My question is a relatively basic one. Say I have some sort of a model which appears to fit my data well, how confident should I be in predictions that my model make for inputs that are very different from what was used to fit the data?
Take the very basic problem below. Here I fit a straight line through a bunch of points that are tightly grouped around 1 on the x-axis. I would expect this model to give reasonable results for values of the independent variable that lie reasonably close to the ones in the sample used to fit the data, say between 0.8 and 1.2, but would be much less confident for a model reading where x=20.
Similarly, for the example below, I would be more worried about model predictions at points on the x-axis where my sample is sparsely populated or unpopulated (say at x=1 or x=2), than I would be in cases where there were many observations around that reading for x (say around 0.45).
Although the examples I am using here are a bit crude and much simpler than the problem I am trying to solve, they illustrate the issue. I am trying to understand the pitfalls of trusting my model for a set of independent variables that are atypical relative to the sample that my model was fitted on. I am trying to understand whether there is a way of quantifying the uncertainty of model readings at these points. I would also like to know whether there are any rules of thumb of how to treat these sorts of problems. If you know of any academic articles that discuss this issue I would greatly appreciate it if you could point them out to me.