Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable Systemscal tests for structural causality based on proposals of WL, using tests for G−causalityand conditional exogeneity. Section 7 contains a summary and concluding remarks.2. Pearl’s Causal ModelPearl’s definition of a causal model (Pearl, J., 2000, def. 7.1.1, p. 203) provides a formalstatement of elements supporting causal reasoning. The PCM is a triple M := (u,v, f ),where u := {u 1 ,...,u m } contains “background" variables determined outside the model,v := {v 1 ,...,v n } contains “endogenous" variables determined within the model, and f :={ f 1 ,..., f n } contains “structural" functions specifying how each endogenous variable isdetermined by the other variables of the model, so that v i = f i (v (i) ,u), i = 1,...,n. Here,v (i) is the vector containing every element of v but v i . The integers m and n are finite.The elements of u and v are system “units."Finally, the PCM requires that for each u, f yields a unique fixed point. Thus, theremust be a unique collection g := {g 1 ,...,g n } such that for each u,v i = g i (u) = f i (g (i) (u),u), i = 1,...,n. (1)The unique fixed point requirement is crucial to the PCM, as this is necessary fordefining the potential response function (Pearl, J., 2000, def. 7.1.4). This provides thefoundation for discourse about causal relations between endogenous variables; withoutthe potential response function, causal discourse is not possible in the PCM. A variantof the PCM (Halpern, J., 2000) does not require a fixed point, but if any exist, there maybe multiple collections of functions g yielding a fixed point. We call this a GeneralizedPearl Causal Model (GPCM). As GPCMs also do not possess an analog of the potentialresponse function in the absence of a unique fixed point, causal discourse in the GPCMis similarly restricted.In presenting the PCM, we have adapted Pearl’s notation somewhat to facilitatesubsequent discussion, but all essential elements are present and complete.Pearl, J. (2000) gives numerous examples for which the PCM is ideally suited forsupporting causal discourse. As a simple game-theoretic example, consider a marketin which there are exactly two firms producing similar but not identical products (e.g.,Coke and Pepsi in the cola soft-drink market). Price determination in this market is atwo-player game known as “Bertrand duopoly."In deciding its price, each firm maximizes its profit, taking into account the prevailingcost and demand conditions it faces, as well as the price of its rival. A simplesystem representing price determination in this market isp 1 = a 1 + b 1 p 2p 2 = a 2 + b 2 p 1 .Here, p 1 and p 2 represent the prices chosen by firms 1 and 2 respectively, and a 1 , b 1 ,a 2 , and b 2 embody the prevailing cost and demand conditions.3