Understanding a "fundamentally unidentified question" example in causal inference

Question

This is the example I'm referring to, it is taken from Mostly Harmless Econometrics: An Empiricistís Companion by Angrist and Pischke:

Suppose that we are interested in whether children do better in school by virtue of having started school a little older. Maybe the 7-year-old brain is better prepared for learning than the 6 year old brain. This question has a policy angle coming from the fact that, in an effort to boost test scores, some school districts are now entertaining older start-ages (to the chagrin of many working mothers). To assess the effects of delayed school entry on learning, we might randomly select some kids to start first grade at age 7, while others start at age 6, as is still typical. We are interested in whether those held back learn more in school, as evidenced by their elementary school test scores. To be concrete, say we look at test scores in first grade. The problem with this question - the effects of start age on first grade test scores - is that the group that started school at age 7 is...older. And older kids tend to do better on tests, a pure maturation effect. Now, it might seem we can fix this by holding age constant instead of grade. Suppose we test those who started at age 6 in second grade and those who started at age 7 in first grade so everybody is tested at age 7. But the first group has spent more time in school; a fact that raises achievement if school is worth anything. There is no way to disentangle the start-age effect from maturation and time-in-school effects as long as kids are still in school. The problem here is that start age equals current age minus time in school. [...] The effect of start age on elementary school test scores is most likely FUQ.

by FUQ the authors mean: "fundamentally unidentified question", namely "research questions that cannot be answered by any experiment".

I'm mostly interested in the bold part of the transcript, I tried to understand it but I was a bit puzzled about this example at first, since it seemed to me that the maturation effect was actually what we wanted to estimate, since after all that's what can be considered the cause of higher learning ability. Thinking about it a bit more, I realized that probably the authors are referring to a situation as the one depicted in this (not very graphically pleasing) causal graph I made. I admit I have just a broad understanding of this type of graphs, so I could have made some error in the design:

What the authors have in mind is probably to only estimate the difference in the learning ability between 7 years old children and 6 years old children, but the maturation effect can be considered a confounder which causes both the learning ability and the result of test scores to grow, this last one through other factors relevant to test scores, such as better focus (even if I now realize that better focus could cause better learning ability, but I don't think this changes the matter). The problem resides in the fact that controlling for the maturation effect is impossible since this would mean fixing the age of the children, eliminating doing so the possibility to learn about the improved learning ability due to the age difference.

Is this what the authors have in mind? If not, what's the correct interpretation? Thank you for the help in advance!

Answer 1

What experiment could possibly answer the question?

I am going with your "cannot be answered by any experiment" definition of FUQ. This is a stronger statement than saying that the effect is not identifiable given some data generated by a DAG. Many effects are not identifiable due to a lack of experimental data, but the experiments required may well be possible and, if conducted, would render the effect identifiable. In the case described, the problem appears to be that it is not possible to design an experiment that only affects the variable we are interested.

We cannot change school start time without changing many other important things

For example, one could think of an RCT assigning different school start times, but this would by definition also affect the time spent in school by a certain age. To keep the time spent in school the same, one could let some students take a school year at 6, then take a year break from school. Meanwhile, the other students take their one year at age 7, during the early-starters' break year. This way, by age 8 they have the same one year of school. However, it seems obvious that the gap year would affect test scores as well. There is simply no way of testing an earlier school start age without simultaneously intervening on other variables that also appear very clearly important to the question. As a result, any such experiment would always run the danger of conflating the effects of the variables intervened upon.

(In summary, it seems like evaluating the matter is really difficult as FUQ).