Skip to main content

Model Selection

  • Chapter
  • First Online:

Abstract

We provide an overview of the vast and rapidly growing area of model selection in statistics and econometrics.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   349.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   449.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   449.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Ahmed, S. E. and Basu, A. K. (2000): Least squares, preliminary test and Stein-type estimation in general vector AR(p) models. Statistica Neerlandica 54, 47-66.

    Article  MATH  MathSciNet  Google Scholar 

  • Akaike, H. (1969): Fitting autoregressive models for prediction. Annals of the Institute of Statistical Mathematics 21, 243-247.

    Article  MATH  MathSciNet  Google Scholar 

  • Akaike, H. (1970): Statistical predictor identification. Annals of the Institute of Statistical Mathematics 22, 203-217.

    Article  MATH  MathSciNet  Google Scholar 

  • Akaike, H. (1973): Information theory and an extension of the maximum likelihood principle B.N. Petrov and F. Csaki Second International Symposium on Information Theory. Akadémiai Kiadó, Budapest.

    Google Scholar 

  • Allen, D. M. (1971): Mean square error of prediction as a criterion for selecting variables. Technometrics 13, 469-475.

    Article  MATH  Google Scholar 

  • Allen, D. M. (1974): The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16, 125-127.

    Article  MATH  MathSciNet  Google Scholar 

  • Amemiya, T. (1980): Selection of regressors. International Economic Review 21, 331-354.

    Article  MATH  MathSciNet  Google Scholar 

  • An, H. Z. and Chen, Z. G. (1986): The identification of ARMA processes. Journal of Applied Probability 23A, 75-87.

    MathSciNet  Google Scholar 

  • An, H. Z. and Gu, L. (1985): On the selection of regression variables. Acta Mathematicae Applicatae Sinica 2, 27-36.

    Article  MATH  Google Scholar 

  • Anderson, T. W. (1962): The choice of the degree of a polynomial regression as a multiple decision problem. Annals of Mathematical Statistics 33, 255-265.

    Article  MATH  MathSciNet  Google Scholar 

  • Anderson, T. W. (1963): Determination of the order of dependence in normally distributed time series Rosenblatt, M. (Ed.): Time Series Analysis, 425-446. Wiley, New York.

    Google Scholar 

  • Bancroft, T. A. and Han, C. P. (1977): Inference based on conditional specification: A note and a bibliography. International Statistical Review 45, 117-127.

    MATH  MathSciNet  Google Scholar 

  • Baraud, Y. (2002): Model selection for regression on a random design. ESAIM Probability and Statistics 6, 127-146.

    Article  MATH  MathSciNet  Google Scholar 

  • Barron, A. R. (1991): Complexity regularization with application to artificial neural networks Nonparametric functional estimation and related topics (Spetses, 1990), NATO Advanced Study Institute Series C Mathematical and Physical Sciences 335, 561-576. Kluwer, Dordrecht.

    Google Scholar 

  • Barron, A. R. (1999): Information-theoretic characterization of Bayes performance and the choice of priors in parametric and nonparametric problems Bayesian Statistics, (Alcoceber, 1998) 6, 27-52. Oxford University Press, Oxford.

    Google Scholar 

  • Barron, A. R. and Cover, T. M. (1991): Minimum complexity density estimation. IEEE Transactions on Information Theory 37, 1034-1054. (Corrections: IEEE Transactions on Information Theory 37, 1738.)

    Article  MATH  MathSciNet  Google Scholar 

  • Barron, A. R., Birgé, L. and Massart, P. (1999): Risk bounds for model selection via penalization. Probability Theory and Related Fields 113, 301-413.

    Article  MATH  MathSciNet  Google Scholar 

  • Bauer, P., Pötscher, B. M. and Hackl, P. (1988): Model selection by multiple test procedures. Statistics 19, 39-44.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, R. (1996): Confidence sets centered at C p -estimators. Annals of the Institute of Statistical Mathematics 48, 1-15.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, R. (2000): REACT scatterplot smoothers: Superefficiency through basis economy. Journal of the American Statistical Association 95, 155-171.

    Article  MATH  MathSciNet  Google Scholar 

  • Beran, R. and Dümbgen, L. (1998): Modulation of estimators and confidence sets. Annals of Statistics 26, 1826-1856.

    Article  MATH  MathSciNet  Google Scholar 

  • Berger, J. O. and Pericchi, L. R. (2001): Objective Bayesian methods for model selection: Introduction and comparison. In: Lahiri, P. (Ed.): Model Selection. IMS Lecture Notes Monograph Series 38 135–193.

    Google Scholar 

  • Bhansali, R. J. (1999): Parameter estimation and model selection for multistep prediction of a time series: A review Ghosh, S. (Ed.): Asymptotics, Nonparametrics and Time Series—A Tribute to Madan Lal Puri, 201-225. Dekker, New York.

    Google Scholar 

  • Bhansali, R. J. and Downham, D. Y. (1977): Some properties of the order of an autoregressive model selected by a generalization of Akaike’s FPE criterion. Biometrika 64, 547-551.

    MATH  MathSciNet  Google Scholar 

  • Birgé, L. (2004): Model selection for Gaussian regression with random design. Bernoulli 10, 1039-1051.

    Article  MATH  MathSciNet  Google Scholar 

  • Birgé, L. (2006): Model selection via testing: An alternative to (penalized) maximum likelihood estimators. Annales de l’Institute Henri Poincaré 42, 273-325.

    Article  MATH  Google Scholar 

  • Birgé, L. and Massart, P. (2001): Gaussian model selection. Journal of the European Mathematical Society 3, 203-268.

    Article  MATH  Google Scholar 

  • Breiman, L. (1995): Better subset regression using the nonnegative garrote. Technometrics 37, 373-384.

    Article  MATH  MathSciNet  Google Scholar 

  • Breiman, L. and Freedman, D. (1983): How many variables should be entered in a regression equation? Journal of the American Statistical Association 78, 131-136.

    Article  MATH  MathSciNet  Google Scholar 

  • Brook, R. J. (1976): On the use of a regret function to set significance points in prior tests of estimation. Journal of the American Statistical Association 71, 126-131. (Correction: Journal of the American Statistical Association 71, 1010.)

    Article  MATH  MathSciNet  Google Scholar 

  • Brown, P. J., Vannucci, M. and Fearn, T. (2002): Bayes model averaging with selection of regressors. Journal of the Royal Statistical Society Series B 64, 519-536.

    Article  MATH  MathSciNet  Google Scholar 

  • Boucheron, S., Bousquet, O. and Lugosi, G. (2005): Theory of classification: A survey of some recent advances. ESAIM Probability and Statistics 9, 323-375.

    Article  MATH  MathSciNet  Google Scholar 

  • Buckland S. T., Burnham, K. P. and Augustin, N. H. (1997): Model selection: An integral part of inference. Biometrics 53, 603-618.

    Article  MATH  Google Scholar 

  • Bunea, F. (2004): Consistent covariate selection and post model selection inference in semiparametric regression. Annals of Statistics 32, 898-927.

    Article  MATH  MathSciNet  Google Scholar 

  • Bunea, F., Tsybakov, A. and Wegkamp, M. H. (2007): Aggregation for Gaussian regression. Annals of Statistics 35, 1674-1697.

    Article  MATH  MathSciNet  Google Scholar 

  • Bunea, F., Wegkamp, M. H. and Auguste, A. (2006): Consistent variable selection in high dimensional regression via multiple testing. Journal of Statistical Planning and Inference 136, 4349-4364.

    Article  MATH  MathSciNet  Google Scholar 

  • Burnham, K. P. and Anderson, D. R. (2002): Model Selection and Multimodal Inference (2nd edition). Springer, New York.

    Google Scholar 

  • Cesa-Bianchi, N. and Lugosi, G. (2006): Prediction, Learning, and Games. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Chan, N.-H. (2008): Time series with roots on or near the unit circle Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. Handbook of Financial Time Series, 694-707. Springer, New York.

    Google Scholar 

  • Chen, S. S., Donoho, D. L. and Saunders, M. A. (1998): Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing 20, 33-61.

    Article  MathSciNet  Google Scholar 

  • Chen, Z. G. and Ni, J. Y. (1989): Subset regression time series and its modeling procedures. Journal of Multivariate Analysis 31, 266-288.

    Article  MATH  MathSciNet  Google Scholar 

  • Choi, B. (1992): ARMA Model Identification. Springer, New York.

    MATH  Google Scholar 

  • Claeskens, G. and Hjort, N. L. (2003): The focused information criterion. Journal of the American Statistical Association 98, 900-916.

    Article  MATH  MathSciNet  Google Scholar 

  • Craven, P. and Wahba, G. (1979): Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numerische Mathematik 31, 377-403.

    Article  MATH  MathSciNet  Google Scholar 

  • Danilov, D. and Magnus, J. R. (2004): On the harm that ignoring pretesting can cause. Journal of Econometrics 122, 27-46.

    Article  MathSciNet  Google Scholar 

  • Davisson, L. D. (1965): The prediction error of stationary Gaussian time series of unknown covariance. IEEE Transactions on Information Theory 11, 527-532.

    Article  MATH  MathSciNet  Google Scholar 

  • DeGooijer, J. G., Bovas, A., Gould, A. and Robinson, L. (1985): Methods for determining the order of an autoregressive-moving average process: A survey. International Statistical Review 53, 301-329.

    MathSciNet  Google Scholar 

  • Dijkstra, T. K. and Veldkamp, J. H. (1988): Data-driven selection of regressors and the bootstrap Dijkstra, T. K. (Ed.): Lecture Notes in Economics and Mathematical Systems 307, 17-38, Springer, New York.

    Google Scholar 

  • Draper, N. R. and Smith, H. (1981): Applied Regression Analysis (2nd edition). Wiley, New York.

    MATH  Google Scholar 

  • Droge, B. (1993): On finite-sample properties of adaptive least squares regression estimates. Statistics 24, 181-203.

    Article  MATH  MathSciNet  Google Scholar 

  • Droge, B. and Georg, T. (1995): On selecting the smoothing parameter of least squares regression estimates using the minimax regret approach. Statistics and Decisions 13, 1-20.

    MATH  MathSciNet  Google Scholar 

  • Dufour, J. M., Pelletier, D. and Renault, E. (2006): Short run and long run causality in time series: Inference. Journal of Econometrics 132, 337-362.

    Article  MathSciNet  Google Scholar 

  • Fan, J. and Li, R. (2001): Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348-1360.

    Article  MATH  MathSciNet  Google Scholar 

  • Findley, D. F. (1985): On the unbiasedness property of AIC for exact or approximating linear stochastic time series models. Journal of Time Series Analysis 6, 229-252.

    Article  MATH  MathSciNet  Google Scholar 

  • Findley, D. F. (1991): Model selection for multistep-ahead forecasting. American Statistical Association Proceedings of the Business and Economic Statistics Section 243-247.

    Google Scholar 

  • Findley, D. F. and Wei, C. Z. (2002): AIC, overfitting principles, and the boundedness of moments of inverse matrices for vector autoregressions and related models. Journal of Multivariate Analysis 83, 415-450.

    Article  MATH  MathSciNet  Google Scholar 

  • Foster, D. P. and George, E. I. (1994): The risk inflation criterion for multiple regression. Annals of Statistics 22, 1947-1975.

    Article  MATH  MathSciNet  Google Scholar 

  • Frank, I. E. and Friedman, J. H. (1993): A statistical view of some chemometrics regression tools (with discussion). Technometrics 35, 109-148.

    Article  MATH  Google Scholar 

  • Francq, C., Roussignol, M. and Zakoïan, J. M. (2001): Conditional heteroskedasticity driven by hidden Markov chains. Journal of Time Series Analysis 22, 197-220.

    Article  MATH  MathSciNet  Google Scholar 

  • George, E. I. and Foster, D. P. (2000): Calibration and empirical Bayes variable selection. Biometrika 87, 731-747.

    Article  MATH  MathSciNet  Google Scholar 

  • Geweke, J. and Meese, R. (1981): Estimating regression models of finite but unknown order. International Economic Review 22, 55-70.

    Article  MATH  MathSciNet  Google Scholar 

  • Giles, J. A. and Giles, D. E. A. (1993): Pre-test estimation and testing in econometrics: recent developments. Journal of Economic Surveys 7, 145-197.

    Article  Google Scholar 

  • Guyon, X. and Yao, J. (1999): On the underfitting and overfitting sets of models chosen by order selection criteria. Journal of Multivariate Analysis 70, 221-249.

    Article  MATH  MathSciNet  Google Scholar 

  • Hall, A. R. and Peixe, F. P. M. (2003): A consistent method for the selection of relevant instruments. Econometric Reviews 22, 269-287.

    Article  MATH  MathSciNet  Google Scholar 

  • Hannan, E. J. (1980): The estimation of the order of an ARMA process. Annals of Statistics 8, 1071-1081.

    Article  MATH  MathSciNet  Google Scholar 

  • Hannan, E. J. (1981): Estimating the dimension of a linear system. Journal of Multivariate Analysis 11, 459-473.

    Article  MATH  MathSciNet  Google Scholar 

  • Hannan, E. J. and Deistler, M. (1988): The Statistical Theory of Linear Systems. Wiley, New York.

    MATH  Google Scholar 

  • Hannan, E. J. and Quinn, B. G. (1979): The determination of the order of an autoregression. Journal of the Royal Statistical Society Series B 41, 190-195.

    MATH  MathSciNet  Google Scholar 

  • Hansen, M. H. and Yu, B. (2001): Model selection and the principle of minimum description length. Journal of the American Statistical Association 96, 746-774.

    Article  MATH  MathSciNet  Google Scholar 

  • Haughton, D. (1991): Consistency of a class of information criteria for model selection in nonlinear regression. Communications in Statistics. Theory and Methods 20, 1619-1629.

    Article  MATH  MathSciNet  Google Scholar 

  • Hemerly, E. M. and Davis, M.H.A. (1989): Strong consistency of the PLS criterion for order determination of autoregressive processes. Annals of Statistics 17, 941-946.

    Article  MATH  MathSciNet  Google Scholar 

  • Hidalgo, J. (2002): Consistent order selection with strongly dependent data and its application to efficient estimation. Journal of Econometrics 110, 213-239.

    Article  MATH  MathSciNet  Google Scholar 

  • Hjort, N. L. and Claeskens, G. (2003): Frequentist model average estimators. Journal of the American Statistical Association 98, 879-899.

    Article  MATH  MathSciNet  Google Scholar 

  • Hocking, R. R. (1976): The analysis and selection of variables in linear regression. Biometrics 32, 1-49.

    Article  MATH  MathSciNet  Google Scholar 

  • Hoerl, A. E. and Kennard, R. W. (1970): Ridge regression: Biased estimation for nonorthogonal problems. Technometrics 12, 55-67.

    Article  MATH  Google Scholar 

  • Hoeting, J. A., Madigan, D., Raftery, A. and Volinsky, C. T. (1999): Bayesian model averaging: A tutorial. Statistical Science 14, 382-401. (Corrections: Statistical Science 15, 193-195.)

    Article  MATH  MathSciNet  Google Scholar 

  • Hosoya, Y. (1984): Information criteria and tests for time series models Anderson, O. D. (Ed.): Time Series Analysis: Theory and Practice 5, 39-52. North-Holland, Amsterdam.

    Google Scholar 

  • Hosoya, Y. (1986): A simultaneous test in the presence of nested alternative hypotheses. Journal of Applied Probability 23A, 187-200.

    Article  MathSciNet  Google Scholar 

  • Hurvich, M. M. and Tsai, C. L. (1989): Regression and time series model selection in small samples. Biometrika 76, 297-307.

    Article  MATH  MathSciNet  Google Scholar 

  • Ing, C. K. (2004): Selecting optimal multistep predictors for autoregressive processes of unknown order. Annals of Statistics 32, 693-722.

    Article  MATH  MathSciNet  Google Scholar 

  • Ing, C. K. (2007): Accumulated prediction errors, information criteria and optimal forecasting for autoregressive time series. Annals of Statistics 35, 1238-1277.

    Article  MATH  MathSciNet  Google Scholar 

  • Ing, C. K. and Wei, C. Z. (2005): Order selection for same-realization predictions in autoregressive processes. Annals of Statistics 33, 2423-2474.

    Article  MATH  MathSciNet  Google Scholar 

  • Ing, C. K. and Wei, C. Z. (2006): A maximal moment inequality for long range dependent time series with applications to estimation and model selection. Statistica Sinica 16, 721-740.

    MATH  MathSciNet  Google Scholar 

  • James, W. and Stein, C. (1961): Estimation with quadratic loss Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 361-379. California University Press, Berkeley.

    Google Scholar 

  • Judge, G. G. and Bock, M. E. (1978): The Statistical Implications of Pre-test and Stein-Rule Estimators in Econometrics. North-Holland, Amsterdam.

    MATH  Google Scholar 

  • Juditsky, A. and Nemirovski, A. (2000): Functional aggregation for nonparametric regression. Annals of Statistics 28, 681-712.

    Article  MATH  MathSciNet  Google Scholar 

  • Kabaila, P. (1995): The effect of model selection on confidence regions and prediction regions. Econometric Theory 11, 537-549.

    Article  MathSciNet  Google Scholar 

  • Kabaila, P. (1996): The evaluation of model selection criteria: Pointwise limits in the parameter space Dowe, D. L., Korb, K. B. and Oliver, J. J. Information, Statistics and Induction in Science, 114-118. World Scientific, Singapore.

    Google Scholar 

  • Kabaila, P. (1998): Valid confidence intervals in regression after variable selection. Econometric Theory 14, 463-482.

    Article  MathSciNet  Google Scholar 

  • Kabaila, P. (2002): On variable selection in linear regression. Econometric Theory 18, 913-925.

    Article  MATH  MathSciNet  Google Scholar 

  • Kabaila, P. and Leeb, H. (2006): On the large-sample minimal coverage probability of confidence intervals after model selection. Journal of the American Statistical Association 101, 619-629.

    Article  MATH  MathSciNet  Google Scholar 

  • Kempthorne, P. J. (1984): Admissible variable-selection procedures when fitting regression models by least squares for prediction. Biometrika 71, 593-597.

    Article  MATH  MathSciNet  Google Scholar 

  • Kennedy, W. J. and Bancroft, T. A. (1971): Model building for prediction in regression based upon repeated significance tests. Annals of Mathematical Statistics 42, 1273-1284.

    Article  MATH  MathSciNet  Google Scholar 

  • Keribin, C. and Haughton, D. (2003): Asymptotic probabilities of over-estimating and under-estimating the order of a model in general regular families. Communications in Statistics. Theory and Methods 32, 1373-1404.

    Article  MATH  MathSciNet  Google Scholar 

  • Kneip, A. (1994): Ordered linear smoothers. Annals of Statistics 22, 835-866.

    Article  MATH  MathSciNet  Google Scholar 

  • Knight, K. (1989): Consistency of Akaike's information criterion for infinite variance autoregressive processes. Annals of Statistics 17, 824-840.

    Article  MATH  MathSciNet  Google Scholar 

  • Knight, K. and Fu, W. (2000): Asymptotics for lasso-type estimators. Annals of Statistics 28, 1356-1378.

    Article  MATH  MathSciNet  Google Scholar 

  • Kohn, R. (1983): Consistent estimation of minimal subset dimension. Econometrica 51, 367-376.

    Article  MATH  MathSciNet  Google Scholar 

  • Konishi, S. and Kitagawa, G. (1996): Generalized information criteria in model selection. Biometrika 83, 875-890.

    Article  MATH  MathSciNet  Google Scholar 

  • Konishi, S. and Kitagawa, G. (2008): Information Criteria and Statistical Modeling. Springer, New York.

    MATH  Google Scholar 

  • Leeb, H. (2005): The distribution of a linear predictor after model selection: Conditional finite-sample distributions and asymptotic approximations. Journal of Statistical Planning and Inference 134, 64-89.

    Article  MATH  MathSciNet  Google Scholar 

  • Leeb, H. (2006a): The distribution of a linear predictor after model selection: Unconditional finite-sample distributions and asymptotic approximations Rojo, J. (Ed.): IMS Lecture Notes Monograph Series 49, 291-311. Institute of Mathematical Statistics, Beachwood.

    Google Scholar 

  • Leeb, H. (2006b): Evaluation and selection of models for out-of-sample prediction when the sample size is small relative to the complexity of the data-generating process. Bernoulli, forthcoming.

    Google Scholar 

  • Leeb, H. (2007): Conditional predictive inference post model selection. Manuscript, Department of Statistics, Yale University.

    Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2003): The finite-sample distribution of post-model-selection estimators and uniform versus nonuniform approximations. Econometric Theory 19, 100-142.

    Article  MATH  MathSciNet  Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2005): Model selection and inference: Facts and fiction. Econometric Theory 21, 21-59.

    Article  MATH  MathSciNet  Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2006a): Performance limits for estimators of the risk or distribution of shrinkage-type estimators, and some general lower risk-bound results. Econometric Theory 22, 69-97. (Corrigendum. Econometric Theory 24, 581-583.)

    Article  MATH  Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2006b): Can one estimate the conditional distribution of post-model-selection estimators? Annals of Statistics 34, 2554-2591.

    Article  MATH  Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2008a): Sparse estimators and the oracle property, or the return of Hodges' estimator. Journal of Econometrics 142, 201-211.

    Article  Google Scholar 

  • Leeb, H. and Pötscher, B. M. (2008b): Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24, 338-376.

    MATH  Google Scholar 

  • Leung, G. and Barron, A. R. (2006): Information theory and mixing least-squares regressions. IEEE Transactions on Information Theory 52, 3396-3410.

    Article  MathSciNet  Google Scholar 

  • Li, K. C. (1987): Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: Discrete index set. Annals of Statistics 15, 958-975.

    Article  MATH  MathSciNet  Google Scholar 

  • Linhart, H. and Zucchini, W. (1986): Model Selection. Springer, New York.

    MATH  Google Scholar 

  • Lütkepohl, H. (1990): Asymptotic distributions of impulse response functions and forecast error variance decompositions of vector autoregressive models. Review of Economics and Statistics 72, 116-125.

    Article  Google Scholar 

  • Magnus, J. R. (1999): The traditional pretest estimator. Teoriya Veroyatnostei i Ee Primeneniya 44, 401-418; translation in Theory of Probability and Its Applications 44, (2000), 293-308.

    MathSciNet  Google Scholar 

  • Magnus, J. R. (2002): Estimation of the mean of a univariate normal distribution with known variance. The Econometrics Journal 5, 225-236.

    Article  MATH  MathSciNet  Google Scholar 

  • Mallows, C. L. (1965): Some approaches to regression problems. Unpublished manuscript.

    Google Scholar 

  • Mallows, C. L. (1967): Choosing a subset regression. Bell Telephone Laboratories unpublished report.

    Google Scholar 

  • Mallows, C. L. (1973): Some comments on C p . Technometrics 15, 661-675.

    Article  MATH  Google Scholar 

  • Mallows, C. L. (1995): More comments on C p . Technometrics 37, 362-372.

    Article  MATH  MathSciNet  Google Scholar 

  • McKay, R. J. (1977): Variable selection in multivariate regression: An application of simultaneous test procedures. Journal of the Royal Statistical Society Series B 39, 371-380.

    MATH  MathSciNet  Google Scholar 

  • McQuarrie, A. D. R. and Tsai, C. L. (1998): Regression and time series model selection. World Scientific, River Edge.

    MATH  Google Scholar 

  • Miller, A. (2002): Subset Selection in Regression (2nd edition). Chapman and Hall, Boca Raton.

    MATH  Google Scholar 

  • Nishii, R. (1984): Asymptotic properties of criteria for selection of variables in multiple regression. Annals of Statistics 12, 758-765.

    Article  MATH  MathSciNet  Google Scholar 

  • Nishii, R. (1988): Maximum likelihood principle and model selection when the true model is unspecified. Journal of Multivariate Analysis 27, 392-403.

    Article  MATH  MathSciNet  Google Scholar 

  • Paulsen, J. (1984): Order determination of multivariate autoregressive time series with unit roots. Journal of Time Series Analysis 5, 115-127.

    Article  MATH  MathSciNet  Google Scholar 

  • Phillips, P. C. B. (2005): Automated discovery in econometrics. Econometric Theory 21, 3-20.

    MATH  MathSciNet  Google Scholar 

  • Polyak, B. T. and Tsybakov, A. B. (1990): Asymptotic optimality of the C p -test for the orthogonal series estimation of regression. Theory of Probability and Its Applications 35, 293-306.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1983): Order estimation in ARMA-models by Lagrangian multiplier tests. Annals of Statistics 11, 872-885.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1985): The behaviour of the Lagrangian multiplier test in testing the orders of an ARMA-model. Metrika 32, 129-150.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1989): Model selection under nonstationarity: Autoregressive models and stochastic linear regression models. Annals of Statistics 17, 1257-1274.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1990): Estimation of autoregressive moving average order given an infinite number of models and approximation of spectral densities. Journal of Time Series Analysis 11, 165-179.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1991): Effects of model selection on inference. Econometric Theory 7, 163-185.

    Article  MathSciNet  Google Scholar 

  • Pötscher, B. M. (1995): Comment on ‘The effect of model selection on confidence regions and prediction regions’ by P. Kabaila. Econometric Theory 11, 550-559.

    Article  Google Scholar 

  • Pötscher, B. M. (2006): The distribution of model averaging estimators and an impossibility result regarding its estimation Ho, H.-C., Ing, C.-K. and Lai, T.-L. Time Series and Related Topics: In Memory of Ching-Zong Wei. IMS Lecture Notes and Monograph Series 52, 113-129. Institute of Mathematical Statistics, Beachwood.

    Chapter  Google Scholar 

  • Pötscher, B. M. (2007): Confidence sets based on sparse estimators are necessarily large. Working paper, Department of Statistics, University of Vienna. arXiv:0711.1036.

    Google Scholar 

  • Pötscher, B. M. and Leeb, H. (2007): On the distribution of penalized maximum likelihood estimators: The LASSO, SCAD, and thresholding. Working paper, Department of Statistics, University of Vienna. arXiv:0711.0660.

    Google Scholar 

  • Pötscher, B. M. and Novak, A. J. (1998): The distribution of estimators after model selection: Large and small sample results. Journal of Statistical Computation and Simulation 60, 19-56.

    Article  MATH  MathSciNet  Google Scholar 

  • Pötscher, B. M. and Schneider, U. (2007): On the distribution of the adaptive LASSO estimator. Working paper, Department of Statistics, University of Vienna. arXiv:0801.4627.

    Google Scholar 

  • Pötscher, B. M. and Srinivasan, S. (1994): A comparison of order estimation procedures for ARMA models. Statistica Sinica 4, 29-50.

    MATH  MathSciNet  Google Scholar 

  • Quinn, B. G. (1980): Order determination for a multivariate autoregression. Journal of the Royal Statistical Society Series B 42, 182-185.

    MATH  MathSciNet  Google Scholar 

  • Quinn, B. G. (1988): A note on AIC order determination for multivariate autoregressions. Journal of Time Series Analysis 9, 241-245.

    Article  MATH  MathSciNet  Google Scholar 

  • Rao, C. R. and Wu, Y. (1989): A strongly consistent procedure for model selection in a regression problem. Biometrika 76, 369-374.

    Article  MATH  MathSciNet  Google Scholar 

  • Rao, C. R. and Wu, Y. (2001): On model selection Lahiri, P. (Ed.): Model Selection. IMS Lecture Notes Monograph Series 38, 1-57. Institute of Mathematical Statistics, Beachwood.

    Google Scholar 

  • Rao, C. R. and Wu, Y. (2005): Linear model selection by cross-validation. Journal of Statistical Planning and Inference 128, 231-240.

    Article  MATH  MathSciNet  Google Scholar 

  • Reschenhofer, E. (1999): Improved estimation of the expected Kullback-Leibler discrepancy in case of misspecification. Econometric Theory 15, 377-387.

    Article  MATH  MathSciNet  Google Scholar 

  • Rissanen, J. (1978): Modeling by shortest data description. Automatica 14, 465-471.

    Article  MATH  Google Scholar 

  • Rissanen, J. (1983): A universal prior for integers and estimation by minimum description length. Annals of Statistics 11, 416-431.

    Article  MATH  MathSciNet  Google Scholar 

  • Rissanen, J. (1986a): Stochastic complexity and modeling. Annals of Statistics 14, 1080-1100.

    Article  MATH  MathSciNet  Google Scholar 

  • Rissanen, J. (1986b): A predictive least squares principle. IMA Journal of Mathematical Control and Information 3, 211-222.

    Article  MATH  Google Scholar 

  • Rissanen, J. (1987): Stochastic complexity (with discussion). Journal of the Royal Statistical Society Series B 49, 223-265.

    MATH  MathSciNet  Google Scholar 

  • Rissanen, J. (1989): Stochastic Complexity and Statistical Inquiry. World Scientific, Tea-neck.

    Google Scholar 

  • Sakai, H. (1981): Asymptotic distribution of the order selected by AIC in multivariate autoregressive model fitting. International Journal of Control 33, 175-180.

    Article  MATH  MathSciNet  Google Scholar 

  • Saleh, A. K. M. E. (2006): Theory of Preliminary Test and Stein-Type Estimation with Applications. Wiley, Hoboken.

    Book  MATH  Google Scholar 

  • Saleh, A. K. M. E. and Sen, P. K. (1983): Asymptotic properties of tests of hypothesis following a preliminary test. Statistics and Decisions 1, 455-477.

    MATH  MathSciNet  Google Scholar 

  • Sawa, T. and Hiromatsu, T. (1973): Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41, 1093-1101.

    Article  MATH  Google Scholar 

  • Schwarz, G. (1978): Estimating the dimension of a model. Annals of Statistics 6, 461-464.

    Article  MATH  MathSciNet  Google Scholar 

  • Sclove, S. L., Morris, C. and Radhakrishnan, R. (1972): Non-optimality of preliminary-test estimators for the mean of a multivariate normal distribution. Annals of Mathematical Statistics 43, 1481-1490.

    Article  MATH  MathSciNet  Google Scholar 

  • Sen, P. K (1979): Asymptotic properties of maximum likelihood estimators based on conditional specification. Annals of Statistics 7, 1019-1033.

    Article  MATH  MathSciNet  Google Scholar 

  • Sen, P. K and Saleh, A. K. M. E. (1987): On preliminary test and shrinkage M-estimation in linear models. Annals of Statistics 15, 1580-1592.

    Article  MATH  MathSciNet  Google Scholar 

  • Shao, J. (1993): Linear model selection by cross-validation. Journal of the American Statistical Association 88, 486-494.

    Article  MATH  MathSciNet  Google Scholar 

  • Shao, J. (1997): An asymptotic theory for linear model selection (with discussion) Statistica Sinica 7, 221-264.

    MATH  MathSciNet  Google Scholar 

  • Shao, J. (1998): Convergence rates of the generalized information criterion. Journal of Nonparametric Statistics 9, 217-225.

    Article  MATH  MathSciNet  Google Scholar 

  • Shen, X., Huang, H. C. and Ye, J. (2004): Inference after model selection. Journal of the American Statistical Association 99, 751-762.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1976): Selection of the order of an autoregressive model by Akaike’s information criterion. Biometrika 63, 117-126.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1980): Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Annals of Statistics 8, 147-164.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1981a): An optimal autoregressive spectral estimate. Annals of Statistics 9, 300-306.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1981b): An optimal selection of regression variables. Biometrika 68, 45-54. (Correction: Biometrika 69, 492).

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1984): Approximate efficiency of a selection procedure for the number of regression variables. Biometrika 71, 43-49.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1986a): Consistency of model selection and parameter estimation. Journal of Applied Probability 23A, 127-141.

    Article  MathSciNet  Google Scholar 

  • Shibata, R. (1986b): Selection of the number of regression variables; a minimax choice of generalized FPE. Annals of the Institute of Statistical Mathematics 38, 459-474.

    Article  MATH  MathSciNet  Google Scholar 

  • Shibata, R. (1989): Statistical aspects of model selection. In: J. C. Willems (Ed.): From Data to Model, 215-240. Springer, New York.

    Google Scholar 

  • Shibata, R. (1997): Bootstrap estimate of Kullback-Leibler information for model selection. Statistica Sinica 7, 375-394.

    MATH  MathSciNet  Google Scholar 

  • Söderström, T. (1977): On model structure testing in system identification. International Journal of Control 26, 1-18.

    Article  MATH  MathSciNet  Google Scholar 

  • Stone, C. (1981): Admissible selection of an accurate and parsimonious normal linear regression model. Annals of Statistics 9, 475-485.

    Article  MATH  MathSciNet  Google Scholar 

  • Stone, C. (1982): Local asymptotic admissibility of a generalization of Akaike’s model selection rule. Annals of the Institute of Statistical Mathematics 34, 123-133.

    Article  MATH  MathSciNet  Google Scholar 

  • Stone, M. (1974): Cross-validatory choice and assessment of statistical prediction. Journal of the Royal Statistical Society Series B 36, 111-133.

    MATH  Google Scholar 

  • Stone, M. (1977): An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. Journal of the Royal Statistical Society Series B 39, 44-47.

    MATH  Google Scholar 

  • Strawderman, W. E. (1971): Proper Bayes minimax estimators of the multivariate normal mean. Annals of Mathematical Statistics 42, 385-388.

    Article  MATH  MathSciNet  Google Scholar 

  • Sugiura, N. (1978): Further analysis of the data by Akaike’s information criterion and the finite corrections. Communications in Statistics A7, 13-26.

    Article  MathSciNet  Google Scholar 

  • Takada, Y. (1982): Admissibility of some variable selection rules in linear regression model. Journal of the Japanese Statistical Society 12, 45-49.

    MATH  MathSciNet  Google Scholar 

  • Takeuchi, K. (1976): Distribution of informational statistics and a criterion of model fitting. Suri-Kagaku 153, 12-18. (In Japanese.)

    Google Scholar 

  • Teräsvirta, T. and Mellin, I. (1986): Model selection criteria and model selection tests in regression models. Scandinavian Journal of Statistics 13, 159-171.

    MATH  Google Scholar 

  • Theil, H. (1961): Economic Forecasts and Policy (2nd edition). North-Holland, Amsterdam.

    Google Scholar 

  • Thompson, M. L. (1978a): Selection of variables in multiple regression: part I. A review and evaluation. International Statistical Review 46, 1-19.

    MATH  Google Scholar 

  • Thompson, M. L. (1978b): Selection of variables in multiple regression: part II. Chosen procedures, computations and examples. International Statistical Review 46, 129-146.

    Article  MATH  Google Scholar 

  • Tibshirani, R. (1996): Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society Series B 58, 267-288.

    MATH  MathSciNet  Google Scholar 

  • Toro-Vizcarrondo, C. and Wallace, T. D. (1968): A test of the mean square error criterion for restrictions in linear regression. Journal of the American Statistical Association 63, 558-572.

    Article  MATH  MathSciNet  Google Scholar 

  • Toyoda, T. and Wallace, T. D. (1976): Optimal critical values for pre-testing in regression. Econometrica 44, 365-375.

    Article  MATH  MathSciNet  Google Scholar 

  • Tsay, R. S. (1984): Order selection in nonstationary autoregressive models. Annals of Statistics 12, 1425-1433.

    Article  MATH  MathSciNet  Google Scholar 

  • Venter, J. H. and Steele, S. J. (1992): Some contributions to selection and estimation in the normal linear model. Annals of the Institute of Statistical Mathematics 44, 281-297.

    Article  MATH  MathSciNet  Google Scholar 

  • Vuong, Q. H. (1989): Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57, 307-333.

    Article  MATH  MathSciNet  Google Scholar 

  • Wallace, T. D. (1972): Weaker criteria and tests for linear restrictions in regression. Econometrica 40, 689-698.

    Article  MATH  MathSciNet  Google Scholar 

  • Wei, C. Z. (1992): On predictive least squares principles. Annals of Statistics 20, 1-42.

    Article  MATH  MathSciNet  Google Scholar 

  • Wegkamp, M. (2003): Model selection in nonparametric regression. Annals of Statistics 31, 252-273.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, Y. (1999): Model selection for nonparametric regression. Statistica Sinica 9, 475-499.

    MATH  MathSciNet  Google Scholar 

  • Yang, Y. (2001): Adaptive regression by mixing. Journal of the American Statistical Association 96, 574-588.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, Y. (2003): Regression with multiple candidate models: Selecting or mixing? Statistica Sinica 13, 783-809.

    MATH  MathSciNet  Google Scholar 

  • Yang, Y. (2005): Can the strengths of AIC and BIC be shared? A conflict between model identification and regression estimation. Biometrika 92, 937-950.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, Y. (2007): Prediction/estimation with simple linear models: Is it really that simple? Econometric Theory 23, 1-36.

    Article  MathSciNet  Google Scholar 

  • Yang, Y. and Barron, A. R. (1998): An asymptotic property of model selection criteria. IEEE Transactions on Information Theory 44, 95-116.

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, Y. and Barron, A. R. (1999): Information-theoretic determination of minimax rates of convergence. Annals of Statistics 27, 1564-1599.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, P. (1992): Inference after variable selection in linear regression models. Biometrika 79, 741-746.

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, P. (1993a): Model selection via multifold cross validation. Annals of Statistics 21, 299-313.

    Article  Google Scholar 

  • Zhang, P. (1993b): On the convergence rate of model selection criteria. Communications in Statistics. Theory and Methods 22, 2765-2775.

    Article  MATH  Google Scholar 

  • Zheng, X. and Loh, W. Y. (1995): Consistent variable selection in linear models. Journal of the American Statistical Association 90, 151-156.

    Article  MATH  MathSciNet  Google Scholar 

  • Zheng, X. and Loh, W. Y. (1997): A consistent variable selection criterion for linear models with high-dimensional covariates. Statistica Sinica 7, 311-325.

    MATH  MathSciNet  Google Scholar 

  • Zou, H. (2006): The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101, 1418-1429.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hannes Leeb .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Leeb, H., Pötscher, B.M. (2009). Model Selection. In: Mikosch, T., Kreiß, JP., Davis, R., Andersen, T. (eds) Handbook of Financial Time Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71297-8_39

Download citation

Publish with us

Policies and ethics