  In multivariate time series analysis, spectral coherence measures the linear
dependency between two time series at different frequencies. However, real data
applications often exhibit nonlinear dependency in the frequency domain.
Conventional coherence analysis fails to capture such dependency. The quantile
coherence, on the other hand, characterizes nonlinear dependency by defining
the coherence at a set of quantile levels based on trigonometric quantile
regression. This paper introduces a new estimation technique for quantile
coherence. The proposed method is semi-parametric, which uses the parametric
form of the spectrum of a vector autoregressive (VAR) model to approximate the
quantile coherence, combined with nonparametric smoothing across quantiles. At
a given quantile level, we compute the quantile autocovariance function (QACF)
by performing the Fourier inverse transform of the quantile periodograms.
Subsequently, we utilize the multivariate Durbin-Levinson algorithm to estimate
the VAR parameters and derive the estimate of the quantile coherence. Finally,
we smooth the preliminary estimate of quantile coherence across quantiles using
a nonparametric smoother. Numerical results show that the proposed estimation
method outperforms nonparametric methods. We show that quantile coherence-based
bivariate time series clustering has advantages over the ordinary VAR
coherence. For applications, the identified clusters of financial stocks by
quantile coherence with a market benchmark are shown to have an intriguing and
more informative structure of diversified investment portfolios that may be
used by investors to make better decisions.
