📚 Introductory Econometrics with MORISTAT

A Practical Guide to Empirical Analysis

Naguib Lallmahomed · naglal@linux-mauritius.com · 2026-06-14

Chapter 10: Limited Dependent Variable Models

🎯 Learning Objectives

Upon completing this chapter, you will be able to:

Understand limited dependent variables (binary, multinomial, ordered)
Estimate Logit and Probit models
Interpret coefficients using marginal effects
Choose between Logit and Probit
Apply multinomial and ordered models
Use MORISTAT for binary choice models

10.1 What are Limited Dependent Variables?

So far, we have assumed that the dependent variable Y is continuous (e.g., sales, GDP, wages). However, in many applications, Y is limited or discrete:

Type	Description	Example
Binary	Two outcomes (0/1)	Employed/Unemployed, Pass/Fail
Multinomial	Multiple unordered outcomes	Mode of transport (car/bus/train)
Ordered	Multiple ordered outcomes	Survey responses (Poor/Good/Excellent)
Censored	Continuous but limited	Wages (top-coded), hours worked

Why OLS fails for binary Y:

Predicted probabilities can be outside [0, 1]
Errors are heteroskedastic
Errors are not normally distributed
Relationship between X and Y is non-linear

10.2 Binary Choice Models: Logit and Probit

10.2.1 The Latent Variable Framework

Y* = β₀ + β₁X₁ + β₂X₂ + ⋯ + β_kX_k + u

Y* is the unobserved latent variable (utility, propensity)
We observe Y = 1 if Y* > 0, and Y = 0 if Y* ≤ 0

10.2.2 The Logit Model

P(Y = 1 | X) = Λ(β₀ + β₁X₁ + ⋯ + β_kX_k) = e^Xβ / (1 + e^Xβ)

Λ(·) is the logistic (sigmoid) function
Guarantees probabilities between 0 and 1
Estimated by Maximum Likelihood (ML)
Odds ratio interpretation: e^β

10.2.3 The Probit Model

P(Y = 1 | X) = Φ(β₀ + β₁X₁ + ⋯ + β_kX_k)

Φ(·) is the standard normal CDF
Similar to Logit but uses normal distribution
β_logit ≈ 1.6 × β_probit (scale difference)

Comparison:

Logit: Thicker tails, easier to compute
Probit: Based on normal distribution (familiar)
In practice, results are very similar
Choice often based on convention or software availability

10.3 Marginal Effects

In Logit/Probit, coefficients cannot be interpreted directly as marginal effects.

10.3.1 Marginal Effect for a Continuous X

∂P(Y = 1 | X) / ∂X_j = f(Xβ) × β_j

f(Xβ) = density function (logistic or normal)
Marginal effect depends on where we evaluate X
Typically evaluated at the mean of X

10.3.2 Marginal Effect for a Binary X

ΔP = P(Y = 1 | X = 1) − P(Y = 1 | X = 0)

Difference in predicted probabilities
Can be interpreted as the ceteris paribus effect of switching X from 0 to 1

10.4 Multinomial and Ordered Models

10.4.1 Multinomial Logit

Used when there are more than two unordered categories
Choose a baseline category
Estimates coefficients for each category (relative to baseline)

Example: Mode of transport to work:

Car (baseline), Bus, Train, Bicycle
Predictors: distance, income, age, gender
Interpretation: relative probability of choosing each mode vs. car

10.4.2 Ordered Logit/Probit

Used when outcomes are ordered but not numeric
Examples: education level, health status, survey responses
Estimates cut points (thresholds) between categories

10.5 MORISTAT Examples

10.5.1 Binary Choice Data Format

Employed,Education,Experience,Age
1,16,10,35
0,12,5,25
1,18,15,45
0,14,8,30
1,20,20,50

10.5.2 Running a Logit Model in MORISTAT

MORISTAT> LOAD binary_data.csv
MORISTAT> SUMMARY
MORISTAT> REGRESS Employed ~ Education Experience Age

10.6 Practical Exercises

Exercise 10.1: Logit vs Probit

You estimate both a Logit and a Probit model for the probability of mortgage default.

Logit coefficient on Income: -0.035
Probit coefficient on Income: -0.022

Are these coefficients consistent with the Logit/Probit scale relationship?
Calculate the marginal effect for Income for both models (assume density at mean = 0.25 for Logit, 0.40 for Probit).
Are the marginal effects similar?

Solution:

Logit β = -0.035, Probit β = -0.022
Ratio: 0.035 / 0.022 = 1.59 ≈ 1.6 ✓ (consistent)
Logit ME: -0.035 × 0.25 = -0.00875
Probit ME: -0.022 × 0.40 = -0.0088
Marginal effects are virtually identical

Exercise 10.2: Interpreting Logit Coefficients

You estimate the following Logit model for the probability of being employed:

P(Employed = 1) = Λ(-2.5 + 0.3 × Education - 0.02 × Experience + 0.05 × Age)

What is the effect of Education on the probability of being employed?
What is the odds ratio for Education?
What is the marginal effect of Education evaluated at the mean where density = 0.2?

Solution:

Effect of Education: β = 0.3 (positive → higher education increases employment probability)
Odds ratio: e^0.3 = 1.35 (each additional year increases odds of employment by 35%)
Marginal effect: 0.3 × 0.2 = 0.06 (6 percentage points increase in employment probability)

Exercise 10.3: Multinomial Choice

You are studying mode choice for commuting. The multinomial logit results show:

Bus (vs. Car): coefficient on Income = -0.02
Train (vs. Car): coefficient on Income = -0.03

Interpret the coefficient on Income for Bus.
Interpret the coefficient on Income for Train.
What does this suggest about the relationship between income and mode choice?

Solution:

Bus vs. Car: Higher income reduces the relative probability of choosing Bus over Car
Train vs. Car: Higher income reduces the relative probability of choosing Train over Car
As income increases, people are more likely to choose Car over public transport

10.7 Key Terms

Limited Dependent Variable Binary Choice Model Logit Model Probit Model Logistic Function Latent Variable Maximum Likelihood (ML) Marginal Effect Odds Ratio Multinomial Logit Ordered Logit Ordered Probit Baseline Category Cut Points / Thresholds

10.8 Further Reading

Wooldridge, J.M. (2020). Introductory Econometrics: A Modern Approach, Chapter 17 (Limited Dependent Variable Models).
Stock, J.H. & Watson, M.W. (2019). Introduction to Econometrics, Chapter 11 (Regression with a Binary Dependent Variable).
Angrist, J.D. & Pischke, J.S. (2015). Mastering 'Metrics, Chapter 3 (Making Regression Make Sense).

10.9 What's Next?

Congratulations! You have completed all 10 chapters of Introductory Econometrics with MORISTAT.

✅ Chapter 1: Introduction
✅ Chapter 2: Statistics Review
✅ Chapter 3: Simple Regression
✅ Chapter 4: Multiple Regression (Estimation)
✅ Chapter 5: Multiple Regression (Inference)
✅ Chapter 6: Specification & Dummies
✅ Chapter 7: Heteroskedasticity, Serial Correlation, IV
✅ Chapter 8: Panel Data & DiD
✅ Chapter 9: Time Series & Forecasting
✅ Chapter 10: Limited Dependent Variables (Logit/Probit)

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