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.
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