  A nonparametric method is proposed for estimating the quantile spectra and
cross-spectra introduced in Li (2012; 2014) as bivariate functions of frequency
and quantile level. The method is based on the quantile discrete Fourier
transform (QDFT) defined by trigonometric quantile regression and the quantile
series (QSER) defined by the inverse Fourier transform of the QDFT. A
nonparametric spectral estimator is constructed from the autocovariance and
cross-covariance functions of the QSER using the lag-window (LW) approach.
Various quantile smoothing techniques are employed further to reduce the
statistical variability of the estimator across quantiles, among which is a new
technique called spline quantile regression (SQR). The performance of the
proposed estimation method is evaluated through a simulation study.
