  A statistical functional, such as the mean or the median, is called
elicitable if there is a scoring function or loss function such that the
correct forecast of the functional is the unique minimizer of the expected
score. Such scoring functions are called strictly consistent for the
functional. The elicitability of a functional opens the possibility to compare
competing forecasts and to rank them in terms of their realized scores. In this
paper, we explore the notion of elicitability for multi-dimensional functionals
and give both necessary and sufficient conditions for strictly consistent
scoring functions. We cover the case of functionals with elicitable components,
but we also show that one-dimensional functionals that are not elicitable can
be a component of a higher order elicitable functional. In the case of the
variance this is a known result. However, an important result of this paper is
that spectral risk measures with a spectral measure with finite support are
jointly elicitable if one adds the `correct' quantiles. A direct consequence of
applied interest is that the pair (Value at Risk, Expected Shortfall) is
jointly elicitable under mild conditions that are usually fulfilled in risk
management applications.
