{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T20:04:28Z","timestamp":1780430668483,"version":"3.54.1"},"reference-count":50,"publisher":"Oxford University Press (OUP)","issue":"6","license":[{"start":{"date-parts":[[2025,1,8]],"date-time":"2025-01-08T00:00:00Z","timestamp":1736294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,11,10]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We provide a finite sample inference method for the structural parameters of Manski's semiparametric binary response model under a conditional median restriction. This is achieved by exploiting distributional properties of observable outcomes conditional on the observed sequence of exogenous variables. Moment inequalities conditional on the size n sequence of exogenous covariates are constructed, and the proposed test statistic is a monotone function of violations of the corresponding sample moment inequalities. The critical value used for inference is provided by the appropriate quantile of a known function of n independent Bernoulli random variables and does not require the use of a cube root asymptotic approximation employing a point estimator of the target parameter. Simulation studies demonstrate favourable finite sample performance of the test in comparison to several existing approaches. Empirical use is illustrated with an application to the classical setting of transportation choice.<\/jats:p>","DOI":"10.1093\/restud\/rdaf001","type":"journal-article","created":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T15:19:57Z","timestamp":1735831197000},"page":"4117-4151","source":"Crossref","is-referenced-by-count":1,"title":["Finite Sample Inference for the Maximum Score Estimand"],"prefix":"10.1093","volume":"92","author":[{"given":"Adam M","family":"Rosen","sequence":"first","affiliation":[{"name":"Duke University, USA and CeMMAP and IFS ,","place":["UK"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Takuya","family":"Ura","sequence":"additional","affiliation":[{"name":"University of California , Davis ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,1,8]]},"reference":[{"key":"2025111022014024400_rdaf001-B1","doi-asserted-by":"crossref","first-page":"1175","DOI":"10.1111\/j.1468-0262.2005.00613.x","article-title":"On the Bootstrap of the Maximum Score Estimator","volume":"73","author":"Abrevaya","year":"2005","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B2","doi-asserted-by":"crossref","first-page":"1141","DOI":"10.3982\/ECTA16907","article-title":"Finite-Sample Optimal Estimation and Inference on Average Treatment Effects Under Unconfoundedness","volume":"89","author":"Armstrong","year":"2021","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B3","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/0166-218X(95)00026-N","article-title":"Reverse Search for Enumeration","volume":"65","author":"Avis","year":"1996","journal-title":"Discrete Applied Mathematics"},{"key":"2025111022014024400_rdaf001-B4","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1111\/ectj.12048","article-title":"Non-standard Rates of Convergence of Criterion-Function-Based Set Estimators for Binary Response Models","volume":"18","author":"Blevins","year":"2015","journal-title":"The Econometrics Journal"},{"key":"2025111022014024400_rdaf001-B5","doi-asserted-by":"crossref","first-page":"2203","DOI":"10.3982\/ECTA17950","article-title":"Bootstrap-Based Inference for Cube Root Asymptotics","volume":"88","author":"Cattaneo","year":"2020","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B6","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.jeconom.2018.05.001","article-title":"Best Subset Binary Prediction","volume":"206","author":"Chen","year":"2018","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B7","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/j.jeconom.2018.12.024","article-title":"Breaking the Curse of Dimensionality in Conditional Moment Inequalities for Discrete Choice Models","volume":"210","author":"Chen","year":"2019","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B8","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.jeconom.2009.01.004","article-title":"Finite Sample Inference for Quantile Regression Models","volume":"152","author":"Chernozhukov","year":"2009","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B9","doi-asserted-by":"crossref","first-page":"667","DOI":"10.3982\/ECTA8718","article-title":"Intersection Bounds: Estimation and Inference","volume":"81","author":"Chernozhukov","year":"2013","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B10","doi-asserted-by":"crossref","first-page":"959","DOI":"10.3982\/ECTA12223","article-title":"Generalized Instrumental Variable Models","volume":"85","author":"Chesher","year":"2017","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B11","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1093\/biomet\/26.4.404","article-title":"The Use of Confidence or Fiducial Limits Illustrated in the Case of the Binomial","volume":"26","author":"Clopper","year":"1934","journal-title":"Biometrika"},{"key":"2025111022014024400_rdaf001-B12","doi-asserted-by":"crossref","first-page":"404","DOI":"10.2307\/1912157","article-title":"Distribution-Free Maximum Likelihood Estimator of the Binary Choice Model","volume":"51","author":"Cosslett","year":"1983","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B13","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1109\/PGEC.1965.264137","article-title":"Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition","volume":"EC-14","author":"Cover","year":"1965","journal-title":"IEEE Transactions on Electrical Computers"},{"key":"2025111022014024400_rdaf001-B14","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/S0165-1765(01)00494-3","article-title":"Subsampling Inference in Cube Root Asymptotics with an Application to Manski\u2019s Maximum Score Estimator","volume":"73","author":"Delgado","year":"2001","journal-title":"Economics Letters"},{"key":"2025111022014024400_rdaf001-B15","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1007\/BF02592055","article-title":"Truncated-Newton Algorithms for Large-Scale Unconstrained Optimization","volume":"26","author":"Dembo","year":"1983","journal-title":"Mathematical Programming"},{"key":"2025111022014024400_rdaf001-B16","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.ecosta.2017.03.005","article-title":"A Hyperplanes Intersection Simulated Annealing Algorithm for Maximum Score Estimation","volume":"8","author":"Florios","year":"2018","journal-title":"Econometrics and Statistics"},{"key":"2025111022014024400_rdaf001-B17","author":"Florios","year":"2022"},{"key":"2025111022014024400_rdaf001-B18","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jeconom.2008.05.018","article-title":"Exact Computation of Max Weighted Score Estimators","volume":"146","author":"Florios","year":"2008","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B19","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1080\/01621459.2020.1802284","article-title":"Nonparametric Maximum Likelihood Methods for Binary Response Models With Random Coefficients","volume":"117","author":"Gu","year":"2022","journal-title":"Journal of the American Statistical Association"},{"key":"2025111022014024400_rdaf001-B20","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1016\/j.jeconom.2022.04.011","article-title":"Partial Identification in Nonseparable Binary Response Models with Endogenous Regressors","volume":"235","author":"Gu","year":"2023","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B21","author":"Gu","year":"2022"},{"key":"2025111022014024400_rdaf001-B22","author":"Gurobi Optimization, LLC","year":"2023"},{"key":"2025111022014024400_rdaf001-B23","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/0304-4076(87)90030-3","article-title":"Non-parametric Analysis of a Generalized Regression Model: The Maximum Rank Correlation Estimator","volume":"35","author":"Han","year":"1987","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B24","doi-asserted-by":"crossref","first-page":"505","DOI":"10.2307\/2951582","article-title":"A Smoothed Maximum Score Estimator for the Binary Response Model","volume":"60","author":"Horowitz","year":"1992","journal-title":"Econometrica: Journal of the Econometric Society"},{"key":"2025111022014024400_rdaf001-B25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0304-4076(93)90113-J","article-title":"Semiparametric Estimation of a Work-Trip Mode Choice Model","volume":"58","author":"Horowitz","year":"1993","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B26","first-page":"505","article-title":"Bootstrap Critical Values for Tests Based on the Smoothed Maximum Score Estimator","volume":"141\u2013167","author":"Horowitz","year":"2002","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B27","author":"Johnson","year":"2007\u20132019"},{"key":"2025111022014024400_rdaf001-B28","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.jeconom.2015.01.005","article-title":"Classical Laplace Estimation for n3-Consistent Estimators: Improved Convergence Rates and Rate-Adaptive Inference","volume":"187","author":"Jun","year":"2015","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B29","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1017\/S0266466616000463","article-title":"Integrated Score Estimation","volume":"33","author":"Jun","year":"2017","journal-title":"Econometric Theory"},{"key":"2025111022014024400_rdaf001-B30","author":"Kaido","year":"2019"},{"key":"2025111022014024400_rdaf001-B31","author":"Khan","year":"2024"},{"key":"2025111022014024400_rdaf001-B32","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1214\/aos\/1176347498","article-title":"Cube Root Asymptotics","volume":"18","author":"Kim","year":"1990","journal-title":"Annals of Statistics"},{"key":"2025111022014024400_rdaf001-B33","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1016\/j.jeconom.2013.05.005","article-title":"Binary Choice Models with Discrete Regressors: Identification and Misspecification","volume":"177","author":"Komarova","year":"2013","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B34","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1002\/cjs.5550340104","article-title":"On the Bootstrap in Cube Root Asymptotics","volume":"34","author":"L\u00e9ger","year":"2006","journal-title":"The Canadian Journal of Statistics \/ La Revue Canadienne de Statistique"},{"key":"2025111022014024400_rdaf001-B35","volume-title":"Testing Statistical Hypotheses","author":"Lehmann","year":"2005","edition":"3rd edn"},{"key":"2025111022014024400_rdaf001-B36","author":"Li","year":"2022"},{"key":"2025111022014024400_rdaf001-B37","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/0304-4076(75)90032-9","article-title":"Maximum Score Estimation of the Stochastic Utility Model of Choice","volume":"3","author":"Manski","year":"1975","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B38","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/0304-4076(85)90009-0","article-title":"Semiparametric Analysis of Discrete Response: Asymptotic Properties of the Maximum Score Estimator","volume":"27","author":"Manski","year":"1985","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B39","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1111\/j.1468-2354.2007.00467.x","article-title":"Partial Identification of Counterfactual Choice Probabilities","volume":"48","author":"Manski","year":"2007","journal-title":"International Economic Review"},{"key":"2025111022014024400_rdaf001-B40","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/0304-4076(86)90013-8","article-title":"Operational Characteristics of Maximum Score Estimation","volume":"32","author":"Manski","year":"1986","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B41","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/0304-4076(89)90032-8","article-title":"Estimation of Best Predictors of Binary Response","volume":"40","author":"Manski","year":"1989","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B42","doi-asserted-by":"crossref","first-page":"3324","DOI":"10.1214\/21-AOS2085","article-title":"Optimal Linear Discriminators for the Discrete Choice Model in Growing Dimensions","volume":"49","author":"Mukherjee","year":"2021","journal-title":"Annals of Statistics"},{"key":"2025111022014024400_rdaf001-B43","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1016\/j.jeconom.2018.04.001","article-title":"A Consistent Bootstrap Procedure for the Maximum Score Estimator","volume":"205","author":"Patra","year":"2018","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B44","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0304-4076(93)90118-O","article-title":"On the Computation of Semiparametric Estimates in Limited Dependent Variable Models","volume":"58","author":"Pinkse","year":"1993","journal-title":"Journal of Econometrics"},{"key":"2025111022014024400_rdaf001-B45","author":"Powell","year":"1994"},{"key":"2025111022014024400_rdaf001-B46","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1137\/15M1027930","article-title":"A New Algorithm for Enumeration of Cells of Hyperplane Arrangements and a Comparison with Avis and Fukuda\u2019s Reverse Search","volume":"32","author":"Rada","year":"2018","journal-title":"Siam Journal of Discrete Mathematics"},{"key":"2025111022014024400_rdaf001-B47","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1214\/17-AOS1552","article-title":"Local M-Estimation with Discontinuous Criterion for Dependent and Limited Observations","volume":"46","author":"Seo","year":"2018","journal-title":"Annals of Statistics"},{"key":"2025111022014024400_rdaf001-B48","author":"Sleumer","year":"1998"},{"key":"2025111022014024400_rdaf001-B49","author":"Syrgkanis","year":"2018"},{"key":"2025111022014024400_rdaf001-B50","author":"Ypma","year":"2018"}],"container-title":["Review of Economic Studies"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/restud\/advance-article-pdf\/doi\/10.1093\/restud\/rdaf001\/61372796\/rdaf001.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/restud\/article-pdf\/92\/6\/4117\/61372796\/rdaf001.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/restud\/article-pdf\/92\/6\/4117\/61372796\/rdaf001.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,11]],"date-time":"2025-11-11T03:02:02Z","timestamp":1762830122000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/restud\/article\/92\/6\/4117\/7945332"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,8]]},"references-count":50,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2025,1,8]]},"published-print":{"date-parts":[[2025,11,10]]}},"URL":"https:\/\/doi.org\/10.1093\/restud\/rdaf001","relation":{},"ISSN":["0034-6527","1467-937X"],"issn-type":[{"value":"0034-6527","type":"print"},{"value":"1467-937X","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025,11]]},"published":{"date-parts":[[2025,1,8]]}}}