Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable SystemsThen we say Q does not finite-order G−cause Y with respect to S . Otherwise, we sayQ finite-order G−causes Y with respect to S.We call max(k,l − 1) the “order" of the finite-order G non-causality.Observe that Q t replaces Q t−1 in the classical definition, that Y t−1 replaces Y t−1 , andthat S t replaces S t−1 . Thus, in addition to dropping all but a finite number of lags in Q t−1and Y t−1 , this version includes Q t . As WL discuss, however, the appearance of Q t neednot involve instantaneous causation. It suffices that realizations of Q t precede thoseof Y t , as in the case of contemporaneous causation discussed above. The replacementof S t−1 with S t entails first viewing S t as representing a finite history, and second therecognition that since S t plays purely a conditioning role, there need be no restrictionwhatever on its timing. We thus call S t “covariates." As WL discuss, the covariatescan even include leads relative to time t. When covariate leads appear, we call this the“retrospective" case.In what follows, when we refer to G−causality, it will be understood that we arereferring to finite-order G−causality, as just defined. We will always refer to the conceptof Definition 5.1 as classical G−causality to avoid confusion.5.2. A Dynamic Structural SystemWe now specify a canonical settable system that will enable us to examine the relationbetween G−causality and direct structural causality. As described above, in suchsystems “predecessors" structurally determine “successors," but not vice versa. In particular,future variables cannot precede present or past variables, enforcing the causaldirection of time. We write Y ⇐ X to denote that Y succeeds X (X precedes Y ). When Yand X have identical time indexes, Y ⇐ X rules out instantaneous causation but allowscontemporaneous causation.We now specify a version of the causal data generating structures analyzed by WLand White, H. and P. Kennedy (2009). We let N denote the integers {0,1,...} and define¯N := N ∪ {∞}. For given l,m,∈ N, l ≥ 1, we let Y t−1 := (Y t−l ,...,Y t−1 ) as above; we alsodefine Z t := (Z t−m ,...,Z t ). For simplicity, we keep attributes implicit in what follows.Assumption A.1 Let {U t ,W t ,Y t ,Z t ; t = 0,1,...} be a stochastic process on (Ω,F , P), acomplete probability space, with U t ,W t ,Y t , and Z t taking values in R k u,R k w,R k y, andR k zrespectively, where k u ∈ ¯N and k w ,k y ,k z ∈ N, with k y > 0. Further, suppose thatY t ⇐ (Y t−1 ,U t ,W t ,Z t ), where, for an unknown measurable k y × 1 function q t , and forgiven l,m,∈ N,l ≥ 1, {Y t } is structurally generated assuch that, with Y t := (Y ′ 1,t ,Y′ 2,t )′ and U t := (U ′ 1,t ,U′ 2,t )′ ,Y t = q t (Y t−1 , Z t ,U t ), t = 1,2,..., (4)Y 1,t = q 1,t (Y t−1 , Z t ,U 1,t ) Y 2,t = q 2,t (Y t−1 , Z t ,U 2,t ).Such structures are well suited to representing the structural evolution of time-seriesdata in economic, biological, or other systems. Because Y t is a vector, this covers the17