Price Levels, Stationarity, and Fractional Differencing in Bitcoin Analysis
Summary
The question concerns regressions linking Bitcoin prices with daily Reddit post counts and sentiment. The author reports that price appears integrated while the social measures appear stationary, yet the series show strong correlation, high regression fit, and stationary residuals. The response cautions that long-run co-movement may reflect shared growth over time, such as increasing adoption, rather than a relationship that predicts prices. Thus, a high R-squared or stationary residuals alone do not establish useful forecasting power.
As an alternative to using only log returns, the answer suggests fractional differentiation. It generalizes ordinary differencing by applying a fractional power of the backshift operator, potentially retaining more of the price series’ information while reducing nonstationarity. Numerical use requires truncating the infinite expansion to a finite lag length. The reply is brief: it does not provide a full econometric diagnosis, specify tests or parameter selection, or demonstrate predictive performance. The suggested transformation should therefore be treated as a candidate for further analysis, not as evidence of a signal.
Key ideas
- Strong correlation and high regression fit can arise from shared long-term growth and need not imply predictive value.
- Mixing an integrated price series with stationary social measures requires careful treatment of time-series properties.
- Fractional differencing generalizes ordinary differencing and may preserve more information than log returns.
- A practical fractional-differencing calculation truncates the operator expansion to a finite number of lags.
- The response does not provide empirical validation or a complete modeling procedure.
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# Price vs log returns - stationarity issues
# Price vs log returns - stationarity issues
I am trying to analyze the price of Bitcoin versus the number of Reddit posts about Bitcoin and the sentiment of those posts (daily).
The price is I(1) while the sentiment and the number of posts are I(0). Surprisingly, they seem cointegrated with the maximum number of cointegrating relationships possible (Johansen). I would like to use prices because simple OLS regressions such as price = const sent(-1) count(-1) are giving a very high R^2, probably due to the fact that the price and the number of comments are correlated at about 0.8. Running such a regression produces stationary residuals too!
For all of my university career they taught me to use log returns (which are of course I(0)), but i cannot seem to find any meaningful relationship with returns.
What are the problems with my work (if any)? And what I could use to make a more professional analysis? This is for my master thesis and we touched subjects like VAR and VECM, even if I don't really know how to look into the results properly.
## Answer by Sebastian (score 1)
https://quant.stackexchange.com/a/69693
It depends on the size of your price timeseries. I suspect that if you would take the complete history starting 2007 until now the connection between the price and the number of posts will be significant. In the beginning both numbers were small but with the rising adoption of bitcoin both have risen. That would be however no meaningful relationship to predict prices.
I would recommend to check out fractionally differentiation which is a generalization of ordinary differentiation. Suppose that $B$ is the backshift opperator, e.g. $BX_t=X_{t-1}$ for a timeseries $X_1,...,X_T$. Then we can represent numerical differentiation as follows:
We can generalize that concept to arbitrary numbers $d\geq 0$ with the binomial formula
$$ \begin{matrix} (1-B)^d & = & \sum_{k=0}^\infty\begin{pmatrix}d\\k\end{pmatrix}(-B)^k \\ & = & 1 - dB + \frac{d(d-1)}{2!}B^2 ... + (-1)^k\prod_{i=0}^{k-1}\frac{d-i}{i} + ... \end{matrix} $$
In case that there is no meaningful information contained in log returns you can try some value $0<d<1$. This resulting series will share features of the log returns and the original prices.
Note: To numerically use the above formula you must crop it to a fixed finite length.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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