One of the reasons why it’s important to think about explicitly about downward causation in society is how it interacts with considerations of social and economic justice.
Purely bottom-up effects can seem to have a different social valence than top-down effects.
One example, as noted by David Massad, has to do with segregation in housing. Famously, the Schelling segregation model shows how segregation in housing could be the result of autonomous individual decisions by people with a small preference for being with others like themselves (homophily). But historically in the United States, one factor influencing segregation was redlining, a top-down legal process.
Today, there is no question that there is great inequality in society. But the mechanism behind that inequality is unknown (at least to me, in my current informal investigation of the topic). One explanation, no doubt overly simplified, would be to say that wealth distribution is just a disorganized heavy tail distribution. A more specific account from Piketty would frame the problem as an organized heavy tail distribution based on the feedback effect of the relative difference in rate of return on capital versus labor. Naidu would argue that this difference in the rate of return is due to political agency on the part of capitalists, which would imply a downward causation mechanism from capitalist class interest to individual wealth distributions.
The key thing to note here is that the mere fact of inequality does not give us a lot to distinguish empirically between these competing hypotheses.
It is possible that the specific distribution (i.e cumulative density function) of inequality can shed light on which, if any, of these hypotheses hold. To work this out, we would need to come up with a likelihood function for the probability of the wealth distributions occurring under each hypothesis. Likely the result would be subtle: the difference in the likelihood functions would be about not that but how much inequality results, and whether and in what ways the wealth distribution is stratified.
Of course, another approach would be to collect other data besides the wealth distribution that bears on the problem. But what would that be? The legal record of the tax code, perhaps. But this does not straightforwardly solve our problem. Whatever the laws are and however they have changed, we cannot be sure of their effect on economic outcomes without testing them somehow against the empirical distribution again.
Another challenge to teasing these hypotheses apart is that they are not entirely distinct from each other. A disorganized heavy tail distribution posits a large number of contributing factors. Difference in rate of return on capital may be one important factor. But is it everything? Need it be everything to be an important social scientific theory?
A principled way of going about the problem would be to regress the total distribution against a number of potential factors, including capital returns and income and whatever other factors come to mind. This is the approach naturally taken in data science and machine learning. The result would be the identification of a vector of coefficients that would indicate the relative importance of different factors on total wealth.
Suppose there are 20 such factors, any one of which can be removed with minimal impact on the overall outcome. What then?