{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T06:27:05Z","timestamp":1762064825106,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,2]],"date-time":"2022-09-02T00:00:00Z","timestamp":1662076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the National Sciences and Engineering Research Council of Canada","award":["RGPIN-2017-04064"],"award-info":[{"award-number":["RGPIN-2017-04064"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Analysis of instrumental variables is an effective approach to dealing with endogenous variables and unmeasured confounding issue in causal inference. We propose using the piecewise linear model to fit the relationship between the continuous instrumental variable and the continuous explanatory variable, as well as the relationship between the continuous explanatory variable and the outcome variable, which generalizes the traditional linear instrumental variable models. The two-stage least square and limited information maximum likelihood methods are used for the simultaneous estimation of the regression coefficients and the threshold parameters. Furthermore, we study the limiting distribution of the estimators in the correctly specified and misspecified models and provide a robust estimation of the variance-covariance matrix. We illustrate the finite sample properties of the estimation in terms of the Monte Carlo biases, standard errors, and coverage probabilities via the simulated data. Our proposed model is applied to an education-salary data, which investigates the causal effect of children\u2019s years of schooling on estimated hourly wage with father\u2019s years of schooling as the instrumental variable.<\/jats:p>","DOI":"10.3390\/e24091235","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T23:35:57Z","timestamp":1662420957000},"page":"1235","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Simultaneous Maximum Likelihood Estimation for Piecewise Linear Instrumental Variable Models"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8396-4515","authenticated-orcid":false,"given":"Shuo Shuo","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Statistics, The Pennsylvania State University, University Park, PA 16801, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yeying","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Sokolovska, N., and Wuillemin, P.H. 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