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I have an $n\times 1$ vector $\vec{v}$ and an $m\times 1$ vector $\vec{w}$, where in general $m \ne n$. Each component of the vectors represents a variable that I am observing. These vectors evolve throughout time, and I store them in two time series: $V$ and $W$. Both time series are of length $k$.

What I am looking to do is to obtain a covariance matrix type object (honestly, it may just be a covariance matrix and I'm over thinking this, so apologies if so.) I want to know the covariance between all of the variables in $\vec{v}$ with respect to all of the variables in $\vec{w}$.

I know that if I were interested only in the covariance matrix of one of the vectors, I'd just take the time series for that vector and find the covariance matrix. This would be a matrix whose entry at location $(i,j)$ represents the covariance between variables $i$ and $j$ (I'm stating this in case my understanding is wrong, so please correct me if so).

However, since $\vec{v}$ and $\vec{w}$ are vectors of different length, I'm not sure how to proceed.

Michael Stachowsky
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  • What's wrong with the $(m+n)\times(m+n)$ covariance matrix you get by looking at the $m+n$ dimensional vector you get by sticking $V$ and $W$ together? – kimchi lover Jan 27 '18 at 15:33
  • Fair point. So then that would give me block matrices, where the block I'm looking for would be in the upper left/lower right? – Michael Stachowsky Jan 28 '18 at 12:50
  • The $V$/$W$ interaction would show up in the upper right and lower left corner blocks. The diagonal blocks are about $V$ alone and about $W$ alone. I assume you are taking time lags into account, so these matrices are matrices of cross-correlation functions? – kimchi lover Jan 28 '18 at 17:53
  • Correct. $V$ and $W$ are matrices whose rows are individual observations of the variables in $\vec{v}$ and $\vec{w}$, respectively – Michael Stachowsky Jan 29 '18 at 11:44

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