17.2 Marginal Effects in Different Contexts
- Linear Regression Models
For a simple linear regression:
\[ E[Y|X] = \beta_0 + \beta_1 X, \]
the marginal effect is constant and equal to \(\beta_1\). This makes interpretation straightforward.
- Logit and Probit Models
In logistic regression, the expected value of \(Y\) is modeled as:
\[ E[Y|X] = P(Y=1|X) = \frac{1}{1 + e^{-\beta_0 - \beta_1 X}}. \]
The marginal effect is given by:
\[ \frac{\partial E[Y|X]}{\partial X} = \beta_1 P(Y=1|X) (1 - P(Y=1|X)). \]
Unlike linear models, the effect varies with \(X\), requiring evaluation at specific values (e.g., means or percentiles).
- Interaction Effects and Nonlinear Terms
When models include interactions (e.g., \(X_1 X_2\)) or transformations (e.g., \(\log(X)\)), marginal effects become more complex. For example, in:
\[ E[Y|X] = \beta_0 + \beta_1 X + \beta_2 X^2, \]
the marginal effect of \(X\) is:
\[ \frac{\partial E[Y|X]}{\partial X} = \beta_1 + 2\beta_2 X. \]
This means the marginal effect depends on the value of \(X\).