Causality in Time Series - ClopiNet

Linking Granger Causality and the Pearl Causal Model with Settable SystemsA common focus of interest when applying structural VARs is to learn the coefficientvector a. In applications, it is typically assumed that the realizations {y t } areobserved, whereas {u t } is unobserved. The least squares estimator for a sample of sizeT, say â T , is commonly used to learn (estimate) a in such cases. This estimator is astraightforward function of y T , say â T = r a,T (y T ). If {u t } is generated as a realizationof a sequence of mean zero finite variance independent identically distributed (IID)random variables, then â T generally converges to a with probability one as T → ∞,implying that a can be fully learned in the limit. Viewing â T as causally determined byy T , we see that we require a countable number of units to treat this learning problem.As these examples demonstrate, the PCM exhibits a number of features that limitits applicability to systems involving optimization, equilibrium, and learning. Theselimitations motivate a variety of features of settable systems, extending the PCM inways that permit straightforward treatment of such systems. We now turn to a morecomplete description of the SS framework.3.2. Formal Settable SystemsWe now provide a formal description of settable systems that readily accommodatescausal discourse in the foregoing examples and that also suffices to establish the desiredlinkage between Granger causality and causal notions in the PCM. The material thatfollows is adapted from Chalak, K. and H. White (2010). For additional details, seeWC.A stochastic settable system is a mathematical framework in which a countablenumber of units i, i = 1,...,n, interact under uncertainty. Here, n ∈ ¯N + := N + ∪{∞}, whereN + denotes the positive integers. When n = ∞, we interpret i = 1,...,n as i = 1,2,....Units have attributes a i ∈ A; these are fixed for each unit, but may vary across units.Each unit also has associated random variables, defined on a measurable space (Ω,F ).It is convenient to define a principal space Ω 0 and let Ω := × n i=0 Ω i, with each Ω i acopy of Ω 0 . Often, Ω 0 = R is convenient. A probability measure P a on (Ω,F ) assignsprobabilities to events involving random variables. As the notation suggests, P a candepend on the attribute vector a := (a 1 ,...,a n ) ∈ A := × n i=1 A.The random variables associated with unit i define a settable variable X i for thatunit. A settable variable X i has a dual aspect. It can be set to a random variabledenoted by Z i (the setting), where Z i : Ω i → S i . S i denotes the admissible setting valuesfor Z i , a multi-element subset of R. Alternatively, the settable variable can be freeto respond to settings of other settable variables. In the latter case, it is denoted bythe response Y i : Ω → S i . The response Y i of a settable variable X i to the settingsof other settable variables is determined by a response function, r i . For example, r ican be determined by optimization, determining the response for unit i that is best insome sense, given the settings of other settable variables. The dual role of a settablevariable X i : {0,1} × Ω → S i , distinguishing responses X i (0,ω) := Y i (ω) and settingsX i (1,ω) := Z i (ω i ), ω ∈ Ω, permits formalizing the directional nature of causal relations,whereby settings of some variables (causes) determine responses of others.9