Estimating elasticity using different regression models My question is how to estimate elasticity based on a Probit model. I know the following formulas are used to estimate elasticity based on OLS (1) and logit (2) models (Ewing & Cervero, 2001): 
(1) ϵ=β(X/Y)
(2) ϵ=βX(1-P)
where ϵ is the elasticity, β is the coefficient of the independent variable of interest, X is the sample mean of the independent variable, Y is the sample mean of the dependent variable, and P is the mean estimated probability of occurrence. Given this, how should I estimate elasticity when I have a probit regression?
Reference:
Ewing, R., & Cervero, R. (2001). Travel and the built environment: a synthesis. Transportation Research Record: Journal of the Transportation Research Board, (1780), 87-114.     
 A: You know that the conditional mean of the dependent variable in a probit model is
$$E[y=1 \vert x,z]=\Phi(\alpha + \beta \cdot x +z \cdot \pi).$$
This is also you predicted probability, $\Pr(y=1 \vert x,z)$. Here $\Phi(.)$ is the standard normal cdf, and $\varphi(.)$ is is the standard normal pdf, which will be used below.
The standard elasticity formula is
$$ \epsilon = \frac{\partial E[y \vert x,z]}{\partial x} \cdot \frac{x}{E[y \vert x, z]}$$
The marginal effect for a continuous variable $x$ in a probit model is
$$\frac{\partial E[y=1 \vert x,z]}{\partial x}=\varphi(\alpha + \beta \cdot x +z \cdot \pi)\cdot \beta .$$
This comes from applying the chain rule for derivatives to the expectation and the fact that $\varphi()$ is the derivative of $\Phi()$. You can plug that in for the first term in the elasticity formula. The denominator in the second term is just the predicted probability.
For a discrete $z$, a semi-elasticity makes more sense (percent change in probability when $z$ goes from 0 to 1). You can approximate a small percentage change in the probability by taking differences of logged predictions with $z$ set to one less the prediction with $z$ set to zero:
$$\ln \Phi(\alpha + \beta \cdot x +1 \cdot \pi) - \ln \Phi(\alpha + \beta \cdot x +0 \cdot \pi).$$
Some people multiply this by 100 to convert to percent.
For both the elasticity and the semi-elasticity, you can average these in some sample or calculate them at representative/interesting values of the covariates. You can also plot curves or surfaces of these functions.
