I'm working through a textbook (Regression and Other Stories) and have come across a particular problem that I am having difficulty convincing myself I understand.
I am specifically interested in part (b), but I include (a) as context.
It is as follows
'Before-after comparisons: The folder Sesame contains data from an experiment in which a randomly selected group of children was encouraged to watch the television program Sesame Street and the randomly selected control group was not.
(a) The goal of the experiment was to estimate the effect on child cognitive development of watching more Sesame Street. In the experiment, encouragement but not actual watching was randomized. Briefly explain why you think this was done. Think of practical as well as statistical reasons.
(b) Suppose that the investigators instead had decided to test the effectiveness of the program simply by examining how test scores changed from before the intervention to after. What assumption would be required for this to be an appropriate causal inference? Use data on just the control group from this study to examine how realistic this assumption would have been.'
I think this question is hinting at the problem of attaining an unbiased estimate for ATE from a difference in means from pre-post scores with systematic differences in pre-treatment variables between the treated and control groups, leading to differences in the outcome that is dependent on these differences rather than the effect of treatment - either through sampling bias. For example, more intelligent children are encouraged to watch the TV program because their parents are aware of its supposed effects, or through heterogeneous treatment effects such as socioeconomic factors.
The question suggests looking at control group data - which I have done, but I am not entirely sure what I am looking for in the data to test the above assumptions given the question suggests only looking at control data.