Lesson 17

Linear Probability Model

Big question

How can OLS model a probability?

Lesson progress

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Big question
Concept
Activity
Quiz

Learning objectives

  • Explain linear probability model in plain language.
  • Use intercept shift correctly in an interpretation.
  • Connect the lesson idea to a formula, graph, Python result, or real example.

Simple explanation

The linear probability model uses OLS with a zero-one outcome. A coefficient is a change in probability measured in points.

Key terms

Intercept shift
A group difference that changes the expected outcome level while leaving slopes unchanged.
Robust standard error
A standard error designed to remain valid under heteroskedasticity.
Binary variable
A variable that equals one when a condition is present and zero otherwise.

Core formula

P(yi=1∣xi)=β0+β1x1i+⋯+βkxkiP(y_i=1\mid x_i) = \beta_0 + \beta_1x_{1i}+\cdots+\beta_kx_{ki}

Use plain-language interpretation before algebra.

Example

MODULE7_STUDENT_COMPLETION_SYNTHETIC is an original Ceteris Lab synthetic teaching dataset for linear probability model. It lets students practice LPM without presenting fabricated real-world empirical findings.

Interactive visual

ProbabilityEffectCalculator: use the controls to code a group, choose a base group, and write one correct interpretation.

Original Module 7 visual for Linear Probability Model.

wage_sample.csv

y variable

wage

The dependent variable. It is the outcome students want to explain.

x variable

education

The explanatory variable. It is used to describe changes in wage.

Live Python

Linear Probability Model Python example

Linear Probability Model Python example

Stdout

Run Python to see results here.

Status / stderr

Ready to run Python in your browser.

Line-by-line guide

  1. Line 1Load a Python library needed for data work or regression.
  2. Line 2Load a Python library needed for data work or regression.
  3. Line 4Load the dataset into a pandas DataFrame.
  4. Line 5Add an intercept column to the regression design matrix.
  5. Line 6Estimate an ordinary least squares regression.
  6. Line 7Display a result so students can inspect the output.
  7. Line 8Display a result so students can inspect the output.

Python walkthrough

  1. 1Load pandas and statsmodels so the workflow is reproducible.
  2. 2Read the installed Ceteris Lab synthetic CSV from the public data folder.
  3. 3Create or inspect dummy variables before estimating the model.
  4. 4Estimate OLS with an intercept and the selected regressors.
  5. 5Print coefficients or summaries, then interpret them as associations unless the design supports causality.

Live notebook

Run this lesson as a notebook

Open an editable notebook cell-by-cell, run Python in the browser, and download the `.ipynb` file for later.

Open live notebookDownload .ipynbDownload .py

Related dataset

MODULE7_STUDENT_COMPLETION_SYNTHETIC

Estimated time

25 to 40 min

Packages

pandas, numpy, statsmodels, patsy

Expected output

Printed Python results that can be compared with the lesson explanation.

Learning goals

  • Load and inspect MODULE7_STUDENT_COMPLETION_SYNTHETIC.
  • Run the Python cells connected to Linear Probability Model.
  • Interpret the output using dummy variables and qualitative information.

Common errors

  • File not found: check that module7_student_completion_synthetic.csv is installed or use the course data folder.
  • Package import error: use the browser notebook first, then download for local Jupyter if your local packages differ.
  • Column name error: compare your variable names with the dataset variables listed for this notebook.

Dataset path helper

import pandas as pd

df = pd.read_csv("/data/module-7/module7_student_completion_synthetic.csv")
df.head()

Interactive activity

ProbabilityEffectCalculator

Linear Probability Model

ProbabilityEffectCalculator: choose the coding rule, base group, or probability interpretation before reading the coefficient.

MODULE7_STUDENT_COMPLETION_SYNTHETIC

Inputs

Try it yourself

Write one plain-English sentence explaining the main idea from this lesson.

Common mistakes

Check these before you move on.

A regression coefficient describes a pattern unless the assumptions or research design support a causal interpretation.

Quick quiz

Which interpretation is most careful for Linear Probability Model?

Quick quiz

What should students check before trusting the result in Linear Probability Model?

Quick quiz

Why is CHARITY a reasonable practice dataset here?

Key takeaway

Linear Probability Model helps students convert qualitative information into transparent, testable regression comparisons.

Download dataDownload resultsDownload interpretation