Quantile-Crossing Spectrum and Spline Autoregression Estimation

TL;DR

This paper introduces a novel spline autoregression method for estimating the quantile-crossing spectrum, capturing richer serial dependence information across frequencies and quantiles, with improved accuracy when the spectrum is smooth.

Contribution

The paper proposes a new spline autoregression approach to estimate the quantile-crossing spectrum as a smooth bivariate function of frequency and quantile level.

Findings

The method outperforms alternatives when the spectrum is smooth in quantile level.

Numerical experiments demonstrate improved estimation accuracy.

The approach captures complex serial dependence structures in time series.

Abstract

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.