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.

翻译：我们考虑由具有恒定尺度与任意自由度的Student-t Lévy过程驱动的线性回归模型的拟似然分析。该模型在扩展周期上以高频观测，据此可量化采样频率对估计精度的影响。在此框架下，同时估计趋势、尺度与自由度参数是一个非平凡问题。其瓶颈在于Student-t分布对卷积不封闭，导致难以基于高频时间尺度完全估计所有参数。为从理论与计算角度有效处理这种复杂性质，我们提出两步拟似然分析方法：首先利用柯西拟似然估计回归系数向量与尺度参数；随后构建单位周期累积残差序列以估计剩余自由度。特别地，第一步使用全部数据会因短时柯西近似引发问题，这表明需要进行数据稀释处理。