Yale University
Department of Statistics
Seminar

Friday, April 13, 2001


Hannes Leeb
Institute of Statistics
University of Vienna, Austria


The distribution of post-model-selection estimators: finite-sample formulas, simple approximations, and estimators


 

We consider estimators for the parameter of a linear regression model after preliminary tests for exclusion of regressors have been conducted.  We first derive the finite-sample distribution of such post-model-selection estimators, which turns out to be quite complicated, and which depends on the unknown regression parameters even after centering. Yet, this distribution can be estimated consistently,e.g., by the (estimated) asymptotic distribution (Poetscher (1991), Sen (1979)), provided that the selected model is correct. But as pointed out in Poetscher (1991), the convergence of the finite-sample distribution to the asymptotic distribution is not uniform over the parameter space, which indicates potential problems with the accuracy of this approximation. This is confirmed by a Monte-Carlo study (Poetscher and Novak (1998)).

Next, we give a uniform asymptotic approximation to the finite-sample distribution, which is quite simple and easy to interpret while  it captures all the essential features of the true finite-sample distribution (however, this uniform asymptotic approximation still depends on unknown regression parameters). From this, we obtain, inter alia, the large-sample limit distribution of the post-model-selection estimator if the selected model is incorrect.

Finally, we show that the non-uniformity observed in Poetscher and Novak (1998) is not a consequence of using the (estimated) asymptotic distribution but rather is an intrinsic problem. More precisely, we show that no uniformly consistent estimator for the finite-sample distribution of the post-model-selection estimator exists. Furthermore, we provide finite-sample lower bounds for appropriate measures of the approximation error. Extensions to more general models and model selection procedures are also discussed.



Seminar begins at 12:30 pm, refreshments will be served at 12:00 noon
24 Hillhouse Avenue, Room 107