interpreting the coefficient on a logged independent variable from a probit model

Question

I am using a double model with log transformed independent variables and have calculated average partial effects. Now I am now not sure how to interpret the coefficients; particularly those from the first stage which is supposed to be capturing the probability of participation using a probit model.

Answer 1

There's some hand-wavy math in what follows, but hopefully the emphasis on intuition over rigor is worthwhile.

Your first stage model is essentially a probit: $$\Pr(y=1 \vert x)=\Phi(\alpha + \beta \cdot \ln x),$$ where $\Phi()$ is the standard normal CDF. You want to know the marginal effect of $x$. Taking the derivative with respect to $x$ and applying the chain rule yields $$\frac{\partial\Pr(y=1 \vert x)}{\partial x}=\frac{\partial \Phi(\alpha + \beta \cdot \ln x)}{\partial x}=\varphi(\alpha + \beta \cdot \ln x)\cdot\frac{\beta}{x},$$ where $\varphi()$ is the standard normal PDF. This is approximately equivalent to $$\frac{\Delta \Pr (y=1 \vert x)}{100 \cdot\frac{\Delta x}{x}}=\frac{\varphi(\alpha + \beta \cdot \ln x)\cdot\beta}{100}$$ The left hand side is an elasticity: it's the change in the conditional probability of a success for a 1% change in $x$. This is equal to the partial effect that I am guessing you calculated, divided by 100. Of course, you can divide by numbers other than 100, including 1, which corresponds to doubling $x$.