*4.2. Bayesian VAR-SV*

In the following section, the models with time-varying volatility are described in detail by differentiating between SV and GARCH. First, the Bayesian VAR(3) with stochastic volatility is similar to the previous model, however there is a difference in the innovations term. This allows the model to take different approaches over time, for example in times of high uncertainty there could be a higher variance in the innovations. For this reason, one should use stochastic volatility, since the model adapts to the movement and volatility of the time series.

The Bayesian VAR-SV(3) model is described in the following way:

$$\begin{aligned} y\_t &= \beta\_1 y\_{t-1} + \beta\_2 y\_{t-2} + \beta\_3 y\_{t-3} + \epsilon\_{t\prime} \\ \varepsilon\_t &= A^{-1} \Lambda\_t^{0.5} \varepsilon\_t, \varepsilon\_t \sim N(0, I\_k), \Lambda\_t \equiv \text{diag}(\lambda\_{1t\prime} \cdots \lambda\_{kt}), \\ \log(\lambda\_t) &= \log(\lambda\_{t-1}) + \nu\_{t\prime} \\ \nu\_t &= (\nu\_{1t}, \nu\_{2t\prime} \cdots \nu\_{kt})^\prime \sim N(0, \Phi), \text{ for } t = 1, \cdots, T \end{aligned}$$

with *T* the number of total days of the data and where *A* is a lower triangular matrix with non-zero coefficients below the diagonal, which are ones. Λ*t* is a diagonal matrix which contains the time-varying variances of shocks. This model implies that the reduced form variance-covariance matrix of innovations to the VAR is *var*( *t*) ≡ Σ*t* = *<sup>A</sup>*−1Λ*t*(*<sup>A</sup>*−<sup>1</sup>) (Clark and Ravazzolo (2015)).
