Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable SystemsParallel to our comment above about estimating direct structural effects of interest,we note that given A.1, A.2, and some further mild regularity conditions, such effectscan be identified and estimated from a neural network regression of the formY 1,t = α 0 + Y ′ 1,t−1 ρ 0 + Y ′ 2,t−1 β 0 + X ′ tβ 1r∑︁+ ψ(Y ′ 1,t−1 γ 1, j + Y ′ 2,t−1 γ 2, j + X ′ tγ 3, j )β j+1 + ε t .j=1Observe that this regression includes Y 2,t−1 inside the hidden units. With r chosen sufficientlylarge, this permits the regression to achieve a sufficiently close approximationto E(Y 1,t |Y t−1 , X t ) and its derivatives (see Hornik, K. M. Stinchcombe, and H. White,1990; Gallant, A.R. and H. White, 1992) that regression misspecification is not such anissue. In this case, the derivative of the estimated regression with respect to Y 2,t−1 wellapproximates(∂/∂y 2 )E(Y 1,t | Y 1,t−1 ,Y 2,t−1 = y 2 , X t )= E[ (∂/∂y 2 )q 1,t (Y 1,t−1 , y 2 , Z t ,U 1,t ) | Y 1,t−1 , X t ].This quantity is the covariate conditioned expected marginal direct effect of Y 2,t−1 onY 1,t .Although it is possible to base a test for GN on these estimated effects, we do notpropose this here, as the required analysis is much more involved than that associatedwith GN Test Regression 2.Finally, to gain additional power WL propose tests using transformations of Y 1,t ,Y 1,t−1 , and Y 2,t−1 , as Y 1,t ⊥ Y 2,t−1 | Y 1,t−1 , X t implies f (Y 1,t ) ⊥ g(Y 2,t−1 ) | Y 1,t−1 , X t forall measurable f and g. One then tests β 1,0 = 0 inψ 1 (Y 1,t ) = α 1,0 + ψ 2 (Y 1,t−1 ) ′ ρ 1,0 + ψ 3 (Y 2,t−1 ) ′ β 1,0 + X ′ tβ 1,1r∑︁+ ψ(Y ′ 1,t−1 γ 1,1, j + X ′ tγ 1, j )β 1, j+1 + η t . (GN Test Regression 3)j=1We take ψ 1 and the elements of the vector ψ 3 to be GCR, e.g., ridgelets or the logisticcdf. The choices of γ,r, and ψ are as described above. Here, ψ 2 can be the identity(ψ 2 (Y 1,t−1 ) = Y 1,t−1 ), its elements can coincide with ψ 1 , or it can be a different GCRfunction.6.2.2. Testing Conditional ExogeneityTesting conditional exogeneity requires testing A.2, i.e., Y 2,t−1 ⊥ U 1,t | Y 1,t−1 , X t . SinceU 1,t is unobservable, we cannot test this directly. But with separability (which holdsunder the null of direct structural non-causality), Proposition 5.9 shows that Y 2,t−1 ⊥U 1,t | Y 1,t−1 , X t implies Y 2,t−1 ⊥ ε t | Y 1,t−1 , X t , where ε t := Y 1,t − E(Y 1,t |Y t−1 , X t ). Withcorrect specification of E(Y 1,t |Y t−1 , X t ) in either GN Test Regression 1 or 2 (or someother appropriate regression), we can estimate ε t and use these estimates to test Y 2,t−1 ⊥25