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There is an abundance of literature about VAR-models, which teaches how to test preconditions, specify and estimate VAR-models for stationary and also cointegrated time-series.

However, I'm still a bit puzzled by the meaning of the model's parameters. Namely the coefficients $\phi_{ijk}$ of the phi-matrices in the standard VAR(p)-model: $\boldsymbol{x}_t = \Phi_1 \boldsymbol{x}_{t-1} + \dots + \Phi_k \boldsymbol{x}_{t-k} + \dots + \Phi_p \boldsymbol{x}_{t-p}+ \boldsymbol{\epsilon}_t$

From what I understand, in the univariate case, the coefficient $\phi_{11,k}$ provides an estimate of the PAC (partial auto correlation) of the $k$-th Lag of $x_t$. I understand this to be the (auto-)correlation of $x_t$ with its $k$-th Lag $x_{t-k}$, when all other correlations $\phi_{11,t-l}$ with $l \ne k$ are controlled for.

  1. What I am not clear about is how to interpret the coefficients $\phi_{ijk}$ in the more general multivariate case: Considering a bivariate VAR(p)-model, does $\phi_{11k}$ still have the same interpretation as in the univariate case or does it now, in fact denote the partial (auto-)correlation of $x_t$ with its $k$-th Lag $x_{t-k}$, when also all other correlations $\phi_{11,l}$, $\phi_{12,l}$ and $\phi_{21,l}$ ($l \ne k$) are controlled for?
    If indeed, those other correlations for lags $l \ne k$ are controlled for, what about the correlation $\phi_{12,k}$ with the same lag as $\phi_{11k}$, is it controlled for, as well?
  2. I am also interested in the partial correlations between all components of $\boldsymbol{x}_t$ without time-lag, so in the bivariate case between $x_{1,t}$ and $x_{2,t}$ after controlling for all the lagged correlations. This should in principle be given by $\Sigma_{\boldsymbol{\epsilon}}=\text{Cov}(\boldsymbol{\epsilon}_t)$. Did I miss something, here?
  3. What is a good (standard) method of determining the significance of the partial-correlations within this setting (with and without time-lag)?

Background to my questions is, that I have a dataset of 12 time-series, originating from couple-interactions (6 series per person). I would like to examine the series (total length = 180), which represent affectional variables which, hypothetically are closely related to one another. One simple way of obtaining an overview of dynamical structure of the dyadic system as a whole would be to estimate partial-correlations between the series (with and without lag). To avoid spurious correlations, I'm actually using cointegration techniques, thus gaining an ECM(p), which then can be reformulated as a VAR(p). I assume, that $p=2$ will suffice for the system in question.

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hello, I'm still struggling with this question and I'm not sure if I spelled it out clearly enough - I'd assume, that a lot of statistics-gurus would know an answer from the top of their head - it really boils down to the question whether or not the coefficients $\phi_{ij}$ of the \Phi-matrices in a VAR-model are (partial) correlation-coefficients or not. They clearly have to be related to (estimated) correlation-coefficients, but I can't find a clear reference to what kind of relationship exists? –  Ronald Jul 17 '12 at 9:02
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