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?


2 Answers 2


Hi: At a quick glance, the Geweke, 1982 and 1984 references on the last page of this link are probably necessary reading for understanding it.

I realize that this doesn't answer your question but I'm trying to see if I'm able to send a link without actually typing the link. and I don't know how to do that in a comment. Hopefully it works here.

  • $\begingroup$ I've read through those before, and many others like it. I've yet to come across a reference that seeks to explain the transfer function/matrix in conceptual terms. $\endgroup$ Sep 30, 2019 at 15:28
  • $\begingroup$ Okay. I thought since those papers seems to be the start of it, Geweke would explain the basics. My bad. $\endgroup$
    – mark leeds
    Oct 1, 2019 at 19:19

If you take the Fourier Transform of your AR equation, you get

$$\begin{align} Y(f) &= \sum_{k=1}^p A(k)e^{-j2\pi f k} Y(f) + U(f)\\ \left(I - \sum_{k=1}^p A(k)e^{-j2\pi f k}\right) Y(f) &= U(f) \end{align} $$ Let $\bar{A}(f) = \left(I - \sum_{k=1}^p A(k)e^{-j2\pi f k}\right)$. This should tell you why

$$\bar{a}_{i,j}(f) = \begin{cases} \displaystyle 1 - \sum_{k=1}^p a_{i,j}(k)e^{-2\pi f k}, &\text{if} \ i = j \\ \displaystyle - \sum_{k=1}^p a_{i,j}(k)e^{-2\pi f k}, &\text{otherwise}. \end{cases}$$

Now, $$Y(f) = \bar{A}^{-1}(f)U(f) \\ Y(f) = H(f) U(f)$$

This simply means that to recover your signal, you should multiply your innovation signal with the matrix $H$. You can see it as a transfer function matrix. The transfer function matrix tells you what the linear system output will be for a given input. Each $i$th elemenf of $Y(f)$ is,

$$y_i(f) = \sum_{j=1}^Q h_{ij}(f)u_j(f)$$

Therefore, the matrix $H(f)$ gives you the relationship between the innovation noise (system input) and the observed output. For example, in speech modeling, the innovation noise is white noise or a periodic pulse train (depending on sibilant or vowel) which gets shaped by the matrix $H$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.