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.
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.