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)
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)