# Partial/marginal effects after probit regression

Is it plausible to get positive coefficients after running a probit but negative partial/marginal effects? If so, what is the intuition?

Yes, this can be the case. There is a widely cited paper in Economics Letters (2003) by Chunrong Ai and Edward C. Norton discussing this issue for interaction terms. The problem is the following, assume $x_1$, $x_2$ are continuous and this is your probit model (translates of course to a logit model etc.)

$E[\,y\,|\,x_1,x_2,X\,]=\Phi(\,\beta_1x_1+\beta_2x_2+\beta_{12}x_1x_2+X\beta$ )

Here, you are interested in the effect of the interaction term $x_1x_2$.

In a linear model you would just look at the coefficient $\beta_{12}$. But in a nonlinear model the (total) effect of the interaction term is the following cross derivative (see the original paper):

$\frac{\partial^2\Phi}{\partial x_1 \partial x_2}=\beta_{12}\Phi' + (\beta_1+\beta_{12}x_2)(\beta_2+\beta_{12}x_1)\Phi''$

and this cross derivative depends on $x_1$ and $x_2$. Therefore you can get different results for the coefficient or the marginal effect at the mean (and also different signs).

The original paper includes also an example. Interesting are also subsequent papers discussing this topic such as Puhani (2012), here a working paper version.

• While this in true, most software does not implement this method (unless you code it yourself), so perhaps the real answer to the question is no. Of course without OP specifying the data and model more clearly, it is anybody's guess. – Repmat Sep 3 '16 at 10:52