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Published online by Cambridge University Press:  22 July 2022

Ruixuan Liu
Chinese University of Hong Kong
Zhengfei Yu*
University of Tsukuba
Address correspondence to Zhengfei Yu, Faculty of Humanities and Social Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan; e-mail:


We propose two simple semiparametric estimation methods for ordered response models with an unknown error distribution. The proposed methods do not require users to choose any tuning parameters, and they automatically incorporate the monotonicity restriction of the unknown distribution function. Fixing finite-dimensional parameters in the model, we construct nonparametric maximum likelihood estimates for the error distribution based on the related binary choice data or the entire ordered response data. We then obtain estimates for finite-dimensional parameters based on moment conditions given the estimated distribution function. Our semiparametric approaches deliver root-n consistent and asymptotically normal estimators of the regression coefficient and threshold parameter. We also develop valid bootstrap procedures for inference. The advantages of our methods are borne out in simulation studies and a real data application.

© The Author(s), 2022. Published by Cambridge University Press

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We are grateful to the Editor (Peter C.B. Phillips), the Co-Editor (Simon Lee), and two anonymous referees for their comments and suggestions that greatly improved the paper. We would like to thank Jason Abrevaya, Bo Honoré, Hidehiko Ichimura, Hiroyuki Kasahara, Toru Kitagawa, Brendan Kline, Zhongjian Lin, Oliver Linton, Essie Maasoumi, Taisuke Otsu, Aureo de Paula, Adam Rosen, Katsumi Shimotsu, and Haiqing Xu; seminar participants at UT Austin and SUNY Albany; and participants at the 2018 North American Econometric Society Meetings, the 2018 CEME Conference for Young Econometricians, and the Workshop on Advances in Econometrics 2018 for their insightful comments and suggestions. Yu gratefully acknowledges the support of JSPS KAKENHI Grant Numbers 19K13666 and 21K01419. All errors are ours.



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