{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T21:57:17Z","timestamp":1768427837781,"version":"3.49.0"},"reference-count":42,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2016,2,15]],"date-time":"2016-02-15T00:00:00Z","timestamp":1455494400000},"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":[[2017,2]]},"abstract":"<jats:p>We study the consequences of substituting an error-laden proxy<jats:italic>W<\/jats:italic>for an instrument<jats:italic>Z<\/jats:italic>on the interpretation of Wald, local instrumental variable (LIV), and instrumental variable (IV) estimands in an ordered discrete choice structural system with heterogeneity. A proxy<jats:italic>W<\/jats:italic>need only satisfy an exclusion restriction and that the treatment and outcome are mean independent from<jats:italic>W<\/jats:italic>given<jats:italic>Z<\/jats:italic>. Unlike<jats:italic>Z<\/jats:italic>,<jats:italic>W<\/jats:italic>need not satisfy monotonicity and may, under particular specifications, fail exogeneity. For example,<jats:italic>W<\/jats:italic>could code<jats:italic>Z<\/jats:italic>with error, with missing observations, or coarsely. We show that Wald, LIV, and IV estimands using<jats:italic>W<\/jats:italic>identify weighted averages of local or marginal treatment effects (LATEs or MTEs). We study a necessary and sufficient condition for nonnegative weights. Further, we study a condition under which the Wald or LIV estimand using<jats:italic>W<\/jats:italic>identifies the same LATE or MTE that would have been recovered had<jats:italic>Z<\/jats:italic>been observed. For example, this holds for binary<jats:italic>Z<\/jats:italic>and therefore the Wald estimand using<jats:italic>W<\/jats:italic>identifies the same \u201caverage causal response,\u201d or LATE for binary treatment, that would have been recovered using<jats:italic>Z<\/jats:italic>. Also, under this condition, LIV using<jats:italic>W<\/jats:italic>can be used to identify MTE and average treatment effects for e.g., the population, treated, and untreated.<\/jats:p>","DOI":"10.1017\/s0266466615000390","type":"journal-article","created":{"date-parts":[[2016,2,15]],"date-time":"2016-02-15T09:48:11Z","timestamp":1455529691000},"page":"69-104","source":"Crossref","is-referenced-by-count":15,"title":["INSTRUMENTAL VARIABLES METHODS WITH HETEROGENEITY AND MISMEASURED INSTRUMENTS"],"prefix":"10.1017","volume":"33","author":[{"given":"Karim","family":"Chalak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2016,2,15]]},"reference":[{"key":"S0266466615000390_ref12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.1540-5982.2010.01622.x","article-title":"An extended class of instrumental variables for the estimation of causal 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