Given

• Orders: $𝑝=1$, $𝑞=1$, $𝐾=2$

• Coefficient matrices:

  $${𝚽}_1=\left[\begin{array}{cc}0.4& 0.2\\ 0.2& 0.4\end{array}\right]{𝚯}_1=\left[\begin{array}{cc}0.4& 0.2\\ 0.2& 0.4\end{array}\right]$$
• Intercept: $𝒄=\left[\begin{array}{c}0\\ 0\end{array}\right]$
• Initial conditions: ${𝒙}_0=\left[\begin{array}{c}0\\ 0\end{array}\right]$, ${𝜺}_0=\left[\begin{array}{c}0\\ 0\end{array}\right]$
• Two series stacked into ${𝒙}_{𝑡}=\left[\begin{array}{c}{𝑥}_{1,𝑡}\\ {𝑥}_{2,𝑡}\end{array}\right]$:

Step 1 — formula

Substitute $𝑝=1$, $𝑞=1$ into the VARMA recursion:

Forecast (set ${𝜺}_{𝑡}=\mathrm{𝟎}$):

Componentwise:

Innovation:

Step 2 — apply at $𝑡=1$

With ${𝒙}_0=\mathrm{𝟎}$, ${𝜺}_0=\mathrm{𝟎}$, every product is zero:

Step 2b — apply at $𝑡=2$

Plug in ${𝒙}_1=\left[\begin{array}{c}12\\ 8\end{array}\right]$, ${𝜺}_1=\left[\begin{array}{c}12\\ 8\end{array}\right]$:

Step 3 — iterate

Two contributions per row: AR part ${𝚽}_1{𝒙}_{𝑡-1}$ and MA part ${𝚯}_1{𝜺}_{𝑡-1}$. Values rounded to 4 decimal places.