  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.
