Probit Model in Python

ECON 3209 · Week 15, Lecture 3 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you should be able to:

  1. write the probit probability model using the standard normal CDF
  2. compare probit intuition with logit and understand why coefficients differ in scale
  3. estimate probit and logit models in Python using statsmodels
  4. evaluate classification quality with a ROC curve and AUC
  5. decide when logit and probit give practically similar conclusions

What is the Probit Model?

  • Probit is another binary response model.
  • Instead of the logistic CDF, it uses the standard normal CDF \(\Phi(\cdot)\):

\[P(Y_i=1|X_i)=\Phi(\beta_0+\beta_1X_{1i}+\cdots+\beta_kX_{ki})\]

  • The probability still stays in \([0,1]\).
  • The latent-variable story is often intuitive: a hidden propensity crosses a threshold.

Logit versus Probit

  • Both models are nonlinear and fitted by maximum likelihood.
  • Both usually give very similar predicted probabilities in practice.
  • Logit has slightly fatter tails; probit is linked to the normal distribution.
  • Coefficients are on different scales, so compare signs, significance, marginal effects, and predictions, not raw magnitudes alone.

Statsmodels versus scikit-learn

  • In applied econometrics, statsmodels is preferred when you want coefficients, standard errors, hypothesis tests, and marginal effects.
  • In machine learning workflows, scikit-learn focuses on prediction pipelines and classification accuracy.
  • In this browser-based course, statsmodels is our main tool for probit.
  • The econometric logic remains the same even if software differs.

Estimate Probit and Logit Together

Comparing Predicted Probabilities

How to Interpret Probit

  • Like logit, the sign of a coefficient shows the direction of the effect.
  • Raw coefficients are changes in the latent index, not direct probability changes.
  • Marginal effects depend on the value of the normal density at the relevant index value.
  • In class reporting, focus on predicted probabilities and marginal effects.

ROC Curve Logic

  • A classifier turns predicted probabilities into decisions using a cutoff.
  • The ROC curve plots:
    • true positive rate (sensitivity),
    • false positive rate (1 - specificity), across many thresholds.
  • A curve closer to the top-left corner indicates better classification performance.
  • The area under the curve (AUC) summarises overall discrimination ability.

ROC Curve in Python

Kerala Interpretation

Suppose your model predicts a higher default probability for borrowers in districts with weaker repayment climate, lower income, and no collateral.

For a bank officer, the value of logit/probit is not only estimation — it is better screening, pricing, and monitoring of risk.

Visual Comparison of Logit and Probit

Exercise

Fit both a logit and a probit model. Which model gives the higher predicted default probability for a low-income borrower without collateral? Does the ranking of risky borrowers change?

Summary

  • ✅ Probit uses the normal CDF and logit uses the logistic CDF, but both usually tell a similar story.
  • ✅ Raw coefficients should not be compared across models without caution because scales differ.
  • ✅ Predicted probabilities, marginal effects, ROC curves, and AUC are more useful for interpretation.
  • ✅ statsmodels is the main tool for binary econometric models in this course.

Next Lecture

  • We shift to model specification problems, starting with omitted variable bias.
  • You will learn how missing regressors distort OLS coefficients.
  • Keep binary-model intuition, but return to the logic of causal interpretation.