Bitcoin data is very different from temperature data. Temperatures have more or less the same value for the same month of each year. This indicates that the distribution of temperatures does not change over time. Time series that exhibit this behavior are called stationary. This allows for relatively easy modeling with time series analysis tools, such as auto regressive (AR), moving average (MA), and auto regressive integrated moving average (ARIMA) models. Financial data is usually non-stationary, as seen in the daily Bitcoin close data, depicted in figure. This means that the data does not exhibit the same behavior throughout its entire history, but instead its behavior varies.
There are clear trends in the data (time intervals where the price, on average, increases or decreases), as well as heteroskedasticity (variable variance over time). One way to identify stationarity is to study the ACF function.If there is a very strong correlation between lags of a very high order that do not decay, the time series is most probably non-stationary. The ACF for the BTC data is also provided, showing weakly decaying correlations:
The following figure depicts the ACF for BTC. We can clearly see that the correlations do not drop for very high lag values:
Take a look at the following formula:
Where p is the percentage change, tn is the price at time n, and tn-1 is the price at time n-1. By applying the transformation to the data, we get a time series that is stationary, but less correlated.
The following figure shows the plots for the data, and the ACF and the average 30-day standard deviation are provided:
