Before revision:  Based on the frequency decomposition of covariance between indicator functions  , we define the quantile cross-spectral beta of an assetcapturing tail-specific  as well as horizon-, or frequency-specific  risks.
After revision:  Based on the decomposition of covariance between indicator functions , we define the quantile cross-spectral beta of an assetcapturing tail-specific as well as horizon-, or frequency-specific risks.