  Bootstrap smoothed (bagged) estimators have been proposed as an improvement
on estimators found after preliminary data-based model selection. Efron, 2014,
derived a widely applicable formula for a delta method approximation to the
standard deviation of the bootstrap smoothed estimator. He also considered a
confidence interval centered on the bootstrap smoothed estimator, with width
proportional to the estimate of this standard deviation. Kabaila and
Wijethunga, 2019, assessed the performance of this confidence interval in the
scenario of two nested linear regression models, the full model and a simpler
model, for the case of known error variance and preliminary model selection
using a hypothesis test. They found that the performance of this confidence
interval was not substantially better than the usual confidence interval based
on the full model, with the same minimum coverage. We extend this assessment to
the case of unknown error variance by deriving a computationally convenient
exact formula for the ideal (i.e. in the limit as the number of bootstrap
replications diverges to infinity) delta method approximation to the standard
deviation of the bootstrap smoothed estimator. Our results show that, unlike
the known error variance case, there are circumstances in which this confidence
interval has attractive properties.
