Quasi-Likelihood Analysis for Student-L\'evy Regression

TL;DR

This paper develops a quasi-likelihood method for high-frequency linear regression driven by Student-t Lévy processes, addressing the challenges of joint parameter estimation with a two-step approach.

Contribution

It introduces a novel two-step quasi-likelihood analysis that efficiently estimates regression, scale, and degrees of freedom in Student-t Lévy driven models.

Findings

Effective estimation of regression and scale parameters using Cauchy quasi-likelihood.

Accurate estimation of degrees of freedom via cumulative residuals.

Data thinning is necessary for reliable small-time approximations.

Abstract

We consider the quasi-likelihood analysis for a linear regression model driven by a Student-t L\'{e}vy process with constant scale and arbitrary degrees of freedom. The model is observed at high frequency over an extending period, under which we can quantify how the sampling frequency affects estimation accuracy. In that setting, joint estimation of trend, scale, and degrees of freedom is a non-trivial problem. The bottleneck is that the Student-t distribution is not closed under convolution, making it difficult to estimate all the parameters fully based on the high-frequency time scale. To efficiently deal with the intricate nature from both theoretical and computational points of view, we propose a two-step quasi-likelihood analysis: first, we make use of the Cauchy quasi-likelihood for estimating the regression-coefficient vector and the scale parameter; then, we construct the sequence of the unit-period cumulative residuals to estimate the remaining degrees of freedom. In particular, using full data in the first step causes a problem stemming from the small-time Cauchy approximation, showing the need for data thinning.