{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T02:30:02Z","timestamp":1791426602148,"version":"4.3.3"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T00:00:00Z","timestamp":1287360000000},"content-version":"unspecified","delay-in-days":9331,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econ Theory"],"published-print":{"date-parts":[[1985,4]]},"abstract":"<jats:p>Starting from the conditional density of the instrumental variable (IV) estimator given the right-hand-side endogenous variables, we provide an alternative derivation of Phillips' result on the joint density of the IV estimator for the endogenous coefficients, and derive an expression for the marginal density of a linear combination of these coefficients. In addition, we extend Phillips' approximation to the joint density to 0(<jats:italic>T<\/jats:italic><jats:sup>\u22122<\/jats:sup>,) and show how this result can be used to improve the approximation to the marginal density. Explicit formulae are given for the special case of no simultaneity, and the case of an equation with just three endogenous variables. The classical assumptions of independent normal reduced-form errors are employed throughout.<\/jats:p>","DOI":"10.1017\/s0266466600010999","type":"journal-article","created":{"date-parts":[[2010,10,18]],"date-time":"2010-10-18T09:12:12Z","timestamp":1287393132000},"page":"53-72","source":"Crossref","is-referenced-by-count":21,"title":["On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation"],"prefix":"10.1017","volume":"1","author":[{"given":"Grant H.","family":"Hillier","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2010,10,18]]},"reference":[{"key":"S0266466600010999_ref019","doi-asserted-by":"publisher","DOI":"10.2307\/1912937"},{"key":"S0266466600010999_ref023","doi-asserted-by":"publisher","DOI":"10.2307\/1912516"},{"key":"S0266466600010999_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02480302"},{"key":"S0266466600010999_ref015","doi-asserted-by":"publisher","DOI":"10.2307\/1912027"},{"key":"S0266466600010999_ref009","doi-asserted-by":"publisher","DOI":"10.2307\/1911467"},{"key":"S0266466600010999_ref024","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1968.10480921"},{"key":"S0266466600010999_ref018","doi-asserted-by":"publisher","DOI":"10.2307\/2297109"},{"key":"S0266466600010999_ref025","doi-asserted-by":"publisher","DOI":"10.2307\/1913972"},{"key":"S0266466600010999_ref004","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177703863"},{"key":"S0266466600010999_ref022","doi-asserted-by":"publisher","DOI":"10.2307\/1912287"},{"key":"S0266466600010999_ref010","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177728846"},{"key":"S0266466600010999_ref011","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177705994"},{"key":"S0266466600010999_ref012","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177704980"},{"key":"S0266466600010999_ref014","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177703550"},{"key":"S0266466600010999_ref002","first-page":"53","volume-title":"Multivariate Analysis","author":"Chikuse","year":"1980"},{"key":"S0266466600010999_ref003","volume-title":"Some properties of the invariant polynomials with matrix arguments and their application in econometrics","author":"Chikuse","year":"1982"},{"key":"S0266466600010999_ref001","volume-title":"Frontiers of Quantitative Economics","author":"Basmann","year":"1974"},{"key":"S0266466600010999_ref016","doi-asserted-by":"publisher","DOI":"10.2307\/2526371"},{"key":"S0266466600010999_ref013","doi-asserted-by":"publisher","DOI":"10.2307\/1970291"},{"key":"S0266466600010999_ref017","doi-asserted-by":"publisher","DOI":"10.1016\/0047-259X(75)90046-9"},{"key":"S0266466600010999_ref020","volume-title":"In Handbook of Econometrics","author":"Phillips","year":"1983"},{"key":"S0266466600010999_ref006","first-page":"278","article-title":"Invariant polynomials with two matrix arguments extending the zonal polynomials","volume":"V","author":"Davis","year":"1980","journal-title":"Multivariate Analysis"},{"key":"S0266466600010999_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02480943"},{"key":"S0266466600010999_ref021","unstructured":"Phillips, P.C.B. (1983). Marginal densities of instrumental variable estimators in the general single equation case. Advances in Econometrics (forthcoming)."},{"key":"S0266466600010999_ref008","doi-asserted-by":"publisher","DOI":"10.2307\/1969810"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466600010999","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T05:01:53Z","timestamp":1636434113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466600010999\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,4]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,4]]}},"alternative-id":["S0266466600010999"],"URL":"https:\/\/doi.org\/10.1017\/s0266466600010999","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,4]]}}}