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  • Hannes Leeb (a1) and Benedikt M. Pötscher (a2)

Model selection has an important impact on subsequent inference. Ignoring the model selection step leads to invalid inference. We discuss some intricate aspects of data-driven model selection that do not seem to have been widely appreciated in the literature. We debunk some myths about model selection, in particular the myth that consistent model selection has no effect on subsequent inference asymptotically. We also discuss an “impossibility” result regarding the estimation of the finite-sample distribution of post-model-selection estimators.

Corresponding author
Address correspondence to Benedikt Pötscher, Department of Statistics, University of Vienna, Universitätsstrasse 5, A-1010, Vienna, Austria; e-mail:
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Ahmed, S.E. & A.K. Basu (2000) Least squares, preliminary test and Stein-type estimation in general vector AR(p) models. Statistica Neerlandica 54, 4766.
Altissimo, F. & V. Corradi (2002) Bounds for inference with nuisance parameters present only under the alternative. Econometrics Journal 5, 494519.
Altissimo, F. & V. Corradi (2003) Strong rules for detecting the numbers of breaks in a time series. Journal of Econometrics 117, 207244.
Andrews, D.W.K. (1986) Complete consistency: A testing analogue of estimator consistency. Review of Economic Studies 53, 263269.
Bauer, P., B.M. Pötscher, & P. Hackl (1988) Model selection by multiple test procedures. Statistics 19, 3944.
Bunea, F. (2004) Consistent covariate selection and post model selection inference in semiparametric regression. Annals of Statistics 32, 898927.
Bunea, F., X. Niu, & M.H. Wegkamp (2003) The Consistency of the FDR Estimator. Working paper, Department of Statistics, Florida State University at Tallahassee.
Chen, S.S., D.L. Donoho, & M.A. Saunders (1998) Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing 20, 3361.
Corradi, V. (1999) Deciding between I(0) and I(1) via FLIL-based bounds. Econometric Theory 15, 643663.
Danilov, D. & J.R. Magnus (2004) On the harm that ignoring pretesting can cause. Journal of Econometrics 122, 2746.
Dijkstra, T.K. & J.H. Veldkamp (1988) Data-driven selection of regressors and the bootstrap. Lecture Notes in Economics and Mathematical Systems 307, 1738.
Dufour, J.M., D. Pelletier, & E. Renault (2003) Short run and long run causality in time series: Inference. Journal of Econometrics (forthcoming).
Dukić, V.M. & E.A Peña (2002) Estimation after Model Selection in a Gaussian Model. Manuscript, Department of Statistics, University of Chicago.
Ensor, K.B. & H.J. Newton (1988) The effect of order estimation on estimating the peak frequency of an autoregressive spectral density. Biometrika 75, 587589.
Fan, J. & R. Li (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 13481360.
Frank, I.E. & J.H. Friedman (1993) A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109148.
Giles, J.A. & D.E.A. Giles (1993) Pre-test estimation and testing in econometrics: Recent developments. Journal of Economic Surveys 7, 145197.
Hajek, J. (1971) Limiting properties of likelihoods and inference. In V.P. Godambe & D.A. Sprott (eds.), Foundations of Statistical Inference: Proceedings of the Symposium on the Foundations of Statistical Inference, University of Waterloo, Ontario, March 31–April 9, 1970, pp. 142159. Holt, Rinehart and Winston.
Hajek, J. & Z. Sidak (1967) Theory of Rank Tests. Academic Press.
Hall, A.R. & F.P.M. Peixe (2003) A consistent method for the selection of relevant instruments. Econometric Reviews 22, 269287.
Hannan, E.J. & B.G. Quinn (1979) The determination of the order of an autoregression. Journal of the Royal Statistical Society, Series B 41, 190195.
Hansen, P.R. (2003) Regression Analysis with Many Specifications: A Bootstrap Method for Robust Inference. Working paper, Department of Economics, Brown University.
Hidalgo, J. (2002) Consistent order selection with strongly dependent data and its application to efficient estimation. Journal of Econometrics 110, 213239.
Hjort, N.L. & G. Claeskens (2003) Frequentist model average estimators. Journal of the American Statistical Association 98, 879899.
Hosoya, Y. (1984) Information criteria and tests for time series models. In O.D. Anderson (ed.), Time Series Analysis: Theory and Practice, vol. 5, pp. 3952. North-Holland.
Judge, G.G. & M.E. Bock (1978) The Statistical Implications of Pre-test and Stein-Rule Estimators in Econometrics. North-Holland.
Judge, G.G. & T.A. Yancey (1986) Improved Methods of Inference in Econometrics. North-Holland.
Kabaila, P. (1995) The effect of model selection on confidence regions and prediction regions. Econometric Theory 11, 537549.
Kabaila, P. (1996) The evaluation of model selection criteria: Pointwise limits in the parameter space. In D.L. Dowe, K.B. Korb, & J.J. Oliver (eds.), Information, Statistics and Induction in Science, pp. 114118. World Scientific.
Kabaila, P. (1998) Valid confidence intervals in regression after variable selection. Econometric Theory 14, 463482.
Kabaila, P. & H. Leeb (2004) On the Large-Sample Minimal Coverage Probability of Confidence Intervals after Model Selection. Working paper, Department of Statistics, Yale University.
Kapetanios, G. (2001) Incorporating lag order selection uncertainty in parameter inference for AR models. Economics Letters 72, 137144.
Kempthorne, P.J. (1984) Admissible variable-selection procedures when fitting regression models by least squares for prediction. Biometrika 71, 593597.
Kilian, L. (1998) Accounting for lag order uncertainty in autoregressions: The endogenous lag order bootstrap algorithm. Journal of Time Series Analysis 19, 531548.
Knight, K. (1999) Epi-convergence in Distribution and Stochastic Equi-semicontinuity. Working paper, Department of Statistics, University of Toronto.
Knight, K. & W. Fu (2000) Asymptotics of lasso-type estimators. Annals of Statistics 28, 13561378.
Koul, H.L. & W. Wang (1984) Local asymptotic normality of randomly censored linear regression model. Statistics & Decisions, supplement 1, 1730.
Kulperger, R.J. & S.E. Ahmed (1992) A bootstrap theorem for a preliminary test estimator. Communications in Statistics: Theory and Methods 21, 20712082.
Leeb, H. (2003a) The distribution of a linear predictor after model selection: Conditional finite-sample distributions and asymptotic approximations. Journal of Statistical Planning and Inference (forthcoming).
Leeb, H. (2003b) The Distribution of a Linear Predictor after Model Selection: Unconditional Finite-Sample Distributions and Asymptotic Approximations. Working paper, Department of Statistics, University of Vienna.
Leeb, H. & B.M. Pötscher (2002) Performance Limits for Estimators of the Risk or Distribution of Shrinkage-Type Estimators, and Some General Lower Risk-Bound Results. Working paper, Department of Statistics, University of Vienna.
Leeb, H. & B.M. Pötscher (2003a) The finite-sample distribution of post-model-selection estimators and uniform versus nonuniform approximations. Econometric Theory 19, 100142.
Leeb, H. & B.M. Pötscher (2003b) Can One Estimate the Conditional Distribution of Post-Model-Selection Estimators? Working paper, Department of Statistics, University of Vienna. (Also available as Cowles Foundation Discussion paper 1444.)
Leeb, H. & B.M. Pötscher (2004) Can One Estimate the Unconditional Distribution of Post-Model-Selection Estimators? Manuscript, Department of Statistics, Yale University.
Lehmann, E.L. & G. Casella (1998) Theory of Point Estimation. Springer Texts in Statistics. Springer-Verlag.
Lütkepohl, H. (1990) Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Review of Economics and Statistics 72, 116125.
Magnus, J.R. (1999) The traditional pretest estimator. Teoriya Veroyatnost. i Primenen. 44, 401418; translation in Theory of Probability and Its Applications 44 (2000), 293–308.
Nickl, R. (2003) Asymptotic Distribution Theory of Post-Model-Selection Maximum Likelihood Estimators. Master's thesis, Department of Statistics, University of Vienna.
Nishii, R. (1984) Asymptotic properties of criteria for selection of variables in multiple regression. Annals of Statistics 12, 758765.
Phillips, P.C.B. (2005) Automated discovery in econometrics. Econometric Theory (this issue).
Pötscher, B.M. (1981) Order Estimation in ARMA-Models by Lagrangian Multiplier Tests. Research report 5, Department of Econometrics and Operations Research, University of Technology, Vienna.
Pötscher, B.M. (1983) Order estimation in ARMA-models by Lagrangian multiplier tests. Annals of Statistics 11, 872885.
Pötscher, B.M. (1991) Effects of model selection on inference. Econometric Theory 7, 163185.
Pötscher, B.M. (1995) Comment on “The effect of model selection on confidence regions and prediction regions.” Econometric Theory 11, 550559.
Pötscher, B.M. (2002) Lower risk bounds and properties of confidence sets for ill-posed estimation problems with applications to spectral density and persistence estimation, unit roots, and estimation of long memory parameters. Econometrica 70, 10351065.
Pötscher, B.M. & A.J. Novak (1998) The distribution of estimators after model selection: Large and small sample results. Journal of Statistical Computation and Simulation 60, 1956.
Rao, C.R. & Y. Wu (2001) On model selection. IMS Lecture Notes Monograph Series 38, 157.
Sargan, D.J. (2001) The choice between sets of regressors. Econometric Reviews 20, 171186.
Sclove, S.L., C. Morris, & R. Radhakrishnan (1972) Non-optimality of preliminary-test estimators for the mean of a multivariate normal distribution. Annals of Mathematical Statistics 43, 14811490.
Sen, P.K (1979) Asymptotic properties of maximum likelihood estimators based on conditional specification. Annals of Statistics 7, 10191033.
Sen, P.K & A.K.M.E. Saleh (1987) On preliminary test and shrinkage M-estimation in linear models. Annals of Statistics 15, 15801592.
Shibata, R. (1986) Consistency of model selection and parameter estimation. Journal of Applied Probability, special volume 23A, 127141.
Söderström, T. (1977) On model structure testing in system identification. International Journal of Control 26, 118.
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B 58, 267288.
Yang, Y. (2003) Can the Strengths of AIC and BIC Be Shared? Working paper, Department of Statistics, Iowa State University.
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Econometric Theory
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