The Finite-Sample Distribution of Post-Model-Selection Estimators, and Uniform Versus Non-Uniform Approximations

In Poetscher [Econometric Theory (1991), 7, pp 163 - 185] the asymptotic distribution of a post-model-selection estimator, both unconditional and conditional on selecting a correct model, 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 w.r.t. the underlying parameters. In the present paper we first obtain the unconditional as well as the conditional finite-sample distribution of the post-model-selection estimator which turns 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 Poetscher [Econometric Theory (1991), 7, pp 163 - 185], but at the same time are close to the finite-sample distributions uniformly w.r.t. the underlying parameters. As a by-product, we also obtain the asymptotic distribution conditional on selecting an incorrect model.