Probit — Normal binary regression

Probit models the probability of a binary outcome through the standard normal cumulative distribution function $\Phi(\cdot)$. Empirically, Probit and Logit usually lead to similar conclusions; they differ in the assumed error distribution (normal vs logistic).

Logit or Probit?

Results are usually very close. Logit is convenient for its odds ratios; Probit is preferred when a normal error assumption is more reasonable or within extended models (Heckman, biprobit). Always compare through marginal effects.

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Model specification

$$P(Y_i = 1 \mid X_i) = \Phi(\beta_0 + \beta_1 X_{1i} + \dots + \beta_k X_{ki})$$

where $\Phi$ is the standard normal CDF. Estimated by MLE.

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Interpretation

• Coefficients $\beta_j$ are not read directly; use marginal effects (AME/MEM).
• Model fit: Pseudo-$R^2$, classification, AUC.

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Running in EcoLab

1. Modeling module → Limited dependent variable family → Probit.
2. Select the binary $Y$ and the $X$ variables.
3. Run; read marginal effects; compare with Logit; export the replication code.

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Replication code

Stata

* ===== Probit — Normal binary regression =====
* Estimate the probit model
probit y x1 x2 x3

* Average marginal effects (AME)
margins, dydx(*)

* Predicted probabilities
predict phat, pr

* Classification table
estat classification

* Pseudo-R² is shown in the estimation output

R

# ===== Probit — Normal binary regression =====
# Estimate the probit model
model <- glm(y ~ x1 + x2 + x3,
             data   = df,
             family = binomial(link = "probit"))

summary(model)

# Average marginal effects (AME)
library(margins)
mfx <- margins(model)
summary(mfx)

# Predicted probabilities
df$phat <- predict(model, type = "response")

# McFadden Pseudo-R²
null_model <- glm(y ~ 1, data = df, family = binomial(link = "probit"))
1 - logLik(model) / logLik(null_model)

Python

# ===== Probit — Normal binary regression =====
import statsmodels.api as sm

# Prepare data
X = sm.add_constant(df[["x1", "x2", "x3"]])
y = df["y"]

# Estimate the probit model
model = sm.Probit(y, X).fit()
print(model.summary())

# Average marginal effects (AME)
mfx = model.get_margeff()
print(mfx.summary())

# Predicted probabilities
df["phat"] = model.predict(X)

# McFadden Pseudo-R² is shown in model.summary()
print("Pseudo R²:", model.prsquared)

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Limitations

• No odds interpretation like Logit.
• Same underlying assumptions (exogeneity, correct specification) as other binary-choice models.

Video tutorial

Video Tutorial: Guide to running Probit in EcoLab

See also

• Logit · Heckman · Catalog