Hostname: page-component-6898cfcb55-zsp5v Total loading time: 0 Render date: 2026-11-12T00:40:43.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Kernel Regularized Least Squares: Reducing Misspecification Bias with a Flexible and Interpretable Machine Learning Approach

Published online by Cambridge University Press:  04 January 2017

Jens Hainmueller and
Chad Hazlett
Jens Hainmueller*
Affiliation:
Department of Political Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139
Chad Hazlett
Affiliation:
Department of Political Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. e-mail: hazlett@mit.edu
*
e-mail: jhainm@mit.edu (corresponding author)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We propose the use of Kernel Regularized Least Squares (KRLS) for social science modeling and inference problems. KRLS borrows from machine learning methods designed to solve regression and classification problems without relying on linearity or additivity assumptions. The method constructs a flexible hypothesis space that uses kernels as radial basis functions and finds the best-fitting surface in this space by minimizing a complexity-penalized least squares problem. We argue that the method is well-suited for social science inquiry because it avoids strong parametric assumptions, yet allows interpretation in ways analogous to generalized linear models while also permitting more complex interpretation to examine nonlinearities, interactions, and heterogeneous effects. We also extend the method in several directions to make it more effective for social inquiry, by (1) deriving estimators for the pointwise marginal effects and their variances, (2) establishing unbiasedness, consistency, and asymptotic normality of the KRLS estimator under fairly general conditions, (3) proposing a simple automated rule for choosing the kernel bandwidth, and (4) providing companion software. We illustrate the use of the method through simulations and empirical examples.

Information

Type
Research Article
Information
Political Analysis , Volume 22 , Issue 2 , Spring 2014, pp. 143 - 168
Copyright
Copyright © The Author 2013. Published by Oxford University Press on behalf of the Society for Political Methodology 

Footnotes

Authors' note: The authors are listed in alphabetical order and contributed equally. We thank Jeremy Ferwerda, Dominik Hangartner, Danny Hidalgo, Gary King, Lorenzo Rosasco, Marc Ratkovic, Teppei Yamamoto, our anonymous reviewers, the editors, and participants in seminars at NYU, MIT, the Midwest Political Science Conference, and the European Political Science Association Conference for helpful comments. Companion software written by the authors to implement the methods proposed in this article in R, Matlab, and Stata can be downloaded from the authors' Web pages. Replication materials are available in the Political Analysis Dataverse at http://dvn.iq.harvard.edu/dvn/dv/pan. The usual disclaimer applies. Supplementary materials for this article are available on the Political Analysis Web site.

References

Beck, N., King, G., and Zeng, L. 2000. Improving quantitative studies of international conflict: A conjecture. American Political Science Review 94: 21–36.CrossRefGoogle Scholar
Brambor, T., Clark, W., and Golder, M. 2006. Understanding interaction models: Improving empirical analyses. Political Analysis 14(1): 63–82.Google Scholar
De Vito, E., Caponnetto, A., and Rosasco, L. 2005. Model selection for regularized least-squares algorithm in learning theory. Foundations of Computational Mathematics 5(1): 59–85.CrossRefGoogle Scholar
Evgeniou, T., Pontil, M., and Poggio, T. 2000. Regularization networks and support vector machines. Advances in Computational Mathematics 13(1): 1–50.Google Scholar
Friedrich, R. J. 1982. In defense of multiplicative terms in multiple regression equations. American Journal of Political Science 26(4): 797–833.CrossRefGoogle Scholar
Golub, G. H., Heath, M., and Wahba, G. 1979. Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21(2): 215–23.Google Scholar
Harff, B. 2003. No lessons learned from the Holocaust? Assessing risks of genocide and political mass murder since 1955. American Political Science Review 97(1): 57–73.Google Scholar
Hastie, T., Tibshirani, R., and Friedman, J. 2009. The elements of statistical learning: Data mining, inference, and prediction. 2nd ed. New York, NY: Springer.Google Scholar
Jackson, J. E. 1991. Estimation of models with variable coefficients. Political Analysis 3(1): 27–49.Google Scholar
Kimeldorf, G., and Wahba, G. 1970. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Annals of Mathematical Statistics 41(2): 495–502.Google Scholar
King, G., and Zeng, L. 2006. The dangers of extreme counterfactuals. Political Analysis 14(2): 131–59.Google Scholar
Rifkin, R. M., and Lippert, R. A. 2007. Notes on regularized least squares. Technical report, MIT Computer Science and Artificial Intelligence Laboratory.Google Scholar
Rifkin, R., Yeo, G., and Poggio, T. 2003. Regularized least-squares classification. Nato Science Series Sub Series III Computer and Systems Sciences 190: 131–54.Google Scholar
Saunders, C., Gammerman, A., and Vovk, V. 1998. Ridge regression learning algorithm in dual variables. In Proceedings of the 15th International Conference on Machine Learning. Volume 19980, 515–21. San Francisco, CA: Morgan Kaufmann.Google Scholar
Schölkopf, B., and Smola, A. 2002. Learning with kernels: Support vector machines, regularization, optimization, and beyond. Cambridge, MA: MIT Press.Google Scholar
Tychonoff, A. N. 1963. Solution of incorrectly formulated problems and the regularization method. Doklady Akademii Nauk SSSR 151: 501–4. Translated in Soviet Mathematics 4: 1035–8.Google Scholar
Wood, S. N. 2003. Thin plate regression splines. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65(1): 95–114.Google Scholar
Supplementary material: PDF

Hainmueller and Hazlett supplementary material

Appendix

Download Hainmueller and Hazlett supplementary material(PDF)
PDF 843.9 KB