📚 Introductory Econometrics with MORISTAT

A Practical Guide to Empirical Analysis

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

Chapter 8: Panel Data and Difference-in-Differences

🎯 Learning Objectives

Upon completing this chapter, you will be able to:

Understand what panel data is and why it is useful
Distinguish between pooled OLS, fixed effects, and random effects
Understand the Hausman test for model selection
Implement Difference-in-Differences (DiD) for causal inference
Understand the parallel trends assumption
Use MORISTAT to analyse panel data

8.1 What is Panel Data?

Panel data (also called longitudinal data) combines both:

Cross-sectional data (multiple individuals/units at one point in time)
Time series data (multiple time periods for one individual/unit)

Example: Data on 100 firms over 10 years (100 × 10 = 1,000 observations).

Each firm is observed in each year
Variables: sales, profits, employees, R&D spending, etc.

Panel data notation:

Y_it = β₀ + β₁X_it + u_it

i = individual/unit (e.g., firm, country, person)
t = time period (e.g., year, quarter, month)
X_it = independent variable for individual i at time t

8.1.1 Advantages of Panel Data

More information → more efficient estimates
Controls for unobserved heterogeneity (fixed effects)
Allows for dynamics (lagged variables)
Better for causal inference (DiD)

8.2 Panel Data Models

8.2.1 Pooled OLS

Ignore the panel structure and treat all observations as independent:

Y_it = β₀ + β₁X_it + u_it

Advantage: Simple
Disadvantage: Ignores individual-specific effects → bias if unobserved heterogeneity exists

8.2.2 Fixed Effects (FE)

Allow each individual to have its own intercept (captures time-invariant unobserved heterogeneity):

Y_it = β₀ + β₁X_it + α_i + u_it

α_i = individual-specific fixed effect (captures all time-invariant unobserved factors)
Within estimator: Uses variation within each individual over time
Key assumption: E(u_it|X_it, α_i) = 0

8.2.3 Random Effects (RE)

Assume α_i is random and uncorrelated with X_it:

Y_it = β₀ + β₁X_it + ν_i + u_it

ν_i = random individual-specific error term
Key assumption: E(ν_i|X_it) = 0 (no correlation between ν_i and X_it)
Advantage: More efficient if the assumption holds
Disadvantage: Biased if the assumption is violated

8.2.4 Hausman Test

Tests whether FE or RE is appropriate:

H₀: RE is consistent (ν_i is uncorrelated with X_it)
H₁: RE is inconsistent (use FE)
If p-value < 0.05 → reject H₀ → use Fixed Effects
If p-value ≥ 0.05 → fail to reject H₀ → use Random Effects

8.3 Difference-in-Differences (DiD)

8.3.1 What is DiD?

DiD is a quasi-experimental method for estimating causal effects using panel data with a treatment and control group.

Example: Effect of a minimum wage increase on employment.

Treatment group: States that increased the minimum wage
Control group: States that did not increase the minimum wage
DiD estimate = (Change in treatment) − (Change in control)

8.3.2 The DiD Model

Y_it = β₀ + β₁ × Treat_i + β₂ × Post_t + β₃ × (Treat_i × Post_t) + u_it

Treat_i = 1 if treated, 0 if control
Post_t = 1 after intervention, 0 before
β₃ = DiD estimator (the causal effect of treatment)

8.3.3 Parallel Trends Assumption

Key Assumption: In the absence of treatment, the treatment and control groups would have followed parallel trends over time.

This assumption is not directly testable (counterfactual)
Can be visually assessed by plotting trends before treatment
Pre-treatment trends should be similar (parallel)

8.4 MORISTAT Examples

8.4.1 Panel Data Format

Panel data in CSV format:

Firm,Year,Sales,Profits,Employees
1,2020,100,10,50
1,2021,110,12,55
1,2022,120,14,60
2,2020,200,20,80
2,2021,210,22,85
2,2022,220,24,90

8.4.2 Pooled OLS in MORISTAT

MORISTAT> LOAD panel_data.csv
MORISTAT> REGRESS Sales ~ Profits Employees

8.4.3 DiD Example

MORISTAT> LOAD did_data.csv
MORISTAT> REGRESS Employment ~ Treat Post Treat_Post

8.5 Practical Exercises

Exercise 8.1: Fixed vs Random Effects

You have panel data on 50 countries over 10 years. You want to estimate the effect of trade openness on GDP growth.

If unobserved country-specific factors (e.g., culture, institutions) are correlated with trade openness, which method would you use?
If these factors are uncorrelated with trade openness, which method would you use?
How would you decide which method to use?

Solution:

Fixed Effects – if unobserved factors are correlated with trade openness
Random Effects – if unobserved factors are uncorrelated with trade openness
Use the Hausman test to decide: p < 0.05 → FE; p ≥ 0.05 → RE

Exercise 8.2: Interpreting DiD

You estimate the following DiD model:

Employment̂ = 100 + 5 × Treat + 10 × Post + 15 × (Treat × Post)

What is the employment level in the control group before the treatment?
What is the employment level in the treatment group before the treatment?
What is the causal effect of the treatment on employment?

Solution:

Control before: 100
Treatment before: 100 + 5 = 105
Causal effect: β₃ = 15 (employment increased by 15 units due to the treatment)

Exercise 8.3: Parallel Trends

You are evaluating a job training programme. You plot employment trends for the treatment and control groups before the programme and see:

Treatment group: employment increasing by 2% per year
Control group: employment increasing by 3% per year

Is the parallel trends assumption violated?
What does this imply for the DiD estimate?
What could you do to address this?

Solution:

Yes – the trends are not parallel (2% vs 3% per year)
The DiD estimate will be biased (it will pick up the difference in trends, not just the treatment effect)
Possible solutions: include time trends in the model, use synthetic control, or use event study methods

8.6 Key Terms

Panel Data Longitudinal Data Pooled OLS Fixed Effects (FE) Random Effects (RE) Within Estimator Hausman Test Difference-in-Differences (DiD) Treatment Group Control Group Parallel Trends Assumption Causal Inference Counterfactual

8.7 Further Reading

Wooldridge, J.M. (2020). Introductory Econometrics: A Modern Approach, Chapter 13 (Pooling Cross-Sections Across Time) and Chapter 14 (Panel Data).
Stock, J.H. & Watson, M.W. (2019). Introduction to Econometrics, Chapter 10 (Regression with Panel Data) and Chapter 13 (Experiments and Quasi-Experiments).
Angrist, J.D. & Pischke, J.S. (2015). Mastering 'Metrics, Chapter 5 (Differences-in-Differences).

8.8 What's Next?

In Chapter 9, we will cover Time Series Analysis and Forecasting:

Stationarity and unit roots
ARIMA models
Forecasting methods
Vector Autoregression (VAR)

→ Proceed to Chapter 9 (Coming Soon)

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