Solving, Estimating and Selecting Nonlinear Dynamic Economic Models Without the Curse of Dimensionality

A welfare analysis of policies for risk averse preferences is handicapped within linear or linearized models and their certainty equivalence property. I calculate the optimal policy in a nonlinear stochastic dynamic general equilibrium model. Then I estimate the posterior density of structural parameters and the marginal likelihood within a nonlinear state space model. The computational challenges are considerable and I concentrate on the numerics and statistics for a simple model of dynamic consumption and labor choice. It is estimated on simulated data in order to test the routines with known true parameters. The code is written for a general model class. The policy function is approximated by Smolyak Chebyshev polynomials and the rational expectation integral by Smolyak Gaussian quadrature. The Smolyak operator is used to extend univariate approximation and integration operators to many dimensions. It is also called boolean method, hyperbolic cross points, discrete or sparse grids, fully symmetrical integration rules or complete polynomials. It reduces the curse of dimensionality from exponential to polynomial growth. The likelihood integrals are evaluated by a Gaussian quadrature and Gaussian quadrature particle filter. The bootstrap or sequential importance resampling particle filter, based on a Monte Carlo integration, is used as an accuracy benchmark. The posterior is estimated by the Gaussian filter and a Metropolis-Hastings algorithm. I propose a genetic extension of the standard Metropolis-Hastings algorithm by parallel random walk sequences. The candidate parameter vectors are constructed by innovation from the parallel sequences in addition to the usual random walk shocks. This improves the robustness of start values and the global maximization properties. Moreover it simplifies a cluster implementation and the random walk variances decision is reduced to only two parameters so that almost no trial sequences are needed. Finally the marginal likelihood is calculated as a criterion for nonnested and quasi-true models in order to select between the nonlinear estimates and a first order perturbation solution combined with the Kalman filter