Second-order robustness for time series inference

This paper studies the second-order asymptotics of maximum likelihood estimator (MLE) and Whittle estimator under $$\varepsilon $$ ε -contaminated model for Gaussian stationary processes. We evaluate the robustness of MLE and Whittle estimator based on the second-order Edgeworth expansion with an $$ \varepsilon $$ ε -disturbance spectral density. The measures of second-order robustness of MLE and Whittle estimator are investigated for concrete models with numerical study. The findings show that the MLE of Gaussian autoregressive process is robust in second-order term to a disturbance in spectral density under the middle level of spectral frequency, while it is more sensitive to a contamination under a too low frequency spectral mass. The Whittle estimator is robust to a moving average contamination when the Gaussian autoregressive process is not near unit root case, while it is sensitive to the disturbance under a nonregular situation in the case of near unit root.