Spline Autoregression Method for Estimation of Quantile Spectrum

Based on trigonometric quantile regression, the quantile spectrum was introduced in Li (2008, 2012) as an alternative tool for spectral analysis of time series. It has been demonstrated to have 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 article for estimating the quantile spectrum as a bivariate 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 model is fitted to the QSER on a grid of quantile levels by penalized least-squares, where the autoregressive coefficients are represented as spline functions of the quantile level. While the ordinary autoregressive (AR) model is widely used for conventional spectral estimation, the simulation study in this article confirms that the proposed SAR method provides an effective way of estimating the quantile spectrum as a bivariate function in comparison with the alternatives. Supplementary materials for this article are available online.