*4.3. Bayesian VAR-GARCH*

The Bayesian VAR(3) with GARCH(1,1) innovations is almost the same as the VAR-SV 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 errors. It also has a memory over time so it can compare the observations with the past to ge<sup>t</sup> a better estimate of the predictions. For this reason, one should use GARCH over SV, because of the memory over time.

The Bayesian VAR(3) with GARCH(1,1) innovations is described in the following way:

$$\begin{aligned} y\_t &= \beta\_1 y\_{t-1} + \beta\_2 y\_{t-2} + \beta\_3 y\_{t-3} + \epsilon\_{t\prime} \\ \epsilon\_t &= H\_t^{0.5} \eta\_t, \eta\_t \sim N(0, l\_k), H\_t = D\_t R\_t D\_t, D\_t = \text{diag}(h\_{1t}^{0.5}, \cdots, h\_{kt}^{0.5}), \\ h\_t &= \omega + B \epsilon\_{t-1}^{(2)} + G h\_{t-1}, \text{ for } t = 1, \cdots, T, \end{aligned}$$

with *T* the number of total days of the data. *R* is the conditional correlation matrix. *ht* follows a GARCH(1,1) model where *ht* = [*h*1*t*, *h*2*t*, ··· , *hkt*] and (2) *t* = [21*<sup>t</sup>*, 22*<sup>t</sup>*, ··· , <sup>2</sup>*kt*] are conditional variances and squared errors, respectively. *ω* and *B* and *G* are matrices of coefficients (Carnero and Eratalay (2014)).

## *4.4. Bayesian VAR-SVt*

The following model description is similar to the VAR-SV, but now with a student-t distribution. This model, referred to as VAR-SVt, is described as:

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

with *T* the number of total days of the data and *η* the degrees of freedom. *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)).
