  In this paper I introduce quantile spectral densities that summarize the
cyclical behavior of time series across their whole distribution by analyzing
periodicities in quantile crossings. This approach can capture systematic
changes in the impact of cycles on the distribution of a time series and allows
robust spectral estimation and inference in situations where the dependence
structure is not accurately captured by the auto-covariance function. I study
the statistical properties of quantile spectral estimators in a large class of
nonlinear time series models and discuss inference both at fixed and across all
frequencies. Monte Carlo experiments illustrate the advantages of quantile
spectral analysis over classical methods when standard assumptions are
violated.
