**1. Introduction**

The idea of cryptocurrency and the related technology, Blockchain, was suggested in 2009 by an anonymous user known as Satoshi Nakamoto. He posted a paper to a cryptographic mailing list introducing a new electronic cash system with very low transaction costs able to avoid the presence of a central bank: the Bitcoin, see Nakamoto (2009). In the last ten years, cryptocurrencies have become more and more popular among researchers and investors, with around 2000 cryptocurrencies available at the time of writing. In recent months, the Bitcoin has experienced a dramatic price increase and consequently, the global interest in cryptocurrencies has spiked substantially. Despite the price increase, there are other numerous reasons for this intensified interest, just to mention a few: Japan and South Korea have recognised Bitcoin as a legal method of paymen<sup>t</sup> (Bloomberg 2017a; Cointelegraph 2017); some central banks are exploring the use of the cryptocurrencies (Bloomberg 2017b); a large number of companies and banks created the Enterprise Ethereum Alliance<sup>1</sup> to make use of the cryptocurrencies and the related technology called blockchain (Forbes 2017). Finally, the Chicago Mercantile Exchange (CME) started the Bitcoin futures on 18 December 2017, see Group (2017), Nasdaq and the Tokyo Financial Exchange will follow, see Bloomberg (2017b).

Although Bitcoin is a relatively new currency, there have already been some studies on this topic: Hencic and Gourieroux (2015) applied a non-causal autoregressive model to detect the presence of bubbles in the Bitcoin/USD exchange rate. The study of Cheah and Fry (2015) focused on the same issue. Fernández-Villaverde and Sanches (2016) analysed the existence of price equilibria among privately issued fiat currencies and Yermack (2015) wondered whether the cryptocurrency can be considered a real currency. Sapuric and Kokkinaki (2014) measured the volatility of the Bitcoin exchange rate against six major currencies. Chu et al. (2015) provided a statistical analysis of the log–returns of the exchange rate of Bitcoin versus the USD. Catania and Grassi (2018) analysed the main characteristics of cryptocurrency volatility.

<sup>1</sup> Source: https://entethalliance.org/members/.

Moreover Bianchi (2018) tried to investigate some of the key features of cryptocurrency returns and volatilities, such as their relationship with traditional asset classes, as well as the main driving factors behind the market activity. He found that returns on cryptocurrencies are moderately correlated with commodities and a few more assets.

Other studies have analyzed cryptocurrency manipulation and predictability. For instance, Hotz-Behofsits et al. (2018) applied a time-varying parameter VAR with t-distributed measurement errors and stochastic volatility. Griffin and Shams (2018) investigated whether Tether (another cryptocurrency backed by USD) is directly manipulating the price of Bitcoin, increasing its predictability. Catania et al. (2019) studied cryptocurrencies' predictability using several alternative univariate and multivariate models. They found statistically significant improvements in point forecasting when using combinations of univariate models and in density forecasting when relying on a selection of multivariate models.

Many institutions tried to investigate the relationship between Bitcoin and the stock market. In some articles, it was speculated that the Bitcoin can improve stock market's predictability, in this case, Bitcoin could be used as a leading indicator. In an article by Bloomberg (2018), Morgan Stanley's analysts stated that "big investors may be dragging Bitcoin toward Market correlation": the increasing risk of this cryptocurrency may have had an attraction for investors who were seeking for high gains. Stavroyiannis and Babalos (2019) examine the dynamic properties of Bitcoin and the Standard & Poor's 500 (S&P500) index. They study whether Bitcoin can be classified as a possible hedge, diversifier, or safe-haven with respect to the US markets. They found that it does not hold any of the hedge, diversifier, or safe-haven properties and it exhibits intrinsic attributes not related to US markets.

To the best of our knowledge, there are still no studies to confirm that Bitcoin is a good stock market predictor. This paper tries to fill this gap, analyzing whether Bitcoin could be used as a leading indicator for the S&P500.

To answer this question, we allow for parameter and model uncertainty, avoiding Markov Chain Monte Carlo (MCMC) estimation at the same time. This is accomplished using the forgetting factors methodology (also known as discount factors) which have been recently proposed by Raftery et al. (2010) and found to be useful in economic and financial applications, see Dangl and Halling (2012) and Koop and Korobilis (2012) (KK). Another advantage of this methodology is to provide, in close form, both the marginal and predictive likelihood (PL), which are useful in model selection.

The rest of the paper proceeds as follows: Section 2 presents the general model and the estimation strategy; Section 3 presents the Dataset; Section 4 discusses the empirical results; finally, Section 5 reports some conclusions.

#### **2. Models and Estimation Strategy**

Let **y***t* ≡ (*y*1, ... , *yt*) denote the time series of interest and **x***t* ≡ (*<sup>x</sup>*1, ... , *xt*) the series of exogenous variables, then the model can be written as:

$$\begin{aligned} \mathbf{y}\_t &= \mathbf{z}\_t \boldsymbol{\gamma}\_t + \boldsymbol{\varepsilon}\_t, \quad \mathbf{z}\_t \sim \mathbf{N}(\mathbf{0}, \mathbf{H}\_t), \\ \boldsymbol{\gamma}\_t &= \boldsymbol{\gamma}\_{t-1} + \boldsymbol{\eta}\_{t\prime}, \quad \boldsymbol{\beta}\_t \sim \mathbf{N}(\mathbf{0}, \mathbf{Q}\_t), \end{aligned} \tag{1}$$

where **y***t* is a scalar representing the observed time series at time *t*, **z***t* = {*yt*−1, ... , *yt*−*p*, *xt*−1, ... , *xt*−*<sup>q</sup>*} is a 1 × *m* vector ( *m* = *p* + *q*) stacking all the lags of the series of interest and of the exogenous variable; *γt* = {*<sup>γ</sup>*1,*t*, ... , *<sup>γ</sup>j*,*<sup>t</sup>*} is an *m* × 1 vector containing the time varying states *γ*s, which are assumed to follow a random-walk dynamic. Finally, the errors, *εt* and *β<sup>t</sup>*, are assumed to be mutually independent at all leads and lags. The H*t* contains the time-varying volatilities of the series. The state space model (SSM) of Equation (1) has been used in several recent papers, see among others, Primiceri (2005) and Koop and Korobilis (2012).

In order to estimate the quantities of interest, maximum likelihood or Bayesian estimation based on MCMC can be used. However, these two estimation approaches end up being computationally complex and, most of the time, infeasible. To reduce the computational burden, KK proposed two main adjustments to the usual MCMC.

The first is to replace the variance-covariance matrix Q*t* with an approximation. Latent states— *γt*—can then be obtained with a closed-form expression avoiding maximum likelihood or MCMC, see Supplementary Materials. The second adjustment is to replace the measurement error variance matrix H*t* with an Exponential Weighted Moving Average (EWMA) type filter.

As discussed in Supplementary Materials, this methodology requires the specification of the hyperparameters *λ*, *α* and *κ* and the specification of the initial condition of the states *γ*0 and Σ0. Refer to KK for an extensive discussion of the problem.

## **3. Dataset Description**

Table 1 reports the dataset used for the analysis, with the transformation and the data source. The sample goes from 11 August 2015 to 19 July 2018 and consists of 740 daily observations. The crypto–market is open 24 h a day, seven days a week; hence, for computing returns we use the closing price at midnight (UTC). As discussed in Catania et al. (2019), the data are available from https://coinmarketcap.com/ with daily frequency; unfortunately, hourly data that could allow for a more precise analysis are not freely available. To investigate non-stationary issues, three Unit-Root tests have been performed: Augmented Dickey-Fuller (ADF) Test, Philips-Perron (PP) Test and Kwiatkowsky, Phillips, Schmidt and Shin (KPSS) Test. All of them confirm the stationarity of each transformed series, results are available from the authors upon request.


**Table 1.** Data overview and transformation. The table reports the series divided by type, Financial predictors, Commodity predictors and Crypto predictors. The series are available for the period 11 August 2015 to 19 July 2018. For each variable the table reports the abbreviation code, the full name, the data source and the transformation applied.

Figure 1 reports Bitcoin closing price (BTC) which shows a steep rise in 2017 reaching the value of almost 20,000 US dollars in December 2017. This ascending trend was severely interrupted at the beginning of 2018, when price quickly dropped down to \$6000. At the time of writing, BTC's price is fluctuating between 5000 and 6000 dollars.

The series reported in Table 1 are divided in: financial predictors, such as VIX; commodity predictors, such as GOLD and crypto predictors such as BTC. Among the financial predictors, the VIX, see Figure S1 in Supplementary Materials (Panel (c)), is the most volatile, as expected. It displays a very steep peak between January and February 2018, the same period in which BTC's price started to fall.

Table S1 in Supplementary Materials reports the correlation matrix of the predictors. As the table shows the BTC appears to be highly positively correlated with all the financial indexes: S&P500, EF300 and NASDAQ.

BTC Daily Closing Price

**Figure 1.** The figure reports the Bitcoin (BTC) closing price from August 2015 to July 2018. The plot clearly shows the steep rise in the price 2017 and the sharp drop in 2018.
