**4. Methodology**

Studies have provided strong evidence of time-varying volatility in macroeconomic variables, however VARs with constant volatility are used in this paper. By using constant volatility, the performance of point forecasting should not be affected that much by conditional heteroscedasticity, which is the case for heteroscedastic models such as GARCH and stochastic volatility. Heteroscedasticity is a major concern in the regression analysis, as well as in the analysis of variance, as it can invalidate statistical tests. These tests assume that the errors, obtained by modelling, are uniform and uncorrelated. For example, the ordinary least squares (OLS) estimator is still unbiased in the case of heteroscedasticity, thus is inefficient because the actual variance and covariance are underestimated.

In this paper, three types of specifications are analysed: the standard VAR model, VAR with stochastic volatility and VAR with GARCH. The reason for multiple specifications of the model is to really see if the forecasting performance of a more complex model is better than a simple model. The Bayesian approach gives some advantages, as the parameter uncertainty can be mitigated. The probabilistic statements can be computed without assumption. Another advantage is that the estimation of complex nonlinear models with many parameters is feasible. For the stochastic volatility, two different models are investigated: one where the normal distribution is used and the other where the student-t distribution is used. These procedures by using these models are not the same, thus could end up with different results. This way, there can also be a conclusion about which distribution would give more accurate forecasts between all the models.

As stated in Catania et al. (2019), the number of lags of the VAR models is selected equal to three based on the BIC. The lag of interest of the cryptopredictors is the first lag. Thus, eight models are discussed and used in this paper: Bayesian VAR(3), Bayesian VARX(3), Bayesian VAR(3)-SV, Bayesian VARX(3)-SV, Bayesian VAR(3)-GARCH, Bayesian VARX(3)-GARCH, Bayesian VAR(3)-SVt and Bayesian VARX(3)-SVt. These models are constant parameter vector autoregressive and among the most common models applied in financial and macroeconomic forecasting (see Koop and Korobilis (2010); Lutkepohl (2007)). Regarding time-varying parameters, we left this issue as future research. To compare the models with each other, the Bayesian VAR(3) is chosen to be the benchmark. In the next subsections, the models used for the in-sample analysis and the forecasting exercise are explained briefly.
