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Assume that I have two time series $Y_{1t}$ and $Y_{2t}$ that are sampled at the same frequency. Is there a way to quantify the uncertainty in their difference $Y_{1t} - Y_{2t}$? That is, can we get confidence bands on $Y_{1t} - Y_{2t}$?
My thought is that some sort of a dependent bootstrap should apply.
This question also seems related to this question.
This is kind of a non-answer, but it might invite its own discussion and is a bit long for a comment:
To get a confidence interval of the different at each time point, you'd need a point estimate of the difference at each time point. Bootstrapping only helps with standard errors. I have no idea how you'd compute such a point estimate unless you have many observations of each variable at each time point. Note that this is the situation in the question you linked. Otherwise, my impression is you might have to fit a model to get a prediction at each point. Then maybe you could construct a CI based on a difference-in-means t-test or something like that.
The difference $Y_{1t}-Y_{2t}$ is known with certainty; after all, you just measured it (unless you are measuring with uncertainty), so confidence bands around the difference seem to make little sense without an underlying model. I assume I may be misunderstanding your question - perhaps you want to edit it?
That said, you could fit, say, a state space model to the difference (e.g., with ets() in the R forecast package) and then simulate it to get the distribution of the underlying states at particular time points. Unfortunately, simulate.ets() only works for out-of-sample simulations, but you could probably extract what you need from the fitted model. Or you could fit the state space model to initial segments of the difference, then simulate out from there, sort of a "rolling origin" forecasting approach.
