  Quantile forecasts made across multiple horizons have become an important
output of many financial institutions, central banks and international
organisations. This paper proposes misspecification tests for such quantile
forecasts that assess optimality over a set of multiple forecast horizons
and/or quantiles. The tests build on multiple Mincer-Zarnowitz quantile
regressions cast in a moment equality framework. Our main test is for the null
hypothesis of autocalibration, a concept which assesses optimality with respect
to the information contained in the forecasts themselves. We provide an
extension that allows to test for optimality with respect to larger information
sets and a multivariate extension. Importantly, our tests do not just inform
about general violations of optimality, but may also provide useful insights
into specific forms of sub-optimality. A simulation study investigates the
finite sample performance of our tests, and two empirical applications to
financial returns and U.S. macroeconomic series illustrate that our tests can
yield interesting insights into quantile forecast sub-optimality and its
causes.
