  Finding parametric models that accurately describe the dependence structure
of observed data is a central task in the analysis of time series. Classical
frequency domain methods provide a popular set of tools for fitting and
diagnostics of time series models, but their applicability is seriously
impacted by the limitations of covariances as a measure of dependence.
Motivated by recent developments of frequency domain methods that are based on
copulas instead of covariances, we propose a novel graphical tool that allows
to access the quality of time series models for describing dependencies that go
beyond linearity. We provide a thorough theoretical justification of our
approach and show in simulations that it can successfully distinguish between
subtle differences of time series dynamics, including non-linear dynamics which
result from GARCH and EGARCH models. We also demonstrate the utility of the
proposed tools through an application to modeling returns of the S&P 500 stock
market index.
