{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,19]],"date-time":"2026-09-19T14:37:17Z","timestamp":1789828637562,"version":"4.0.1"},"reference-count":57,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T00:00:00Z","timestamp":1649116800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["020M680225"],"award-info":[{"award-number":["020M680225"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["BX20200011"],"award-info":[{"award-number":["BX20200011"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["NSFC 11771028"],"award-info":[{"award-number":["NSFC 11771028"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates the problem of selecting instrumental variables relative to a target causal influence X\u2192Y from observational data generated by linear non-Gaussian acyclic causal models in the presence of unmeasured confounders. We propose a necessary condition for detecting variables that cannot serve as instrumental variables. Unlike many existing conditions for continuous variables, i.e., that at least two or more valid instrumental variables are present in the system, our condition is designed with a single instrumental variable. We then characterize the graphical implications of our condition in linear non-Gaussian acyclic causal models. Given that the existing graphical criteria for the instrument validity are not directly testable given observational data, we further show whether and how such graphical criteria can be checked by exploiting our condition. Finally, we develop a method to select the set of candidate instrumental variables given observational data. Experimental results on both synthetic and real-world data show the effectiveness of the proposed method.<\/jats:p>","DOI":"10.3390\/e24040512","type":"journal-article","created":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T11:26:07Z","timestamp":1649157967000},"page":"512","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Testability of Instrumental Variables in Linear Non-Gaussian Acyclic Causal Models"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7229-3955","authenticated-orcid":false,"given":"Feng","family":"Xie","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"},{"name":"School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yangbo","family":"He","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhi","family":"Geng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3839-5269","authenticated-orcid":false,"given":"Zhengming","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Computer, Guangdong University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ru","family":"Hou","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Peking University, Beijing 100871, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kun","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213, USA"},{"name":"Machine Learning Department, Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi 7909, United Arab Emirates"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,5]]},"reference":[{"key":"ref_1","unstructured":"Wright, P.G. 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