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We consider the quasi-likelihood analysis for a linear regression modeldriven by a Student-t L\'evy process with constant scale and arbitrary degreesof freedom. The model is observed at a high frequency over an extending period,under which we can quantify how the sampling frequency affects estimationaccuracy. In that setting, joint estimation of trend, scale, and degrees offreedom is a non-trivial problem. The bottleneck is that the Student-tdistribution is not closed under convolution, making it difficult to estimateall the parameters fully based on the high-frequency time scale. To efficientlydeal with the intricate nature from both theoretical and computational pointsof view, we propose a two-step quasi-likelihood analysis: first, we make use ofthe Cauchy quasi-likelihood for estimating the regression-coefficient vectorand the scale parameter; then, we construct the sequence of the unit-periodcumulative residuals to estimate the remaining degrees of freedom. Inparticular, using full data in the first step causes a problem stemming fromthe small-time Cauchy approximation, showing the need for data thinning. Alsopresented is the implementation in a computer through the yuima package andsome numerical examples.

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