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Nonparametric instrumental variable estimation under monotonicity


  • Denis Chetverikov

    () (Institute for Fiscal Studies and UCLA)

  • Daniel Wilhelm

    () (Institute for Fiscal Studies and cemmap and UCL)


The ill-posedness of the nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the function to be estimated is monotone and consider a sieve estimator that enforces this monotonicity constraint. We define a constrained measure of ill-posedness that is relevant for the constrained estimator and show that, under a monotone IV assumption and certain other mild regularity conditions, this measure is bounded uniformly over the dimension of the sieve space. This finding is in stark contrast to the well known result that the unconstrained sieve measure of ill-posedness that is relevant for the unconstrained estimator grows to in nity with the dimension of the sieve space. Based on this result, we derive a novel non-asymptotic error bound for the constrained estimator. The bound gives a set of data-generating processes for which the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimator. The form of the bound implies that the regularization effect can be strong even in large samples and even if the function to be estimated is steep, particularly so if the NPIV model is severely ill-posed. Our simulation study con rms these findings and reveals the potential for large performance gains from imposing the monotonicity constraint.

Suggested Citation

  • Denis Chetverikov & Daniel Wilhelm, 2017. "Nonparametric instrumental variable estimation under monotonicity," CeMMAP working papers CWP14/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:14/17

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    References listed on IDEAS

    1. Xiaohong Chen & Timothy M. Christensen, 2013. "Optimal uniform convergence rates for sieve nonparametric instrumental variables regression," CeMMAP working papers CWP56/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Mammen, Enno & Thomas-Agnan, C., 1996. "Smoothing Splines And Shape Restrictions," SFB 373 Discussion Papers 1996,87, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    3. Guido W. Imbens & Whitney K. Newey, 2009. "Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity," Econometrica, Econometric Society, vol. 77(5), pages 1481-1512, September.
    4. S. Darolles & Y. Fan & J. P. Florens & E. Renault, 2011. "Nonparametric Instrumental Regression," Econometrica, Econometric Society, vol. 79(5), pages 1541-1565, September.
    5. Richard Blundell & Joel L. Horowitz & Matthias Parey, 2012. "Measuring the price responsiveness of gasoline demand: Economic shape restrictions and nonparametric demand estimation," Quantitative Economics, Econometric Society, vol. 3(1), pages 29-51, March.
    6. Horowitz, Joel L. & Lee, Sokbae, 2012. "Uniform confidence bands for functions estimated nonparametrically with instrumental variables," Journal of Econometrics, Elsevier, vol. 168(2), pages 175-188.
    7. Brendan Kline, 2016. "Identification of the Direction of a Causal Effect by Instrumental Variables," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 34(2), pages 176-184, April.
    8. Kasy, Maximilian, 2011. "Identification In Triangular Systems Using Control Functions," Econometric Theory, Cambridge University Press, vol. 27(03), pages 663-671, June.
    9. Jason Abrevaya & Jerry A. Hausman & Shakeeb Khan, 2010. "Testing for Causal Effects in a Generalized Regression Model With Endogenous Regressors," Econometrica, Econometric Society, vol. 78(6), pages 2043-2061, November.
    10. Joel L. Horowitz, 2011. "Applied Nonparametric Instrumental Variables Estimation," Econometrica, Econometric Society, vol. 79(2), pages 347-394, March.
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    Cited by:

    1. Denis Chetverikov & Dongwoo Kim & Daniel Wilhelm, 2018. "Nonparametric instrumental-variable estimation," Stata Journal, StataCorp LP, vol. 18(4), pages 937-950, December.
    2. Daniel Wilhelm, 2018. "Testing for the presence of measurement error," CeMMAP working papers CWP45/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    3. Olivier Scaillet, 2016. "On ill‐posedness of nonparametric instrumental variable regression with convexity constraints," Econometrics Journal, Royal Economic Society, vol. 19(2), pages 232-236, June.
    4. repec:gnv:wpaper:unige:79975 is not listed on IDEAS
    5. Denis Chetverikov & . ., 2016. "On cross-validated Lasso," CeMMAP working papers CWP47/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Victor Chernozhukov & Whitney K. Newey & Andres Santos, 2015. "Constrained conditional moment restriction models," CeMMAP working papers CWP59/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. repec:eee:econom:v:201:y:2017:i:1:p:95-107 is not listed on IDEAS
    8. Denis Chetverikov & Daniel Wilhelm & Dongwoo Kim, 2018. "An adaptive test of stochastic monotonicity," CeMMAP working papers CWP24/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Christoph Breunig, 2016. "Specification Testing in Nonparametric Instrumental Quantile Regression," SFB 649 Discussion Papers SFB649DP2016-032, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    10. Daniel Wilhelm, 2015. "Identification and estimation of nonparametric panel data regressions with measurement error," CeMMAP working papers CWP34/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Kohtaro Hitomi & Masamune Iwasawa & Yoshihiko Nishiyama, 2018. "Rate Optimal Specification Test When the Number of Instruments is Large," KIER Working Papers 986, Kyoto University, Institute of Economic Research.
    12. Young Jun Lee & Daniel Wilhelm, 2018. "Testing for the presence of measurement error in Stata," CeMMAP working papers CWP51/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.


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