{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T14:13:42Z","timestamp":1791641622384,"version":"4.3.4"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2009,2,11]],"date-time":"2009-02-11T00:00:00Z","timestamp":1234310400000},"content-version":"unspecified","delay-in-days":4455,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[1996,12]]},"abstract":"<jats:p>This paper focuses on the estimation of an approximated function and its derivatives. Let us assume that the data-generating process can be described by a family of regression models <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S0266466600007143_inline1\"\/>, where a is a multi-index of differentiation such that D<jats:sub>\u03b1<\/jats:sub>\u03d5(x<jats:sub>i<\/jats:sub>) is the \u03b1th derivative of \u03d5(<jats:italic>x<jats:sub>i<\/jats:sub><\/jats:italic>) with respect to x<jats:sub>i<\/jats:sub>. The estimated model is characterized by a family D<jats:sup>\u03b1<\/jats:sup>f(X<jats:sub>i<\/jats:sub>|\u03b8), where D<jats:sup>\u03b1<\/jats:sup>f(X<jats:sub>i<\/jats:sub>|\u03b8) is the \u03b1th derivative of f(x<jats:sub>i<\/jats:sub>,|\u03b8) and \u03b8 is an unknown parameter. The model is in general misspecified; that is, there is no \u03b8 such that D<jats:sup>\u03b1<\/jats:sup>f(X<jats:sub>i<\/jats:sub>|6) is equal to D<jats:sup>\u03b1<\/jats:sup>\u03d5(X<jats:sub>i<\/jats:sub>). Three different problems are discussed. First, the asymptotic behavior of the seemingly unrelated regression estimator of \u03b8 is shown to achieve the best approximation, in the Sobolev norm sense, of \u03d5 by an element of (f(X<jats:sub>i<\/jats:sub>|\u03b8)|\u03b8 \u03b5 \u0398). Second, in the case of polynomial approximations, the expected derivatives of the limit of the estimated regression and of the true regression are proved to be equal if and only if the set of explanatory variables has a normal distribution. Third, different sets of \u03b1 are introduced, and the different limits of estimated regressions characterized by these sets are proved to be equal if and only if the explanatory variables have a normal distribution. This result leads to a specification test.<\/jats:p>","DOI":"10.1017\/s0266466600007143","type":"journal-article","created":{"date-parts":[[2009,3,10]],"date-time":"2009-03-10T15:38:09Z","timestamp":1236699489000},"page":"753-772","source":"Crossref","is-referenced-by-count":7,"title":["Sobolev Estimation of Approximate Regressions"],"prefix":"10.1017","volume":"12","author":[{"given":"Jean-Pierre","family":"Florens","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc","family":"Ivaldi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sophie","family":"Larribeau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2009,2,11]]},"reference":[{"key":"S0266466600007143_ref004","volume-title":"Probability","author":"Breiman","year":"1968"},{"key":"S0266466600007143_ref016","volume-title":"The Approximation of Functions","author":"Rice","year":"1964"},{"key":"S0266466600007143_ref017","doi-asserted-by":"publisher","DOI":"10.2307\/2526245"},{"key":"S0266466600007143_ref009","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-9892.1985.tb00395.x"},{"key":"S0266466600007143_ref001","volume-title":"Sobolev Spaces","author":"Adams","year":"1975"},{"key":"S0266466600007143_ref005","doi-asserted-by":"publisher","DOI":"10.2307\/2526125"},{"key":"S0266466600007143_ref002","volume-title":"Advanced Econometrics","author":"Amemiya","year":"1985"},{"key":"S0266466600007143_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/1912526"},{"key":"S0266466600007143_ref010","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316719"},{"key":"S0266466600007143_ref013","doi-asserted-by":"publisher","DOI":"10.1017\/CCOL0521382483"},{"key":"S0266466600007143_ref012","volume-title":"Statistique et Modeles Econometriques","author":"Gouri\u00e9rioux","year":"1989"},{"key":"S0266466600007143_ref014","doi-asserted-by":"publisher","DOI":"10.2307\/1913827"},{"key":"S0266466600007143_ref008","unstructured":"Florens J.P. & Larribeau S. (1993) Derivative Consistent Estimation. WP Gremaq 93.31.318, University of Toulouse I."},{"key":"S0266466600007143_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/S0266466600010975"},{"key":"S0266466600007143_ref007","doi-asserted-by":"publisher","DOI":"10.2307\/1911990"},{"key":"S0266466600007143_ref015","doi-asserted-by":"publisher","DOI":"10.1017\/CCOL0521344301.002"},{"key":"S0266466600007143_ref011","volume-title":"A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models","author":"Gallant","year":"1988"},{"key":"S0266466600007143_ref006","doi-asserted-by":"publisher","DOI":"10.2307\/2526127"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600007143","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T19:45:38Z","timestamp":1557690338000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600007143\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,12]]},"references-count":18,"journal-issue":{"issue":"5","published-print":{"date-parts":[[1996,12]]}},"alternative-id":["S0266466600007143"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600007143","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,12]]}}}