Summary
- A dummy variable takes the value 1 when a condition holds and 0 otherwise.
It converts a category into something a regression can use, and “a one-unit
increase” becomes “switching groups”.
- With one dummy and nothing else, the intercept is the base group mean and
the coefficient is the difference in means. On the wage data these are
9.818 and 1.076, matching the two group means exactly.
- A regression on a single dummy is the two-sample \(t\)-test of Unit
5. The \(t\) statistics agree to every digit. Unit
5 was a special case of regression all along.
- Adding controls turns that raw comparison into a comparison among people alike
in other respects. The urban premium falls from 1.076 to 0.805.
- With \(g\) categories, include \(g-1\) dummies. Including all \(g\) alongside an
intercept is the dummy variable trap — a violation of CLRM7, since the
dummies sum to the intercept’s column of ones.
- The omitted category is the base group. Changing it changes every
coefficient and no part of the fit. A dummy coefficient means nothing until
the reader knows the base.
- Never code an unordered category as 1, 2, 3. That imposes equal spacing and an
ordering the variable does not have.
- An interaction term \(x \times D\) lets the slope differ across groups.
Without one the two group lines are forced parallel.
- In the interacted model, \(\beta_2\) is the difference in intercepts and
\(\beta_3\) the difference in slopes, so the two groups get entirely separate
lines: 8.0% per year of schooling in rural areas against 6.9% in urban.
- With an interaction present, the coefficient on \(x\) alone is the effect in
the base group, not the effect of \(x\).
- The Chow test asks whether one line fits both groups:
\(H_0: \beta_2 = \beta_3 = 0\). It is the \(F\) test of Section 8.6
applied to a group split.
- Separate regressions and one interacted regression are the same model. Their
sums of squared residuals are identical, and both give \(F = 1198.5\) here. The
interacted version is more informative and easier to control.
- A rejection means the groups are detectably different, not that the difference
matters, and not that the difference is causal.