Squares projection.6 This observation is somehow puzzling as a priori we would assume that it is easier to solve a minimization problem than a non-linear equations problem. If we choose complete polynomials for the approximation of the policy function, we run into problems with the standard collocation method, where we set the residual function equal to zero at a number of points that is equal to the number of coefficients (and solve a system of non-linear equations). The Smolyak algorithm is a device how to pick the collocation points optimally This algorithm suffers from its lack of universal applicability as it only works for certain combinations of the state space dimension and the degree of the complete polynomial in the approximating function.
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