  This paper proposes a novel asymmetric continuous probabilistic score (ACPS)
for evaluating and comparing density forecasts. It extends the proposed score
and defines a weighted version, which emphasizes regions of interest, such as
the tails or the center of a variable's range. A test is also introduced to
statistically compare the predictive ability of different forecasts. The ACPS
is of general use in any situation where the decision maker has asymmetric
preferences in the evaluation of the forecasts. In an artificial experiment,
the implications of varying the level of asymmetry in the ACPS are illustrated.
Then, the proposed score and test are applied to assess and compare density
forecasts of macroeconomic relevant datasets (US employment growth) and of
commodity prices (oil and electricity prices) with particular focus on the
recent COVID-19 crisis period.
