Dummy Interactions and the Chow Test
Interacting a dummy with a continuous regressor lets the slope differ across groups, not just the intercept. In , the group with has intercept and slope . Dummy-by-dummy interactions capture combined category effects. To test whether an entire regression differs across two groups, the Chow test is an test of the joint null that all intercept and slope coefficients are equal, equivalent to testing every interaction at once.
Try it yourself
One model, many specifications. See how functional form, a squared term, and a dummy variable each change what a coefficient means and how the fitted line bends. The same seeded data sit under all three views.
group D: ŷ = 15.0 + 1.05·x
δ₀ shifts the intercept by 7.0. With the interaction off the slopes match, so the lines stay parallel.
Discussion. With the interaction off, δ₀ only lifts the line; turn it on and δ₁ tilts the slope. Which question does each parameter answer, and why must the base group and the main effect stay in the model for δ₀ and δ₁ to be readable?
y = β₀ + δ₀D + β₁x + δ₁(D·x) + u. The base group (D = 0) has intercept β₀ and slope β₁; group D has intercept β₀ + δ₀ and slope β₁ + δ₁. With the interaction off, δ₁ = 0 and the lines are parallel.
Why it matters
A plain dummy moves a line up or down but keeps it parallel. Sometimes groups differ in steepness too: maybe each year of schooling pays off faster for one group than another. Interacting the dummy with the continuous variable lets each group have its own slope. If you want a single yes-or-no answer about whether the whole relationship is different across groups, the Chow test bundles all those differences into one joint hypothesis test.
Formulas
Worked examples
Does the wage-education relationship differ for men and women in both level and slope?
Run `regress lwage i.female##c.educ`. Stata reports the female intercept shift, the base education slope, and the `female#c.educ` interaction (the slope difference). Then `testparm i.female i.female#c.educ` performs the joint Chow-style test that the equation is identical across genders.
Test whether a wage equation has the same coefficients before and after a policy year, using a `post` dummy.
Interact `post` with every regressor, for example `regress lwage i.post##(c.educ c.exper)`, then `testparm i.post i.post#c.educ i.post#c.exper`. A significant joint statistic rejects coefficient stability, indicating a structural break between the two periods.
Common mistakes
- ✗Believing a single dummy already allows different slopes. A lone dummy only shifts the intercept; you need the interaction term for the slope to differ.
- ✗Forgetting the main effects. Both the dummy and the continuous variable must appear on their own alongside the interaction, or the interaction coefficient is not interpretable.
- ✗Thinking the Chow test checks only intercepts. It is a joint test of all coefficients, slopes included, equal across groups.
- ✗Applying the standard Chow test under heteroskedasticity. It assumes a common error variance across groups, so use a robust Wald test (`testparm` after robust estimation) when that fails.
Revision bullets
- •Dummy times continuous lets the slope differ by group.
- •Group slope is ; intercept is .
- •Keep all main effects in alongside the interaction.
- •The Chow test is an test that all coefficients are equal across groups.
- •Use a robust Wald version when error variances differ across groups.
Quick check
In `regress lwage i.female##c.educ`, the coefficient on the interaction female#c.educ measures:
The Chow test is best described as:
Connected topics
Sources
- Wooldridge (2019), §7.4Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.Covers interactions between dummies and continuous variables and different slopes across groups.
- Wooldridge (2019), §7.4 (Chow test)Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.Presents the Chow test as an F test for equality of regression functions across groups or periods.