Causality in Time Series - ClopiNet

White Chalak Luvariables. In practice, researchers typically use parametric methods. These are convenient,but they may lack power against important alternatives. To provide convenientprocedures for testing GN and CE with power against a wider range of alternatives,WL propose augmenting standard tests with neural network terms, motivated by the“QuickNet" procedures introduced by White, H (2006b) or the extreme learning machine(ELM) methods of Huang, G.B., Q.Y. Zhu, and C.K. Siew (2006). We now provideexplicit practical methods for testing GN and CE for a leading class of structuresobeying A.1.6.2.1. Testing Granger Non-CausalityStandard tests for finite-order G−causality (e.g., Stock, J. and M. Watson, 2007, p. 547)typically assume a linear regression, such as 3E(Y 1,t |Y t−1 , X t ) = α 0 + Y ′ 1,t−1 ρ 0 + Y ′ 2,t−1 β 0 + X ′ tβ 1 .For simplicity, we let Y 1,t be a scalar here. The extension to the case of vector Y 1,t iscompletely straightforward. Under the null of GN, i.e., Y 1,t ⊥ Y 2,t−1 | Y 1,t−1 , X t , we haveβ 0 = 0. The standard procedure therefore tests β 0 = 0 in the regression equationY 1,t = α 0 + Y ′ 1,t−1 ρ 0 + Y ′ 2,t−1 β 0 + X ′ tβ 1 + ε t . (GN Test Regression 1)If we reject β 0 = 0, then we also reject GN. But if we don’t reject β 0 = 0, care is needed,as not all failures of GN will be indicated by β 0 0.Observe that when CE holds and if GN Test Regression 1 is correctly specified, i.e.,the conditional expectation E(Y 1,t |Y t−1 , X t ) is indeed linear in the conditioning variables,then β 0 represents precisely the direct structural effect of Y 2,t−1 on Y 1,t . Thus,GN Test Regression 1 may not only permit a test of GN, but it may also provide aconsistent estimate of the direct structural effect of interest.To mitigate specification error and gain power against a wider range of alternatives,WL propose augmenting GN Test Regression 1 with neural network terms, as inWhite’s (2006b, p. 476) QuickNet procedure. This involves testing β 0 = 0 inY 1,t = α 0 + Y ′ 1,t−1 ρ 0 + Y ′ 2,t−1 β 0 + X ′ tβ 1 +r∑︁ψ(Y ′ 1,t−1 γ 1, j + X ′ tγ j )β j+1 + ε t .j=1(GN Test Regression 2)Here, the activation function ψ is a generically comprehensively revealing (GCR) function(see Stinchcombe, M. and H. White, 1998). For example, ψ can be the logistic cdfψ(z) = 1/(1 + exp(−z)) or a ridgelet function, e.g., ψ(z) = (−z 5 + 10z 3 − 15z)exp(−.5z 2 )(see, for example, Candès, E. (1999)). The integer r lies between 1 and ¯r, the maximumnumber of hidden units. We randomly choose (γ 0 j ,γ j ) as in White, H (2006b, p. 477).3. For notational convenience, we understand that all regressors have been recast as vectors containingthe referenced elements.24