Given

• Order: $𝑞=1$, $𝐾=2$

• Coefficient matrix:

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

Step 1 — formula

Substitute $𝑞=1$ into the VMA$\left(𝑞\right)$ recursion:

Forecast (set ${𝜺}_{𝑡}=\mathrm{𝟎}$):

Componentwise:

Innovation:

Step 2 — apply at $𝑡=1$

Plug in $𝒄=\mathrm{𝟎}$, ${𝚯}_1$, ${𝜺}_0=\left[\begin{array}{c}0\\ 0\end{array}\right]$:

Step 3 — iterate

Each row uses the previous innovation ${𝜺}_{𝑡-1}$. Values rounded to 4 decimal places at every step (computation continues from the rounded value).