Questions: Interaction Terms in Regression

Question 1

In the model Wage = β₀ + β₁*Education + β₂*Female + β₃*(Education × Female) + ε, what does β₁ represent?

• A. The average return to education across all workers in the sample
• B. The return to education for male workers specifically (the reference group when Female = 0)
• C. The return to education for female workers specifically
• D. The difference in education returns between men and women

Answer: (B)

When an interaction is present, β₁ is the effect of Education on Wage *when all other interacted variables equal zero* — here, when Female = 0, meaning for male workers. It is not the average effect for all workers; that average would require integrating over the distribution of Female. β₃ is the *difference* in returns (how much the female education return differs from the male return), and the female return to education is β₁ + β₃. This is the critical interpretation trap: β₁ alone no longer summarizes the effect of Education for anyone other than the reference group.

Question 2

A researcher centers Education by subtracting its mean before including it in an interaction model. Which of the following correctly describes the effect of centering?

• A. Centering changes the interaction coefficient β₃ and improves model fit
• B. Centering changes the interpretation of main effect coefficients but leaves β₃, model fit, and predicted values unchanged
• C. Centering eliminates multicollinearity between the main effects and the interaction term
• D. Centering changes predicted values for observations near the mean of Education

Answer: (B)

Centering is a reparameterization, not a change in model. β₃ (the interaction coefficient) is invariant to centering. Model fit (R², residuals) is unchanged. Predicted values for any observation are identical. What changes is that the main effect coefficients now refer to the effect at the mean of Education rather than at zero — a more interpretable and estimable quantity when zero is outside the data range. Option C is a common misconception: centering reduces the *sample* correlation between X and X*Z, but does not eliminate the structural collinearity between them.

Question 3

In a regression with an interaction term X*Z, the marginal effect of X on Y is a function of Z rather than a single constant.

• T. True
• F. False

Answer: True

The marginal effect ∂Y/∂X = β₁ + β₃Z. When Z = 0, the effect is β₁; when Z takes other values, the effect changes accordingly. This is precisely what the interaction term captures: the relationship between X and Y is not fixed but depends on the level of another variable. Failing to recognize this leads to the common error of reporting β₁ as 'the effect of X' and ignoring that this is only correct at Z = 0.

Question 4

If you estimate a model with an interaction term and find that β₁ is small and statistically insignificant, this tells you that X has no effect on Y.

• T. True
• F. False

Answer: False

With an interaction term, β₁ is the effect of X *only when Z = 0*. If Z = 0 is not a meaningful or common value in the data, β₁ is essentially extrapolation and an insignificant β₁ says nothing about X's effect at typical values of Z. The full marginal effect is β₁ + β₃Z, so X can have a large, significant effect at common values of Z even when β₁ alone is near zero. Evaluating 'the effect of X' requires computing the marginal effect at substantively relevant values of Z.

Question 5

Why is a plot of predicted values of Y against X at multiple representative values of Z more informative than simply reporting the interaction coefficient β₃?

Model answer: β₃ alone tells you whether the relationship between X and Y changes with Z and by how much per unit of Z, but it does not convey the practical magnitude of those differences at the values of Z that actually appear in the data. A plot shows the full conditional relationship: whether the lines diverge, cross, or remain nearly parallel across the data range. Crossing lines signal that X has opposite effects at different levels of Z — a substantively important finding that a single coefficient cannot communicate. The plot forces the analyst to think about the entire function β₁ + β₃Z rather than a single-number summary.

This visualization approach is the recommended diagnostic for interaction models because it translates the abstract coefficient into concrete predictions. If the plotted lines are parallel (same slope at all Z values), there is no meaningful interaction even if β₃ is technically nonzero. If lines diverge substantially, the interaction matters. If they cross within the data range, the sign of X's effect reverses — a finding that policy analysis must take seriously. The plot also reveals whether any of those representative Z values are in-sample, preventing misleading extrapolations.