  We address estimation of parametric coefficients of a pure-jump L\'evy driven
univariate stochastic differential equation (SDE) model, which is observed at
high frequency over a fixed time period. It is known from the previous study
Masuda (2013) that adopting the conventional Gaussian quasi-maximum likelihood
estimator then leads to an inconsistent estimator. In this paper, under the
assumption that the driving L\'evy process is locally stable, we extend the
Gaussian framework into a non-Gaussian counterpart, by introducing a novel
quasi-likelihood function formally based on the small-time stable approximation
of the unknown transition density. The resulting estimator turns out to be
asymptotically mixed normally distributed without ergodicity and finite moments
for a wide range of the driving pure-jump L\'evy process, showing much better
theoretical performance compared with the Gaussian quasi-maximum likelihood
estimator. Extensive simulations are carried out to show good estimation
accuracy. The case of large-time asymptotics under ergodicity is briefly
mentioned as well, where we can deduce an analogous asymptotic normality
result.
