  When providing probabilistic forecasts for uncertain future events, it is
common to strive for calibrated forecasts, that is, the predictive distribution
should be compatible with the observed outcomes. Several notions of calibration
are available in the case of a single forecaster alongside with diagnostic
tools and statistical tests to assess calibration in practice. Often, there is
more than one forecaster providing predictions, and these forecasters may use
information of the others and therefore influence one another. We extend common
notions of calibration, where each forecaster is analysed individually, to
notions of cross-calibration where each forecaster is analysed with respect to
the other forecasters in a natural way. It is shown theoretically and in
simulation studies that cross-calibration is a stronger requirement on a
forecaster than calibration. Analogously to calibration for individual
forecasters, we provide diagnostic tools and statistical tests to assess
forecasters in terms of cross-calibration. The methods are illustrated in
simulation examples and applied to probabilistic forecasts for inflation rates
by the Bank of England.
