{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,16]],"date-time":"2026-01-16T08:59:20Z","timestamp":1768553960250,"version":"3.49.0"},"reference-count":47,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2018,2,14]],"date-time":"2018-02-14T00:00:00Z","timestamp":1518566400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["www.cambridge.org"],"crossmark-restriction":true},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2018,12]]},"abstract":"<jats:p>This paper considers nonparametric instrumental variable regression when the endogenous variable is contaminated with classical measurement error. Existing methods are inconsistent in the presence of measurement error. We propose a wavelet deconvolution estimator for the structural function that modifies the generalized Fourier coefficients of the orthogonal series estimator to take into account the measurement error. We establish the convergence rates of our estimator for the cases of mildly\/severely ill-posed models and ordinary\/super smooth measurement errors. We characterize how the presence of measurement error slows down the convergence rates of the estimator. We also study the case where the measurement error density is unknown and needs to be estimated, and show that the estimation error of the measurement error density is negligible under mild conditions as far as the measurement error density is symmetric.<\/jats:p>","DOI":"10.1017\/s0266466617000469","type":"journal-article","created":{"date-parts":[[2018,2,14]],"date-time":"2018-02-14T09:51:51Z","timestamp":1518601911000},"page":"1256-1280","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":12,"title":["NONPARAMETRIC INSTRUMENTAL REGRESSION WITH ERRORS IN VARIABLES"],"prefix":"10.1017","volume":"34","author":[{"given":"Karun","family":"Adusumilli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Taisuke","family":"Otsu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2018,2,14]]},"reference":[{"key":"S0266466617000469_ref36","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1006\/jeem.1998.1029","article-title":"Measurement error in recreation demand models: The joint estimation of participation, site choice, and site characteristics","volume":"35","author":"Morey","year":"1998","journal-title":"Journal of Environmental Economics and Management"},{"key":"S0266466617000469_ref44","doi-asserted-by":"crossref","first-page":"749","DOI":"10.3982\/QE275","article-title":"Estimating nonseparable models with mismeasured endogenous variables","volume":"6","author":"Song","year":"2015","journal-title":"Quantitative Economics"},{"key":"S0266466617000469_ref29","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":"S0266466617000469_ref10","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":"S0266466617000469_ref31","doi-asserted-by":"crossref","unstructured":"Hu, Y . (2016) Microeconomic Models with Latent Variables: Applications of Measurement Error Models in Empirical Industrial Organization and Labor Economics. Working paper, John Hopkins University.","DOI":"10.1920\/wp.cem.2015.0315"},{"key":"S0266466617000469_ref46","doi-asserted-by":"crossref","DOI":"10.1002\/9780470317020","volume-title":"Statistical Modeling by Wavelets","author":"Vidakovic","year":"1999"},{"key":"S0266466617000469_ref17","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1111\/j.2517-6161.1995.tb02032.x","article-title":"Wavelet shrinkage: Asymptopia?","volume":"57","author":"Donoho","year":"1995","journal-title":"Journal of the Royal Statistical Society: Series B"},{"key":"S0266466617000469_ref37","doi-asserted-by":"crossref","first-page":"616","DOI":"10.1162\/003465301753237704","article-title":"Flexible simulated moment estimation of nonlinear errors-in-variables models","volume":"83","author":"Newey","year":"2001","journal-title":"Review of Economics and Statistics"},{"key":"S0266466617000469_ref26","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1214\/009053607000000028","article-title":"A ridge-parameter approach to deconvolution","volume":"35","author":"Hall","year":"2007","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref35","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-87557-4","volume-title":"Deconvolution Problems in Nonparametric Statistics","author":"Meister","year":"2009"},{"key":"S0266466617000469_ref25","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":"S0266466617000469_ref39","doi-asserted-by":"crossref","first-page":"2033","DOI":"10.1214\/aos\/1017939249","article-title":"Adaptive wavelet estimator for nonparametric density deconvolution","volume":"27","author":"Pensky","year":"1999","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref47","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1007\/BF01845640","article-title":"Some limit theorems for the empirical process indexed by functions","volume":"74","author":"Yukich","year":"1987","journal-title":"Probability Theory and Related Fields"},{"key":"S0266466617000469_ref12","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":"S0266466617000469_ref30","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.jeconom.2011.09.023","article-title":"Specification testing in nonparametric instrumental variable estimation","volume":"167","author":"Horowitz","year":"2012","journal-title":"Journal of Econometrics"},{"key":"S0266466617000469_ref13","doi-asserted-by":"crossref","first-page":"280","DOI":"10.1198\/016214507000001355","article-title":"Using SIMEX for smoothing-parameter choice in errors-in-variables problems","volume":"103","author":"Delaigle","year":"2008","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466617000469_ref6","doi-asserted-by":"crossref","first-page":"S219","DOI":"10.1086\/259996","article-title":"Schooling and inequality from generation to generation","volume":"80","author":"Bowles","year":"1972","journal-title":"Journal of Political Economy"},{"key":"S0266466617000469_ref28","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/0304-4076(94)01602-V","article-title":"Nonlinear errors in variables estimation of some Engel curves","volume":"65","author":"Hausman","year":"1995","journal-title":"Journal of Econometrics"},{"key":"S0266466617000469_ref42","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1017\/CBO9781139060035.009","volume-title":"Advances in Economics and Econometrics","volume":"vol. 3","author":"Schennach","year":"2013"},{"key":"S0266466617000469_ref45","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/02331889008802238","article-title":"Deconvoluting kernel density estimators","volume":"21","author":"Stefanski","year":"1990","journal-title":"Statistics"},{"key":"S0266466617000469_ref24","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.3982\/ECTA7937","article-title":"Nonparametric instrumental variable estimation of structural quantile effect","volume":"80","author":"Gagliardini","year":"2012","journal-title":"Econometrica"},{"key":"S0266466617000469_ref5","doi-asserted-by":"crossref","first-page":"816","DOI":"10.1080\/01621459.1973.10481429","article-title":"Response bias in reports of father\u2019s education and socioeconomic status","volume":"68","author":"Borus","year":"1973","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466617000469_ref16","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1214\/aos\/1024691081","article-title":"Minimax estimation via wavelet shrinkage","volume":"26","author":"Donoho","year":"1998","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref1","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1080\/01621459.1994.10476873","article-title":"Wavelet methods for curve estimation","volume":"89","author":"Antoniadis","year":"1994","journal-title":"Journal of the American Statistical Association"},{"key":"S0266466617000469_ref38","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1111\/1468-0262.00459","article-title":"Instrumental variable estimation of nonparametric model","volume":"71","author":"Newey","year":"2003","journal-title":"Econometrica"},{"key":"S0266466617000469_ref41","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1017\/S0266466604206028","article-title":"Nonparametric regression in the presence of measurement error","volume":"20","author":"Schennach","year":"2004b","journal-title":"Econometric Theory"},{"key":"S0266466617000469_ref32","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0167-7152(92)90231-S","article-title":"Density estimation in Besov spaces","volume":"13","author":"Kerkyacharian","year":"1992","journal-title":"Statistics and Probability Letters"},{"key":"S0266466617000469_ref4","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 curve","volume":"75","author":"Blundell","year":"2007","journal-title":"Econometrica"},{"key":"S0266466617000469_ref9","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"},{"key":"S0266466617000469_ref3","volume-title":"Measurement Errors in Surveys","author":"Biemer","year":"1991"},{"key":"S0266466617000469_ref21","doi-asserted-by":"crossref","first-page":"1900","DOI":"10.1214\/aos\/1176349402","article-title":"Nonparametric regression with errors in variables","volume":"21","author":"Fan","year":"1993","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref27","first-page":"271","article-title":"Measurement errors in polynomial regression models","volume":"50","author":"Hausman","year":"1991","journal-title":"Journal of Econometrics"},{"key":"S0266466617000469_ref20","doi-asserted-by":"crossref","first-page":"734","DOI":"10.1109\/18.986021","article-title":"Wavelet deconvolution","volume":"48","author":"Fan","year":"2002","journal-title":"IEEE Transactions on Information Theory"},{"key":"S0266466617000469_ref34","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1006\/jmva.1998.1741","article-title":"Nonparametric estimation of the measurement error model using multiple indicators","volume":"65","author":"Li","year":"1998","journal-title":"Journal of Multivariate Analysis"},{"key":"S0266466617000469_ref15","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1006\/acha.1995.1008","article-title":"Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition","volume":"2","author":"Donoho","year":"1995","journal-title":"Applied and Computational Harmonic Analysis"},{"key":"S0266466617000469_ref40","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1111\/j.1468-0262.2004.00477.x","article-title":"Estimation of nonlinear models with measurement error","volume":"72","author":"Schennach","year":"2004a","journal-title":"Econometrica"},{"key":"S0266466617000469_ref19","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1214\/aos\/1176348248","article-title":"On the optimal rates of convergence for nonparametric deconvolution problems","volume":"19","author":"Fan","year":"1991","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0304-4076(02)00120-3","article-title":"Robust and consistent estimation of nonlinear errors-in-variables models","volume":"110","author":"Li","year":"2002","journal-title":"Journal of Econometrics"},{"key":"S0266466617000469_ref2","first-page":"1157","article-title":"Estimates of the economic returns to schooling from a new sample of twins","volume":"84","author":"Ashenfelter","year":"1994","journal-title":"American Economic Review"},{"key":"S0266466617000469_ref8","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1257\/jel.49.4.901","article-title":"Nonlinear models of measurement errors","volume":"49","author":"Chen","year":"2011","journal-title":"Journal of Economic Literature"},{"key":"S0266466617000469_ref43","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/j.jeconom.2011.09.041","article-title":"Local indirect least squares and average marginal effects in nonseparable structural systems","volume":"166","author":"Schennach","year":"2012","journal-title":"Journal of Econometrics"},{"key":"S0266466617000469_ref14","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1214\/009053607000000884","article-title":"On deconvolution with repeated measurements","volume":"36","author":"Delaigle","year":"2008","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref18","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1214\/aos\/1032894451","article-title":"Density estimation by wavelet thresholding","volume":"24","author":"Donoho","year":"1996","journal-title":"Annals of Statistics"},{"key":"S0266466617000469_ref7","volume-title":"Measurement Error in Nonlinear Models: A Modern Perspective","author":"Carroll","year":"2012"},{"key":"S0266466617000469_ref22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1086\/298021","article-title":"Longitudinal analysis of the effects of trade unions","volume":"2","author":"Freeman","year":"1984","journal-title":"Journal of Labor Economics"},{"key":"S0266466617000469_ref23","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316665","volume-title":"Measurement Error Models","author":"Fuller","year":"1987"},{"key":"S0266466617000469_ref11","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.jmva.2015.04.004","article-title":"Density deconvolution from repeated measurements without symmetry assumption on the errors","volume":"140","author":"Comte","year":"2015","journal-title":"Journal of Multivariate Analysis"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466617000469","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T05:53:12Z","timestamp":1719813192000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466617000469\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,14]]},"references-count":47,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2018,12]]}},"alternative-id":["S0266466617000469"],"URL":"https:\/\/doi.org\/10.1017\/s0266466617000469","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,14]]},"assertion":[{"value":"Copyright \u00a9 Cambridge University Press 2018\u00a0","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}