  In this paper, conditional data augmentation (DA) is investigated for the
degrees of freedom parameter $\nu$ of a Student-$t$ distribution. Based on a
restricted version of the expected augmented Fisher information, it is
conjectured that the ancillarity DA is progressively more efficient for MCMC
estimation than the sufficiency DA as $\nu$ increases; with the break even
point lying at as low as $\nu\approx4$. The claim is examined further and
generalized through a large simulation study and a application to U.S.
macroeconomic time series. Finally, the ancillarity-sufficiency interweaving
strategy is empirically shown to combine the benefits of both DAs. The proposed
algorithm may set a new standard for estimating $\nu$ as part of any model.
