Dummy variables, the dummy trap and interactions

A dummy variable (also called an indicator or a binary variable) is 1 when an observation has some property and 0 when it does not. Put it in a regression next to a quantitative regressor and it works like a switch that adds a fixed amount to the prediction when the property holds:

For a base-group member the prediction is a + bx. For a member of the other group it is a + bx + d. The two lines have the same slope b and differ by d everywhere, so they are parallel and d is called the differential intercept.

In the survey, 244 of 600 heads of household live in a town. Their earnings average 19.13 (₹ thousand a month) against 11.78 in villages, a gap of 7.35. Regress earnings on the town dummy alone and the coefficients are exactly these averages: intercept 11.78, dummy 7.35. The derivation shows why this must happen.

Now hold other things constant. Add schooling and the head’s ability to the regression and the town coefficient falls to 3.04 (we wrote the model with a true town effect of 3). Most of the raw gap of 7.35 was town heads being better schooled and abler, not towns paying more for the same person. A dummy coefficient in a multiple regression is a gap at the same values of everything else in the model, and which controls are included changes what it measures.

Three small readings the exams ask for:

• Averages from a dummy regression. If Ŵ = 400 − 60D for 300 people, 200 of them men (D = 1), women average 400 and men 340, and the sample average is 400 − 60 × 200 ÷ 300 = 360. The sample average is the intercept plus the dummy coefficient times the share in the group.
• Precision. The variance of the estimated gap is σ²(1 ÷ n₀ + 1 ÷ n₁), the two-sample formula. With 300 people it is smallest when the groups are equal in size: 1 ÷ 150 + 1 ÷ 150 = 0.0133 against 1 ÷ 200 + 1 ÷ 100 = 0.015 for a 200/100 split. Lopsided groups give a less precise gap.
• A dummy for a characteristic. In a house-price regression, a coefficient of −42.5 on a “flood-prone” dummy, with price in thousands of rupees, means a flood-prone house sells for about 42.5 thousand rupees (₹42,500) less than an otherwise identical house. The standard error and t test work as for any coefficient.

A dummy alone forces the two groups to have the same slope. If the payoff to schooling may differ between towns and villages, multiply the dummy by the quantitative variable and include the product, an interaction term:

In the lab, choose between the dummy alone and the dummy with its interaction, code the dummy 0/1 or −1/+1, and move the slider to build a real extra payoff to schooling into town earnings. Watch the lines, and watch which coefficient carries which change.

Reading the interaction model:

• g is the difference in slopes. A t test on g tests whether the payoff to x differs between the groups. In the survey with the slider at 0 the true difference is zero, because we wrote none, yet the t statistic comes out above 2: a false alarm. A 5% test gives one about one time in twenty, and the survey’s floor on earnings (nobody earns less than ₹1 thousand) makes it a little likelier. Raise the slider and the interaction picks up the extra payoff you build in. A t of 2.5 alone is not proof of a different slope.
• d is the gap at x = 0 only. Once the interaction is in, the gap between the groups is d + g·x, which changes with x. The coefficient on D alone is the gap at x = 0, which may be a value nobody has (zero years of experience, zero income). Centre x on its mean if you want d to mean the gap at a typical x.
• The coefficient on a variable that appears in an interaction is its effect for the base group. In y = α + βx + γUrban + θImmigrant + δUrban×Immigrant, θ is the difference between immigrants and non-immigrants who do not live in a city (Urban = 0). For city dwellers it is θ + δ.
• To test whether the groups have the same line at all, test d = 0 and g = 0 together with an F test (lesson 3). That is the dummy-variable form of the Chow test for a structural break.

A variable with m categories (four seasons, five regions) needs m − 1 dummies when the regression has a constant. The omitted category is the base, and each dummy coefficient is a gap from it. The survey’s heads of household fall into three education groups, by years of schooling:

Take primary as the base, with a constant and two dummies: the constant is 6.028, the secondary dummy is 8.234 (14.262 − 6.028) and the higher dummy is 17.725 (23.753 − 6.028). Now take secondary as the base: the constant becomes 14.262, and the other two coefficients become −8.234 and 9.491. Everything is rewritten, and nothing has changed: both regressions give the same fitted values, the same residuals and the same R², 0.6721. Try it, and then try to include all three dummies together with the constant.

There are two ways out, and both are standard:

• Constant and m − 1 dummies. Each coefficient is a gap from the omitted group. R² is the usual R².
• m dummies and no constant. Each coefficient is simply that group’s average (here 6.028, 14.262, 23.753). The regression is fine (the model y = β₁D₁ + β₂D₂ + ε with D₂ = 1 − D₁ can be estimated), but the R² a package prints is the uncentred one, 0.9327 here, which is not comparable with the R² of regressions that have a constant.

Two papers print “quantitative variable” where they mean a qualitative one with m categories; the rule is for qualitative variables, and with a constant the answer is m − 1 dummies. A regression that has a constant plus both a Pass and a Fail dummy cannot be estimated, for the same reason: Pass + Fail = 1.

Recoding a dummy rewrites the same line in new coefficients. Suppose y = a + bx + dD with D = 0 or 1. Define a new variable E = 2D − 1, which is −1 for the base group and +1 for the other group, so D = (E + 1) ÷ 2. Substitute:

In every row the fitted values, the residuals, R² and the t statistic of the dummy (up to sign) are identical. Only what the numbers are called changes. The same is true of changing the base category with m groups (chapter 3), and of rescaling x (lesson 6), which changes b but not the dummy’s coefficient. Switch the lab above to −1/+1 coding and compare the lines.

Suppose the slope of spending on income changes once income passes a threshold X*, say 400. Two ways to write it:

• Differential slope, jumps allowed: y = a + bX + cD·X, where D = 1 when X ≥ X*. The two lines have different slopes and the lines meet at X = 0, not at the threshold, so the prediction can jump at X*.
• Piecewise linear (a kink): y = a + bX + c·D·(X − X*). The extra term is zero at X = X*, so the two pieces join there, and the slope is b below the threshold and b + c above it.

Take Ŷ = 100 + 0.5X − 0.2(X − 400)D. Below 400 the slope is 0.5; above, 0.5 − 0.2 = 0.3. At X = 400 both pieces give 100 + 0.5 × 400 = 300, so the line bends there without a gap. At X = 500 the prediction is 100 + 250 − 20 = 330. The coefficient on D·(X − X*) is the change in slope, and a t test on it tests whether the kink is real. This is the econometrics behind kinked Engel curves, tax schedules and piecewise-linear spline models.

• Is there a gender difference, after controlling for the rest? Include a gender dummy next to the controls and test its coefficient (a t test). That is the simplest correct way to test it, and it is the survey’s town coefficient of 3.04 with its t statistic of 12.9.
• ANOVA models have only dummy variables as regressors, so they compare group averages. ANCOVA models mix dummies with quantitative regressors, which is what the lessons above have been doing: the quantitative variable controls for something while the dummies compare groups.
• Seasonal dummies. For quarterly data with a constant, three dummies for three quarters, the fourth being the base. Each coefficient is the average seasonal gap from the base quarter after the other regressors are held fixed.
• A dummy as the dependent variable is a different model (a probability), covered with logit and probit in lesson 14.

The Delhi School’s questions on dummies (a dummy for gender and the most precise estimate of the gap, a poor-condition dummy in a home-sales regression, a kinked Engel curve) are in the hard-problems lesson at the end of the module. The case where the dependent variable is a dummy is in lesson 14, on the linear probability model, logit and probit.