  Based on a novel dynamic Whittle likelihood approximation for locally
stationary processes, a Bayesian nonparametric approach to estimating the
time-varying spectral density is proposed. This dynamic frequency-domain based
likelihood approximation is able to depict the time-frequency evolution of the
process by utilizing the moving periodogram previously introduced in the
bootstrap literature. The posterior distribution is obtained by updating a
bivariate extension of the Bernstein-Dirichlet process prior with the dynamic
Whittle likelihood. Asymptotic properties such as sup-norm posterior
consistency and L2-norm posterior contraction rates are presented.
Additionally, this methodology enables model selection between stationarity and
non-stationarity based on the Bayes factor. The finite-sample performance of
the method is investigated in simulation studies and applications to real-life
data-sets are presented.
