{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,28]],"date-time":"2025-09-28T06:46:38Z","timestamp":1759041998633},"reference-count":49,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T00:00:00Z","timestamp":1622764800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2022,8]]},"abstract":"<jats:p>This study examines identification and estimation in a correlated random coefficients (CRC) model with an unknown transformation of the dependent variable, namely<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline1.png\"\/><jats:tex-math>$\\lambda \\left (Y^{*}\\right)=B_{0}+X^{\\prime }B$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where the latent outcome<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline2.png\"\/><jats:tex-math>$Y^{*}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>may be subject to a certain kind of censoring mechanism,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline3.png\"\/><jats:tex-math>$\\lambda (\\cdot)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is an unknown, one-to-one monotone function, and the random coefficients<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline4.png\"\/><jats:tex-math>$\\left (B_{0},B\\right)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>are allowed to be correlated with one or several components of<jats:italic>X<\/jats:italic>. Under a conditional median independence plus a conditional median zero restriction, the mean of<jats:italic>B<\/jats:italic>is shown to be identified up to scale. Moreover, we show the derivative of the median structural function (MSF) is point identified. This derivative of MSF resembles the marginal treatment effect introduced by Heckman and Vytlacil (2005,<jats:italic>Econometrica<\/jats:italic>73, 669\u2013738).<\/jats:p><jats:p>It generalizes the usual average treatment effect in a linear CRC model and coincides with<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline5.png\"\/><jats:tex-math>$E(B)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>when<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0266466621000281_inline6.png\"\/><jats:tex-math>$\\lambda $<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is equal to the identity function; it is invariant to both location and scale normalization on the coefficients. We develop estimators for the identified parameters and derive asymptotic properties for the derivative of MSF. An empirical example using the U.K. Family Expenditure Survey is provided.<\/jats:p>","DOI":"10.1017\/s0266466621000281","type":"journal-article","created":{"date-parts":[[2021,6,4]],"date-time":"2021-06-04T09:09:12Z","timestamp":1622797752000},"page":"621-688","source":"Crossref","is-referenced-by-count":2,"title":["IDENTIFICATION AND ESTIMATION IN A CORRELATED RANDOM COEFFICIENTS TRANSFORMATION MODEL"],"prefix":"10.1017","volume":"38","author":[{"given":"Zhengyu","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zequn","family":"Jin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Beili","family":"Mu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2021,6,4]]},"reference":[{"key":"S0266466621000281_r27","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.3982\/ECTA7108","article-title":"Identification and estimation of triangular simultaneous equations models without additivity","volume":"77","author":"Imbens","year":"2009","journal-title":"Econometrica"},{"key":"S0266466621000281_r10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jeconom.2009.09.020"},{"key":"S0266466621000281_r5","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1111\/j.2517-6161.1964.tb00553.x","article-title":"An analysis of transformations","volume":"26","author":"Box","year":"1964","journal-title":"Journal of the Royal Statistical Society"},{"key":"S0266466621000281_r24","doi-asserted-by":"publisher","DOI":"10.2307\/2951582"},{"key":"S0266466621000281_r49","volume-title":"Handbook of Econometrics","volume":"4","author":"Newey","year":"1994"},{"key":"S0266466621000281_r36","unstructured":"Moon, J.M. 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