Quantile-Crossing Spectrum and Spline Autoregression Estimation

The quantile-crossing spectrum is the spectrum of quantile-crossing processes created from a time series by the indicator function that shows whether or not the time series lies above or below a given quantile at a given time. This bivariate function of frequency and quantile level provides a richer view of serial dependence than that offered by the ordinary spectrum. We propose a new method for estimating the quantile-crossing spectrum as a bivariate function of frequency and quantile level. The proposed method, called spline autoregression (SAR), jointly fits an AR model to the quantile-crossing series across multiple quantiles; the AR coefficients are represented as spline functions of the quantile level and penalized for their roughness. Numerical experiments show that when the underlying spectrum is smooth in quantile level the proposed method is able to produce more accurate estimates in comparison with the alternative that ignores the smoothness.

💡 Research Summary

The paper introduces a novel estimator for the quantile‑crossing spectrum, a two‑dimensional object defined over frequency (ω) and quantile level (α). The quantile‑crossing process uₜ(α)=α−I{yₜ≤q(α)} has zero mean and variance α(1−α). Its autocovariance R(τ,α) leads, via Fourier transform, to the spectrum S(ω,α). While prior work has treated S(ω,α) as a collection of one‑dimensional spectra (fixed α) estimated independently by lag‑window or AR methods, the authors argue that when S(ω,α) varies smoothly in α, a joint estimation across quantiles can improve accuracy, much like spectral smoothing across frequencies.

Three baseline approaches are reviewed: (i) a lag‑window estimator that smooths over τ with a bandwidth M, (ii) a parametric AR(p) estimator that fits an AR model separately for each α, and (iii) an AR‑S estimator that first fits separate AR models and then smooths the resulting coefficient curves with splines. The new method, called spline autoregression (SAR), integrates smoothing directly into the AR fitting. Specifically, the AR coefficients a_j(α) are modeled as spline functions of α, and a penalized least‑squares criterion is minimized: ∑{ℓ=1}^L ∑{t=p+1}^n