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I have been measuring Granger Causality between pairs of vectors processes (i.e. 2 vectors consisting of multiple time-series variables). Most of the equations I find in references utilize a determinant when comparing "reduced" and "full" model error ratios in the time domain, or use a determinant when comparing total power and intrinsic power in the frequency domain. I am wondering--when it comes to measuring GC across pairs of vector processes, why can't one simply sum across signle variable errors (power) rather than taking the determinant? Would a sum also be an appropriate approach?

For additional background info, more about GC can be found in these references:

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