Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable SystemsFirst, we specify the sense in which conditional exogeneity is necessary for theequivalence of G−causality and direct structural causality.DProposition 5.7 Given A.1, suppose that Y 2,t−1 S Y 1,t , t = 1,2,... . If A.2 does nothold, then for each t there exists q 1,t such that Y 1,t ⊥ Y 2,t−1 | Y 1,t−1 , X t does not hold.That is, if conditional exogeneity does not hold, then there are always structures thatgenerate data exhibiting G−causality, despite the absence of direct structural causality.Because q 1,t is unknown, this worst case scenario can never be discounted. Further, asWL show, the class of worst case structures includes precisely those usually assumedin applications, namely separable structures (e.g., Y 1,t = q 1,t (Y 1,t−1 , Z t ) + U 1,t ), as wellas the more general class of invertible structures. Thus, in the cases typically assumedin the literature, the failure of conditional exogeneity guarantees G−causality in theabsence of structural causality. We state this formally as a corollary.Corollary 5.8 Given A.1 with Y 2,t−1DS Y 1,t , t = 1,2,..., suppose that q 1,t is invertiblein the sense that Y 1,t = q 1,t (Y 1,t−1 , Z t ,U 1,t ) implies the existence of ξ 1,t such that U 1,t =ξ 1,t (Y 1,t−1 , Z t ,Y 1,t ), t = 1,2,... . If A.2 fails, then Y 1,t ⊥ Y 2,t−1 | Y 1,t−1 , X t fails, t = 1,2,....Together with Theorem 5.6, this establishes that in the absence of direct causality andfor the class of invertible structures predominant in applications, conditional exogeneityis necessary and sufficient for G non-causality.Tests of conditional exogeneity for the general separable case follow from:Proposition 5.9 Given A.1, suppose that E(Y 1,t ) < ∞ and thatq 1,t (Y t−1 , Z t ,U 1,t ) = ζ t (Y t−1 , Z t ) + υ t (Y 1,t−1 , Z t ,U 1,t ),where ζ t and υ t are unknown measurable functions. Let ε t := Y 1,t − E(Y 1,t |Y t−1 , X t ). IfA.2 holds, thenε t = υ t (Y 1,t−1 , Z t ,U 1,t ) − E(υ t (Y 1,t−1 , Z t ,U 1,t ) | Y 1,t−1 , X t )E(ε t |Y t−1 , X t ) = E(ε t |Y 1,t−1 , X t ) = 0andY 2,t−1 ⊥ ε t | Y 1,t−1 , X t . (5)Tests based on this result detect the failure of A.2, given separability. Such tests arefeasible because even though the regression error ε t is unobserved, it can be consistentlyestimated, say as ˆε t := Y 1,t − Ê(Y 1,t |Y t−1 , X t ), where Ê(Y 1,t |Y t−1 , X t ) is a parametric ornonparametric estimator of E(Y 1,t |Y t−1 , X t ). These estimated errors can then be used totest (5). If we reject (5), then we must reject A.2. We discuss a practical procedure inthe next section. WL provide additional discussion.WL also discuss dropping the separability assumption. For brevity, we maintainseparability here. Observe that under the null of direct non-causality, q 1,t is necessarilyseparable, as then ζ t is the zero function.21