Interactions in Regression Models

Section one

Key concepts

The two interaction models and how to read their coefficients.

Key concepts

An interaction is a product term, and it always asks whether the effect differs by group.

Dummy × dummy — the difference-in-difference model
Dummy × continuous — varying program impact (slopes) by group

Difference-in-differences: interacting two dummies.

Y = b₀ + b₁ · Treat + b₂ · Post + b₃ · (Treat · Post) + e

Coefficient	Label	Meaning
b₀	A	comparison group, before
b₁	B	treatment group offset, before
b₂	C	change over time common to both groups
b₃	D	the program effect

Comparison group, after: A + C
Treatment group, before: A + B
Treatment group counterfactual, after: A + B + C
Treatment group, actually observed after: A + B + C + D

In the worked example b₀ = 20, b₁ = 15, b₂ = 10, b₃ = 20: the counterfactual is 45, the observed outcome is 65, and the program effect is the 20 between them.

The comparison group supplies the counterfactual trend.

Both groups would have moved by b₂ anyway. The interaction term isolates the extra movement in the treatment group — the part the common trend does not explain.

Dummy × continuous: letting the slope vary by group.

height = b₀ + b₁dumA + b₂dumB + b₃fertilizer + b₄(dumA · fertilizer) + b₅(dumB · fertilizer) + e

With Type C as the omitted reference category:

b₀ = height of Type C at fertilizer = 0 — the reference intercept
b₁ = how much higher Type A starts than Type C — an intercept shift
b₃ = the effect of fertilizer for Type C — the reference slope
b₄ = the difference in fertilizer slope between Type A and Type C
Fertilizer slope for Type A = b₃ + b₄

The reference category is arbitrary; the fit is not.

The deck runs the same model with different groups omitted. Every coefficient changes, and R² stays at 0.59. You are re-describing the same fitted lines from a different baseline.

Section two

Key relationships

The mental map. How the pieces connect, and how to read the deck.

Main effects shift intercepts; interactions shift slopes

This is the sentence that organises the whole deck.

A dummy on its own moves a group’s line up or down. Same slope, different starting height.
A dummy interacted with a continuous variable changes that group’s slope. The lines are no longer parallel.

In the corn example you can see both at once: Type A starts 8.98 units above Type C (b₁, an intercept shift) and responds to fertilizer differently (b₄, a slope shift). Type A’s line is flat, Type B’s rises steeply, Type C’s falls. Without the interaction terms the model would be forced to give all three the same slope and would fit none of them.

So the diagnostic question is: do I think the groups start at different levels, respond at different rates, or both? Each answer maps to a specific term.

Every coefficient is a comparison to the omitted group

Interaction models are hard to read because nothing means what it appears to mean in isolation. Everything is relative to whichever category you left out.

b₃ is not “the effect of fertilizer.” It is the effect of fertilizer for Type C.
b₄ is not “the effect of fertilizer for Type A.” It is the difference between Type A’s effect and Type C’s.
To get Type A’s actual slope you must add: b₃ + b₄.

The four-model table drives this home. Same data, same fitted lines, same R² = 0.59 — and completely different numbers, because each model uses a different reference group. If you ever find yourself surprised that a coefficient changed sign when you re-coded a factor, this is why.

Practical habit: before interpreting any interaction model, say out loud which group is the baseline. Every number in the output is an answer to “compared to that.”

Difference-in-differences is the same trick with a purpose

The DiD model is just dummy × dummy — but the two dummies are chosen so the interaction has a causal reading.

The setup gives you four cells: treated and untreated, before and after. Three of them tell you what you need to construct a counterfactual:

The comparison group’s change over time (b₂) says what would have happened anyway.
The treatment group’s starting point (b₀ + b₁) says where they began.
Add the two and you have where the treatment group would have ended up without the program: A + B + C.

The fourth cell is what actually happened. The gap is b₃.

This is why the counterfactual line on the slide is dashed: it is not data. It is constructed from the comparison group’s trend, transplanted onto the treatment group’s starting level.

The assumption hiding in the dashed line

Worth being explicit, because the slides show the mechanics rather than the caveat: DiD assumes the two groups would have moved in parallel absent the program. That is what licenses using the comparison group’s change as the treatment group’s counterfactual.

Notice what this buys you. The two groups are allowed to differ in level — that is exactly what b₁ absorbs, and it is why DiD is useful when treatment was not randomly assigned. What they are not allowed to differ in is trend.

Look again at the first example: the treatment group starts at 55 and the comparison at 75. They are not comparable groups. DiD does not care, as long as both were on track to improve by the same amount.

How this connects to the rest of the sequence

p-05 asked whether a program worked. This deck asks whether it worked differently for different groups — the same coefficient plot, split.
p-07 warned that omitted variables bias your estimate. DiD is one of the standard responses: instead of measuring the confounders, difference them out. Anything fixed about a group cancels when you take the before-and-after difference.

That last point is worth holding onto. The comparison-group design is not just a convenience — it is a way of neutralising unmeasured differences without ever naming them, which is the same ambition as randomization, achieved by a different route.

How to read the slides

Slides	What they are doing	What to take away
1–2	Framing	Two kinds of interaction, one idea.
3	The two-group, two-period plot	75→95 and 55→85. Compute the DiD yourself before turning the page.
4	The counterfactual added	The dashed line is constructed, not observed.
5–6	The regression version	Map A, B, C, D onto b₀ through b₃.
7	Framing part two	Now the slopes vary.
8	Intercept terms	b₀ is Type C’s intercept; b₁ is A’s offset from it.
9	Slope terms	b₃ is C’s slope; b₄ is the difference; A’s slope is b₃ + b₄.
10	Four models, four baselines	Same R², different coefficients. The reference category is a choice.

Section three

What should be clear in my mind?

This deck has no closing checklist, so these are the things to be able to do.

Read a difference-in-difference table. Given b₀ through b₃, state all four cell means and identify which one is the counterfactual.
Say what the interaction coefficient means. It is the difference in differences — the extra change in the treatment group beyond the common trend.
Recover a group’s slope from an interaction model. Reference group: b₃. Any other group: b₃ plus its own interaction term.
Name the omitted category before interpreting anything. Every coefficient is relative to it.
State the parallel-trends assumption and explain why differing levels between groups are acceptable but differing trends are not.

Key takeaways

An interaction is a product term that asks whether an effect differs across groups.
Dummies alone move intercepts; interactions move slopes. You can use both at once, and the corn example does.
Every coefficient is a comparison to the omitted reference group. Nothing is absolute.
A group’s slope is the reference slope plus its interaction term, not the interaction term alone.
Changing the reference category changes every number and no fitted line. R² is the tell: it does not move.
In DiD, b₃ is the program effect and A+B+C is the counterfactual.
The comparison group’s trend is the counterfactual trend, which is an assumption, not a measurement.
DiD tolerates level differences between groups but not trend differences.