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和任意自由度的学生t莱维过程驱动的线性回归模型的拟似然分析。该模型在较长时期内以高频进行观测，在此设定下，我们可以量化抽样频率如何影响估计精度。在这种情况下，趋势、尺度和自由度的联合估计是一个非平凡问题。瓶颈在于学生t分布对卷积不封闭，这使得完全基于高频时间尺度估计所有参数变得困难。为了从理论和计算角度有效处理这种复杂性质，我们提出了一种两步拟似然分析方法：首先，我们利用柯西拟似然来估计回归系数向量和尺度参数；然后，我们构建单位周期累积残差序列来估计剩余的自由度。特别地，在第一步中使用全部数据会引发由小时间柯西近似产生的问题，这表明需要进行数据稀释。