How to compute Average partial effects?

Question

If we have a model like this: $$\hat {Prob}(Y=1|X) = F(\hat \beta_0 + \hat \beta_1 age + \hat \beta_2 education + \hat \beta_3 salary)$$ (suppose education is a scalar variable here) where $F$ is a cdf of some distribution, then do we compute AVERAGE PARTIAL EFFECT of e.g. variable age by averaging the effects for every observation in the sample as: $$\frac{1}{n} \sum_{i=1}^n f(\hat \beta_0 + \hat \beta_1 age_i + \hat \beta_2 education_i + \hat \beta_3 salary_i) * \hat \beta_1 $$

($f$ is the density) or do we keep other variables (education, salary) at their means during the computation as: $$\frac{1}{n} \sum_{i=1}^n f(\hat \beta_0 + \hat \beta_1 age_i + \hat \beta_2 \overline {education} + \hat \beta_3 \overline {salary}) * \hat \beta_1 $$

$\overline {education} = \frac{1}{n} \sum_{i=1}^n education_i$ and the same for salary.

Which one of the two is used when computing Average partial effects?

Answer 1

It is the first one. In general, the average partial effect of a continuous variable $x_k$ is $$\frac{\beta_k}{n}\sum_{i=1}^nf(\textbf{x}_i\beta),$$ where $\textbf{x}_i$ is the observed vector of explanatory variables corresponding to observation $i$. To estimate the average partial effect of $x_k$ you can thus compute $$\frac{\hat\beta_k}{n}\sum_{i=1}^nf(\textbf{x}_i\hat\beta).$$ Thus, since $age$ is continuous, it is the first expression you need to compute.