  We propose an original two-part, duration-severity approach for backtesting
Expected Shortfall (ES). While Probability Integral Transform (PIT) based ES
backtests have gained popularity, they have yet to allow for separate testing
of the frequency and severity of Value-at-Risk (VaR) violations. This is a
crucial aspect, as ES measures the average loss in the event of such
violations. To overcome this limitation, we introduce a backtesting framework
that relies on the sequence of inter-violation durations and the sequence of
severities in case of violations. By leveraging the theory of (bivariate)
orthogonal polynomials, we derive orthogonal moment conditions satisfied by
these two sequences. Our approach includes a straightforward, model-free Wald
test, which encompasses various unconditional and conditional coverage
backtests for both VaR and ES. This test aids in identifying any mis-specified
components of the internal model used by banks to forecast ES. Moreover, it can
be extended to analyze other systemic risk measures such as Marginal Expected
Shortfall. Simulation experiments indicate that our test exhibits good finite
sample properties for realistic sample sizes. Through application to two stock
indices, we demonstrate how our methodology provides insights into the reasons
for rejections in testing ES validity.
