The Curse of Dimensionality in Solving, Estimating and Comparing Non-Linear Rational Expectation Models

This paper presents an attempt to solve and estimate a structural dynamic non-linear rational expectation model. The main contribution of this paper is to explore the Smolyak operator for numerical approximation and integration in a generic model class which do not suffer exponentially but only polynomially from the curse of dimensionality. The approximation of the policy function is done by Smolyak Chebyshev polynomials in the first order conditions f(s,x,Eh(s,x,e,s',x'))=0 with rational expectations about next period state s' and policy x'. Start values are generated by a linear approximation. The solution $x(s)$ forms a non-linear state space model analyzed by the unscented and particle filter. The rational expectation integration is done with an adaptive Smolyak scheme. For the estimation of posterior densities of structural parameters I propose a genetic extension of the Metropolis-Hastings algorithm to overcome the covariance choice problem in the random walk variant. Linearization is finally compared to the non-linear solution by a Bayesian model choice criterium for non-nested models