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1\\ Thx to @Kripfganz for the discussion on several relevant statistical properties of timeseries after the panel discussion. Now, I'll have to write a follow up article that again invalidates my Medium article: https://medium.com/burgercrypto-com/reviewing-modelling-bitcoins-value-with-scarcity-part-ii-the-hunt-for-cointegration-66a8dcedd7ef
2\\ In this article I made a mistake with the Johansen test in two ways.
- For 2 variables the rank of the matrix can be at max 1 (max 1 significant eigenvalue, while I found 2).
- even more important, the Johansen test should not have been used at all
- For 2 variables the rank of the matrix can be at max 1 (max 1 significant eigenvalue, while I found 2).
- even more important, the Johansen test should not have been used at all
3\\ Let's have a closer look at that last point:
The reason not to use the Johansen test or for any cointegration test at all, is because cointegration requires two timeseries with identical integration order to begin with.
The reason not to use the Johansen test or for any cointegration test at all, is because cointegration requires two timeseries with identical integration order to begin with.
4\\ S2F timeseries is often called deterministic, but in reality it is stochastic. Versus blockheight it would indeed be fully deterministic, but due to the varying interarrival time (IAT) (see stats below) of the blocks, the series becomes stochastic.
5\\ So if the series would be fully stochastic, there wouldn't be an issue and we could just test for cointegration. But...
S2F timeseries is not fully stochastic though; It has a deterministic trend between every halving.
S2F timeseries is not fully stochastic though; It has a deterministic trend between every halving.
6\\ If we would correct (what would be common practice, as I learned), a stationary ( I(0) ) series remains. And I(0) can never be cointegrated with I(1).
7\\ So I wondered, what if we ignore that S2F has a deterministic trend. What would that mean?
It would mean that we do have an I(1) series, albeit under 'false premises' but we could test for cointegration bc both series are I(1) again. (this is just a thought experiment)
It would mean that we do have an I(1) series, albeit under 'false premises' but we could test for cointegration bc both series are I(1) again. (this is just a thought experiment)
8\\ This would mean that bitcoin price possibly cointegrates with S2F series that shows a deterministic trend.
That's something that makes every timeseries expert start laughing: "Why would you consider cointegration without adjusting for deterministic elements first?"
That's something that makes every timeseries expert start laughing: "Why would you consider cointegration without adjusting for deterministic elements first?"
9\\ Even though it might be very strange and unheard of, does it then mean it's completely impossible and nonsensical?
That's something to further explore, but don't expect too much. It's more likely to end here than that you'll find support for the other approach.
That's something to further explore, but don't expect too much. It's more likely to end here than that you'll find support for the other approach.
10\\ Maybe there's hope:
“Cointegration can be defined as the onset of long-term equilibrium (stationary) between two or more time series which evolve individually under the influence of time (in a deterministic or stochastic way).”
Last part matters! https://www.cairn.info/revue-cahiers-d-economie-politique-2015-1-page-91.htm#no2
“Cointegration can be defined as the onset of long-term equilibrium (stationary) between two or more time series which evolve individually under the influence of time (in a deterministic or stochastic way).”
Last part matters! https://www.cairn.info/revue-cahiers-d-economie-politique-2015-1-page-91.htm#no2
11\\ Structural breaks are still there though...
12\\ To be continued.....
13\\ I'll have to work through the work of Perron and Campbell who seemed to have addressed how to deal with cointegration analysis in case of presence of trends: https://www.academia.edu/22235240/A_note_on_Johansens_cointegration_procedure_when_trends_are_present
