📚 Introductory Econometrics with MORISTAT

A Practical Guide to Empirical Analysis

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

Chapter 6: Specification Issues and Dummy Variables

🎯 Learning Objectives

Upon completing this chapter, you will be able to:

Understand and identify omitted variable bias
Detect and address multicollinearity
Use logarithmic, quadratic, and interaction terms
Create and interpret dummy variables
Use MORISTAT to estimate models with dummy variables

6.1 Omitted Variable Bias

When a relevant variable is omitted from the regression, the estimated coefficients may be biased.

Suppose the true model is:

Y = β₀ + β₁X₁ + β₂X₂ + u

But we estimate:

Y = β₀ + β₁X₁ + v

Then the bias of β̂₁ is:

Bias(β̂₁) = β₂ × Corr(X₁, X₂) × (σ_X₂ / σ_X₁)

Key Insight:

If β₂ > 0 and Corr(X₁, X₂) > 0 → Upward bias
If β₂ > 0 and Corr(X₁, X₂) < 0 → Downward bias
If Corr(X₁, X₂) = 0 → No bias

Example: Omitted Ability

True model:

Wage = β₀ + β₁ × Education + β₂ × Ability + u

If Ability is omitted, the estimated β₁ will be biased upward if Ability and Education are positively correlated.

6.2 Multicollinearity

Multicollinearity occurs when independent variables are highly correlated with each other.

Consequences:

Large standard errors (imprecise estimates)
Difficulty identifying individual effects
Coefficients may be unstable

Detection:

Use CORR to check correlations
Use VIF (Variance Inflation Factor)

Rule of Thumb:

VIF > 10 → Problematic multicollinearity
VIF > 5 → Worth investigating

6.3 Functional Form

6.3.1 Logarithmic Models

Model	Interpretation
log(Y) = β₀ + β₁log(X)	Elasticity: 1% change in X → β₁% change in Y
log(Y) = β₀ + β₁X	Semi-elasticity: 1 unit X → β₁×100% change in Y
Y = β₀ + β₁log(X)	1% change in X → β₁/100 units change in Y

6.3.2 Quadratic Terms

Y = β₀ + β₁X + β₂X² + u

If β₂ > 0 → U-shaped (minimum at X = −β₁/(2β₂))
If β₂ < 0 → Inverted U (maximum at X = −β₁/(2β₂))

6.3.3 Interaction Terms

Y = β₀ + β₁X₁ + β₂X₂ + β₃(X₁ × X₂) + u

Effect of X₁ on Y depends on X₂
∂Y/∂X₁ = β₁ + β₃X₂

6.4 Dummy Variables

A dummy variable (indicator variable) takes values 0 or 1:

D = 1 if condition is true
D = 0 if condition is false

Example: Gender Dummy

Wage = β₀ + β₁ × Education + β₂ × Female + u

Female = 1 for female, 0 for male
β₂ = average wage difference between females and males (holding education constant)
If β₂ < 0, females earn less than males, ceteris paribus

6.4.1 Multiple Categories

For a variable with m categories, use m−1 dummy variables.

Category	D₁	D₂	Interpretation
North	1	0	β₁ = difference (North vs South)
South	0	1	β₂ = difference (South vs South) = baseline
East	0	0	Baseline category

Dummy Variable Trap:

Never include all m dummy variables – this creates perfect collinearity with the intercept. Always omit one category as the baseline.

6.5 MORISTAT Examples

6.5.1 Creating Dummy Variables

Create a dataset wage_gender.csv with a gender dummy:

Wage,Education,Gender
25,12,0
30,14,1
35,16,0
40,18,1
28,13,0
45,20,1
32,15,0
38,17,1

(Gender = 0 for Male, 1 for Female)

Run regression:

MORISTAT> LOAD wage_gender.csv
MORISTAT> REGRESS Wage ~ Education Gender

6.5.2 Interaction Term

MORISTAT> REGRESS Wage ~ Education Gender Education_Gender

6.6 Practical Exercises

Exercise 6.1: Omitted Variable Bias

You want to study the effect of class size on test scores. You estimate:

TestScore = β₀ + β₁ × ClassSize + u

But you omit TeacherQuality from the model.

If smaller classes attract better teachers, what is the sign of Corr(ClassSize, TeacherQuality)?
If TeacherQuality improves test scores (β₂ > 0), what is the sign of the bias?
Is β̂₁ biased upward or downward?

Solution:

Corr(ClassSize, TeacherQuality) < 0 (negative)
β₂ > 0 (positive)
Bias = β₂ × Corr × (σ_Teacher/σ_Class) = negative bias
β̂₁ is biased downward (underestimates the true effect)

Exercise 6.2: Interpreting Dummy Variables

You estimate the following regression:

Salarŷ = 30,000 + 2,500 × Experience + 5,000 × Female

where Female = 1 if female, 0 if male.

What is the predicted salary for a male with 10 years of experience?
What is the predicted salary for a female with 10 years of experience?
What is the gender wage gap (holding experience constant)?
Is the coefficient on Female positive or negative? What does this imply?

Solution:

Male (Female = 0): 30,000 + 2,500×10 = 55,000
Female (Female = 1): 30,000 + 2,500×10 + 5,000 = 60,000
Gender gap = 5,000 (females earn more than males)
Coefficient is positive → females earn more than males, ceteris paribus

Exercise 6.3: Quadratic Model

You estimate the following model:

Wagê = 10 + 2 × Experience − 0.05 × Experience²

What is the effect of an additional year of experience at Experience = 10?
At Experience = 20?
At Experience = 30?
What is the "peak" experience level (where wage is maximized)?

Hint: ∂Wage/∂Experience = 2 − 0.10 × Experience

Solution:

At Exp = 10: ∂Wage/∂Exp = 2 − 0.10×10 = 1.00
At Exp = 20: ∂Wage/∂Exp = 2 − 0.10×20 = 0.00
At Exp = 30: ∂Wage/∂Exp = 2 − 0.10×30 = -1.00
Peak: 2 − 0.10×Exp = 0 → Exp = 20 years

Exercise 6.4: Interaction Term

You estimate:

Wagê = 20 + 2 × Education + 3 × Experience + 0.5 × (Education × Experience)

What is the effect of an additional year of education when Experience = 5?
When Experience = 10?
When Experience = 15?
Interpret the interaction term coefficient.

Solution:

∂Wage/∂Education = 2 + 0.5×Experience
At Exp = 5: 2 + 0.5×5 = 4.5
At Exp = 10: 2 + 0.5×10 = 7.0
At Exp = 15: 2 + 0.5×15 = 9.5
The return to education increases with experience (positive interaction)

Exercise 6.5: Real Data – Wage Discrimination

Using wage_gender.csv, run a regression of Wage on Education and Gender.

What is the coefficient on Gender?
Is the coefficient statistically significant?
What is the estimated gender wage gap?
What assumptions must hold for this coefficient to measure discrimination?

6.7 Key Terms

Omitted Variable Bias Multicollinearity VIF Logarithmic Model Quadratic Model Interaction Term Dummy Variable Indicator Variable Dummy Variable Trap Baseline Category

6.8 Further Reading

Wooldridge, J.M. (2020). Introductory Econometrics: A Modern Approach, Chapter 6 (Multiple Regression Analysis: Further Issues)
Stock, J.H. & Watson, M.W. (2019). Introduction to Econometrics, Chapter 8 (Nonlinear Regression Functions)
Angrist, J.D. & Pischke, J.S. (2015). Mastering 'Metrics, Chapter 3 (The Regression Discontinuity Design)

6.9 What's Next?

In Chapter 7, we will address:

Heteroskedasticity – Violation of constant variance
Serial Correlation – Correlation over time
Instrumental Variables – When X is endogenous

→ Proceed to Chapter 7 (Coming Soon)

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