**1. Introduction**

Nowadays it is more common to handle your affairs online. According to the World Payments Report (Capgemini and BNP Paribas 2017), electronic payments are expected to increase by almost 11% each year worldwide from 2015 to 2020. The world is becoming more online accessible due to innovations and modern technology. Online investing on the open market is due to technology much easier to do, for example there are applications such as eToro, Robinhood and Plus500 where people can invest money with their mobile devices.

In the last decades, a new type of currency is launched on the financial market and has gained importance. In particular, it is a virtual currency of which the main feature is the total absence of any intrinsic value. In 2009, Nakamoto (Nakamoto 2008) documented the creation of the first decentralised cryptocurrency, called Bitcoin. Since its introduction, it has been gaining more attention from the media, the finance industry, and academics. There are several reasons for this interest: Firstly Japan and South Korea have recognised Bitcoin as a legal method of paymen<sup>t</sup> (Bloomberg 2017a; Cointelegraph 2017). Second, some central banks are exploring the use of cryptocurrencies (Bloomberg 2017b). Third, the Enterprise Ethereum Alliance was created by a large number of companies and banks to make use of cryptocurrencies and the related technology called blockchain (Forbes 2017). These are just three of the many reasons the interest in cryptocurrencies has spiked. After the introduction of Bitcoin, many cryptocurrencies (around 1000) were created and became a new investment opportunity for trades. Hereafter, a short overlook of the four most important cryptocurrencies is described.

Bitcoin (BTC) is based on decentralisation, which means that it is controlled and owned by its users. This decentralisation is often criticised due to the lack of control over the whole system. Despite this criticism, Bitcoin increased in value from a couple of cents in the beginning (2009) to about 20,000 US dollar at the end of 2017. Ethereum (ETH, Ethereum 2014) is also decentralised and features smart contract functionality. Due to this contractual agreement, there is no possibility of fraud,

**<sup>\*</sup>** Correspondence: luca.rossini87@gmail.com

downtime, third party interference or censorship. The researcher and programmer Vitalik Buterin proposed it in late 2013 and Ethereum went live at the end of July in 2015.

Ripple (XRP, Ripple 2012) is founded by Ryan Fugger in 2004. It is a blockchain network that incorporates both a currency system known as XRP and a paymen<sup>t</sup> system. This enables real-time international payments and is therefore currently used by multiple banks. Litecoin (LTC, Litecoin 2014) was created in 2011 by Charles Lee and is based on the same peer to peer protocol used by Bitcoin. It is often considered Bitcoin's rival due to its improvements in transactions; these transactions are significantly faster than Bitcoin. Therefore it could be particularly attractive in certain situations to invest in.

Recently, researchers have started to study cryptocurrencies by applying different models and techniques. However, apart from Catania et al. (2019), a forecasting analysis of cryptocurrencies has not been strongly used and proposed. This paper tries to continue the analysis initialised by Catania et al. (2019) and to improve it by comparing different multivariate models for point and density forecasting of the four most capitalised cryptocurrencies previously described.

To study and forecast the cryptocurrencies, vector autoregressive models and moreover its extension to time-varying volatility have been introduced. Vector autoregressions (VARs) are used in models for empirical macroeconomic applications. VARs were introduced by Sims (1980) and have been widely adopted for forecasting and analysis of macroeconomic variables. The formulation of VARs is simple, however they tend to forecast well and are often used as the benchmark to compare the performance of forecasts among models. Sims and Zha (2006) emphasised the value of volatility modelling for improving efficiency. Accordingly, taking time variation in volatility into account should improve the estimation of a VAR-based model and inference common in analysis of macroeconomic variables. Modelling changes in volatility of VARs should also improve the accuracy of density forecasts. Forecast densities are potentially either too wide or too narrow, due to shifts in volatility. D'Agostino et al. (2013) showed that the combination of time-varying parameters and stochastic volatility improves the accuracy of point and density forecasts. One application of these regressions on a macroeconomic level is investing in assets, stocks and, as the purpose for this paper, in cryptocurrencies, as mentioned above.

VAR models can have many parameters if they include many lags, however using non-data information and turning it into priors is found to greatly improve the forecast performance. In Bayesian estimation algorithms, the stochastic volatility specification is computationally tractable, while in frequentist estimation it is captured with a single model. This is one of the reasons, in this paper, the Bayesian approach is used. Another reason is that the Bayesian approach gives some advantages in parameter uncertainty, computing of probabilistic statements and estimation with many parameters. As a standard procedure, the normal distribution is often used as a distribution of the so called "noise". For this paper, not only the normal distribution, but also the student-t distribution is used for modelling the errors.

A strong improvement of our paper is the introduction of time-varying specifications for multivariate models for better forecasting the cryptocurrencies behaviour. In particular, the use of time-varying volatility jointly with the multivariate time series is of interest for capturing the possible heteroscedasticity of the shocks and non-linearities in the simultaneous relations among the different cryptocurrencies in the models. Moreover, taking into account the time variation in volatility improves the VAR-based estimation and inference that have been shown in the preliminary cryptocurrencies analyses.

Our results show that including time-varying volatility and in particular stochastic volatility provides forecasting gains in terms of point and density forecasting relative to the multivariate autoregressive model. The inclusion of cryptopredictors can lead to better forecasting with respect to the benchmark but not strong improvements with respect to time-varying volatility models with only lags of the cryptocurrencies included. Directional predictability indicates that using stochastic volatility with heavy tails can be used to create profitable investment strategies.

The content of this paper is structured as follows. In Section 2, some literature used as research background is reviewed, especially research in the field of Bayesian VARs and cryptocurrencies. Section 3 describes the data. Section 4 presents our models, estimation methodology and metrics used to assess our results, which are discussed in Section 5 together with the major findings. Finally, Section 6 concludes.

## **2. Literature Review**

Cryptocurrency is becoming a hot topic in academia and outside of it. In particular, in the last years, the interest in cryptocurrencies has exploded from around 19 billion Dollars in February 2018 to around 800 billion Dollars in December 2017, thus much research has been done about this subject. Although Bitcoin is a relatively new currency, there have already been some studies on this topic.

Hencic and Gourieroux (2015) investigated the presence of bubbles in Bitcoin/US Dollar exchange rate by applying a non-causal AR model; the dynamics of the daily Bitcoin/USD exchange rate shows episodes of local trends, which can be modelled and interpreted as speculative bubbles. Cheah and Fry (2015) focused on the same issue; as with many asset classes, they showed that Bitcoin exhibits bubbles. They found empirical evidence that the fundamental price of Bitcoin is zero. The volatility of six major currencies against the volatility of Bitcoin was measured by Sapuric and Kokkinaki (2014), the results indicate a high volatility for Bitcoin exchange rate. Then, Chu et al. (2015) did a statistical analysis of the log-returns of the exchange rate of Bitcoin against the US Dollar and the generalised hyperbolic distribution is shown to give the best fit. Yermack (2015) wondered whether Bitcoin can be considered a real currency on the financial market.

Fernández-Villaverde and Sanches (2016) analysed privately issued fiat currencies, checked the existence of price equilibria and showed that there exists an equilibrium in which price stability is consistent with competing private monies. However, they also concluded that the value of private currencies monotonically converges to zero by equilibrium trajectories. Dyhrberg (2016) showed that the movements of the volatility of Bitcoin has several similarities to gold and the dollar. Bianchi (2018) investigated if there is a relationship between returns on cryptocurrencies and traditional asset classes. There was a mild correlation with some commodities, but not that many macroeconomic variables.

Catania et al. (2018) showed that predicting volatility can be improved by using leverage and time-varying skewness at different forecast horizons. Hotz-Behofsits et al. (2018) used time-varying parameter VAR with *t*-distributed measurement errors and stochastic volatility to model three cryptocurrencies: Bitcoin, Ethereum and Litecoin. Griffin and Shams (2018) investigated whether the cryptocurrency called Tether is directly manipulating the price of Bitcoin, increasing its predictability. By using algorithms to analyse the data, they found that purchases with Tether go along with sizeable increases in Bitcoin prices.

In 2019, there are more studies done on cryptocurrencies. Muglia et al. (2019) investigated the predictability of the S&P 500 by the movement of Bitcoin, showing that Bitcoin does not have any direct impact on the predictability of the S&P 500. Catania et al. (2019) found that point forecasting is statistically significant for Bitcoin and Ethereum when using combinations of univariate models. They also concluded that density forecasting for all four cryptocurrencies is significant when relying on time-varying multivariate models.

The exercise in this paper is generalised to multivariate models where the four cryptocurrencies are predicted jointly using Bayesian VAR models with stochastic volatility as in Koop and Korobilis (2013). Johannes et al. (2014) predicted stock prices using time-varying parameter and stochastic volatility VAR models and found statistically and economically significant portfolio benefits for an investor who uses models of return predictability.

Many institutions tried to investigate the relationship between Bitcoin and the stock market. An article by Bloomberg (2018) stated that "big investors may be dragging Bitcoin toward Market correlation", thus investors looking for high gains may be attracted to the increasing risk of this cryptocurrency. Stavroyiannis et al. (2019) studied the relation between Bitcoin and the S&P500 and

found that it does not hold any of the hedge, diversifier, or safe-haven properties and the intrinsic value is not related to US markets.

There are still no studies that can confirm that Bitcoin is a good stock market predictor. This paper tries to fill the gap, analysing whether Bitcoin, Ethereum, Litecoin and Ripple can be forecasted by its lags and other macroeconomic variables.
