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Bayesian Versus Maximum Likelihood Estimation of Treatment Effects in Bivariate Probit Instrumental Variable Models

  • Florian M. Hollenbach, Jacob M. Montgomery and Adriana Crespo-Tenorio

Bivariate probit models are a common choice for scholars wishing to estimate causal effects in instrumental variable models where both the treatment and outcome are binary. However, standard maximum likelihood approaches for estimating bivariate probit models are problematic. Numerical routines in popular software suites frequently generate inaccurate parameter estimates and even estimated correctly, maximum likelihood routines provide no straightforward way to produce estimates of uncertainty for causal quantities of interest. In this note, we show that adopting a Bayesian approach provides more accurate estimates of key parameters and facilitates the direct calculation of causal quantities along with their attendant measures of uncertainty.

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Florian M. Hollenbach is an Assistant Professor in the Department of Political Science, Texas A&M University, 2010 Allen Building, 4348 TAMU, College Station, TX 77843-4348 ( Jacob M. Montgomery, Associate Professor, Department of Political Science, Washington University in St. Louis, Campus Box 1063, One Brookings Drive, St. Louis, MO 63130 ( Adriana Crespo-Tenorio, Lead Researcher, Facebook, 1 Hacker Way, Menlo Park, CA 94025 ( Previous versions of this paper were presented at the 2013 Annual Meeting of the Midwest Political Science Association in Chicago, the 2013 Summer Meeting of the Society of Political Methodology at the University of Virginia, and the 2016 Annual Meeting of the Southern Political Science Association in San Juan, Puerto Rico. Portions of this research were conducted with high performance research computing resources provided by Texas A&M University ( The authors are grateful for helpful comments from Kosuke Imai, Kevin Quinn, Marc Ratkovic, Justin Esarey, and a helpful audience at Washington University in St. Louis. To view supplementary material for this article, please visit

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Angrist, Joshua D., Imbens, Guido W., and Rubin, Donald B.. 1996. ‘Identification of Causal Effects Using Instrumental Variables’. Journal of the American Statistical Association 91(434):444455.
Angrist, Joshua D., and Pischke, Jörn-Steffen. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton, NJ: Princeton University Press.
Arceneaux, Kevin, and Nickerson, David W.. 2009. ‘Who is Mobilized to Vote? A Re-Analysis of 11 Field Experiments’. American Journal of Political Science 53(1):116.
Barnett, W. Steven. 2010. ‘Benefit-Cost Analysis of the Perry Preschool Program and its Policy Implications’. Educational Evaluation and Policy Analysis 7(4):333342.
Berinsky, Adam J., and Lenz, Gabriel S.. 2011. ‘Education and Political Participation: Exploring the Causal Link’. Political Behavior 33(3):357373.
Cederman, Lars-Erik, Hug, Simon, Schadel, Andreas, and Wucherpfennig, Julian. 2015. ‘Territorial Autonomy in the Shadow of Conflict: Too Little, Too Late?American Political Science Review 109(2):354370.
Chib, Siddhartha. 2003. ‘On Inferring Effects of Binary Treatments with Unobserved Confounders’. In José M. Bernardo, M.J. Bayarri, A. Philip Dawid, James O. Berger, David Heckerman, Adrian F.M. Smith and Mike West (eds), Bayesian Statistics, vol. 7 6584. Oxford: Oxford University Press.
Chib, Siddhartha, and Greenberg, Edward. 2007. ‘Semiparametric Modeling and Estimation of Instrumental Variable Models’. Journal of Computational and Graphical Statistics 16(1):86114.
Chib, Siddhartha, Greenberg, Edward, and Jeliazkov, Ivan. 2009. ‘Estimation of Semiparametric Models in the Presence of Endogeneity and Sample Selection’. Journal of Computational and Graphical Statistics 18(2):321348.
Freedman, David A. and Sekhon, Jasjeet S.. 2010. ‘Endogeneity in Probit Response Models’. Political Analysis 18(2):138150.
Fuhrmann, Matthew, and Sechser, Todd S.. 2014. ‘Signaling Alliance Commitments: Hand-Tying and Sunk Costs in Extended Nuclear Deterrence’. American Journal of Political Science 58(4):919935.
Gerber, Alan S., and Green, Donald P.. 2000. ‘The Effects of Canvassing, Telephone Calls, and Direct Mail on Voter Turnout: A Field Experiment’. American Political Science Review 94(3):653663.
Hanushek, Eric A. 1999. ‘Some Findings From an Independent Investigation of the Tennessee STAR Experiment and From Other Investigations of class size effects’. Educational Evaluation and Policy Analysis 21(2):143163.
Heckman, James J., Moon, Seong Hyeok, Pinto, Rodrigo, Savelyev, Peter A., and Yavitz, Adam. 2010. ‘The Rate of Return to the HighScope Perry Preschool Program’. Journal of Public Economics 94(1):114128.
Hoffman, Matthew D., and Gelman, Andrew. 2014. ‘The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo’. The Journal of Machine Learning Research 15(1):15931623.
Kahne, Joseph, and Bailey, Kim. 1999. ‘The Role of Social Capital in Youth Development: The Case of “I Have a Dream” Programs’. Educational Evaluation and Policy Analysis 21(3):321343.
Maves, Jessica, and Braithwaite, Alex. 2013. ‘Autocratic Institutions and Civil Conflict Contagion’. Journal of Politics 75(2):478490.
Rosenbaum, Paul R., and Rubin, Donald B.. 1983. ‘Assessing Sensitivity to an Unobserved Binary Covariate in an Observational Study with Binary Outcome’. Journal of the Royal Statistical Society. Series B (Methodological) 45(2):212218.
Rubin, Donald B. 1974. ‘Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies’. Journal of Educational Psychology 66(5):688701.
Sondheimer, Rachel Milstein, and Green, Donald P.. 2010. ‘Using Experiments to Estimate the Effects of Education on Voter Turnout’. American Journal of Political Science 54(1):174189.
Sovey, Allison J., and Green, Donald P.. 2010. ‘Instrumental Variables Estimation in Political Science: A Readers’ Guide’. American Journal of Political Science 55(1):188200.
Wucherpfennig, Julian, Hunziker, Philipp, and Cederman, Lars-Erik. 2016. ‘Who Inherits the State? Colonial Rule and Postcolonial Conflict’. American Journal of Political Science 60(4):882898.
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Political Science Research and Methods
  • ISSN: 2049-8470
  • EISSN: 2049-8489
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