# Computing (lagged) correlations (or similar) between multiple time-series from a VECM or its levels-VARM

Ok, after trying to find a way to get all lagged and non-lagged - correlations from multiple time-series (behavioral series with length around 180) I've confined myself to the following:

I have multiple (5 so far, more coming) sets of 12 time-series with equidistant measurements, same sampling-times $t_1, t_2, ... ,t_{180}$. Not all of them are stationary, but I'm assuming for the moment, most if not all of those will be participating in some cointegration-relation. (Will have to check that, but just assuming for the time being...)

For each set (containg 12 parallel time-series) I would like to estimate a VECM and, based on that, would like to get some unique measure of intra- and inter-series correlations for lags=0..p (where p is the order of the VECM - i expect p to turn out 2..3, 4 at the max).

Now, all I'm really interested in are unique parameters of "connection", it doesn't have to be actual correlation, as long as it is unique and thus comparable between different sets of the 12 series. (I would like to compare a certain subset of the parameters of one of the sets of series with the parameters of the other sets of series). For now, I also don't really care, wheter the parameters are partial (correlations) or not.

My idea is to estimate the VECM and (supposing I can get a good fit for the model by some criterion) reformulate the VEVM(p) as a levels-VAR(p+1) and then using the VAR's Matrix-Parameters as measures of (lagged) "connections" between the series. Only problem (I see) is: How can I ensure uniqueness of the estimated VAR-model's parameters - I know the cointegration vectors in the VECM are unique only wrt. scalar multiplication by some constant, so I could just normalize the cointegration vectors to length=1. Would this yield a unique VAR? Also, I've found this post, which gave me the idea to Z-transform the series beforehand, I'm quite new to the cointegration/VECM/VAR-business, but I would assume this (combined with normalized cointegration vectors) should give me unique VAR-parameters.

Does anyone have some advice on this, confirmation or criticism?

My answer to your question "Would this yield a unique VAR?" is yes.

A vector error correction (VECM) model has an equivalent vector autoregression (VAR) representation.

(VECM) $\;\;\;\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-(p-1)}+\varepsilon_t$

(VAR) $\;\;\;\;\;\;\;\; y_t=A_1 y_{t-1}+...+A_p y_{t-p}+\varepsilon_t$

where on one hand

(A) $\;\;\Pi=-(I-A_1-...-A_p) \;$ and $\;\;\Gamma_i=-(A_{i+1}+...+A_p)$

while on the other hand

(B) $\;\; A_1=\Pi+I+\Gamma_1$, $\;A_i=\Gamma_i-\Gamma_{i-1}$ for $i=2,...,p-1$, and $A_p=-\Gamma_{p-1}$.

The problem the of lack of uniqueness is "inside" the matrix $\Pi$ (if you factor it into a loading matrix $\alpha$ and a matrix containing the cointegrating vectors $\beta$, then you could ask what happens if you multiply $\alpha$ by a constant and divide $\beta$ by the same constant). However, there does not seem to be a problem once you go to the VAR representation; the possible factoring of $\Pi$ does not matter there. The VECM-equivalent VAR representation is unique.