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PROPERTIES OF DOUBLY ROBUST ESTIMATORS WHEN NUISANCE FUNCTIONS ARE ESTIMATED NONPARAMETRICALLY

Published online by Cambridge University Press:  03 December 2018

Christoph Rothe  and
Sergio Firpo
Christoph Rothe*
Affiliation:
University of Mannheim
Sergio Firpo
Affiliation:
Insper Institute of Education and Research
*
*Address correspondence to Christoph Rothe, Department of Economics, University of Mannheim, D-68131 Mannheim, Germany; e-mail: rothe@vwl.uni-mannheim.de.
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Abstract

An estimator of a finite-dimensional parameter is said to be doubly robust (DR) if it imposes parametric specifications on two unknown nuisance functions, but only requires that one of these two specifications is correct in order for the estimator to be consistent for the object of interest. In this article, we study versions of such estimators that use local polynomial smoothing for estimating the nuisance functions. We show that such semiparametric two-step (STS) versions of DR estimators have favorable theoretical and practical properties relative to other commonly used STS estimators. We also show that these gains are not generated by the DR property alone. Instead, it needs to be combined with an orthogonality condition on the estimation residuals from the nonparametric first stage, which we show to be satisfied in a wide range of models.

Type
ARTICLES
Information
Econometric Theory , Volume 35 , Issue 5 , October 2019 , pp. 1048 - 1087
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

An earlier working version of this article was circulated under the title “Semiparametric Estimation and Inference Using Doubly Robust Moment Conditions”. We would like to thank Matias Cattaneo, Michael Jansson, Marcelo Moreira, Ulrich Müller, Whitney Newey, Cristine Pinto, and audiences at numerous seminar and conference presentations for their helpful comments; and Bernardo Modenesi for excellent research assistance. Christoph Rothe gratefully acknowledges financial support from German Scholars Organization & Carl-Zeiss-Stiftung. Sergio Firpo gratefully acknowledges financial support from CNPq-Brazil.

References

REFERENCES

Ai, C. & Chen, X. (2003) Efficient estimation of models with conditional moment restrictions containing unknown functions. Econometrica 71, 1795–1843.CrossRefGoogle Scholar
Andrews, D. (1994) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 43–72.CrossRefGoogle Scholar
Bang, H. & Robins, J.M. (2005) Doubly robust estimation in missing data and causal inference models. Biometrics 61, 962–973.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., Fernández-Val, I., & Hansen, C. (2017) Program evaluation with high-dimensional data. Econometrica 85, 233–298.CrossRefGoogle Scholar
Belloni, A., Chernozhukov, V., & Hansen, C. (2014) Inference on treatment effects after selection among high-dimensional controls. Review of Economic Studies 81, 608–650.CrossRefGoogle Scholar
Bickel, P.J. & Ritov, Y. (2003) Nonparametric estimators which can be “plugged-in”. Annals of Statistics 31, 1033–1053.Google Scholar
Bravo, F. & Jacho-Chávez, D.T. (2010) Empirical likelihood for efficient semiparametric average treatment effects. Econometric Reviews 30, 1–24.CrossRefGoogle Scholar
Cattaneo, M. (2010) Efficient semiparametric estimation of multi-valued treatment effects under ignorability. Journal of Econometrics 155, 138–154.CrossRefGoogle Scholar
Cattaneo, M., Crump, R., & Jansson, M. (2013) Generalized jackknife estimators of weighted average derivatives. Journal of the American Statistical Association 108, 1243–1268.CrossRefGoogle Scholar
Cattaneo, M. & Jansson, M. (2018) Kernel-Based Semiparametric Estimators: Small bandwidth asymptotics and bootstrap consistency. Econometrica 86, 955–995.CrossRefGoogle Scholar
Chen, X., Hong, H., & Tarozzi, A. (2008) Semiparametric efficiency in GMM models with auxiliary data. Annals of Statistics 36, 808–843.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Chen, X. & Shen, X. (1998) Sieve extremum estimates for weakly dependent data. Econometrica 66, 289–314.CrossRefGoogle Scholar
Escanciano, J.C., Jacho-Chávez, D., & Lewbel, A. (2014) Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing. Journal of Econometrics 178, 426–443.CrossRefGoogle Scholar
Escanciano, J.C., Jacho-Chávez, D., & Lewbel, A. (2016) Identification and estimation of semiparametric two-step models. Quantitative Economics 7, 561–589.CrossRefGoogle Scholar
Fan, J. (1993) Local linear regression smoothers and their minimax efficiencies. Annals of Statistics 21, 196–216.CrossRefGoogle Scholar
Fan, J., Heckman, N., & Wand, M. (1995) Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. Journal of the American Statistical Association 90, 141–150.CrossRefGoogle Scholar
Farrell, M.H. (2015) Robust inference on average treatment effects with possibly more covariates than observations. Journal of Econometrics 189, 1–23.CrossRefGoogle Scholar
Firpo, S. (2007) Efficient semiparametric estimation of quantile treatment effects. Econometrica 75, 259–276.CrossRefGoogle Scholar
Frölich, M., Huber, M., & Wiesenfarth, M. (2015) The Finite Sample Performance of Semi-and Nonparametric Estimators for Treatment Effects and Policy Evaluation. Working paper.Google Scholar
Goldstein, L. & Messer, K. (1992) Optimal Plug-in estimators for nonparametric functional estimation. Annals of Statistics 20, 1306–1328.CrossRefGoogle Scholar
Graham, B., Pinto, C., & Egel, D. (2012) Inverse probability tilting for moment condition models with missing data. Review of Economic Studies 79, 1053–1079.CrossRefGoogle Scholar
Hahn, J. (1998) On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66, 315–331.CrossRefGoogle Scholar
Hall, P. & Marron, J.S. (1987) Estimation of integrated squared density derivatives. Statistics & Probability Letters 6, 109–115.CrossRefGoogle Scholar
Hirano, K., Imbens, G., & Ridder, G. (2003) Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71, 1161–1189.CrossRefGoogle Scholar
Ichimura, H. & Lee, S. (2010) Characterization of the asymptotic distribution of semiparametric M-estimators. Journal of Econometrics 159, 252–266.CrossRefGoogle Scholar
Ichimura, H. & Linton, O. (2005) Asymptotic expansions for some semiparametric program evaluation estimators. In Andrews, D. & Stock, J. (eds.), Identifcation and Inference for Econometric Models: A Festschrift in Honor of Thomas J. Rothenberg, pp. 149–170. Cambridge University Press.CrossRefGoogle Scholar
Ichimura, H. & Newey, W. (2015) The Influence Function of Semiparametric Estimators. Working paper. arXiv:1508.01378Google Scholar
Kang, J. & Schafer, J. (2007) Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical Science 22, 523–539.CrossRefGoogle Scholar
Kong, E., Linton, O., & Xia, Y. (2010) Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econometric Theory 26, 1529–156.CrossRefGoogle Scholar
Linton, O. (1995) Second order approximation in the partially linear regression model. Econometrica 63, 1079–1112.CrossRefGoogle Scholar
Mammen, E., Rothe, C., & Schienle, M. (2016) Semiparametric estimation with generated covariates. Econometric Theory 32, 1140–1177.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Naimi, A.I. & Kennedy, E.H. (2017) Nonparametric Double Robustness. Working paper.Google Scholar
Newey, W. (1994) The asymptotic variance of semiparametric Estimators. Econometrica 62, 1349–1382.CrossRefGoogle Scholar
Newey, W., Hsieh, F., & Robins, J. (2004) Twicing kernels and a small bias property of semiparametric estimators. Econometrica 72, 947–962.CrossRefGoogle Scholar
Newey, W. & McFadden, D. (1994) Large sample estimation and hypothesis testing. Handbook of Econometrics 4, 2111–2245.CrossRefGoogle Scholar
Newey, W. & Stoker, T. (1993) Efficiency of weighted average derivative estimators and index models. Econometrica 61, 1199–223.CrossRefGoogle Scholar
Ogburn, E.L., Rotnitzky, A., & Robins, J.M. (2015) Doubly robust estimation of the local average treatment effect curve. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77, 373–396.CrossRefGoogle Scholar
Powell, J.L., Stock, J.H., & Stoker, T.M. (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.CrossRefGoogle Scholar
Powell, J.L. & Stoker, T.M. (1996) Optimal bandwidth choice for density-weighted averages. Journal of Econometrics 75, 291–316.CrossRefGoogle Scholar
Racine, J. & Li, Q. (2004) Nonparametric estimation of regression functions with both categorical and continuous data. Journal of Econometrics 119, 99–130.CrossRefGoogle Scholar
Robins, J. & Ritov, Y. (1997) Toward a curse of dimensionality appropriate (CODA) asymptotic theroy for semi-parametric models. Statistics in Medicine 16, 285–319.3.0.CO;2-#>CrossRefGoogle Scholar
Robins, J. & Rotnitzky, A. (1995) Semiparametric efficiency in multivariate regression models with missing data. Journal of the American Statistical Association 90, 122–129.CrossRefGoogle Scholar
Robins, J., Rotnitzky, A., & Zhao, L. (1994) Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association 89, 846–866.CrossRefGoogle Scholar
Robins, J.M. & Rotnitzky, A. (2001) Comment on “Inference for semiparametric models: Some questions and an answer” by Bickel, P. and Kwon, J.. Statistica Sinica 11, 920–936.Google Scholar
Robinson, P. (1988) Root-N-consistent semiparametric regression. Econometrica 56, 931–954.CrossRefGoogle Scholar
Ruppert, D. & Wand, M. (1994) Multivariate locally weighted least squares regression. Annals of Statistics 22, 1346–1370.CrossRefGoogle Scholar
Scharfstein, D., Rotnitzky, A., & Robins, J. (1999) Adjusting for nonignorable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association 94, 1096–1120.CrossRefGoogle Scholar
Shen, X. (1997) On methods of sieves and penalization. Annals of Statistics 25, 2555–2591.Google Scholar
Stock, J.H. (1989) Nonparametric policy analysis. Journal of the American Statistical Association 84, 567–575.CrossRefGoogle Scholar
Tan, Z. (2006) Regression and weighting methods for causal inference using instrumental variables. Journal of the American Statistical Association 101, 1607–1618.CrossRefGoogle Scholar
Tan, Z. (2010) Bounded, efficient and doubly robust estimation with inverse weighting. Biometrika 97, 661–682.CrossRefGoogle Scholar
Vermeulen, K. & Vansteelandt, S. (2015) Bias-reduced doubly robust estimation. Journal of the American Statistical Association 110, 1024–1036.CrossRefGoogle Scholar
Wooldridge, J. (2007) Inverse probability weighted estimation for general missing data problems. Journal of Econometrics 141, 1281–1301.CrossRefGoogle Scholar