Interaction Terms

ECON 3209 · Week 13, Lecture 2 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

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

1. write and interpret dummy-by-continuous interaction models
2. distinguish intercept shifts from slope shifts
3. compute group-specific marginal effects
4. visualize fitted lines for different categories
5. estimate interaction models in Python with formula syntax

Interaction Model

\[Y_i = \beta_0 + \beta_1 X_i + \beta_2 D_i + \beta_3(D_i \cdot X_i) + \varepsilon_i\]

• \(\beta_0\): intercept for the reference group (\(D=0\))
• \(\beta_1\): slope for the reference group
• \(\beta_2\): intercept shift for the dummy group
• \(\beta_3\): slope difference for the dummy group

Group-Specific Equations

If \(D=0\):

\[Y = \beta_0 + \beta_1 X + \varepsilon\]

If \(D=1\):

\[Y = (\beta_0 + \beta_2) + (\beta_1 + \beta_3)X + \varepsilon\]

So the interaction allows both the intercept and the slope to differ.

Kerala Interpretation

Example: effect of fertilizer on yield may differ between irrigated and rainfed farms.

• D = 1 for irrigated plots
• \(\beta_2\) tells us whether irrigated farms start from a different baseline yield
• \(\beta_3\) tells us whether fertilizer is more or less productive under irrigation

Reading the Coefficients Carefully

• Do not interpret \(\beta_1\) as the effect for everyone.
• Do not interpret \(\beta_2\) as the total effect of the dummy when \(X\) changes.
• Always compute the group-specific line.

In interaction models, coefficients rarely speak alone. They speak as a bundle.

Python Demo: Simulate Slope Differences

Python Demo: Estimate the Interaction Model

Python Demo: Plot Group-Specific Fitted Lines

Python Demo: Compute Marginal Effects by Group

Common Mistakes

• forgetting the main effects when adding the interaction
• comparing groups without computing both intercept and slope
• reading the interaction coefficient as a standalone marginal effect
• using interaction models without plotting them

Alternative Examples

Interaction terms also help with: - wage returns to education by gender - income effects before and after a policy reform - crop-response functions by soil type

They are a flexible way to ask whether one effect depends on another condition.

🏋️ Exercise

1. Simulate crop yield as a function of fertilizer and irrigation.
2. Estimate yield_ ~ fertilizer * irrigated.
3. Compute the fertilizer slope for rainfed and irrigated farms separately.
4. Plot the two fitted lines and interpret the difference.

Summary

✅ Interaction terms allow slopes to differ across groups.

✅ In a dummy-by-continuous interaction, one coefficient shifts the intercept and another shifts the slope.

✅ Group-specific effects must be computed from combinations of coefficients.

✅ Plots are one of the best ways to interpret interaction models.

✅ Interaction models answer whether one relationship depends on context.

Next Lecture

Lecture 3 — Structural Breaks in Python

We will cover: - regime dummies - before-and-after comparisons - Chow-test logic - cooperative credit scheme example