Bayesian VARX

To introduce possible dependence to other variables, it is possible to extend the Bayesian VAR model, by including other variables of interest. The so-called VARX model can be described as:

$$y\_t = \beta\_1 y\_{t-1} + \beta\_2 y\_{t-2} + \beta\_3 y\_{t-3} + \sum\_{j=1}^{8} \gamma\_j \mathbb{W}\_{j,t} + \varepsilon\_{i,t}, \quad \varepsilon\_t \sim \mathcal{N}(0, \Sigma\_{\mathfrak{c}\_t}), \text{ for } t = 1, \dots, T\_{\mathfrak{c}\_t}$$

with *T* the number of total days of the data and where *γj* and *Wj*,*<sup>t</sup>* are the parameter and cryptopredictor, respectively. Since this model is for every cryptocurrency, the equation above can be rewritten in stacked form:

$$\begin{aligned} Y\_t &= Z\_t \boldsymbol{\beta} + \epsilon\_t, \quad \boldsymbol{\beta} = \nu \boldsymbol{\alpha} (\boldsymbol{\beta}\_1, \boldsymbol{\beta}\_2, \boldsymbol{\beta}\_3, \boldsymbol{\gamma}\_{1'} \cdot \boldsymbol{\cdot}, \boldsymbol{\gamma}\_8)\_{\ast}, \\ Z\_t &= (I\_N \otimes X\_t)\_{\ast} \end{aligned}$$

with *T* the number of total days of the data and where *Xt* = [*yt*−1, *yt*−2, *yt*−3, *W*1*t*, ··· , *<sup>W</sup>*8*t*] , for every cryptocurrency.
