I am attempting to gain a better mathematical understanding for how autoregressive models can be used to infer frequency-domain granger causality. All freq. domain measures of causality that utilize autoregressive models convert the time-domain parameters to a frequency-based representation (i.e. take the Fourier transform of the original parameters), and then take the inverse of that representation to get the so-called "transfer matrix". This transfer matrix is referenced in many causality-related papers, and is the backbone of all freq. domain measures. Here is the derivation for the transfer matrix:
$a_{i,j}(k)$ - Autoregressive coefficient describing influence of node $j$ on node $i$ at lag=$k$
$p$ - Model-order (i.e. how many past time-points to use to predict the present)
$\textbf{A}(k)$ - $Q$-by-$Q$ autoregressive parameter matrix at lag=$k$. This matrix is estimated via model fitting.
$U(t)$ - Innovation noise (i.e. the difference between the model's predictions and observed data at time $t$.)
\begin{align} {Y(t)} = \sum_{k=1}^{p}\textbf{A}(k)Y(t-k) + U(t) \end{align}
$\bar{\textbf{A}}(f)$ - Frequency representation of $\textbf{A}(k)$
$\bar{{a}}_{i,j}(f)$ - Frequency representation of autoregression describing influence of node $j$ on node $i$ at freqeuncy $f$
$\bar{a}_{i,j}(f) = \begin{cases} 1-\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k} & \text{if $i=j$} \\ -\sum_{k=1}^{p} a_{i,j}(k) e^{-\textbf{j}2\pi f k)} & \text{otherwise} \end{cases} $
\begin{align} \text{where } \textbf{j}=\sqrt{-1} \end{align}
Note that above piecewise equation stems from taking the FT of model coefs. At least as far as I understand it. I'm not actually positive as to why this equation is piecewise and not identical for all $i$/$j$ combinations. I haven't worked with FTs very much.
$\textbf{H}(f)$ - Transfer matrix
\begin{align} \textbf{H}(f)=\bar{\textbf{A}}^{-1}(f) \end{align}
Now for my question--does anyone have an intuitive definition for what the transfer matrix, or a given element of the transfer matrix ($h_{i,j}(f)$), actually represents? I can easily and intuitively understand what $a_{i,j}(k)$ represents, but I've yet to fully grasp what $h_{i,j}(f)$ represents. How does one interpret such a value in layman's terms?
