Bayesian copula spectral analysis for stationary time series

Recently, quantile-based spectral analysis has drawn much attention due to that it can capture serial dependence more than covariance-related. One of typical quantile-based spectra is the copula spectral density kernel (CSDK) proposed by Dette et al. (2015), which is more informative than the traditional spectral density. To avoid smoothing all CSDKs at different pairs of quantiles in the same way in the classical method, we propose a Bayesian approach that uses Markov Chain Monte Carlo scheme to fit smoothing splines to many different CSDKs automatically at a time. By replacing the spectral matrix with its modified Cholesky decomposition and rearranging it in a summation, a Whittle-type likelihood function is expressed in a product-form, by which the coefficients of spline basis and smoothing parameters are grouped independently. Then our approach produces an automatically smoothed estimator for CSDKs, along with samples from the posterior distributions of the parameters via a Hamiltonian Monte Carlo (HMC) step. The parameter grouping scheme reduces the encoding workload, and the HMC reduces the computation complexity. Both of them allow the method to be applicable to estimate a large number of CSDKs simultaneously.