I am trying to understand multivariate statistical models, but unfortunately my linear algebra is poor. I saw this example of converting three models into matrix notation:
Formulating regression model in matrix notation
Is there a reason why in matrix form, it looks like that as opposed to just the sum of a bunch of column vectors? For example, we could have a column vector of [1θ, 2θ, 1θ] and another column vector of [0ϕ, -1ϕ, 2ϕ]?
Also for this one...would both equations have lagged X1 and X2 terms?
I think you just need to review how matrix multiplication works. Note that $$\begin{bmatrix} 1 & 0 \\ 2 & -1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} \theta \\ \phi \end{bmatrix} = \begin{bmatrix} \theta \\ 2 \theta - \phi \\ \theta + 2 \phi \end{bmatrix} = \theta \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} + \phi \begin{bmatrix} 0 \\ -1 \\ 2 \end{bmatrix}$$ which shows you how the matrix notation is the same as your "sum of vectors" interpretation.