Causality in Time Series - ClopiNet

Popescuy i =K∑︁A k y i−k + Bu + Dw ik=0D −1 (1 − A 0 )y i =K∑︁D −1 A k y i−k + D −1 Bu + wk=1Taking the singular value decomposition of the (nonsingular) matrix coefficient on theLHS:V T 0 y i = S −1 U T 0U 0 S V T 0 y i =K∑︁D −1 A k y i−k + D −1 Bu + w ik=1K∑︁D −1 A k y i−k + S −1 U0 T D−1 Bu + S −1 U0 T w ik=1Using the coordinate transformation z = V0 T y. The unitary transformation UT 0can beignored due closure properties of the Gaussian. This leaves us with the mixed-outputform:z i =K∑︁S −1 U0 T D−1 A k V 0z i−k + S −1 U0 T D−1 Bu + S −1 w ′ ik=1y = V 0 zSo far we’ve shown that for every zero-lag SVAR there is at least one mixed-outputVAR. Let us for a moment consider the covariate noise SVAR (after pre-multiplication)D −1w y i =K∑︁k=1D −1w θA k y i−k + D −1w θBu + w iWe can easily then write it in terms of zero lag:y i = (︁ )︁I − D −1w yi +However, the entries of I − D −1wdone by scaling by the diagonal:K∑︁k=1diag(D −1w )y i = (diag(D −1w ) − D −1w )y i +D −1w θA k y i−k + D −1w θBu + w iare not zero (as required by definition). This can beK∑︁k=1D 0 diag(D −1w )D −1w θA k y i−k + D −1w θBu + w i46