Linear Probability Model

ECON 3209 · Week 15, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

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

define a binary dependent variable and write the linear probability model (LPM)
estimate an LPM with OLS and interpret coefficients as probability changes
identify why LPM predictions may fall below 0 or above 1
explain heteroskedasticity and other weaknesses of the LPM
use Python to fit and diagnose an LPM for Kerala loan default data

Why Binary Outcomes Need Special Care

Many econometric questions have yes/no outcomes: loan default, insurance enrolment, crop failure, technology adoption.
Here, \(Y_i \in \{0,1\}\), so \(E[Y_i\mid X_i] = P(Y_i=1\mid X_i)\).
If we run OLS with a binary \(Y\), we get the linear probability model:

\[P(Y_i=1\mid X_i) = \beta_0 + \beta_1 X_{1i} + \cdots + \beta_k X_{ki}\]

The fitted value is interpreted as a predicted probability.

Why Teachers Still Start with LPM

OLS is familiar, fast, and easy to explain.
Coefficients are direct: a one-unit increase in \(X_j\) changes probability by \(\beta_j\).
It works as a useful baseline model before logit and probit.
In large samples, LPM can approximate average partial effects reasonably well near the center of the probability range.
But convenience does not remove its weaknesses.

Estimating an LPM in Python

Interpreting the Coefficients

If the income coefficient is \(-0.01\), then one extra unit of income lowers default probability by about 1 percentage point.
A dummy variable coefficient compares two groups directly.
Example: if the collateral coefficient is \(-0.08\), borrowers with collateral have about 8 percentage points lower default risk, holding other factors constant.
This direct interpretation is the main attraction of the LPM.

Predicted Probability Problem

Main Weaknesses of LPM

Nonlinearity ignored: probability often changes more slowly near 0 and 1.
Impossible fitted values: some predictions can be negative or above 1.
Heteroskedastic errors: for binary \(Y\), the variance depends on \(P(Y=1\mid X)\).
Marginal effects forced to be constant: income has the same effect for every borrower.
These limits motivate logit and probit models.

Robust Standard Errors Matter

Kerala Context: Cooperative Bank Default

Suppose a KAU student is analysing whether a cooperative bank borrower defaults on a short-term agricultural loan.

Possible regressors: - borrower income, - collateral ownership, - district dummy for high-risk lending zones, - past repayment record.

The LPM gives a quick first look at which borrower characteristics are associated with higher default probability.

Visualising LPM Predictions

When is LPM Still Useful?

As a teaching model to connect OLS with binary outcomes.
As a first-pass descriptive model.
In some large datasets with many fixed effects, LPM remains computationally convenient.
But for serious probability prediction, we usually prefer logit or probit because they keep fitted values inside \([0,1]\).

Exercise

Use the starter code to estimate an LPM for loan default. Then answer: (i) what is the sign of the income effect, (ii) how many fitted values lie outside \([0,1]\), and (iii) would you trust this model for prediction?

Summary

✅ LPM estimates binary outcomes with OLS and interprets fitted values as probabilities.
✅ Its biggest strengths are simplicity and easy coefficient interpretation.
✅ Its biggest weaknesses are heteroskedasticity, constant marginal effects, and impossible predictions.
✅ Robust standard errors improve inference but do not fix the model’s functional-form problem.

Next Lecture

We move from a linear probability line to the nonlinear logistic curve.
You will learn log-odds, maximum likelihood, odds ratios, and marginal effects.
Bring today’s intuition about binary outcomes into the logit model.