Reading coefficients when there is an interaction term

Question

You fit y = \beta0 + \beta1 x1 + \beta2 x2 + \beta3\,(x1 x2) + \varepsilon by OLS, where x2 is a 0/1 indicator (say, "high-volatility regime"). Interpret \beta1 and \beta3 precisely. Why is it a mistake to read \beta1 as the overall effect of x1?

Accepted Answer

The effect of x1 is no longer a single number. Differentiate: \frac{\partial\,\mathbb{E}[y]}{\partial x1} = \beta1 + \beta3\,x2. The slope on x1 depends on the value of x2. So there is no single "effect of x1"; there is one effect in each regime. Interpreting \beta1. It is the effect of x1 on y when x2 = 0, the baseline (low-volatility) regime. It is emphatically not the average or overall effect of x1; it is the effect only at the reference level of x2. Reading it as the general slope is the classic interaction-term error. Interpreting \beta3. It is how much the x1 slope changes when x2 goes…

Reading coefficients when there is an interaction term

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