  In small area estimation, it is a smart strategy to rely on data measured
over time. However, linear mixed models struggle to properly capture time
dependencies when the number of lags is large. Given the lack of published
studies addressing robust prediction in small areas using time-dependent data,
this research seeks to extend M-quantile models to this field. Indeed, our
methodology successfully addresses this challenge and offers flexibility to the
widely imposed assumption of unit-level independence. Under the new model,
robust bias-corrected predictors for small area linear indicators are derived.
Additionally, the optimal selection of the robustness parameter for bias
correction is explored, contributing theoretically to the field and enhancing
outlier detection. For the estimation of the mean squared error (MSE), a
first-order approximation and analytical estimators are obtained under general
conditions. Several simulation experiments are conducted to evaluate the
performance of the fitting algorithm, the new predictors, and the resulting MSE
estimators, as well as the optimal selection of the robustness parameter.
Finally, an application to the Spanish Living Conditions Survey data
illustrates the usefulness of the proposed predictors.
