Why is time series forecasting even possible?

Question

Mathematics

How do you know if there is an adequate model for time series forecasting? Is there a definition of "series is well predictable" / "series is not predictable"?
For example: exchange rate, electricity consumption, air temperature, human heart rate measurements, output of a sequence of calls to the random ( ) function, the number of tails out of 100 rolls at different times of the day, the number of Somali pirates in different years, etc.
It is intuitively clear that not all such series can be predicted "equally". Point out, pliz, the existing formalism.

Answer 1

I don't think formalism exists. Model selection is more of an art than an algorithm.
"How to find out" - sort through all known models in a row.
If you want to shorten the enumeration, you can first do all the known "analyzes" of the series (seasonality, trend, autocorrelation, etc.)
If you want to shorten the path to the goal, you need to delve into the "physical sense" of the process that generates the series.
For example, electricity consumption (as well as air temperature) rises during the day and falls at night. Obviously, this is a stable pattern, and it is not difficult to predict it.
Further purely research work, without a clear algorithm. We make hypotheses and test them. If it doesn't work, we think further.
In general, I think it does not always make sense to express the "degree of predictability of the series" by some number. For example, the exchange rate is a mixture of many processes, not just one. These processes (and patterns) can be local, overlap, appear and disappear, and so on. It is more logical to evaluate "predictability" for some one narrow pattern, isolated from a series.

Answer 2

Any time series has an extrapolation. But by what criteria will you evaluate?