**6. Discussion and Conclusions**

This study forecasts the next-day Bitcoin price using two univariate models—ARIMA and NNAR. Based on the employed forecast accuracy measures (RMSE, MAPE and MASE), while NNAR models perform better than ARIMA in the first training-sample (500 days) Bitcoin price forecasts, ARIMA models outperform NNAR models in both the test-samples. In line with this, from Figure 4, one could argue than NNAR models perform better than ARIMA (see Table 2) in times of less volatility, but not during extremely volatile test-sample periods of Bitcoin price, particularly in the year 2018. Furthermore, the DM test suggests the same, that is, ARIMA forecast results are more accurate than the NNAR forecasts in the test-sample forecasts.

Meanwhile, existing studies offer interesting insights. In a review of neural network models in forecasting, Adya and Collopy (1998) find that neural networks are not necessarily the best modelling approach for all types of data. Abrahart and See (2000) and Álvarez-Díaz et al. (2018) find that ARIMA and NNAR perform similarly. On the other hand, similar to this study, Alon et al. (2001) and Munim and Schramm (2018) also find that neural networks outperform ARIMA in some training-sample, but the opposite holds for test-sample. The reason for better accuracy of ARIMA models could be that we employ the feed-forward NNAR model, which is found to be inferior by Ho et al. (2002) as well when comparing with ARIMA and recurrent neural network (RNN) models. Thus, future study should attempt the RNN approach to Bitcoin price forecast. Furthermore, according to the DM test results, the forecast of ARIMA models are similar for with or without model re-estimation in each step. However, the NNAR model with re-estimation in each step performs better than without re-estimation. Thus, this unique approach of model re-estimation at each step can be adopted in inter-day forecasts, such as in next-hour and next-minute Bitcoin price (also stock price) forecasts. However, the model re-estimation approach to forecast next-day price increases computational duration slightly. To this end, with the growing market-cap of cryptocurrencies and extreme volatility of cryptocurrency prices, further attention should be paid to modelling their returns.

**Author Contributions:** Conceptualization Z.H.M. and I.A., Data curation Z.H.M., Methodology Z.H.M., Visualization Z.H.M., Introduction M.H.S., Literature review M.H.S. and I.A., Writing–review & editing Z.H.M., M.H.S. and I.A.

**Funding:** The APC was funded by the University of Agder.

**Acknowledgments:** The authors would like to thank the two anonymous reviewers and Qazi Haque for useful suggestions on an earlier draft of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest. *J. Risk Financial Manag.* **2019**, *12*, 103

(**b**) Test-sample (1966 days) forecast performance 

**Figure A1.** Examined NNAR model performance. (**<sup>a</sup>**,**b**) The first training and test-sample forecast performance, respectively.

(**b**) Test-sample (466 days) forecast performance

**Figure A2.** Examined NNAR model performance. (**<sup>a</sup>**,**b**) The second training and test-sample forecast performance, respectively.
