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So I am doing an econometric study on Bitcoin's USD price. As an exogenous regressor, I have the total number of bitcoins in circulation (nbtc). The graph looks like this: Graph of number of bitcoins over time

This variable is:

  1. Exogenous – It can affect my other variables in the model, but they cannot affect it; the number of bitcoins increases as a diminishing rate (halves every 2 years) to a maximum of ~21m.
  2. I(0) – if I include a trend and controlling for the structural break, this is basically just a straight line.

So what we have here is linear trend. Now, this should be in my model and should (theoretically) affect the USD price in the long-run, but it can't form part of the long-term cointegrating relationship between my variables (Johansen tests find one such linear relationship), because it is I(0) and cannot "correct" any disequilibrium itself.

How would this be addressed in a VECM model?

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Some general points:

  1. The variable in the picture does not seem to have a linear trend, but perhaps its logarithm would. In any case, there seems to be a structural break somewhere at the end of 2012.
  2. If there are convincing subject-matter reasons why the variable should not cointegrate with other variables, you do not need to test for cointegration; if you are in doubt, you can of course do that.
  3. Note that trending variables may be mistaken for integrated variables (and vice versa) and care needs to be taken to account for deterministic trends in test specifications (this holds both for unit root tests and the Johansen cointegration test).
  4. Stationary exogenous regressors can be included in a VECM; they need not be used at the stage of cointegration testing, but they may be included at the stage of assembling the model. In your case you would need to stationarize the variable in the picture; you could detrend it and also account for the structural break. The problem appears to be, how to account for the long-run effect of the variable. Perhaps you could try including lags of the variable. If the lag order is quite high, you might want to use regularization (shrinkage) to prevent overfitting.
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  • $\begingroup$ Do you have any examples of where regularization as you suggest here has been successfully applied? When attending a talk at a conference in December I got the impression that it has not been well studied in VECM in general, but if you have examples I'd gladly read them. $\endgroup$ – hejseb Apr 1 '16 at 12:17
  • $\begingroup$ There is some work by Kascha & Trenkler presented at the conference of Computational and Financial Econometrics in London, 2015 (is there any chance that we attended the same conference ?). I don't know if they have published anything yet, though. I have a paper under review where I find VECM with regularization to perform relatively well (actually, best among a bunch of models considered -- but these other models may or may not be very competitive). $\endgroup$ – Richard Hardy Apr 1 '16 at 12:24
  • $\begingroup$ Yes, that's the one! That is funny that we were both in the same session. There was also something in another session chaired by Smeekes. I am thinking about this presentation: CO1534: A comparative study on forecasting performance of high-dimensional time series methods. Is there a working paper or published draft of your submission? $\endgroup$ – hejseb Apr 2 '16 at 8:46
  • $\begingroup$ @hejseb, I have not published the draft. It will take a while to get that publication out since it is just the first round of the review process. Anyway, it was my first attempt at applying regularization in a time series setting, so I do not consider it advanced or anything. It just happened to work OK, but even though it was competitive amongst the candidate models, its performance was not significantly superior (no wonder as I was forecasting a financial time series which is not much different from a random walk). $\endgroup$ – Richard Hardy Apr 2 '16 at 9:18
  • $\begingroup$ 1) once you account for the break, doesn't it effectively become linear? 2) Hmm. Eventually it will become a horizontal line, and is totally exogenous to the model. The problem is that it is exogenous – no other variable an affect it, but it can affect the variables. Does this mean it can be cointegrated? $\endgroup$ – df_256 Apr 3 '16 at 18:14

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