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INFERENCE AFTER MODEL AVERAGING IN LINEAR REGRESSION MODELS

  • Xinyu Zhang (a1) and Chu-An Liu (a2)
Abstract

This article considers the problem of inference for nested least squares averaging estimators. We study the asymptotic behavior of the Mallows model averaging estimator (MMA; Hansen, 2007) and the jackknife model averaging estimator (JMA; Hansen and Racine, 2012) under the standard asymptotics with fixed parameters setup. We find that both MMA and JMA estimators asymptotically assign zero weight to the under-fitted models, and MMA and JMA weights of just-fitted and over-fitted models are asymptotically random. Building on the asymptotic behavior of model weights, we derive the asymptotic distributions of MMA and JMA estimators and propose a simulation-based confidence interval for the least squares averaging estimator. Monte Carlo simulations show that the coverage probabilities of proposed confidence intervals achieve the nominal level.

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Corresponding author
*Address correspondence to Chu-An Liu, Institute of Economics, Academia Sinica, Taipei 115, Taiwan; e-mail: caliu@econ.sinica.edu.tw.
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We thank three anonymous referees, the co-editor Liangjun Su, and the editor Peter C.B. Phillips for many constructive comments and suggestions. We also thank conference participants of SETA 2016, AMES 2016, and CFE 2017 for their discussions and suggestions. Xinyu Zhang gratefully acknowledges the research support from National Natural Science Foundation of China (Grant numbers 71522004, 11471324 and 71631008). Chu-An Liu gratefully acknowledges the research support from the Ministry of Science and Technology of Taiwan (MOST 104-2410-H-001-092-MY2). All errors remain the authors’.

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References
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Andrews, D.W.K. (1991) Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors. Journal of Econometrics 47, 359377.
Breiman, L. (1996) Bagging predictors. Machine Learning 24, 123140.
Buckland, S.T., Burnham, K.P., & Augustin, N.H. (1997) Model selection: An integral part of inference. Biometrics 53, 603618.
Camponovo, L. (2015) On the validity of the pairs bootstrap for lasso estimators. Biometrika 102(4), 981987.
Chatterjee, A. & Lahiri, S.N. (2011) Bootstrapping lasso estimators. Journal of the American Statistical Association 106(494), 608625.
Chatterjee, A. & Lahiri, S.N. (2013) Rates of convergence of the adaptive lasso estimators to the oracle distribution and higher order refinements by the bootstrap. The Annals of Statistics 41(3), 12321259.
Claeskens, G. & Hjort, N.L. (2008) Model Selection and Model Averaging. Cambridge University Press.
DiTraglia, F. (2016) Using invalid instruments on purpose: Focused moment selection and averaging for GMM. Journal of Econometrics 195, 187208.
Hansen, B.E. (2007) Least squares model averaging. Econometrica 75, 11751189.
Hansen, B.E. (2008) Least-squares forecast averaging. Journal of Econometrics 146(2), 342350.
Hansen, B.E. (2014) Model averaging, asymptotic risk, and regressor groups. Quantitative Economics 5(3), 495530.
Hansen, B.E. (2018) Econometrics. Unpublished manuscript, University of Wisconsin.
Hansen, B.E. & Racine, J. (2012) Jacknife model averaging. Journal of Econometrics 167, 3846.
Hansen, P., Lunde, A., & Nason, J. (2011) The model confidence set. Econometrica 79, 453497.
Hjort, N.L. & Claeskens, G. (2003a) Frequentist model average estimators. Journal of the American Statistical Association 98, 879899.
Hjort, N.L. & Claeskens, G. (2003b) Rejoinder to the focused information criterion and frequentist model average estimators. Journal of the American Statistical Association 98(464), 938945.
Hoeting, J.A., Madigan, D., Raftery, A.E., & Volinsky, C.T. (1999) Bayesian model averaging: A tutorial. Statistical Science 14, 382417.
Inoue, A. & Kilian, L. (2008) How useful is bagging in forecasting economic time series? A case study of US consumer price inflation. Journal of the American Statistical Association 103, 511522.
Kabaila, P. (1995) The effect of model selection on confidence regions and prediction regions. Econometric Theory 11(3), 537549.
Kabaila, P. (1998) Valid confidence intervals in regression after variable selection. Econometric Theory 14(4), 463482.
Kim, J. & Pollard, D. (1990) Cube root asymptotics. The Annals of Statistics 18, 191219.
Leeb, H. & Pötscher, B. (2003) The finite-sample distribution of post-model-selection estimators and uniform versus non-uniform approximations. Econometric Theory 19(1), 100142.
Leeb, H. & Pötscher, B. (2005) Model selection and inference: Facts and fiction. Econometric Theory 21(1), 2159.
Leeb, H. & Pötscher, B. (2006) Can one estimate the conditional distribution of post-model-selection estimators? The Annals of Statistics 34(5), 25542591.
Leeb, H. & Pötscher, B. (2008) Can one estimate the unconditional distribution of post-model-selection estimators? Econometric Theory 24(02), 338376.
Leeb, H. & Pötscher, B. (2017) Testing in the presence of nuisance parameters: Some comments on tests post-model-selection and random critical values. In Ahmed, S.E. (ed.), Big and Complex Data Analysis: Methodologies and Applications, pp. 6982. Springer International Publishing.
Li, K.-C. (1987) Asymptotic optimality for Cp, CL, cross-validation and generalized cross-validation: Discrete index set. The Annals of Statistics 15, 958975.
Liang, H., Zou, G., Wan, A.T.K., & Zhang, X. (2011) Optimal weight choice for frequentist model average estimators. Journal of the American Statistical Association 106, 10531066.
Liu, C.-A. (2015) Distribution theory of the least squares averaging estimator. Journal of Econometrics 186, 142159.
Liu, Q. & Okui, R. (2013) Heteroscedasticity-robust Cp model averaging. Econometrics Journal 16, 462473.
Lu, X. (2015) A covariate selection criterion for estimation of treatment effects. Journal of Business and Economic Statistics 33, 506522.
Lu, X. & Su, L. (2015) Jackknife model averaging for quantile regressions. Journal of Econometrics 188(1), 4058.
Magnus, J., Powell, O., & Prüfer, P. (2010) A comparison of two model averaging techniques with an application to growth empirics. Journal of Econometrics 154(2), 139153.
Moral-Benito, E. (2015) Model averaging in economics: An overview. Journal of Economic Surveys 29(1), 4675.
Pötscher, B. (1991) Effects of model selection on inference. Econometric Theory 7(2), 163185.
Pötscher, B. (2006) The distribution of model averaging estimators and an impossibility result regarding its estimation. Lecture Notes-Monograph Series 52, 113129.
Pötscher, B. & Leeb, H. (2009) On the distribution of penalized maximum likelihood estimators: The lasso, scad, and thresholding. Journal of Multivariate Analysis 100(9), 20652082.
Raftery, A.E. & Zheng, Y. (2003) Discussion: Performance of Bayesian model averaging. Journal of the American Statistical Association 98(464), 931938.
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B 58, 267288.
Van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes. Springer Verlag.
Wan, A.T.K., Zhang, X., & Zou, G. (2010) Least squares model averaging by Mallows criterion. Journal of Econometrics 156, 277283.
Yang, Y. (2000) Combining different procedures for adaptive regression. Journal of Multivariate Analysis 74(1), 135161.
Yang, Y. (2001) Adaptive regression by mixing. Journal of the American Statistical Association 96, 574588.
Yuan, Z. & Yang, Y. (2005) Combining linear regression models: When and how? Journal of the American Statistical Association 100, 12021214.
Zhang, X. & Liang, H. (2011) Focused information criterion and model averaging for generalized additive partial linear models. The Annals of Statistics 39, 174200.
Zhang, X., Wan, A.T., & Zhou, S.Z. (2012) Focused information criteria, model selection, and model averaging in a Tobit model with a nonzero threshold. Journal of Business and Economic Statistics 30, 132142.
Zhang, X., Wan, A.T.K., & Zou, G. (2013) Model averaging by jackknife criterion in models with dependent data. Journal of Econometrics 174, 8294.
Zhang, X., Zou, G., & Liang, H. (2014) Model averaging and weight choice in linear mixed-effects models. Biometrika 101, 205218.
Zou, H. (2006) The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101, 14181429.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
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