*3.2. Time-Series Prediction Methods*

Bakar and Rosbi (2017) proposed the autoregressive integrated moving average model (ARIMA) to forecast the exchange rate between Bitcoin and the US dollar. In this method, the upcoming price depends upon autoregression, integration, and moving average, respectively. They believed the ARIMA model could be a reliable model to forecast the volatile characteristic of Bitcoin. Both Roy et al. (2018) and Anupriya and Garg (2018) applied the ARIMA model to predict Bitcoin's price. The experimental result demonstrated the strong forecasting ability of the ARIMA. The mean error between the actual prices and the predicted prices was less than 6% for most values. Roy et al. (2018) also compared the performance of the ARIMA model with the autoregressive model (AR) and moving average model (MA), and the ARIMA model resulted in better accuracy than the other two models. However, the ARIMA model's shortcoming is that this it can give a more accurate prediction for short-term data, based on the research result of Ariyo et al. (2014). Rane and Dhage (2019) introduced nine approaches for Bitcoin price prediction and discussed each methodology in their research. The ARIMA model targets to forecast uncertainty time-series data within a short-term period, but class imbalance can bias it. Linear regression is unsuitable to predict Bitcoin price as the time series data.

The strength of the vector autoregression (VAR) model and the Bayesian vector autoregression (BVAR) model to estimate currency and exchange rate fluctuations have been demonstrated in recent research. VAR has been used widely by financial theorists and economists in predicting time series economic variables in systems that involve supply and demand (Ito and Sato 2006; Wang et al. 2017; Carriero et al. 2009; Alquist et al. 2013; Sims 1993). We found several papers that use VARs to estimate currency and exchange rate fluctuations, notably Koray and Lastrapes, who use a VAR model to estimate the exchange rate on a series of macroeconomic variables (Koray and Lastrapes 1989). Additionally, Ito and Sato performed VAR research on the exchange rate of post-crisis Asia (Ito and Sato 2006). Wang et al. (2017) established a VAR model to analyze the impact of exchange rate volatility on economic growth. Furthermore, there is some research on forecasting using the Bayesian vector autoregression (BVAR) method. For example, Carriero, Kapetanio, and Marcellino demonstrated that the BVAR model produced better forecasting for exchange rates (Wang et al. 2017). In the econometric/finance community, (Catania et al. 2019) and (Bohte and Rossini 2019) have studied the forecasting performance of cryptocurrencies by vector autoregression with and without time-varying volatility. (Bianchi) has investigated the possible relationship between returns on cryptocurrencies and traditional asset classes. Bianchi et al. (2020) discussed the relationship between the returns on stable-coins and major cryptocurrency pairs within the context of a large Bayesian vector autoregression model. The BVAR model extends the classical VAR model by using Bayesian methods to estimate a vector autoregression. The BVAR model treats input parameters as random variables, and prior probabilities are then assigned. Current related work to both VAR and BVAR models in forecasting BTC prices does not focus on selecting the set of endogenous and exogenous variables and drivers that control the BTC market, which is the primary focus of this paper.

### **4. BTC Closing Price Prediction Models**

Both VAR and BVAR models are used in this paper to forecast the Bitcoin price and simulate the BTC market to understand market participants' behavior as well as the market conditions according to the closing price of BTC.

### *4.1. Endogenous and Exogenous Variables*

An autoregressive model is typically used to develop predictions and understand the trend of a time series. However, in financial and economic data, several factors are a ffecting the time series, such as supply, demand, and regulation. The complex nature of any financial market warrants a more sophisticated model. The performance of the VAR and BVAR forecasting models depends on the optimal selection of the set of endogenous variables of interest. Several variables were tested as proxies to represent the price, demand, and supply of the BTC market, respectively, after trying out numerous iterations of VARs and BVARs and using sensitivity analysis with di fferent variables, lags, and time frames. The final set of endogenous variables is defined in Equation (1). Let *Yt* be a vector of the endogenous variable of interest such that:

$$Y\_t = \langle \text{MKPRU}, \text{ MWNUS}, \text{ TOTBC} \rangle \tag{1}$$

### *J. Risk Financial Manag.* **2020**, *13*, 189

$$\mathbf{t}\_1 = \text{[04-01-2009, 22-11-2016]} \tag{2}$$

$$\mathbf{t}\_2 = \begin{bmatrix} 01\text{-}01\text{-}2011, 01\text{-}08\text{-}2020 \end{bmatrix} \tag{3}$$

where MKPRU represents the equilibrium closing price of the market for BTC as denominated by the US dollar (Figures 1 and 2). MWNUS is the number of unique MyWallet users, and TOTBC is the total BTC available in the market to date, as there is a limited amount of BTC available at 21,000,000. Our time frames are across two intervals, the first one is [04-01-2009, 22-11-2016] (Figure 1), and the second period is [01-01-2011, 01-08-2020] (Figure 2). The decision-making process uses reasonable metrics deemed viable drivers of the endogenous variables, where the following were selected as the exogenous variables: Average Block Size in, MB (AVBLS), Bitcoin Di fficulty (DIFF), Number of Transactions per Block (NTRBL), Miner's Revenue (MIREV), Change in the Number of unique addresses (NADDU), Total Output Volume (TRVOU), and Hash Rate (HRATE). A majority of the factors selected were those that had been a result of the BTC network's transaction behavior and how the fundamental mechanics influenced the closing price. The variables AVBLS, DIFF, TRVOU, and HRATE were taken as the variables which dictated the di fficulty of accessing and supplying BTC to the market. NTRBL considers the growing number of transactions occurring per block of BTC as a measure of transaction volume per available block of BTC. The NADDU variable considers the changing number of unique addresses performing BTC transactions to understand behavior trends over time. TRVOU measures the exchange trade volume of USD within the BTC market, which serves as a guideline as to how the market reacts to changes in value when buying or selling BTC. Finally, *Xt*, as the list of exogenous variables, is defined in Equation (4) as:

*Xt* = {MIREV, NTRBL, AVBLS, DIFF, NADDU, TRVOU, HRATE} (4)

**Figure 1.** Bitcoin closing price in USD (MKPRU), [04-01-2009, 22-11-2016].

### *J. Risk Financial Manag.* **2020**, *13*, 189

### *4.2. Vector Autoregression (VAR) Model*

A Vector autoregression (VAR) (Sims 1993), (Kuschnig et al. 2020), and (Kuschnig and Vashold 2019) model was developed to understand the relationship between the system of variables that are of interest (Equations (1) and (4)). Thus, the VAR of interest is as follows:

$$Y\_t = \beta\_0 + \beta\_1 Y\_{t-1} + \beta\_2 Y\_{t-2} + \beta\_3 Y\_{t-3} + \dots + \beta\_n Y\_{t-n} + \beta\_{n+1} X\_t + \epsilon\_t \tag{5}$$

where the betas (β*t<sup>s</sup>*) are vectors of constants and coefficients representative of the relationship between the variables, where *n* is the number of lags used in the VAR model. The purpose of selecting this model is to use the model coefficients to simulate a certain period of BTC endogenous variable (Equation (1)) given the exogenous variables (Equation (4)). Furthermore, one could ideally forecast out the BTC price behavior over time, such that there are verified and validated forecasts of the exogenous variables.
