Bivariate Probit Model
Also known as: Bivariate Binary Probit, Joint Probit Model, Two-Equation Probit, İki Değişkenli Probit
The Bivariate Probit Model, introduced by Ashford and Sowden (1970), jointly estimates two binary outcome equations whose error terms are allowed to be correlated. By modeling both outcomes simultaneously under a bivariate normal distribution, it corrects for the dependence between decisions that separate probit regressions would ignore, producing consistent and efficient parameter estimates for researchers studying interrelated binary choices.
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When to use it
Use Bivariate Probit when you have two binary dependent variables that are plausibly driven by correlated unobservables and you want consistent, efficient joint estimates. It is appropriate when outcomes are genuinely binary (not ordered or multinomial), when sample sizes are large enough for bivariate normal integration, and when the error correlation ρ is expected to differ from zero. If ρ is statistically indistinguishable from zero, two separate probit models are sufficient. For endogenous binary regressors, the recursive Bivariate Probit variant applies. Multinomial Logit or Ordered Probit should be used when outcome categories exceed two.
Strengths & limitations
- Corrects for correlated unobservables across two binary equations, yielding consistent estimates that separate probits cannot provide.
- Full-information MLE is asymptotically efficient, using all information in the joint distribution of the outcomes.
- The correlation parameter ρ is directly interpretable and testable, providing a formal diagnostic for whether joint modeling is warranted.
- Extends naturally to endogenous binary regressors via the recursive bivariate probit specification.
- Relies on the bivariate normality assumption for the error terms; misspecification of this distribution can bias all parameter estimates.
- Numerical evaluation of the bivariate normal CDF adds computational complexity compared to univariate probit, particularly in large datasets.
- Identification in the recursive variant rests on functional-form assumptions unless valid exclusion restrictions are available.
- Marginal effects require averaging over the joint distribution and are less straightforward to compute and interpret than in univariate probit.
Frequently asked
How does Bivariate Probit differ from running two separate probit models?
Separate probit models ignore the correlation ρ between the error terms of the two equations. When ρ ≠ 0, separate estimation is inefficient and standard errors are incorrect. Bivariate Probit estimates both equations simultaneously under the bivariate normal distribution, producing consistent and efficient estimates. A likelihood-ratio test on ρ = 0 formally determines whether joint estimation is necessary.
What is the recursive Bivariate Probit and when should I use it?
The recursive variant allows one binary outcome to appear as a regressor in the other equation, addressing endogeneity of a binary treatment variable. It is used when a binary explanatory variable (e.g., program participation) is itself determined by an equation whose errors correlate with the outcome equation. Credible identification typically requires at least one valid exclusion restriction — a variable that affects treatment but not the outcome directly.
Can I use Bivariate Probit with more than two binary outcomes?
The standard Bivariate Probit is designed for exactly two binary outcomes. Extending to three or more outcomes requires the Multivariate Probit model, which integrates over a higher-dimensional normal distribution. Numerical simulation methods such as the GHK algorithm are typically needed for estimation when the number of outcomes exceeds two, as closed-form likelihoods become computationally prohibitive.
Sources
- Ashford, J. R., & Sowden, R. R. (1970). Multi-variate probit analysis. Biometrics, 26(3), 535–546. DOI: 10.2307/2529107 ↗
How to cite this page
ScholarGate. (2026, June 2). Bivariate Probit Model. ScholarGate. https://scholargate.app/en/econometrics/bivariate-probit
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
- Multinomial LogitEconometrics↔ compare
- Ordered LogitEconometrics↔ compare
- Probit ModelEconometrics↔ compare