Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable SystemsTo define recursive settable systems, for b ≥ 0 define Π [0:b] := Π 0 ∪ ... ∪ Π b−1 ∪ Π b .Definition 3.3 (Recursive Partitioned Settable System) Let S be a partitioned settablesystem. For b = 0,1,..., B, let Z[0:b] Π denote the vector containing the settings ZΠ ifor i ∈ Π [0:b] and taking values in S [0:b] ⊆ Ω 0 × i∈Π[1:b] S i , S [0:b] ∅. For b = 1,..., B andi ∈ Π b , suppose that r Π := {ri Π} is such that the responses YΠ i= X Π i(1,·) are determinedas:= ri Π (ZΠ [0:b−1] ;a).Y Π iThen we say that Π is a recursive partition, that r Π is recursive, and that S :={(A,a),(Ω,F ), (Π,X Π )} is a recursive partitioned settable system or simply that S isrecursive.Example 3.2 is a recursive settable system, as the responses of block 1 depend onthe settings of block 0, and the responses of block 2 depend on the settings of block 1.Canonical settable systems are recursive settable systems in which the settings fora given block equal the responses for that block, i.e.,Z[b] Π = YΠ [b] := rΠ [b] (ZΠ [0:b−1];a), b = 1,..., B.Without loss of generality, we can represent canonical responses and settings solely asa function of ω 0 , so thatZ Π [b] (ω 0) = Y Π [b] (ω 0) := r Π [b] (ZΠ [0:b−1] (ω 0);a), b = 1,..., B.The canonical representation drops the distinction between settings and responses; wewriteY[b] Π = rΠ [b] (YΠ [0:b−1];a), b = 1,..., B.It is easy to see that the structural VAR of Example 3.3 corresponds to the canonicalrepresentation of a canonical settable system. The canonical responses y 0 and {u t }belong to the first block, and canonical responses y t = (y 1,t ,y 2,t ) belong to block t + 1,t = 1,2,... Example 3.3 implements the time partition, where joint responses for a giventime period depend on previous settings.4. Causality in Settable Systems and in the PCMIn this section we examine the relations between concepts of direct causality in settablesystems and in the PCM, specifically the PCM notions of direct cause and controlleddirect effect (Pearl, J. (2000, p. 222); Pearl, J. (2001, definition 1)). The close correspondencebetween these notions for the recursive systems relevant to Granger causalityenables us to take the first step in linking Granger causality and causal notions in thePCM. Section 5 completes the chain by linking direct structural causality and Grangercausality.13