Spline Autoregression Method for Estimation of Quantile Spectrum

The quantile spectrum was introduced in Li (2012; 2014) as an alternative tool for spectral analysis of time series. It has the capability of providing a richer view of time series data than that offered by the ordinary spectrum especially for nonlinear dynamics such as stochastic volatility. A novel method, called spline autoregression (SAR), is proposed in this paper for estimating the quantile spectrum as a bivaraite function of frequency and quantile level, under the assumption that the quantile spectrum varies smoothly with the quantile level. The SAR method is facilitated by the quantile discrete Fourier transform (QDFT) based on trigonometric quantile regression. It is enabled by the resulting time-domain quantile series (QSER) which represents properly scaled oscillatory characteristics of the original time series around a quantile. A functional autoregressive (AR) model is fitted to the QSER on a grid of quantile levels by penalized least-squares with the AR coefficients represented as smoothing splines of the quantile level. While the ordinary AR model is widely used for conventional spectral estimation, the proposed SAR method provides an effective way of estimating the quantile spectrum as a bivariate function in comparison with the alternatives. This is confirmed by a simulation study.

💡 Research Summary

The paper introduces a novel approach for estimating the quantile spectrum—a two‑dimensional object that depends jointly on frequency (ω) and quantile level (α). Traditional methods treat each α separately, estimate an autoregressive (AR) model for the quantile periodogram, and then smooth the resulting AR coefficients across quantiles. This two‑step procedure suffers from high variance at extreme quantiles and from the need to choose a post‑hoc smoothing bandwidth.

The authors first define the Quantile Discrete Fourier Transform (QDFT) by solving a trigonometric quantile regression for each frequency–quantile pair (ω,α). From the QDFT they obtain the Quantile Series (QSER) via an inverse Fourier transform. The QSER is a real‑valued time series whose mean equals the α‑quantile of the original data and whose periodogram coincides with the quantile periodogram. Importantly, the QSER approximates the underlying level‑crossing process, allowing conventional spectral tools to be applied.

Building on this, the paper proposes the Spline Autoregression (SAR) estimator. Instead of estimating AR(p) coefficients Aτ(α) independently for each α, SAR models each coefficient matrix as a smooth spline function of α over a chosen interval