IDC Revision

Recipe

Computing marginal effects (logit / probit)

In LPM the coefficient is the marginal effect, but in logit and probit the raw β^j\hat\beta_j tells you only the direction of the effect — the magnitude depends on where you are on the S-curve. To get a probability change you have to evaluate g(x′β^)⋅β^jg(\mathbf{x}'\hat{\boldsymbol\beta}) \cdot \hat\beta_j, where gg is the PDF of the link distribution. The standard protocol evaluates this at the sample mean (the marginal effect for the average person).

  1. Get the coefficient estimates β^0,β^1,…,β^k\hat\beta_0, \hat\beta_1, \ldots, \hat\beta_k from R output (summary(glm(...))).
  2. Calculate sample means of all explanatory variables: xˉ1,xˉ2,…,xˉk\bar x_1, \bar x_2, \ldots, \bar x_k.
  3. Compute the linear prediction at the mean: xˉ′β^=β^0+β^1xˉ1+⋯+β^kxˉk\bar{\mathbf{x}}'\hat{\boldsymbol\beta} = \hat\beta_0 + \hat\beta_1 \bar x_1 + \cdots + \hat\beta_k \bar x_k.
  4. Evaluate the scaling factor:
    • Probit: g=φ(xˉ′β^)g = \varphi(\bar{\mathbf{x}}'\hat{\boldsymbol\beta}) using dnorm() in R.
    • Logit: gg = dlogis() evaluated at xˉ′β^\bar{\mathbf{x}}'\hat{\boldsymbol\beta}.
  5. Multiply: marginal effect of xjx_j = g×β^jg \times \hat\beta_j.

Common pitfalls

  • Comparing raw β^j\hat\beta_j across LPM / logit / probit. The three are on different scales — compare marginal effects, never raw coefficients. The Logit coefficient on schooly (0.128) being 6× the LPM coefficient (0.021) doesn't mean Logit thinks education matters more; it's the same effect on a different scale.
  • Evaluating at a single non-representative point. The "average" person may not exist (e.g. with binary regressors). The margins package's summary(margins(model)) computes the average marginal effect — the mean of g(xi′β^)⋅β^jg(\mathbf{x}_i'\hat{\boldsymbol\beta}) \cdot \hat\beta_j across all observations — which is often a more honest summary.
  • Forgetting the marginal effect has its own standard error (you can't just use SE(β^j)\text{SE}(\hat\beta_j)). Use the delta method, or rely on margins::summary().

Worked example

Topic 3 women's labour supply with probit β^schooly=0.073\hat\beta_{\text{schooly}} = 0.073. At the sample means the lecture reports φ(xˉ′β^)=0.342\varphi(\bar{\mathbf{x}}'\hat{\boldsymbol\beta}) = 0.342, so ME = 0.342×0.073=0.0250.342 \times 0.073 = 0.025 — the average woman's probability of being employed rises by 2.5 percentage points per extra year of schooling. The corresponding logit ME is 0.032 and the LPM coefficient is 0.021 — all three agree on direction and on order of magnitude.