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.