When building a vector autoregression model is there some theory that would guide me in chosing the number of variable to include? For example, I have about 3000 data points and I would like to get an idea of how many lags of explanatory variables to ues.
The standard practice is to use some sort of information criterion, usually the Akaike Information Criterion (AIC) or the Schwarz Information Criterion (SIC).
The AIC is defined as $2k-2\log(L)$, where $k$ is the number of parameters, and L is the maximized likelihood function of the model (estimated with k parameters).
The SIC is defined as $k\log(n)-2\log(L)$, where $k$ and $L$ are as above, and $n$ is the number of observations of the model.
In general, one adopts the model that minimizes the chosen criterion.