Is cointegration able to change regarding different time period? I'm doing my Bachelor Thesis which is about the correlation between Google Trends and the price of Bitcoin with a VAR model. In a similar work, they find out that there is no cointegration between these 2 variables, regarding the time period from May 2011 to June 2013.
I'm considering the time period from September 2012 to September 2017 and find out, that there is a cointegration relationship between these 2 variables. 
Is it possible that 2 variables can change their cointegration characteristics regarding another time period which is relative near to the other one, or are there any thumb rules or something like that regarding this issue.
 A: You could not possibly rule that out for stochastic processes in general based on any statistical reasoning. I could simply generate a time series from any  model without cointegration, and then another from one with cointegration, and stick them end-to-end.
At best, you may be able to find domain-specific arguments about certain kinds of data, for example if in a specific application where cointegration is representative of a certain fundamental physical link between physical processes which is immutable once the process is started. In finance, linkages aren't quite so permanent and can in general vary depending on market conditions or technological changes (e.g. in an industrial process that transforms commodity A to commodity B, you might find cointegration between companies linked to A and those linked to B which disappear once an innovation cuts A out of the process).
Do you know of any event(s) in the history of Bitcoin that could explain the differences between the two periods? For example, could it be that the earlier BTC price history involves small expert communities and their technological advancements moving prices around, and the later period involves mainstream investors learning about and entering the BTC market? It's likely that the second situation is more tied to google trends around words like "bitcoin".
If not, it's possible that the difference isn't really "real". Maybe the first period is too short to detect cointegration (which is a long-term phenomenon) reliably but that it really does exist there, or that neither model is truly appropriate and test assumptions are violated and the test results are unreliable (for example, standard cointegration tests are notoriously vulnerable to structural breaks).
A: I made the assumptions before testing for cointegration that, both variable are tested in level form and non-stationary of order 1. And I also assume that there is a constant and a trend.
After decomposition I found out that there is a quadratic trend in the series and the remainder has still some irregular behaviour. After differentiating (in purpose to remove the stochastical trend) the mean is still not zero so I may thought that this could implicate that there is constant.

As you can see there is a big difference in the time series of price and Goggle Trends since the beginning/mid of the year 2013 for the shorter period (1st picture), where there was no Cointegration detected.

It seems that this difference tend to disappear looking to the graph of the longer period (2nd picture), after ca. March/April 2014. Maybe this could explain the difference of cointegarion or like you said, the shorter period was just to short to detect cointegration.
In the case of mainstream investors or structural breaks, I don't know any specific events but as you can see the amount of investment increased very strong and thus the price of the bitcoin increased very strong.
Regarding the point of technology the only potentially helpful hint is that the 
cost of mining the next coin increases over time, while the marginal benefit decreases, miners receive less Bitcoins on the margin for each new block added to the chain.
