Interaction Variable (Effect)

Definition

An interaction variable (or interaction term) captures how the effect of one independent variable on the dependent variable depends on the level of another variable.

Purpose

Used when the relationship between $X_1$ and $Y$ changes depending on $X_2$.

Formulation

For two predictors $X_1$ and $X_2$:

$$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+{\beta }_3(X_1\times x_2)+\epsilon$$

• ${\beta }_3$: the interaction coefficient
• $X_1\times X_2$: the interaction term

Interpretation

• If ${\beta }_3\ne 0$, the effect of $X_1$ on $Y$ depends on $X_2$.
• The marginal effect of $X_1$ is:

$$\frac{\delta Y}{\delta X_1}={\beta }_1+{\beta }_3{X}_2$$

$\Rightarrow$ The slope of $X_1$ changes with $X_2$.

Example

Given:

• $X_1$ = Hours studied
• $X_2$ = Tutoring (1=yes, 0=no)
• $X_1\times X_2$: interaction between study hours and tutoring

Interpretation:

The effect of study hours on grades is different for students with tutoring than for those without.

Visualization Tip

Plot $Y$ vs $X_1$ for different fixed values of $X_2$. Non-parallel lines $\to$ interaction.