  We study a class of backtests for forecast distributions in which the test
statistic depends on a spectral transformation that weights exceedance events
by a function of the modeled probability level. The weighting scheme is
specified by a kernel measure which makes explicit the user's priorities for
model performance. The class of spectral backtests includes tests of
unconditional coverage and tests of conditional coverage. We show how the class
embeds a wide variety of backtests in the existing literature, and further
propose novel variants which are easily implemented, well-sized and have good
power. In an empirical application, we backtest forecast distributions for the
overnight P&L of ten bank trading portfolios. For some portfolios, test results
depend materially on the choice of kernel.
