Average marginal effect is not a beta in OLS?

Question

since average marginal effect in a regression model $y = m(X) + e = \alpha + \beta X + e$ should be $\int m'(X)f_X(x)dx$, could it be the case that $\hat{\beta}_{OLS}$ will be equal to AME only in some cases, like when $X$ is normal? Am I right that generally $\hat{\beta}_{OLS}$ is not equal to AME?

Answer 1

OLS parameters and average marginal effects can differ if the model is nonlinear. For example, if you compute AME for the following OLS regression, then AME and beta coefficients will differ: $$y_i=\beta_1+\beta_2 x_i+\beta_3 x_i^2+\varepsilon_i.$$ How could average marginal effects of $x$ be equal to either $\beta_2$ and $\beta_3$, if it just returns one number? They couldn't.

More generally, other statistical models like logistic regression or negative binomial regression are nonlinear by design, so the estimated coefficients generally differ from the average marginal effects.

Indeed, the only case where average marginal effects and coefficients generally coincide is in OLS with exclusively linear terms (unlike the above example).

Answer 2

Yes, you, of course, right, that theoretically AME and $\beta$ coincide in purely linear model. But since $\hat{\beta}_{OLS} = cov(X,y)/var(x)$, then $\hat{m}'(x) \times f_X(x)$ generally not constant times integral over density of $X$, right?