Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable Systemsenables us to link direct structural causality and finite order G−causality. To state thisexogeneity condition, we write Y 1,t−1 := (Y 1,t−l ,...,Y 1,t−1 ), Y 2,t−1 := (Y 2,t−l ,...,Y 2,t−1 ),and, for given τ 1 ,τ 2 ≥ 0, X t := (X t−τ1 ,..., X t+τ2 ), where X t := (W ′ t ,Z′ t )′ .Assumption A.2 For l and m as in A.1 and for τ 1 ≥ m,τ 2 ≥ 0, suppose that Y 2,t−1 ⊥U 1,t | (Y 1,t−1 , X t ), t = 1,...,T − τ 2 .The classical strict exogeneity condition specifies that (Y t−1 , Z t ) ⊥ U 1,t , which impliesY 2,t−1 ⊥ U 1,t | (Y 1,t−1 , Z t ). (Here, W t can be omitted.) Assumption A.2 is a weakerrequirement, as it may hold when strict exogeneity fails. Because of the conditioninginvolved, we call this conditional exogeneity. Chalak, K. and H. White (2010) discussstructural restrictions for canonical settable systems that deliver conditional exogeneity.Below, we also discuss practical tests for this assumption.Because of the finite numbers of lags involved in A.2, this is a finite-order conditionalexogeneity assumption. For convenience and because no confusion will arisehere, we simply refer to this as “conditional exogeneity."Assumption A.2 ensures that expected direct effects of Y 2,t−1 on Y 1,t are identified.As WL note, it suffices for A.2 that U t−1 ⊥ U 1,t | (Y 0 ,Z t−1 , X t ) and Y 2,t−1 ⊥ (Y 0 ,Z t−τ 1−1 ) |(Y 1,t−1 , X t ). Imposing U t−1 ⊥ U 1,t | (Y 0 ,Z t−1 , X t ) is the analog of requiring that serialcorrelation is absent when lagged dependent variables are present. Imposing Y 2,t−1 ⊥(Y 0 ,Z t−τ 1−1 ) | (Y 1,t−1 , X t ) ensures that ignoring Y 0 and omitting distant lags of Z t fromX t doesn’t matter.Our first result linking direct structural causality and G−causality shows that, givenA.1 and A.2 and with proper choice of Q t and S t , G−causality implies direct structuralcausality.Proposition 5.4 Let A.1 and A.2 hold. If Y 2,t−1finite order G−cause Y 1 with respect to X, i.e.,DS Y 1,t , t = 1,2,..., then Y 2 does notY 1,t ⊥ Y 2,t−1 | Y 1,t−1 , X t , t = 1,...,T − τ 2 .In stating G non-causality, we make the explicit identifications Q t = Y 2,t−1 and S t = X t .This result leads one to ask whether the converse relation also holds: does directstructural causality imply G−causality? Strictly speaking, the answer is no. WL discussseveral examples. The main issue is that with suitably chosen causal and probabilisticrelationships, Y 2,t−1 can cause Y 1,t , but Y 2,t−1 and Y 1,t can be independent, conditionallyor unconditionally, i.e. Granger non-causal.As WL further discuss, however, these examples are exceptional, in the sense thatmild perturbations to their structure destroy the Granger non-causality. WL introduce arefinement of the notion of direct structural causality that accommodates these specialcases and that does yield a converse result, permitting a characterization of structuraland Granger causality. Let supp(Y 1,t ) denote the support of Y 1,t , i.e., the smallest setcontaining Y 1,t with probability 1, and let F 1,t (· | Y 1,t−1 , X t ) denote the conditional distributionfunction of U 1,t given Y 1,t−1 , X t . WL introduce the following definition:19