**3. Data**

Daily Bitcoin exchange rate data (USD per Bitcoin) is collected from the *Quandl*<sup>1</sup> database. Data from the same source has been used by others, too (Chu et al. 2015). We use daily Bitcoin price data from 1 January 2012 to 4 October 20182, daily Bitcoin price of 2466 days. Figure 1 presents the (a) original time series along with (b) log-transformed and (c) first di fferenced log operator series. For the e ffectiveness of forecast validation (Adya and Collopy 1998), we divide the dataset into a training-sample (in-sample) and test-sample (out-sample). We consider two training-samples and subsequently two-test samples for cross-validation purposes. The first training sample is from 1 January 2012 until 14 May 2013 (500 days), and the second from 1 January 2012 until 25 June 2017 (2000 days). As a consequence, the first test-sample is from 15 May 2013 to 4 October 2018 (1966 days), and the second from 16 June 2017 to 4 October 2018 (466 days).

At the end of 2014, the price of Bitcoin dropped significantly to USD 302 (www.coindesk.com). The cause of the price decline was the suspension of trading of Bitcoin by Mt. Gox, one of the leading Bitcoin exchanges, which handled 70% of the Bitcoin exchange worldwide at that time. They reported that around 850,000 Bitcoins were hacked, which belongs to customers and are worth around USD 3.5 billion (Roberts 2017). The incident resulted in a lack of confidence in the security system of Bitcoin; thus, the price decline continued until 2016. At the beginning of 2017, the Bitcoin price increased dramatically, and at the end of 2017 the price of Bitcoin surged at USD 19,661.63, but again after five days from 17 December 2017 it dropped to USD 12,616.64 (www.coindesk.com).

<sup>1</sup> www.quandl.com/data/BCHARTS/BITSTAMPUSD-Bitcoin-Markets-bitstampUSD.

<sup>2</sup> Bitcoin price data for three days, that is, 6–8 January 2015 was not available.

(**c**) First difference log operator of daily Bitcoin price in USD

**Figure 1.** (**a**) Original, (**b**) log-transformed and (**c**) first difference log operator bitcoin price in USD.

Stationarity of data is a prerequisite for predictive modelling, particularly when using autoregressive time series models such as ARIMA. Table 1 shows results of the stationary test of the training-data samples using the Augmented Dicky-Fuller test (ADF) (Dickey and Fuller 1979) and Phillips-Perron test (PP) (Phillips and Perron 1988). Data, both in levels and log-transformed series, are not stationary but become stationary at first difference log operator; thus, the ARIMA modelling approach is feasible. It might be noted that the stationarity of data is not essential for neural network models (Hyndman and Athanasopoulos 2018).


**Table 1.** Stationary test of data.

ADF. Augmented Dicky-Fuller test; PP. Phillips-Perron test. *p*-values in parenthesis, *p*-value less than 0.05 confirms stationarity.
