Robustified Gaussian quasi-likelihood inference for volatility

We consider statistical inference for a class of continuous regression models contaminated by finite-activity jumps and spike noises. We propose an $M$-estimator through some easy-to-implement one-parameter robustifications of the conventional Gaussian quasi-likelihood function, and prove its asymptotic mixed normality at the standard rate $\sqrt{n}$. It is theoretically shown that the estimator is simultaneously robust against the contaminations in both the covariate process and the objective process. Additionally, we prove that, under suitable design conditions on the tuning parameter, the proposed estimators can enjoy the same asymptotic distribution as in the case of no contamination. Some illustrative simulation results are presented, highlighting the estimator's insensitivity to fine-tuning.