  It can be argued that optimal prediction should take into account all
available data. Therefore, to evaluate a prediction interval's performance one
should employ conditional coverage probability, conditioning on all available
observations. Focusing on a linear model, we derive the asymptotic distribution
of the difference between the conditional coverage probability of a nominal
prediction interval and the conditional coverage probability of a prediction
interval obtained via a residual-based bootstrap. Applying this result, we show
that a prediction interval generated by the residual-based bootstrap has
approximately 50% probability to yield conditional under-coverage. We then
develop a new bootstrap algorithm that generates a prediction interval that
asymptotically controls both the conditional coverage probability as well as
the possibility of conditional under-coverage. We complement the asymptotic
results with several finite-sample simulations.
