Causality in Time Series - ClopiNet

Popescuy i =K∑︁A k y i−k + Bu + Dw i (7)k=0The difference between this and Equation (6) is the presence of a 0-lag matrix A 0which, for easy tractability has zero diagonal entries and is sometimes present on theLHS. This 0-lag matrix is meant to model the sub-sampling interval dynamic interactionsamong observations, which appear instantaneous, see Moneta et al. (2011) in thisvolume. Let us call this form zero lag SVAR. In electric- and magneto- encephalography(EEG/MEG) we often encounter the following form:x i =K∑︁µA k x i−k + µ Bu + Dw i ,k=1y i = Cx i (8)Where C represents the observation matrix, or mixing matrix and is determinedby the conductivity/permeability of tissue, and accounts for the superposition of theelectromagnetic fields created by neural activity, which happens at nearly the speed oflight and therefore appears instantaneous. Let us call this mixed output SVAR. Finally,in certain engineering applications we may see structured disturbances:y i =K∑︁θA k y i−k + θ Bu + D w w i (9)k=1Which we shall call covariate innovations SVAR (D w is a general nonsingular matrixunlike D which is diagonal). Another final SVAR form to consider would be onein which the 0-lag matrix ⊳A 0 is strictly upper triangular (upper triangular zero lagSVAR):y i = ⊳A 0 y i +K∑︁A k y i−k + ⊳Bu + Dw i (10)k=1Finally, we may consider a upper or lower triangular co-variate innovations SVAR:y i =K∑︁A k y i−k + Bu + ⊳Dw i (11)k=0Where ⊳D is upper/lower triangular. The SVAR forms (6)-(10) may look different,and in fact each of them may uniquely represent physical processes and allow for directinterpretation of parameters. From a statistical point of view, however, all four SVARDGPs introduced above are equivalent since they have identical cover.Lemma 3 The Gaussian covariate innovations SVAR DGP has the same cover as theGaussian mixed output SVAR DGP. Each of these sets has a redundancy of 2 N N! for44