Model selection and model averaging in nonparametric instrumental variables models

Liu, Chu-An and Tao, Jing (2016): Model selection and model averaging in nonparametric instrumental variables models.

Abstract

This paper considers the problem of choosing the regularization parameter and the smoothing parameter in nonparametric instrumental variables estimation. We propose a simple Mallows’ Cp-type criterion to select these two parameters simultaneously. We show that the proposed selection criterion is optimal in the sense that the selected estimate asymptotically achieves the lowest possible mean squared error among all candidates. To account for model uncertainty, we introduce a new model averaging estimator for nonparametric instrumental variables regressions. We propose a Mallows criterion for the weight selection and demonstrate its asymptotic optimality. Monte Carlo simulations show that both selection and averaging methods generally achieve lower root mean squared error than other existing methods. The proposed methods are applied to two empirical examples, the effect of class size question and Engel curve.

• Item Type: MPRA Paper
• Language: English
• Keywords: Ill-posed inverse problem, Mallows criterion, Model averaging, Model selection, Nonparametric instrumental variables, Series estimation
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C26 - Instrumental Variables (IV) Estimation; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection

References

Ai, C. and X. Chen (2003): “Efficient Estimation of Models with Conditional Moment Restrictions Containing Unknown Functions,” Econometrica, 71, 1795–1843.

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, 359–377.

——— (1999): “Consistent Moment Selection Procedures for Generalized Method of Moments Estimation,” Econometrica, 67, 543–563.

——— (2011): “Examples of L2-Complete and Boundedly-Complete Distributions,” Cowles Foundation for Research in Economics.

Andrews, D. W. K. and B. Lu (2001): “Consistent Model and Moment Selection Procedures for GMM Estimation with Application to Dynamic Panel Data Models,” Journal of Econometrics, 101, 123–164.

Angrist, J. D. and V. Lavy (1999): “Using Maimonides’ Rule to Estimate the Effect of Class Size on Scholastic Achievement,” The Quarterly Journal of Economics, 114, 533–575.

Belloni, A., D. Chen, V. Chernozhukov, and C. Hansen (2012): “Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain,” Econometrica, 80, 2369–2429.

Blundell, R., X. Chen, and D. Kristensen (2007): “Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves,” Econometrica, 75, 1613–1669.

Breunig, C. and J. Johannes (2015): “Adaptive Estimation of Functionals in Nonparametric Instrumental Regression,” Econometric Theory, FirstView, 1–43.

Carrasco, M. (2012): “A Regularization Approach to the Many Instruments Problem,” Journal of Econometrics, 170, 383–398.

Carrasco, M., J.-P. Florens, and E. Renault (2007): “Linear Inverse Problems in Structural Econometrics Estimation Based on Spectral Decomposition and Regularization,” in Handbook of Econometrics, ed. by J. J. Heckman and E. E. Leamer, Elsevier, vol. 6, 5633–5751.

Centorrino, S. (2015): “Data driven selection of the regularization parameter in additive nonparametric instrumental regressions,” Working Paper.

Centorrino, S., F. Feve, and J.-P. Florens (2015): “Additive Nonparametric Instrumental Regressions: A Guide to Implementation,” Forthcoming. Journal of Econometric Methods.

Chen, X., V. Chernozhukov, S. Lee, and W. K. Newey (2014): “Local Identification of Nonparametric and Semiparametric Models,” Econometrica, 82, 785–809.

Chen, X. and T. Christensen (2013): “Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression,” Cemmap Working Paper CWP 56/13.

———(2015): “Optimal Sup-normRates, Adaptivity and Inference in Nonparametric Instrumental Variables Estimation,” Cemmap Working Paper CWP 32/15.

Chen, X. and D. Pouzo (2012): “Estimation of Nonparametric Conditional Moment Models with Possibly Nonsmooth Generalized Residuals,” Econometrica, 80, 277–321.

Chen, X. and M. Reiss (2011): “On Rate Optimality for Ill-Posed Inverse Problems in Econometrics,” Econometric Theory, 27, 497–521.

Chernozhukov, V., G. W. Imbens, and W. K. Newey (2007): “Instrumental Variable Estimation of Nonseparable Models,” Journal of Econometrics, 139, 4–14.

Claeskens, G. and N. L. Hjort (2008): Model Selection and Model Averaging, Cambridge: Cambridge University Press. Darolles, S., Y. Fan, J.-P. Florens, and E. Renault (2011): “Nonparametric Instrumental Regression,” Econometrica, 79, 1541–1565.

D’Haultfoeuille, X. (2011): “On the Completeness Condition in Nonparametric Instrumental Problems,” Econometric Theory, 27, 460–471.

Donald, S. G., G. W. Imbens, and W. K. Newey (2009): “Choosing Instrumental Variables in Conditional Moment Restriction Models,” Journal of Econometrics, 152, 28–36.

Donald, S. G. and W. K. Newey (2001): “Choosing the Number of Instruments,” Econometrica, 69, 1161–1191.

Gagliardini, P. and O. Scaillet (2012a): “Nonparametric Instrumental Variable Estimation of Structural Quantile Effects,” Econometrica, 80, 1533–1562.

——— (2012b): “Tikhonov Regularization for Nonparametric Instrumental Variable Estimators,” Journal of Econometrics, 167, 61–75.

Hall, P. and J. L. Horowitz (2005): “Nonparametric Methods for Inference in the Presence of Instrumental Variables,” The Annals of Statistics, 33, 2904–2929.

Hansen, B. E. (2007): “Least Squares Model Averaging,” Econometrica, 75, 1175–1189. ——— (2015): “The Integrated Mean Squared Error of Series Regression and a Rosenthal Hilbert-Space Inequality,” Econometric Theory, 31, 337–361.

Hansen, B. E. and J. Racine (2012): “Jackknife Model Averaging,” Journal of Econometrics, 167, 38–46.

Hanushek, E. A. (1986): “The Economics of Schooling: Production and Efficiency in Public Schools,” Journal of Economic Literature, 24, 1141–1177.

Hjort, N. L. and G. Claeskens (2003): “Frequentist Model Average Estimators,” Journal of the American Statistical Association, 98, 879–899.

Hong, H., B. Preston, and M. Shum (2003): “Generalized Empirical Likelihood-Based Model Selection Criteria for Moment Condition Models,” Econometric Theory, 19, 923–943.

Horowitz, J. L. (2011): “Applied Nonparametric Instrumental Variables Estimation,” Econometrica, 79, 347–394.

——— (2012): “Specification Testing in Nonparametric Instrumental Variable Estimation,” Journal of Econometrics, 167, 383–396.

——— (2014): “Adaptive Nonparametric Instrumental Variables Estimation: Empirical Choice of the Regularization Parameter,” Journal of Econometrics, 180, 158–173.

Horowitz, J. L. and S. Lee (2007): “Nonparametric Instrumental Variables Estimation of a Quantile Regression Model,” Econometrica, 75, 1191–1208.

Ing, C.-K. and C.-Z. Wei (2003): “On Same-Realization Prediction in an Infinite-Order Autoregressive Process,” Journal of Multivariate Analysis, 85, 130–155.

——— (2005): “Order Selection for Same-Realization Predictions in Autoregressive Processes,” The Annals of Statistics, 33, 2423–2474.

Kress, R. (1999): Linear Integral Equations, Springer-Verlag.

Kuersteiner, G. and R. Okui (2010): “Constructing Optimal Instruments by First-Stage Prediction Averaging,” Econometrica, 78, 697–718.

Leeb, H. and B. P¨otscher (2005): “Model Selection and Inference: Facts and Fiction,” Econometric Theory, 21, 21–59.

Li, K.-C. (1987): “Asymptotic Optimality for Cp, CL, Cross-Validation and Generalized Cross-Validation: Discrete Index Set,” The Annals of Statistics, 15, 958–975.

Liu, Q. and R. Okui (2013): “Heteroscedasticity-Robust Cp Model Averaging,” The Econometrics Journal, 16, 463–472.

Newey, W. K. (2013): “Nonparametric Instrumental Variables Estimation,” The American Economic Review, 103, 550–556.

Newey, W. K. and J. L. Powell (2003): “Instrumental Variable Estimation of Nonparametric Models,” Econometrica, 71, 1565–1578.

Okui, R. (2011): “Instrumental Variable Estimation in the Presence of Many Moment Conditions,” Journal of Econometrics, 165, 70–86.

Shao, J. (1997): “An Asymptotic Theory for Linear Model Selection,” Statistica Sinica, 7, 221–242.

Shibata, R. (1980): “Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process,” The Annals of Statistics, 8, 147–164.

——— (1981): “An Optimal Selection of Regression Variables,” Biometrika, 68, 45–54.

Sueishi, N. (2012): “Model Selection Criterion for Instrumental Variable Models,” Working Paper, Kobe University.

Wan, A., X. Zhang, and G. Zou (2010): “Least SquaresModel Averaging byMallows Criterion,” Journal of Econometrics, 156, 277–283.

Whittle, P. (1960): “Bounds for the Moments of Linear and Quadratic Forms in Independent Variables,” Theory of Probability and Its Applications, 5, 302–305.

Zhang, X., A. T. Wan, and G. Zou (2013): “Model Averaging by Jackknife Criterion in Models with Dependent Data,” Journal of Econometrics, 174, 82–94.