  Expectiles define the only law-invariant, coherent and elicitable risk
measure apart from the expectation. The popularity of expectile-based risk
measures is steadily growing and their properties have been studied for
independent data, but further results are needed to use extreme expectiles with
dependent time series such as financial data. In this paper we establish a
basis for inference on extreme expectiles and expectile-based marginal expected
shortfall in a general $\beta$-mixing context that encompasses ARMA, ARCH and
GARCH models with heavy-tailed innovations. Simulations and applications to
financial returns show that the new estimators and confidence intervals greatly
improve on existing ones when the data are dependent.
