In Pötscher (1991, Econometric
Theory 7, 163–185) the asymptotic distribution of a
post-model-selection estimator, both unconditional and conditional on
selecting a correct model (minimal or not), has been derived.
Limitations of these results are (i) that they do not provide
information on the distribution of the post-model-selection estimator
conditional on selecting an incorrect model and (ii) that the quality
of this asymptotic approximation to the finite-sample distribution is
not uniform with respect to the underlying parameters. In the present
paper we first obtain the unconditional and also the conditional
finite-sample distribution of the post-model-selection estimator, which
turn out to be complicated and difficult to interpret. Second, we
obtain approximations to the finite-sample distributions that are as
simple and easy to interpret as the asymptotic distributions obtained
in Pötscher (1991) but at the same time
are close to the finite-sample distributions uniformly with respect to
the underlying parameters. As a by-product, we also obtain the
asymptotic distribution conditional on selecting an incorrect model.We thank the co-editor Richard Smith and the
two referees for helpful comments on a previous version of this paper.
Hannes Leeb's research was supported by the Austrian Science
Foundation (FWF), project P13868-MAT.