Tractable Bayesian Inference For An Unidentified Simple Linear Regression Model

References

• Ashley, R. A., and Parmeter, C. F. (2015), “When Is It Justifiable to Ignore Explanatory Variable Endogeneity in a Regression Model?” Economics Letters, 137, 70–74. DOI: 10.1016/j.econlet.2015.09.029.
   Web of Science ®Google Scholar
• Berger, J. O., and Sun, D. (2008), “Objective Priors for the Bivariate Normal Model,” The Annals of Statistics, 36, 963–982. DOI: 10.1214/07-AOS501.
   Web of Science ®Google Scholar
• Burstyn, I., Barone-Adesi, F., De Vocht, F., and Gustafson, P. (2019), “What to Do When Accumulated Exposure Affects Health but Only Its Duration Was Measured? A Case of Linear Regression,” International Journal of Environmental Research and Public Health, 16, 1896. DOI: 10.3390/ijerph16111896.
   PubMed Web of Science ®Google Scholar
• Chan, J. C., and Tobias, J. L. (2015), “Priors and Posterior Computation in Linear Endogenous Variable Models with Imperfect Instruments,” Journal of Applied Econometrics, 30, 650–674. DOI: 10.1002/jae.2390.
   Web of Science ®Google Scholar
• Cinelli, C., and Hazlett, C. (2020), “Making Sense of Sensitivity: Extending Omitted Variable Bias,” Journal of the Royal Statistical Society, Series B, 82, 39–67. DOI: 10.1111/rssb.12348.
   Google Scholar
• Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B., and Wynder, E. L. (1959), “Smoking and Lung Cancer: Recent Evidence and a Fiscussion of Some Questions,” Journal of the National Cancer Institute, 22, 173–203.
   PubMed Web of Science ®Google Scholar
• Crooks, G. E. (2019), “Field Guide to Continuous Probability Distributions,” Technical Report, Berkeley Institute for Theoretical Sciences. Available at http://threeplusone.com/fieldguide
   Google Scholar
• DiTraglia, F. J., and Garcia-Jimeno, C. (2021), “A Framework for Eliciting, Incorporating, and Disciplining Identification Beliefs in Linear Models,” Journal of Business & Economic Statistics, 39, 1038–1053. DOI: 10.1080/07350015.2020.1753528.
   Web of Science ®Google Scholar
• Dubs, H. H. (1942), “The Principle of Insufficient Reason,” Philosophy of Science, 9, 123–131. DOI: 10.1086/286754.
   Google Scholar
• Florens, J., Mouchart, M., and Rolin, J. (1985), “On Two Definitions of Identification,” Statistics: A Journal of Theoretical and Applied Statistics, 16, 213–218.
   Google Scholar
• Florens, J., Mouchart, M., and Rolin, J. (1990), Elements of Bayesian Statistics, Monographs and Textbooks in Pure and Applied Mathematics, Boca Raton, FL: CRC Press.
   Google Scholar
• Frank, K. A. (2000), “Impact of a Confounding Variable on a Regression Coefficient,” Sociological Methods & Research, 29, 147–194. DOI: 10.1177/0049124100029002001.
   Web of Science ®Google Scholar
• Frank, R. W., and Martínez i Coma, F. (2023), “Correlates of Voter Turnout,” Political Behavior, 45, 607–633. DOI: 10.1007/s11109-021-09720-y.
   Web of Science ®Google Scholar
• Gelman, A. (2006), “Prior Distributions for Variance Parameters in Hierarchical Models,” comment on article by Browne and Draper, Bayesian Analysis, 1, 515–534. DOI: 10.1214/06-BA117A.
   Web of Science ®Google Scholar
• Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian Data Analysis, Boca Raton, FL: Chapman and Hall/CRC. Available at http://www.stat.columbia.edu/∼gelman/book/
   Google Scholar
• Greenland, S. (2005), “Multiple-Bias Modelling for Analysis of Observational Data,” Journal of the Royal Statistical Society, Series A, 168, 267–306. DOI: 10.1111/j.1467-985X.2004.00349.x.
   Web of Science ®Google Scholar
• ——(2021), “Invited Commentary: Dealing with the Inevitable Deficiencies of Bias Analysis - and All Analyses,” American Journal of Epidemiology, 190, 1617–1621.
   PubMed Web of Science ®Google Scholar
• Gustafson, P. (2005), “On Model Expansion, Model Contraction, Identifiability and Prior Information: Two Illustrative Scenarios Involving Mismeasured Variables,” Statistical Science, 20, 111–140. DOI: 10.1214/088342305000000098.
   Web of Science ®Google Scholar
• ——(2009), “What are the Limits of Posterior Distributions Arising from Nonidentified Models, and Why Should We Care?” Journal of the American Statistical Association, 104, 1682–1695.
   Web of Science ®Google Scholar
• ——(2010), “Bayesian Inference for Partially Identified Models,” The International Journal of Biostatistics, 6, Article 17.
   Google Scholar
• ——(2012), “On the Behaviour of Bayesian Credible Intervals in Partially Identified Models,” Electronic Journal of Statistics, 6, 2107–2124.
   Web of Science ®Google Scholar
• ——(2014), “Bayesian Inference in Partially Identified Models: Is the Shape of the Posterior Distribution Useful?,” Electronic Journal of Statistics, 8, 476–496.
   Web of Science ®Google Scholar
• ——(2015), Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data (Vol. 140), Boca Raton, FL: CRC Press.
   Google Scholar
• Gustafson, P., McCandless, L., Levy, A., and Richardson, S. (2010), “Simplified Bayesian Sensitivity Analysis for Mismeasured and Unobserved Confounders,” Biometrics, 66, 1129–1137. DOI: 10.1111/j.1541-0420.2009.01377.x.
   PubMed Web of Science ®Google Scholar
• Gustafson, P., and McCandless, L. C. (2018), “When is a Sensitivity Parameter Exactly That?” Statistical Science, 33, 86–95. DOI: 10.1214/17-STS632.
   Web of Science ®Google Scholar
• Hosman, C. A., Hansen, B. B., Holland, P. W., et al. (2010), “The Sensitivity of Linear Regression Coefficients’ Confidence Limits to the Omission of a Confounder,” The Annals of Applied Statistics, 4, 849–870. DOI: 10.1214/09-AOAS315.
   Web of Science ®Google Scholar
• Imbens, G. W. (2003), “Sensitivity to Exogeneity Assumptions in Program Evaluation,” American Economic Review, 93, 126–132. DOI: 10.1257/000282803321946921.
   Web of Science ®Google Scholar
• Jeffreys, H. (1961), Theory of Probability (3rd ed.), New York: The Clarendon Press, Oxford University Press.
   Google Scholar
• Kadane, J. (1974), “The Role of Identification in Bayesian Theory,” in Bayesian Econometrics and Statistics, eds. S. Fienberg, and A. Zellner, Amsterdam: North Holland.
   Google Scholar
• Kiviet, J. F. (2016), “When Is It Really Justifiable to Ignore Explanatory Variable Endogeneity in a Regression Model?” Economics Letters, 145, 192–195. DOI: 10.1016/j.econlet.2016.06.021.
   Web of Science ®Google Scholar
• Kraay, A. (2012), “Instrumental Variables Regressions with Uncertain Exclusion Restrictions: A Bayesian Approach,” Journal of Applied Econometrics, 27, 108–128. DOI: 10.1002/jae.1148.
   Web of Science ®Google Scholar
• Leamer, E. (1974), “False Models and Post-Data Model Construction,” Journal of the American Statistical Association, 69, 122–131. DOI: 10.1080/01621459.1974.10480138.
   Web of Science ®Google Scholar
• Leamer, E. E. (1983), “Let’s Take the Con Out of Econometrics,” The American Economic Review, 73, 31–43.
   Web of Science ®Google Scholar
• Lindley, D. (1965), Introduction to Probability and Statistics from a Bayesian Viewpoint (Part 2: Inference), Cambridge, UK: Cambridge University Press.
   Google Scholar
• Ly, A., Marsman, M., and Wagenmakers, E.-J. (2018), “Analytic Posteriors for Pearson’s Correlation Coefficient,” Statistica Neerlandica, 72, 4–13. DOI: 10.1111/stan.12111.
   PubMed Web of Science ®Google Scholar
• McCandless, L. C., Gustafson, P., and Levy, A. (2007), “Bayesian Sensitivity Analysis for Unmeasured Confounding in Observational Studies,” Statistics in Medicine, 26, 2331–2347. DOI: 10.1002/sim.2711.
   PubMed Web of Science ®Google Scholar
• McCandless, L. C., Gustafson, P., and Levy, A. (2008), “A Sensitivity Analysis Using Information about Measured Confounders Yelded Improved Uncertainty Assessments for Unmeasured Confounding,” Journal of Clinical Epidemiology, 61, 247–255. DOI: 10.1016/j.jclinepi.2007.05.006.
   PubMed Web of Science ®Google Scholar
• Moon, H. R., and Schorfheide, F. (2012), “Bayesian and Frequentist Inference in Partially Identified Models,” Econometrica, 80, 755–782.
   Web of Science ®Google Scholar
• Novack, G. (2010), “A Defense of the Principle of Indifference,” Journal of Philosophical Logic, 39, 655–678. DOI: 10.1007/s10992-010-9147-1.
   Web of Science ®Google Scholar
• Oster, E. (2019), “Unobservable Selection and Coefficient Stability: Theory and Evidence,” Journal of Business & Economic Statistics, 37, 187–204. DOI: 10.1080/07350015.2016.1227711.
   Web of Science ®Google Scholar
• Pogány, T. K., and Nadarajah, S. (2013), “On the Convolution of Normal and t Random Variables,” Statistics, 47, 1363–1369. DOI: 10.1080/02331888.2012.694447.
   Web of Science ®Google Scholar
• Poirier, D. (1998), “Revising Beliefs in Nonidentified Models,” Econometric Theory, 14, 483–509. DOI: 10.1017/S0266466698144043.
   Web of Science ®Google Scholar
• Rosenbaum, P. R., and Rubin, D. B. (1983), “Assessing Sensitivity to an Unobserved Binary Covariate in an Observational Study with Binary Outcome,” Journal of the Royal Statistical Society, Series B, 45, 212–218. DOI: 10.1111/j.2517-6161.1983.tb01242.x.
   Google Scholar
• Tufte, E. (2006), The Cognitive Style of PowerPoint: Pitching Out Corrupts Within, Cheshire, CT: Graphics Press LLC.
   Google Scholar
• Wechsler, S., Izbicki, R., and Esteves, L. G. (2013), “A Bayesian Look at Nonidentifiability: A Simple Example,” The American Statistician, 67, 90–93. DOI: 10.1080/00031305.2013.778787.
   Web of Science ®Google Scholar
• Wooldridge, J. M. (2010), Econometric Analysis of Cross Section and Panel Data, Cambridge, MA: MIT Press.
   Google Scholar
• Xia, M., and Gustafson, P. (2016), “Bayesian Regression Models Adjusting for Unidirectional Covariate Misclassification,” Canadian Journal of Statistics, 44, 198–218. DOI: 10.1002/cjs.11284.
   Web of Science ®Google Scholar