What Identification Requires
A bivariate probit model for a binary outcome and an endogenous binary treatment uses two linked latent-variable equations. Identification asks whether the distribution of the observed variables uniquely determines the model parameters under the stated assumptions. It is not the same as estimation: a numerical algorithm can return estimates even when the data and assumptions do not pin down a unique answer.
Let \(Y\) be the binary outcome, \(D\) the binary treatment, \(X\) a vector of observed covariates, and \(Z\) one or more instruments excluded from the outcome equation. The recursive model is
The scale of each latent error is normalized to one, because the binary observations reveal the sign of each latent variable, not its scale. The usual bivariate probit specification assumes that, conditional on \(X,Z\), \((\varepsilon,u)\) has a standard bivariate normal distribution with correlation \(\rho\). Thus both marginal error variances equal one. A nonzero \(\rho\) allows unobserved causes of treatment and outcome to be related; that is the model's representation of endogeneity.
A causal interpretation requires more than fitting these equations. Conditional on \(X\), the instrument must be independent of the unobserved determinants represented by \(\varepsilon\) and \(u\), and it must have no direct effect on \(Y\) except through \(D\). The structural equations and the specified conditional error distribution must also be credible. Instrument relevance means that changing \(Z\) changes the treatment distribution for at least some covariate values. None of these conditions follows merely from observing a nonzero fitted coefficient.
How the Exclusion Creates Identifying Variation
The key contribution of an excluded instrument is a change in treatment propensity while the outcome index at a fixed treatment value stays unchanged. To see how the model connects these changes to observed data, first hold \(X\) fixed and write \(a=X\beta\) and \(q=X\gamma+Z\pi\). Let \(\Phi\) denote the standard normal cumulative distribution function and \(\Phi_2(\cdot,\cdot;\rho)\) the standard bivariate normal cumulative distribution function with correlation \(\rho\).
Treatment occurs when \(u>-q\), and the outcome is one when \(\varepsilon>-(a+\tau D)\). By symmetry of the bivariate normal distribution, the probability of observing both treatment and outcome is
For untreated individuals, the outcome threshold is instead \(-a\). Reversing the signs of both errors preserves their correlation, giving
The treatment probability itself is \(P(D=1\mid X,Z)=\Phi(q)\). These equations show why an excluded instrument can help: \(Z\) changes \(q\), and therefore the treatment probability, while leaving \(a\) and the outcome threshold at either treatment value unchanged. Across instrument values, the model must account jointly for the treatment frequencies and the outcome frequencies within treatment groups using the same \(\tau\) and \(\rho\).
A short derivative makes the source of this information visible. The derivative of the joint treated-and-outcome probability with respect to the treatment index is
where \(\phi\) is the standard normal density. For finite \(q\), finite \(a+\tau\), and \(|\rho|<1\), both factors on the right are positive. Thus a change in \(q\) changes this joint probability. The pattern of changes across the observed joint cells helps distinguish treatment effects from selection through correlated latent errors. The derivative is an explanation of the model's curvature, not a guarantee that a particular data set contains enough information for precise identification.
Worked Example: An Excluded Instrument Shifts Treatment Propensity
Suppose, hypothetically, that \(X\) is fixed and a binary instrument \(Z\) has no direct outcome effect. The treatment index is \(q=-0.5244+0.7777Z\). Calculate the model-implied treatment probabilities at each instrument value and describe the identifying variation.
When \(Z=0\), \(q=-0.5244\), so
When \(Z=1\), \(q=-0.5244+0.7777=0.2533\), and
The treatment-probability difference is \(0.6000-0.3000=0.3000\), or 30 percentage points. Under the exclusion assumption, this shift changes the mix of treated and untreated people without changing the outcome index directly. The model can use the associated differences in observed joint outcome frequencies to learn about the parameters governing treatment response and latent-error correlation.
These numbers establish relevance within the hypothetical model, not instrument validity in an application. A researcher would need substantive grounds for the claim that \(Z\) is unrelated to the latent outcome determinants conditional on \(X\), and that it affects \(Y\) only through \(D\). The calculated treatment shift cannot verify those claims.
Why Functional Form Alone Can Be Fragile
If there is no excluded instrument, bivariate probit identification may still be possible in some designs. It then depends on the parametric restrictions—especially the probit link and the assumed bivariate normal error distribution—and on enough covariate variation to distinguish the model parameters. This is often called identification by functional form. It is not a general substitute for an exclusion restriction: whether it identifies the model depends on the covariates and their support, and even when it does, the information can be weak and sensitive to misspecification.
A simple parameter count illustrates why an exclusion can matter. With no covariates and no excluded instrument, the treatment equation has an intercept, the outcome equation has an intercept and treatment coefficient, and the error correlation is unknown. The observed pair \((D,Y)\) has only four cell probabilities that sum to one, hence only three independent probabilities. Four unknown parameters cannot generally be uniquely recovered from three independent quantities. Adding covariates or imposing further restrictions changes the problem, but a parameter count alone does not establish identification.
Worked Example: An Absent Exclusion Leaves Too Little Information
Consider a hypothetical sample with no varying covariates and no excluded instrument. Its observed table is:
| Y=1 | Y=0 | Total | |
|---|---|---|---|
| D=1 | 50 | 30 | 80 |
| D=0 | 40 | 80 | 120 |
| Total | 90 | 110 | 200 |
The four empirical cell proportions are \(50/200=0.2500\), \(30/200=0.1500\), \(40/200=0.2000\), and \(80/200=0.4000\). Since they sum to one, only three are independent. Yet the constant-index bivariate probit has four unknown parameters: the treatment-equation intercept, the outcome-equation intercept, \(\tau\), and \(\rho\). For example, the treatment proportion \(80/200=0.4000\) informs the treatment intercept, but the remaining outcome information must distinguish the outcome intercept, treatment effect, and error correlation. There are only two independent cell probabilities left to do so.
Thus this design does not generally identify all four parameters without additional information or restrictions. The count is a warning, not a complete identification proof: in richer designs, identification depends on the exact mapping from parameters to probabilities. But no optimizer can create the missing independent information. Reporting a converged estimate from this design would not resolve the identification problem.
With covariate variation but no exclusion, the nonlinear probability relationships can provide additional identifying restrictions. Those restrictions come from the chosen functional form, however. If the probit link or normal dependence structure is wrong, apparent identification can be an artifact of the assumptions. In contrast, a credible excluded instrument supplies a distinct source of variation, although it does not eliminate the need for a correctly specified model.
Weak Exclusions, Diagnostics, and Alternatives
An excluded instrument can be valid but weak. If its effect on the treatment probability is very small over the observed covariate range, the joint probabilities change little as \(Z\) varies. Then quite different combinations of \(\tau\) and \(\rho\) may fit the observed distribution similarly well. Identification may exist in the mathematical sense while the data provide little precision. In finite samples this can appear as large standard errors, unstable estimates, sensitivity to modest specification changes, or a nearly flat likelihood over a range of parameter values.
Worked Example: A Small First-Stage Shift
Suppose, hypothetically, that two equally sized groups of 1,000 observations have treatment proportions \(520/1{,}000=0.5200\) and \(550/1{,}000=0.5500\) for \(Z=0\) and \(Z=1\), respectively. The estimated difference is \(0.5500-0.5200=0.0300\), or 3 percentage points.
Using the usual approximate standard error for the difference in two independent proportions,
The estimated difference is \(0.0300/0.0223\approx1.35\) standard errors from zero. A rough 95% interval is \(0.0300\pm1.96(0.0223)\), or approximately \([-0.0137,0.0737]\). This indicates that the observed treatment shift is small and imprecisely measured in this example; it does not establish that the instrument is invalid. Nor is this simple difference-in-proportions calculation a complete weak-instrument analysis for a bivariate probit model. It illustrates why little treatment movement across instrument values can leave the nonlinear model weakly informed.
Useful checks include comparing treatment probabilities across instrument values conditional on \(X\), checking that relevant instrument values have adequate representation, and examining whether estimates are highly sensitive to plausible changes in the covariates or model specification. A flat or irregular likelihood, large uncertainty, or convergence problems can signal weak information or misspecification; convergence by itself is not evidence of identification. Instrument relevance can be assessed from observed treatment variation, but exclusion and conditional independence principally require design knowledge and substantive argument.
The main alternatives make different trade-offs. A linear instrumental-variables analysis can rely on different functional assumptions and targets a different model scale; it does not inherit bivariate probit identification automatically. Control-function approaches model endogeneity through a generated control and have their own assumptions. A bivariate probit specification uses a joint latent-normal model and can be efficient when that structure is credible, but its conclusions may be sensitive to distributional assumptions when exclusions are weak or absent. These approaches are not interchangeable merely because each uses an instrument.
Common Mistakes
- Equating exclusion with relevance. An instrument can have no direct outcome effect and still barely change treatment. Validity and strength are distinct requirements.
- Treating functional-form identification as assumption-free. Identification based on probit curvature depends on the specified link, error distribution, and covariate support.
- Assuming a converged fit proves identification. Numerical convergence says the algorithm stopped; it does not show that the observed distribution uniquely determines the parameters.
- Claiming the data prove exclusion. A strong treatment association does not rule out a direct path to the outcome or dependence on unobserved outcome determinants.
Check Your Understanding
Answer each question, then compare with the solutions.
- In the bivariate probit model, which equation contains an excluded instrument, and what does the exclusion restriction say?
- With fixed \(X\), suppose the treatment index is \(q=-0.2533+0.7777Z\), where \(Z\) is binary. Calculate the treatment probabilities for \(Z=0\) and \(Z=1\), to three significant figures.
- Why does a weak instrument make it difficult to distinguish the treatment effect from selection represented by \(\rho\)?
- Does a statistically clear treatment shift establish that an instrument has no direct outcome effect? Explain.
Solutions. (1) The instrument enters the treatment equation \(D^*=X\gamma+Z\pi+u\), but not the outcome equation \(Y^*=X\beta+\tau D+\varepsilon\). Exclusion says it has no direct outcome effect after accounting for treatment and covariates. (2) For \(Z=0\), \(q=-0.2533\), so \(\Phi(q)\approx0.400\). For \(Z=1\), \(q=-0.2533+0.7777=0.5244\), so \(\Phi(q)\approx0.700\). (3) A weak instrument produces little change in treatment probability and therefore little change in the observed joint outcome probabilities across instrument values. A range of treatment-effect and error-correlation values may then fit those probabilities similarly well. (4) No. The shift supports relevance, not exclusion. The claim of no direct effect requires substantive justification and cannot be established from the first-stage association alone.