We develop a new algorithm to solve large scale dynamic stochastic general equilibrium models over a large transition. The method consists of Taylor expanding the equilibrium conditions of the model not just around the steady state, but sequentially along the entire equilibrium path. The method can be applied to a broad class of models and is orders of magnitudes more accurate than solutions based on local perturbation of the steady state. The method is also able to solve models with strong nonlinearities. Finally, because our policies are locally linear, we can make use of a version of the Kalman filter with time varying coefficients to identify shocks from data. With this tool in hand we are able to evaluate the likelihood and use the algorithm for the estimation of nonlinear models.