Estimation of nonlinear DSGE models through Laplace based solutions

This paper proposes a novel Laplace-based solution to non-linear DSGE models that has a closed-form likelihood. We implicitly use a non-linear approximation to the policy function that is invertible with respect to the shocks, implying that in the approximation the shocks can be recovered uniquely from some of the control variables. Using perturbation methods and a Lagrange inversion formula, we are able to calculate the derivatives of the likelihood and construct the Laplace based solution. In contrast with previous likelihood-based approaches, the method used here requires neither the introduction of linear shocks nor simulation to evaluate the likelihood. Using US data, we estimate linear and nonlinear variants of a well-known neoclassical growth model with and without time-varying variances. We find that a nonlinear heteroscedastic model has a much better empirical performance. Furthermore, our models allow us to ascertain that the monetary policy shock causes most of the time changes in economic uncertainty.