# What are average partial effects?

Does anybody know the meaning of average partial effects? What exactly is it and how can I calculate them? Here is a reference that might help.

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I don't know why anybody downvoted this question, but it may be related to the ease with which Googling "average partial effects" (or, better yet, "average partial effects" definition) turns up great references. Nevertheless, a clear answer by an expert would be very welcome here. –  whuber Jun 24 '11 at 14:36
Unfortunately, that link appears to be broken. –  Macro Jul 17 '12 at 17:48

I don't think there is a consensus on terminology here, but the following is what I think most people have in mind when someone says "average partial effect" or "average marginal effect".

Suppose, for concreteness, that we are analyzing a population of people. Consider the linear model $$Y = \beta X + U,$$ where $(Y,X)$ are observed scalar random variables, and $U$ is an unobserved scalar random variable. Suppose that $\beta$ is an unknown constant. Suppose this is a structural model, meaning that it has a causal interpretation. So, if we could pick a person out of the population and increase their value of $X$ by 1 unit, then their value of $Y$ would increase by $\beta$. Then $\beta$ is called the marginal or causal effect of $X$ on $Y$.

Now, assuming that $\beta$ is a constant means that no matter which person we pick out of the population, a one unit increase in $X$ has the same effect on $Y$---it increases $Y$ by $\beta$. This is clearly restrictive. We can relax this constant effect assumption by supposing that $\beta$ itself a random variable---each person has a different value of $\beta$. Consequently, there is an entire distribution of marginal effects, the distribution of $\beta$. The mean of this distribution, $E(\beta)$, is called the average marginal effect (AME), or average partial effect. If we were to increase everyone's value of $X$ by one unit, then the average change in $Y$ is given by the AME.

Alternatively, consider the nonlinear model $$Y = m(X,U),$$ where again $(Y,X)$ are scalar observables and $U$ is a scalar unobservable, and $m$ is some unkown function (assume it is differentiable for simplicity). Here the causal/marginal effect of $X$ on $Y$ is $\partial m(x,u)/\partial x$. This value may depend on the value of $U$. Thus, even if we look at people who all have the same observed value of $X$, a small increase in $X$ will not necessarily increase $Y$ by the same amount, because each person may have a different value of $U$. Thus there is a distribution of marginal effects, just as in the linear model above. And, again, we can look at the mean of this distribution: $$E_{U \mid X} \left[ \frac{\partial m(x,U)}{\partial x} \mid X=x \right].$$ This mean is called the average marginal effect, given $X=x$. If we assume $U$ is independent of $X$, as is sometimes done, then the AME at $X=x$ is simply $$E_{U} \left[ \frac{\partial m(x,U)}{\partial x} \right].$$ In general, an average marginal effect is just a derivative (or sometimes a finite difference), of a structural function (such as $m(x,u)$ or $\beta x + u$) with respect to an observed variable $X$, averaged over an unobserved variable $U$, perhaps within a particular subgroup of people with $X=x$. The precise form of this effect depends on the specific model under consideration.

Also note that these objects might also be called average treatment effects, especially when considering a finite difference. For example, the difference of the structural function at $X=1$ ('treated') and at $X=0$ ('untreated'), averaged over the unobservables.

Finally, to be clear, note that when I refer to 'distributions' above, I mean distributions over the population of people. Each person in the population has a value of $U$, of $X$, and of $Y$. Hence there is a distribution of these values if I look over all people in the population. The thought experiment here is the following. Take all people with $X=x$. Now take one of these people, and increase their $X$ value by a small amount, but keep their $U$ value the same, and we write down the change in their $Y$ value. We do this for each person with $X=x$, and then average the values. This is what it means to average over $U \mid X=x$.

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