{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,16]],"date-time":"2026-06-16T07:59:52Z","timestamp":1781596792231,"version":"3.54.5"},"reference-count":60,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T00:00:00Z","timestamp":1735603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/100016995","name":"CREST","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100016995","id-type":"DOI","asserted-by":"publisher"}]},{"name":"ENSAI"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,6,6]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We extend nonparametric regression smoothing splines to a context where there is endogeneity and instrumental variables are available. Unlike popular existing estimators, the resulting estimator is one step and relies on a unique regularisation parameter. We derive rates of the convergence for the estimator and its first derivative, which are uniform in the support of the endogenous variable. We also address the issue of imposing monotonicity in estimation and extend the approach to a partly linear model. Simulations confirm the good performances of our estimator compared to two-step procedures. Our method yields economically sensible results when used to estimate Engel curves.<\/jats:p>","DOI":"10.1093\/ectj\/utae024","type":"journal-article","created":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T05:47:01Z","timestamp":1735710421000},"page":"176-197","source":"Crossref","is-referenced-by-count":1,"title":["One-step smoothing splines instrumental regression"],"prefix":"10.1093","volume":"28","author":[{"given":"Jad","family":"Beyhum","sequence":"first","affiliation":[{"name":"KU Leuven Department of Economics, , Naamsestraat 69, 3000 Leuven ,","place":["Belgium"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elia","family":"Lapenta","sequence":"additional","affiliation":[{"name":"ENSAE-CREST , 5 Avenue Henry Le Chatelier, 91120 Palaiseau ,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pascal","family":"Lavergne","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Toulouse Capitole Toulouse School of Economics, , 1 Esplanade de l'Universit\u00e9, 31000 Toulouse ,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,12,31]]},"reference":[{"key":"2025060607015509500_bib1","doi-asserted-by":"crossref","first-page":"1795","DOI":"10.1111\/1468-0262.00470","article-title":"Efficient estimation of models with conditional moment restrictions containing unknown functions","volume":"71","author":"Ai","year":"2003","journal-title":"Econometrica"},{"issue":"4","key":"2025060607015509500_bib2","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1017\/S0266466619000380","article-title":"Honest confidence sets in nonparametric IV regression and other ill-posed models","volume":"36","author":"Babii","year":"2020","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib3","first-page":"3564","article-title":"Deep generalized method of moments for instrumental variable analysis","volume-title":"Advances in Neural Information Processing Systems","author":"Bennett","year":"2019"},{"key":"2025060607015509500_bib4","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/0304-4076(82)90105-1","article-title":"Consistent model specification tests","volume":"20","author":"Bierens","year":"1982","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib5","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.1111\/j.1468-0262.2007.00808.x","article-title":"Semi-nonparametric IV estimation of shape-invariant Engel curves","volume":"75","author":"Blundell","year":"2007","journal-title":"Econometrica"},{"key":"2025060607015509500_bib6","article-title":"Engel95 dataset","author":"Blundell","year":"2007"},{"key":"2025060607015509500_bib7","doi-asserted-by":"crossref","DOI":"10.3982\/ECTA18602","article-title":"Adaptive, rate-optimal hypothesis testing in nonparametric IV models","volume-title":"arXiv: Econometrics","author":"Breunig","year":"2024"},{"issue":"3","key":"2025060607015509500_bib8","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1017\/S0266466614000966","article-title":"Adaptive estimation of functionals in nonparametric instrumental regression","volume":"32","author":"Breunig","year":"2016","journal-title":"Econometric Theory"},{"issue":"6","key":"2025060607015509500_bib9","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1017\/S0266466600166010","article-title":"Generalization of GMM to a continuum of moment conditions","volume":"16","author":"Carrasco","year":"2000","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib10","doi-asserted-by":"crossref","first-page":"5633","DOI":"10.1016\/S1573-4412(07)06077-1","article-title":"Linear inverse problems in structural econometrics estimation based on spectral decomposition and regularization","volume-title":"Handbook of Econometrics","author":"Carrasco","year":"2007"},{"key":"2025060607015509500_bib11","first-page":"65","article-title":"Asymptotic normal inference in linear inverse problems","volume-title":"The Oxford Handbook of Applied Nonparametric and Semiparametric Econometrics and Statistics","author":"Carrasco","year":"2014"},{"issue":"1","key":"2025060607015509500_bib13","doi-asserted-by":"crossref","first-page":"39","DOI":"10.3982\/QE722","article-title":"Optimal sup-norm rates and uniform inference on nonlinear functionals of nonparametric IV regression","volume":"9","author":"Chen","year":"2018","journal-title":"Quantitative Economics"},{"key":"2025060607015509500_bib12","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1093\/restud\/rdae025","article-title":"Adaptive estimation and uniform confidence bands for nonparametric structural functions and elasticities","volume":"92","author":"Chen","year":"2024","journal-title":"Review of Economic Studies"},{"key":"2025060607015509500_bib14","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.jeconom.2009.02.002","article-title":"Efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals","volume":"152","author":"Chen","year":"2009","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.3982\/ECTA7888","article-title":"Estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals","volume":"80","author":"Chen","year":"2012","journal-title":"Econometrica"},{"issue":"3","key":"2025060607015509500_bib16","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.3982\/ECTA10771","article-title":"Sieve Wald and QLR inferences on semi\/nonparametric conditional moment models","volume":"83","author":"Chen","year":"2015","journal-title":"Econometrica"},{"issue":"3","key":"2025060607015509500_bib17","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1017\/S0266466610000381","article-title":"On rate optimality for ill-posed inverse problems in econometrics","volume":"27","author":"Chen","year":"2011","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib18","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.3982\/ECTA13639","article-title":"Nonparametric instrumental variable estimation under mono-tonicity","volume":"85","author":"Chetverikov","year":"2017","journal-title":"Econometrica"},{"key":"2025060607015509500_bib19","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1093\/biomet\/asp035","article-title":"Asymptotic properties of penalized spline estimators","volume":"96","author":"Claeskens","year":"2009","journal-title":"Biometrika"},{"key":"2025060607015509500_bib20","doi-asserted-by":"crossref","first-page":"1541","DOI":"10.3982\/ECTA6539","article-title":"Nonparametric instrumental regression","volume":"79","author":"Darolles","year":"2011","journal-title":"Econometrica"},{"key":"2025060607015509500_bib24","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1017\/S0266466610000368","article-title":"On the completeness condition in nonparametric instrumental problems","volume":"27","author":"D\u2019Haultfoeuille","year":"2011","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib21","first-page":"12248","article-title":"Minimax estimation of conditional moment models","volume-title":"Advances in Neural Information Processing Systems","author":"Dikkala","year":"2020"},{"key":"2025060607015509500_bib22","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/S0167-7152(02)00032-9","article-title":"A note on the characteristic function of the t-distribution","volume":"57","author":"Dreier","year":"2002","journal-title":"Statistics and Probability Letters"},{"key":"2025060607015509500_bib23","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1016\/j.jeconom.2013.06.001","article-title":"Iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression","volume":"178","author":"Dunker","year":"2014","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib25","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-009-1740-8","volume-title":"Regularization of Inverse Problems","author":"Engl","year":"1996"},{"issue":"3","key":"2025060607015509500_bib26","doi-asserted-by":"crossref","first-page":"S1","DOI":"10.1111\/j.1368-423X.2010.00314.x","article-title":"The practice of non-parametric estimation by solving inverse problems: The example of transformation models","volume":"13","author":"F\u00e8ve","year":"2010","journal-title":"Econometrics Journal"},{"issue":"2","key":"2025060607015509500_bib27","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.jeconom.2014.03.009","article-title":"Non parametric analysis of panel data models with endogenous variables","volume":"181","author":"F\u00e8ve","year":"2014","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib28","doi-asserted-by":"crossref","first-page":"284","DOI":"10.1017\/CBO9780511610257.010","article-title":"Inverse problems and structural econometrics: The example of instrumental variables","volume-title":"Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress","author":"Florens","year":"2003"},{"key":"2025060607015509500_bib29","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1017\/S026646661000037X","article-title":"Identification and estimation by penalization in nonparametric instrumental regression","volume":"27","author":"Florens","year":"2011","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib30","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1111\/j.1368-423X.2011.00358.x","article-title":"Instrumental regression in partially linear models","volume":"15","author":"Florens","year":"2012","journal-title":"Econometrics Journal"},{"issue":"2","key":"2025060607015509500_bib31","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1080\/10485252.2018.1428745","article-title":"Nonparametric instrumental variable derivative estimation","volume":"30","author":"Florens","year":"2018","journal-title":"Journal of Nonparametric Statistics"},{"key":"2025060607015509500_bib32","doi-asserted-by":"crossref","first-page":"1629","DOI":"10.3982\/ECTA13304","article-title":"On completeness and consistency in nonparametric instrumental variable models","volume":"85","author":"Freyberger","year":"2017","journal-title":"Econometrica"},{"key":"2025060607015509500_bib33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v094.i14","article-title":"CVXR: An R package for disciplined convex optimization","volume":"94","author":"Fu","year":"2020","journal-title":"Journal of Statistical Software"},{"key":"2025060607015509500_bib34","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.jeconom.2011.08.006","article-title":"Tikhonov regularization for nonparametric instrumental variable estimators","volume":"167","author":"Gagliardini","year":"2012","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib35","doi-asserted-by":"crossref","DOI":"10.1201\/b15710","volume-title":"Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach","author":"Green","year":"1993"},{"key":"2025060607015509500_bib36","doi-asserted-by":"crossref","first-page":"2904","DOI":"10.1214\/009053605000000714","article-title":"Nonparametric methods for inference in the presence of instrumental variables","volume":"33","author":"Hall","year":"2005","journal-title":"Annals of Statistics"},{"key":"2025060607015509500_bib37","doi-asserted-by":"crossref","first-page":"624","DOI":"10.1214\/aos\/1009210683","article-title":"Nonparametric kernel regression subject to monotonicity constraints","volume":"29","author":"Hall","year":"2001","journal-title":"Annals of Statistics"},{"key":"2025060607015509500_bib38","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1093\/biomet\/92.1.105","article-title":"Theory for penalised spline regression","volume":"92","author":"Hall","year":"2005","journal-title":"Biometrika"},{"issue":"5","key":"2025060607015509500_bib39","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v027.i05","article-title":"Nonparametric econometrics: The np package","volume":"27","author":"Hayfield","year":"2008","journal-title":"Journal of Statistical Software"},{"key":"2025060607015509500_bib40","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1111\/j.2517-6161.1986.tb01407.x","article-title":"Spline smoothing in a partly linear model","volume":"48","author":"Heckman","year":"1986","journal-title":"Journal of the Royal Statistical Society: Series B (Methodological)"},{"key":"2025060607015509500_bib41","doi-asserted-by":"crossref","first-page":"347","DOI":"10.3982\/ECTA8662","article-title":"Applied nonparametric instrumental variables estimation","volume":"79","author":"Horowitz","year":"2011","journal-title":"Econometrica"},{"key":"2025060607015509500_bib42","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.jeconom.2014.03.006","article-title":"Adaptive nonparametric instrumental variables estimation: Empirical choice of the regularization parameter","volume":"180","author":"Horowitz","year":"2014","journal-title":"Journal of Econometrics"},{"key":"2025060607015509500_bib43","article-title":"Towards a general large sample theory for regularized estimators","volume-title":"arXiv: Statistics Theory","author":"Jansson","year":"2020"},{"key":"2025060607015509500_bib44","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1017\/S0266466610000393","article-title":"Convergence rates for ill-posed inverse problems with an unknown operator","volume":"27","author":"Johannes","year":"2011","journal-title":"Econometric Theory"},{"key":"2025060607015509500_bib45","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.jspi.2012.07.010","article-title":"Iterative regularisation in nonparametric instrumental regression","volume":"143","author":"Johannes","year":"2013","journal-title":"Journal of Statistical Planning and Inference"},{"key":"2025060607015509500_bib46","volume-title":"Continuous Univariate Distributions","author":"Johnson","year":"1995"},{"key":"2025060607015509500_bib47","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0559-3","volume-title":"Linear Integral Equations","author":"Kress","year":"1999"},{"key":"2025060607015509500_bib48","volume-title":"Introductory Functional Analysis with Applications","author":"Kreyszig","year":"1978"},{"key":"2025060607015509500_bib49","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1093\/biomet\/asn010","article-title":"On the asymptotics of penalized splines","volume":"95","author":"Li","year":"2008","journal-title":"Biometrika"},{"key":"2025060607015509500_bib50","first-page":"8947","article-title":"Provably efficient neural estimation of structural equation models: An adversarial approach","volume-title":"Advances in Neural Information Processing Systems","author":"Liao","year":"2020"},{"key":"2025060607015509500_bib51","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1093\/biomet\/82.1.93","article-title":"A kernel method of estimating structured nonparametric regression based on marginal integration","volume":"82","author":"Linton","year":"1995","journal-title":"Biometrika"},{"key":"2025060607015509500_bib52","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1111\/1467-9469.00147","article-title":"Smoothing splines and shape restrictions","volume":"26","author":"Mammen","year":"1999","journal-title":"Scandinavian Journal of Statistics"},{"key":"2025060607015509500_bib53","first-page":"2710","article-title":"Dual instrumental variable regression","volume-title":"Advances in Neural Information Processing Systems","author":"Muandet","year":"2020"},{"key":"2025060607015509500_bib54","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1111\/1468-0262.00459","article-title":"Instrumental variable estimation of nonparametric models","volume":"71","author":"Newey","year":"2003","journal-title":"Econometrica"},{"key":"2025060607015509500_bib55","doi-asserted-by":"crossref","first-page":"931","DOI":"10.2307\/1912705","article-title":"Root-n consistent semiparametric regression","volume":"56","author":"Robinson","year":"1988","journal-title":"Econometrica"},{"key":"2025060607015509500_bib56","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511618994","volume-title":"Spline Functions: Basic Theory","author":"Schumaker","year":"2007","edition":"3 edn"},{"key":"2025060607015509500_bib57","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1093\/biomet\/asv070","article-title":"A unified framework for spline estimators","volume":"103","author":"Schwarz","year":"2016","journal-title":"Biometrika"},{"key":"2025060607015509500_bib58","first-page":"4593","article-title":"Kernel instrumental variable regression","volume-title":"Advances in Neural Information Processing Systems","author":"Singh","year":"2019"},{"key":"2025060607015509500_bib59","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970128","volume-title":"Spline Models for Observational Data","author":"Wahba","year":"1990"},{"key":"2025060607015509500_bib60","doi-asserted-by":"crossref","first-page":"20220073","DOI":"10.1515\/jci-2022-0073","article-title":"Instrumental variable regression via kernel maximum moment loss","volume":"11","author":"Zhang","year":"2023","journal-title":"Journal of Causal Inference"}],"container-title":["The Econometrics Journal"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/ectj\/advance-article-pdf\/doi\/10.1093\/ectj\/utae024\/61302607\/utae024.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/ectj\/article-pdf\/28\/2\/176\/61302607\/utae024.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/ectj\/article-pdf\/28\/2\/176\/61302607\/utae024.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T11:02:13Z","timestamp":1749207733000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/ectj\/article\/28\/2\/176\/7935515"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,31]]},"references-count":60,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2024,12,31]]},"published-print":{"date-parts":[[2025,6,6]]}},"URL":"https:\/\/doi.org\/10.1093\/ectj\/utae024","relation":{},"ISSN":["1368-4221","1368-423X"],"issn-type":[{"value":"1368-4221","type":"print"},{"value":"1368-423X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,5]]},"published":{"date-parts":[[2024,12,31]]}}}