Marginal effects measure the change in the conditional mean of outcome $y$ when regressors change by one unit.
For a linear model, $E[y | x,z]=\alpha + \beta x + \gamma z$, the partial derivative with respect to $x$ is $$\frac{\partial E[y | x]}{\partial x} = \beta, $$ so that the coefficient has a direct interpretation as a marginal effect of $x$ on $y$.
For nonlinear regression models, this interpretation is no longer possible. For example, if $E[y | x]=\exp \left(\alpha + \beta x + \gamma z \right)$ like in the Poisson model for count data, the marginal effect is a function of both parameters and regressors: $$\frac{\partial E[y | x]}{\partial x} = \exp \left(\alpha + \beta x + \gamma z \right)\cdot \beta$$
It is customary to present such marginal effects
- evaluated at own values of $x$ and $z$ and averaged for all individuals in the sample (average marginal effect or AME)
- evaluated at the mean/median/modal values of $x$ and $z$ (marginal effect at representative values or MER)
- estimated at specific values that are interesting to the analyst
These three measures will generally differ in nonlinear models, whereas they will agree in a linear one. Moreover, the sign of the marginal effect may change at different values of regressors: it may be positive for some values of $x$ and negative for others. Similar complication arise for interactions between variables.
Finally, for binary ($0/1$) regressors in nonlinear models, many prefer the finite difference rather than the derivative:$$ \Delta E[y | x]=E[y | x=1]-E[y | x=0]$$ (where of course all other quantities that might be involved in evaluating the expectations are held constant).
