# Change the partial covariance matrix in a VAR with constant?

The computation of the partial covariance matrix formula in a VAR (without constant and time trend) is very well known. For example, in the context of the conditional Granger causality, they are given as follows (throughout the question, VAR order in numerator and denominator is the same finite fixed integer given in advance, say 5): How to change the partial covariance matrix computation formula in a VAR (without constant and time trend) [

e1 = cov(x) - cov(x, cbind(xlag, zlag)) %*% solve(cov(cbind(xlag, zlag))) %*% cov(cbind(xlag, zlag), x)
e2 = cov(x) - cov(x, cbind(xlag, ylag, zlag)) %*% solve(cov(cbind(xlag, ylag, zlag))) %*% cov(cbind(xlag, ylag, zlag), x)


]

to obtain partial covariance matrix formula for a VAR with constant?

Note that d1 and d2 change as VAR order changes. For simplicity, assume the VAR order is the same everywhere (say, 5),i.e., VAR(5) with constant.

More basically perhaps, how the computation of F statistics is got affected in regression when constant (and time trend perhaps) added?

What I did:
I find some references but those ones do not directly handle the problem:
The "conditional covariance matrix" are mentioned as "partial covariance matrix" in some sources.
1. Rassler2002, "Statistical Matching: A Frequentist Theory, Practical Applications, and...", p.143, (5.8)
2. Hogben, "Handbook of Linear Algebra", p.52-4
3. Schreier, "Statistical Signal Processing of Complex-Valued Data The Theory of Improper...", p.41, (2.59)
RASSLER2002:
HOGBEN2007: SCHREIER2010:

• What is the partial covariance matrix? And of what vector? Is it somehow related to the error variance matrix $\text{Var}(\mathbf{\varepsilon}_t)$? You say it is very well known, but still... (I have studied VAR models a bit but never heard of it.) Could your question be rephrased in some other terms excluding the partial covariance matrix? Regarding how the computation of F statistics is got affected in regression when constant (and time trend perhaps) added?, just treat the constant and the trend as regressors (yet another column in the $X$ matrix), then... – Richard Hardy Nov 16 '16 at 6:59
• ...the general matrix algebra expressions for the model will stay the same. Or are you interested in something more particular? – Richard Hardy Nov 16 '16 at 7:04
• Thx a lot. I will try to do your suggestion: treating the constant and the trend as regressors (yet another column in the X matrix. – Erdogan CEVHER Nov 16 '16 at 12:11