17.2 Marginal Effects in Different Contexts

1. 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.

2. 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).

3. 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 $X$.

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