using the union of contraction functions(read the book of Barnesly"fractal everywhere").

using the union of contraction functions(read the book of Barnesly"fractal everywhere").

While not testing for self-similarity per se, a hypothesis test for nonlinearity, typically used to detect chaos in time series, is the BDS (Brock-Dechert-Scheinkman) test which allows you to reject the null hypothesis of linearity for a given p value. You can use this first to eliminate linearity and then use other tests to try to measure the self-similarity.

Remember though self-similarity and long range dependence are different. Self-similarity is scaling behavior while LRD is the fact that the autocorrelation should never decrease completely to 0 (integration of the autocorrelation plot is infinite).

Imagine your time series data looks like a curve, e.g., stock price. You can derive all the bends of the curve X, as shown in Figure 1 of the following paper:

https://www.researchgate.net/publication/280294316_Scaling_as_a_Design_Principle_for_Cartography

There are far more small bends than large ones, and the large bends are self-similar to all bends as a whole. To see or understand clearly the kind of self-similarity, you need to conduct so called head/tail breaks for bends X, and accordingly derive ht-index for characterizing how many times the scaling pattern of far more small bends than large ones. 

Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541.

Jiang B. (2013), Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution, The Professional Geographer, 65 (3), 482 – 494.

https://www.researchgate.net/publication/236627484_Ht-Index_for_Quantifying_the_Fractal_or_Scaling_Structure_of_Geographic_Features

https://www.researchgate.net/publication/230839837_HeadTail_Breaks_A_New_Classification_Scheme_for_Data_with_A_Heavy-Tailed_Distribution?ev=prf_pub

Research Scaling as a Design Principle for Cartography

Article Ht-Index for Quantifying the Fractal or Scaling Structure of...

Article Head/tail Breaks: A New Classification Scheme for Data with ...