Workshop: A Control Function For Binary Choice

Why Use a Control Function in Binary Choice?

Suppose an outcome is binary, but one of its predictors is chosen or determined together with unobserved factors that also influence the outcome. A probit model that treats this predictor as exogenous can then give misleading estimates. A control function addresses this problem by using a first-stage residual to represent the part of the endogenous predictor associated with those unobserved factors.

This workshop develops the procedure for a continuous endogenous regressor in a recursive binary-choice model. The key steps are to estimate the first stage, include its residual in the probit specification, calculate the average probability effect of the regressor, and test whether the residual term contributes to the outcome equation. The procedure relies on specific assumptions: in particular, a correctly specified first stage and a conditional distribution that makes the residual a sufficient control for the dependence at issue.

Model and assumptions: Let $y_i$ be a binary outcome and $d_i$ a continuous endogenous regressor. Let $x_i$ denote exogenous outcome covariates, and let $z_i$ denote excluded instruments. A recursive model is $$ y_i=1[y_i^*>0], \qquad y_i^*=x_i'\beta+\gamma d_i+u_i, $$ $$ d_i=m(x_i,z_i)+v_i. $$ Assume the conditional mean $m(x_i,z_i)$ and the first-stage specification are correct; the instruments are exogenous and relevant; and, conditional on $(x_i,z_i)$, $(u_i,v_i)$ is jointly normal with mean zero, $\operatorname{Var}(u_i)=1$, $\operatorname{Var}(v_i)=\sigma_v^2>0$, and correlation $\rho$. The variance normalization for $u_i$ is the usual probit scale normalization. Observations are assumed independent for the simple inference described here.

First Stage and the Residual-Inclusion Probit

Estimate the conditional mean of $d_i$ using the exogenous variables $x_i$ and instruments $z_i$. In a linear first stage, this can be written as $d_i=x_i'\delta+z_i'\pi+v_i$. For each observation, form the fitted value $\hat d_i$ and residual $$ \hat v_i=d_i-\hat d_i. $$ The residual measures how far the observed value of the endogenous regressor lies from its fitted conditional mean. It is not itself an instrument; it is included as a control in the outcome equation.

The reason this works under the stated distributional assumption follows from the conditional normal distribution. If $(u_i,v_i)$ are jointly normal, then $$ u_i=\frac{\rho}{\sigma_v}v_i+\varepsilon_i, \qquad \varepsilon_i\mid (x_i,z_i,v_i)\sim N(0,1-\rho^2). $$ The remaining error $\varepsilon_i$ is independent of $v_i$ conditional on the exogenous variables. Replacing the unobserved $v_i$ by its first-stage residual therefore yields a conditional probit specification. Since the probit error has variance $1-\rho^2$, its index is divided by $\sqrt{1-\rho^2}$. In practice, estimating the coefficients on $x_i$, $d_i$, and $\hat v_i$ in a probit absorbs this common scale into the coefficients.

Control-function probit: Estimate the probit $$ P(y_i=1\mid x_i,d_i,\hat v_i) =\Phi\!\left(x_i'\beta^*+\gamma^*d_i+\tau^*\hat v_i\right), $$ where $\Phi$ is the standard normal cumulative distribution function. Under the maintained model, $\gamma^*=\gamma/\sqrt{1-\rho^2}$ and $\tau^*=(\rho/\sigma_v)/\sqrt{1-\rho^2}$. The coefficient $\gamma^*$ is an index coefficient, not a probability effect. A nonzero $\tau^*$ indicates dependence between the outcome disturbance and the first-stage disturbance under the maintained assumptions.

Worked Example: Constructing First-Stage Residuals

Suppose a hypothetical study models program participation as a binary outcome and uses a continuous measure $d_i$ of program intensity. An excluded instrument $z_i$ and an exogenous covariate $x_i$ predict intensity. A fitted first stage is $\hat d_i=0.4+0.5x_i+0.8z_i$. Calculate the residuals for three observations.

Observation	$x_i$	$z_i$	Observed $d_i$	Fitted $\hat d_i$	Residual $\hat v_i$
1	0	0	0.20	0.40	-0.20
2	1	1	1.60	1.70	-0.10
3	0	1	1.10	1.20	-0.10

For observation 2, for example, $\hat d_2=0.4+0.5(1)+0.8(1)=1.70$, so $\hat v_2=1.60-1.70=-0.10$. The negative residual means this observation has less intensity than the first stage predicts. In an actual analysis, the residuals for the full estimation sample—not just these illustrative rows—are included in the probit.

The first stage must use valid exogenous predictors, and the excluded instrument must provide relevant variation in $d_i$ beyond the included covariates. A residual-inclusion model does not repair an invalid or weak instrument, an incorrect first-stage conditional mean, or a failure of the distributional assumptions.

Average Probability Effects

In the fitted control-function probit, the probability for observation $i$ is $\Phi(\hat\eta_i)$, where $\hat\eta_i=x_i'\hat\beta^*+\hat\gamma^*d_i+\hat\tau^*\hat v_i$. To define the effect of changing $d_i$, hold the control value and exogenous covariates fixed. The derivative of this conditional probability with respect to $d_i$ is $\hat\gamma^*\phi(\hat\eta_i)$, where $\phi$ is the standard normal density. Averaging these derivatives over the sample gives the average probability effect (APE).

Average probability effect: For the continuous regressor $d_i$, the sample APE in the residual-inclusion probit is $$ \widehat{\mathrm{APE}}_d =\frac{1}{n}\sum_{i=1}^n \hat\gamma^*\phi(\hat\eta_i). $$ It is measured in probability units per unit of $d_i$. This is a derivative holding the control value fixed; it is not the effect of changing $d_i$ and mechanically recomputing its first-stage residual at the same time.

Worked Example: Calculating an Average Probability Effect

Consider a hypothetical fitted residual-inclusion probit with $\hat\gamma^*=0.7723$. For three observations, suppose the fitted indices are $-0.25$, $0.65$, and $-0.175$. Calculate the sample APE for this three-observation illustration.

The standard normal densities at those indices are $\phi(-0.25)\approx0.3867$, $\phi(0.65)\approx0.3230$, and $\phi(-0.175)\approx0.3929$. Substituting these values gives

\widehat{\mathrm{APE}}_d =\frac{0.7723(0.3867)+0.7723(0.3230)+0.7723(0.3929)}{3} \approx 0.2838.

Thus, in this illustration, a one-unit increase in $d_i$, holding the control value and other covariates fixed, is associated with an average increase of about $0.2838$ in the fitted probability of $y_i=1$, or 28.38 percentage points per unit. The derivative is local: for a finite change, the probability difference need not equal the derivative multiplied by that change.

The index values matter because the density $\phi(\hat\eta_i)$ changes across observations. Consequently, the APE is generally not just the coefficient on $d_i$, nor is it that coefficient multiplied by one density evaluated at the average index.

Testing the Control-Function Term

A standard endogeneity diagnostic tests $H_0:\tau^*=0$ against $H_a:\tau^*\ne0$. Under the maintained model, the null corresponds to zero conditional association between the outcome disturbance and the first-stage disturbance. A significant residual coefficient is evidence against treating $d_i$ as exogenous in this specification. This is an endogeneity test, not a test of whether the instruments are valid: instrument validity requires separate assumptions and evidence.

Because $\hat v_i$ is estimated in a first stage, its estimation uncertainty must be reflected in inference for $\hat\tau^*$. Suitable approaches include a valid joint estimation procedure or a bootstrap that repeats both the first-stage and outcome-stage estimation in each resample. A standard error calculated as though the generated residual were observed can give incorrect inference. When a bootstrap is used, the reported standard error and test should be based on the bootstrap procedure appropriate to the sampling design.

Worked Example: Testing Whether the Residual Matters

Suppose a hypothetical analysis estimates $\hat\tau^*=0.31$. A bootstrap that re-estimates both stages in each resample gives a standard error of $0.12$. Test $H_0:\tau^*=0$ against a two-sided alternative at $\alpha=0.05$, using the large-sample normal approximation.

The Wald statistic is $$ z=\frac{\hat\tau^*}{SE(\hat\tau^*)} =\frac{0.31}{0.12} \approx 2.583. $$

The two-sided p-value is approximately $2[1-\Phi(2.583)]\approx0.0098$. Since $0.0098<0.05$, reject $H_0$ at the 5% level. Under the model assumptions, the result is evidence that the residual term contributes to the conditional outcome model, so the naïve exogeneity specification is not supported. It does not establish instrument validity, and it does not show that every possible source of misspecification has been ruled out.

Comparing Naïve and Adjusted Probit

The naïve specification omits $\hat v_i$ and estimates a probit using $x_i$ and $d_i$ alone. Its coefficient on $d_i$ can differ from the residual-inclusion coefficient because the naïve model does not account for the dependence represented by the control function. Comparing coefficients alone is not a formal test of endogeneity: the coefficients are on a nonlinear index scale, and the two specifications may imply different fitted indices and probability effects.

A useful descriptive comparison reports both specifications’ APEs, computed from each model’s own fitted indices, alongside the residual-term test and appropriate uncertainty measures. If the models produce notably different APEs, that difference may be substantively important, but it should not be described as causal evidence unless the identifying and distributional assumptions are credible.

Worked Example: Comparing Naïve and Adjusted APEs

Suppose a hypothetical naïve probit has a coefficient on $d_i$ of $0.55$, and its fitted indices for a selected three-observation diagnostic calculation are $-0.25$, $0.65$, and $-0.175$. The corresponding densities are $0.3867$, $0.3230$, and $0.3929$. Its APE is

\widehat{\mathrm{APE}}_{\text{naive}} =\frac{0.55(0.3867)+0.55(0.3230)+0.55(0.3929)}{3} \approx 0.2021.

For comparison, suppose the residual-inclusion model has coefficient $0.7723$ on $d_i$ and, for this illustrative calculation, the same three fitted indices. Its APE is approximately $0.2838$, as calculated above. The adjusted-minus-naïve difference is $$ 0.2838-0.2021\approx0.0817 $$ probability units per unit of $d_i$. The unrounded calculation gives the same difference to four significant figures. This is a descriptive contrast in estimated APEs, not a separate hypothesis test. In a full analysis, each APE must use the fitted indices from its own model; identical indices here simply make the arithmetic comparison transparent.

The residual-term test and the APE comparison answer different questions. The test assesses whether the control-function term is distinguishable from zero under the model and its inference procedure. The APE comparison describes how the estimated probability response differs across specifications. Neither result alone verifies the instruments or the conditional-normal model.

Implementation and Interpretation

A defensible analysis makes each modeling choice explicit. The first stage should include the exogenous covariates in the outcome equation as well as the excluded instruments, and should be specified to represent the conditional mean of the continuous endogenous regressor. Instrument relevance should be assessed rather than presumed. Then the first-stage residual is included in the probit, and inference must account for its estimation.

The conditional-normal assumption gives the scalar residual a particular role: it summarizes the dependence between the latent outcome disturbance and the endogenous regressor’s disturbance. If that structure is not a reasonable approximation—for example, if the first-stage disturbances are heteroskedastic in a way the model ignores—the residual may fail to control the relevant dependence. The fitted model can still return coefficients and standard errors, but those outputs do not by themselves validate the control-function interpretation.

Specify the recursive system.
State the binary latent-index equation, continuous first stage, exogenous controls, excluded instruments, and assumptions that justify the control function.

Estimate the first stage.
Predict $d_i$ from the exogenous controls and instruments, then compute $\hat v_i=d_i-\hat d_i$.

Fit the residual-inclusion probit.
Include $d_i$, the exogenous controls, and $\hat v_i$; use inference that reflects first-stage estimation.

Interpret and compare.
Test the control-function coefficient, calculate each model’s APE from its fitted indices, and distinguish descriptive comparisons from tests and causal conclusions.

Common Pitfall

A frequent error is to read the probit coefficient on $d_i$ as a percentage-point effect. That coefficient shifts the latent index, not the probability directly. The probability derivative also depends on the normal density at each observation’s fitted index, so the APE must be calculated by multiplying the coefficient by those densities and averaging. A second error is to treat a significant residual coefficient as proof that the instruments are valid. It is evidence about endogeneity under the maintained specification; instrument validity is a distinct requirement.

Key takeaway: In a recursive binary-choice model with a continuous endogenous regressor, a control-function probit includes a first-stage residual to account for modeled dependence. Interpret probability effects using average derivatives, test the residual term with generated-regressor uncertainty accounted for, and keep endogeneity evidence separate from instrument-validity claims.

Check Your Understanding

Use the model and assumptions in this tutorial to answer the following questions.

In the first stage, $\hat d_i=0.4+0.5x_i+0.8z_i$. If $x_i=1$, $z_i=0$, and $d_i=0.7$, calculate the fitted value and residual.
Why is the coefficient on $d_i$ in a residual-inclusion probit not itself a percentage-point effect?
What null hypothesis is tested by testing whether the control-function coefficient is zero, and why must inference account for the estimated first-stage residual?
Explain why rejecting the zero-residual-coefficient null does not establish that the excluded instrument is valid.
If two specifications have different APEs, what additional information is needed before interpreting their difference as evidence about a causal effect?