The focus of this post involves a simple issue: extrapolation. Let me show two graphs from Manski's chapter:
Suppose one wanted to infer y based on x. Obviously, as the sample size gets larger, the confidence interval gets smaller, for the set of x that we are able to observe. Note that in this instance, however, no x between 4 and 6 is sampled.
Most empirical analysis would simply assume that E(y|x=5) is some smooth function that gives weights to E(y|x=4) and E(y|x=6). Put in English, one would just draw some sort of line between the x,y relation at x=4 and the x,y relation at x=6, and read off an x,y relation for x=5.
But doing this involves an important assumption: that y doesn't go flying off in one direction or another at x=5. We actually cannot know this, because we have no observations at x=5; indeed, maybe the reason we never observe x=5 is because y is highly unstable at that point. Just as problematic (perhaps more so) is predicting y when x > 9.
I am pretty sure that it is hard to go a day without reading something that involves someone extrapolating outside the support of observed data. Sometimes it is necessary to do this, but when we do, we should always say so.