Leeb, Hannes and Pötscher, Benedikt M. and Ewald, Karl (2014): On various confidence intervals postmodelselection.

PDF
MPRA_paper_52858.pdf Download (9MB)  Preview 
Abstract
We compare several confidence intervals after model selection in the setting recently studied by Berk et al. (2013), where the goal is to cover not the true parameter but a certain nonstandard quantity of interest that depends on the selected model. In particular, we compare the PoSIintervals that are proposed in that reference with the `naive' confidence interval, which is constructed as if the selected model were correct and fixed apriori (thus ignoring the presence of model selection). Overall, we find that the actual coverage probabilities of all these intervals deviate only moderately from the desired nominal coverage probability. This finding is in stark contrast to several papers in the existing literature, where the goal is to cover the true parameter.
Item Type:  MPRA Paper 

Original Title:  On various confidence intervals postmodelselection 
Language:  English 
Keywords:  Confidence intervals, model selection 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General 
Item ID:  52858 
Depositing User:  Benedikt Poetscher 
Date Deposited:  14 Jan 2014 07:57 
Last Modified:  01 Oct 2019 09:04 
References:  D. W. K. Andrews and P. Guggenberger. Hybrid and sizecorrected subsampling methods. Econometrica, 77, 721762, 2009. R. Berk, L. Brown, A. Buja, K. Zhang, and L. Zhao. Valid postselection inference. Ann. Statist., 41, 802837, 2013. P. J. Bickel and K. A. Doksum. Mathematical Statistics: Basic Ideas and Selected Topics. HoldenDay, Oakland, 1977. L. D. Brown. The conditional level of Student's t test. Ann. Math. Stat., 38, 10681071, 1967. R. J. Buehler and A. P. Feddersen. Note on a conditional property of Student's t. Ann. Math. Stat., 34, 10981100, 1963. T. K. Dijkstra and J. H. Veldkamp. Datadriven selection of regressors and the bootstrap. Lecture Notes in Econom. and Math. Systems, 307, 1738, 1988. K. Ewald. On the influence of model selection on confidence regions for marginal associations in the linear model. Master's thesis, University of Vienna, 2012. P. Kabaila. Valid confidence intervals in regression after variable selection. Econometric Theory, 14, 463482, 1998. P. Kabaila. The coverage properties of confidence regions after model selection. Int. Statist. Rev., 77, 405414, 2009. P. Kabaila and H. Leeb. On the largesample minimal coverage probability of confidence intervals after model selection. J. Amer. Statist. Assoc., 101, 619629, 2006. H. Leeb. The distribution of a linear predictor after model selection: unconditional finitesample distributions and asymptotic approximations. IMS Lecture Notes  Monograph Series, 49, 291311, 2006. H. Leeb and B. M. Pötscher. The finitesample distribution of postmodelselection estimators, and uniform versus nonuniform approximations. Econometric Theory, 19, 100142, 2003. H. Leeb and B. M. Pötscher. Model selection and inference: Facts and fiction. Econometric Theory, 21,2159, 2005. H. Leeb and B. M. Pötscher. Can one estimate the conditional distribution of postmodelselection estimators? Ann. Statist., 34, 25542591, 2006a. H. Leeb and B. M. Pötscher. Performance limits for estimators of the risk or distribution of shrinkagetype estimators, and some general lower riskbound results. Econometric Theory, 22, 6997, 2006b. H. Leeb and B. M. Pötscher. Can one estimate the unconditional distribution of postmodelselection estimators? Econometric Theory, 24, 338376, 2008a. H. Leeb and B. M. Pötscher. Model selection. In T. G. Andersen, R. A. Davis, J.P. Kreiß, and Th. Mikosch, editors, Handbook of Financial Time Series, pages 785821, New York, NY, 2008b. Springer. R. A. Olshen. The conditional level of the Ftest. J. Amer. Statist. Assoc., 68, 692698, 1973. B. M. Pötscher. Effects of model selection on inference. Econometric Theory, 7, 163185, 1991. B. M. Pötscher. The distribution of model averaging estimators and an impossibility result regarding its estimation. IMS Lecture Notes  Monograph Series, 52, 113129, 2006. B. M. Pötscher and H. Leeb. On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding. J. Multivariate Anal., 100, 20652082, 2009. B. M. Pötscher and U. Schneider. On the distribution of the adaptive LASSO estimator. J. Statist. Plann. Inference, 139, 27752790, 2009. B. M. Pötscher and U. Schneider. Confidence sets based on penalized maximum likelihood estimators in Gaussian regression. Electron. J. Statist., 4, 334360, 2010. B. M. Pötscher and U. Schneider. Distributional results for thresholding estimators in highdimensional Gaussian regression models. Electron. J. Statist., 5, 18761934, 2011. J.O. Rawlings. Applied Regression Analysis: A Research Tool. Springer Verlag, New York, NY, 1998. P. K. Sen. Asymptotic properties of maximum likelihood estimators based on conditional specification. Ann. Statist., 7, 10191033, 1979. P. K. Sen and E. A. K. Md. Saleh. On preliminary test and shrinkage Mestimation in linear models. Ann. Statist., 15, 15801592, 1987. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/52858 