Probit With An Endogenous Continuous Regressor

Why a Continuous Endogenous Regressor Changes Probit

A binary outcome is often modeled with a probit equation, but the usual probit interpretation relies on the included regressors being conditionally unrelated to the outcome disturbance. If a continuous regressor is endogenous, its values contain information about that disturbance. A probit equation that omits this dependence can therefore give misleading conditional probabilities and probability effects.

A control-function approach makes the dependence explicit. It models the endogenous regressor using an instrument and exogenous covariates, then includes the first-stage residual in the probit index. Under joint normality, this residual is a sufficient control for the dependence in the model described below. This is a distributional result, not a general recipe that works regardless of the error assumptions.

Model and Conditional-Normal Derivation

Write the first stage and latent outcome equation as

Here \(m(X,Z)\) is the conditional mean of \(D\) given the exogenous variables and instrument, \(V\) is the first-stage disturbance, and \(U\) is the outcome disturbance. The outcome is observed as \(Y=1\) when the latent index \(Y^*\) is positive. Assume that, conditional on \(X,Z\), \(U\) and \(V\) have mean zero, are jointly normal, and have constant variances and covariance. In particular, \(\operatorname{Var}(V)=\sigma_v^2>0\) and \(\operatorname{Cov}(U,V)=\sigma_{uv}\). Also assume \(\tau^2=\sigma_u^2-\sigma_{uv}^2/\sigma_v^2>0\), so the conditional variance of \(U\) given \(V\) is positive. The instrument is relevant for \(D\), is independent of the structural outcome disturbance conditional on \(X\), and is excluded from the structural outcome equation.

For jointly normal disturbances, the conditional mean of \(U\) given \(V=v\) is linear in \(v\). Define \(\kappa=\sigma_{uv}/\sigma_v^2\). Then

The conditional variance formula follows by subtracting the variance explained by the linear projection of \(U\) on \(V\). Joint normality also makes \(E\) independent of \(V\); under the stated conditional model, its conditional distribution does not depend on \(X,Z,V\). Substituting \(V=D-m(X,Z)\) into the outcome equation gives

A probit model normalizes the scale of its latent disturbance. Thus the coefficients estimated in the conditional probit are the scale-normalized coefficients \(b=\beta/\tau\), \(\theta=\delta/\tau\), and \(\rho=\kappa/\tau\), rather than the unscaled latent-equation coefficients. In practice, estimate the first stage, form the residual \(\hat V=D-\hat m(X,Z)\), and estimate a probit with \(X,D,\hat V\) as regressors. Its index is \(X'b+\theta D+\rho\hat V\).

Worked Examples

Interpreting the Residual Coefficient

The coefficient \(\rho\) on the first-stage residual captures conditional dependence between the outcome disturbance and the part of \(D\) not predicted by \(X,Z\), on the probit's normalized latent scale. Its sign indicates the direction of that dependence under this model. A positive \(\rho\) means that, conditional on the included covariates and \(D\), larger first-stage residuals are associated with a higher latent outcome index. A negative \(\rho\) indicates the opposite direction.

The residual coefficient is not a probability-point effect, and it is not the effect of the instrument on the outcome. Nor is it, by itself, proof of endogeneity: its interpretation depends on the joint-normal control-function specification and valid instruments. The coefficient \(\theta\) is the normalized structural coefficient on \(D\); the probability effect at a given index also depends on the normal density at that index.

Conditions, Estimation, and Common Pitfalls

The control-function probit requires more than a first-stage regression. Its interpretation depends on a correctly specified conditional mean for \(D\), instrument relevance, and an exclusion restriction: \(Z\) affects \(Y\) through \(D\), not directly through the structural outcome equation. The instrument must also be exogenous with respect to the structural outcome disturbance, conditional on \(X\). The derivation above additionally uses joint normality and constant conditional variances and covariance. If these assumptions fail, the residual need not capture the relevant dependence, and a linear residual term may be misspecified.

In estimation, the first-stage residual is estimated, not observed. Inference must therefore account for that estimation; treating \(\hat V\) as fixed in an ordinary second-stage probit generally gives incorrect standard errors. Use an inference method appropriate for estimated control functions, such as a suitable stacked-estimation variance calculation or bootstrap procedure. This issue affects uncertainty estimates even when the fitted two-step coefficients are the quantities of interest.

• Wrong: “The residual coefficient is the change in the probability for a one-unit increase in the residual.” Correction: It is a coefficient on the normalized latent index; probability changes depend on the index through \(\phi(\eta)\).
• Wrong: “Adding the residual makes the coefficient on \(D\) itself a probability effect.” Correction: A local probability effect also requires evaluating the normal density at the relevant index.
• Wrong: “A significant residual coefficient proves that the instrument is valid.” Correction: The residual coefficient concerns conditional dependence under the model; instrument validity requires separate substantive and statistical justification.
• Wrong: “The control function works because any first-stage residual removes endogeneity.” Correction: The scalar residual has this role here because of the specified conditional-normal structure and first stage.