I need to code/write a Vector Autoregressive Model VAR(p) as a function of the parameters. There is no estimation, nor Data involved, this is a "population" exercise:
Assume X and Y are univariate, scalar random variables. I need to write the two following time-series models:
[Model 1] $\quad X_{t} = \sum_{j=1}^{p}a_{j}X_{t-j} + u1_{t,}, \quad\quad\quad\quad\quad\quad$ where $\;\mathbb{E}[u1_{t}]=0, \; Var[u1_{t}]=\Sigma1$
[Model 2] $\quad X_{t} = \sum_{j=1}^{p}b_{j}X_{t-j} + \sum_{j=1}^{p}c_{j}Y_{t-j} + u2_{t}, \quad$ where $\; \mathbb{E}[u2_{t}]=0, \; Var[u2_{t}]=\Sigma2$
In the first model, X only depends only on it's own past, while in the second model X depends on both the past of X and the past of Y.
EXAMPLE: if $p=2$ these models become
[Model 1] $\quad X_{t} = a_{1}X_{t-1} + a_{2}X_{t-2}+ u1_{t,}$
[Model 2] $\quad X_{t} = b_{1}X_{t-1} + b_{2}X_{t-2} + c_{1}Y_{t-1} + c_{2}Y_{t-2} + u2_{t}$
I need to do this in symbolic terms so that I can compute a population measure of Granger Causality:
$C_{Y \to X} = \frac{1}{2}\; \log\frac{\det \Sigma_{1}}{\det \Sigma_{2}}$
$\quad\quad\;\, = \frac{1}{2}\; \log\frac{Var(X_{t}) - Cov(X_{t},Z_{t})Var(Z_{t})^{-1}Cov(Z_{t},X_{t})}{Var(X_{t}) - Cov(X_{t},\tilde{Z}_{t})Var(\tilde{Z}_{t})^{-1}Cov(\tilde{Z}_{t},X_{t})}$
where $\;\;Z_{t} := [X_{t-1},..., X_{t-p}]'$ (vector that has the past X's) and $\quad\quad\quad\tilde{Z}_{t}:= [X_{t-1},..., X_{t-p}, Y_{t-1},..., Y_{t-p} ]'$ (vector that has the past X's and Y's)
MY OBJECTIVE: code $C_{Y \to X}$ as a function of the parameters $p$ and $a,b,c$ so that I can show how it changes for different values of the parameters. (There is no data, no simulation of data either)
NOTE: for people familiar with OLS (Ordinary Least Squares), this is a similar idea to doing population OLS: $\beta = \frac{Cov(X,Y)}{Var(X)}$ where $Y = \alpha + \beta X + \epsilon $, no estimation involved.
Do you have ideas how to start ?
