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I am looking for a metric and the associated statistical test to compare two time series or to determine whether a short series has the same parameters as the long one. The series are likely modeled by a GARCH process or similar.

To give more background: if the points in the series were completely uncorrelated, one could have used the Kolmogorov-Smirnov test. I am looking for something similar.

Update
Another idea is to model series as a random walk or GARCH process, extract the parameters of the process and compare them using Kullback-Leibler divergence. It is not clear how to construct the statistical test and what are the limitations, if one of the series is very short.

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    $\begingroup$ Is Kullback-Leibler divergence applicable to comparisons of parameters? Or did I misunderstand your action plan? Also, is the time-series setting essentially different from a cross-sectional setting? If not, then you may look up some other threads which discuss the cross-sectional case. $\endgroup$ – Richard Hardy Jun 3 '20 at 8:06
  • $\begingroup$ I don't know about the KL - if you can confirm that it is unusable for comparison, it is important for me. The time series are uncorrelated: more precisely, some of them are real data, whereas the others are generated with the purpose of bootstrapping. $\endgroup$ – Vadim Jun 3 '20 at 8:15

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