  We propose to construct copulas from the inversion of nonlinear state space
models. These allow for new time series models that have the same serial
dependence structure of a state space model, but with an arbitrary marginal
distribution, and flexible density forecasts. We examine the time series
properties of the copulas, outline serial dependence measures, and estimate the
models using likelihood-based methods. Copulas constructed from three example
state space models are considered: a stochastic volatility model with an
unobserved component, a Markov switching autoregression, and a Gaussian linear
unobserved component model. We show that all three inversion copulas with
flexible margins improve the fit and density forecasts of quarterly U.S. broad
inflation and electricity inflation.
