Causality in Time Series - ClopiNet

Causal Search in SVARTo test Granger noncausality, researchers often specify linear vector autoregressive(VAR) models:Y t = A 1 Y t−1 + ... + A p Y t−p + u t , (3)in which Y t is a k × 1 vector of time series variables (Y 1,t ,...,Y k,t ) ′ , where () ′ is thetranspose, the A j ( j = 1,..., p) are k × k coefficient matrices, and u t is the k × 1 vectorof random disturbances. In this framework, testing the hypothesis that {Y i,t } does notGranger-cause {Y j,t }, reduces to test whether the ( j,i) entries of the matrices A 1 ,...,A pare vanishing simultaneously. Granger noncausality tests have been extended to nonlinearsettings by Baek and Brock (1992), Hiemstra and Jones (1994), and Su and White(2008), using nonparametric tests of conditional independence (more on this topic insection 4).The concept of Granger causality has been criticized for failing to capture ‘structuralcausality’ (Hoover, 2008). Suppose one finds that a variable A Granger-causesanother variable B. This does not necessarily imply that an economic mechanism existsby which A can be manipulated to affect B. The existence of such a mechanism inturn does not necessarily imply Granger causality either (for a discussion see Hoover2001, pp. 150-155). Indeed, the analysis of Granger causality is based on coefficientsof reduced-form models, like those incorporated in equation (3), which are unlikely toreliably represent actual economic mechanisms. For instance, in equation (3) the simultaneouscausal structure is not modeled in order to facilitate estimation. (However, notethat Eichler (2007) and White and Lu (2010) have recently developed and formalizedricher structural frameworks in which Granger causality can be fruitfully analyzed.)1.2. The SVAR frameworkStructural vector autoregressive (SVAR) models constitute a middle way between theCowles Commission approach and the Granger-causality approach. SVAR models aimat recovering the concept of structural causality, but eschew at the same time the strong‘apriorism’ of the Cowles Commission approach. The idea is, like in the Cowles Commissionapproach, to articulate an unobserved structural model, formalized as a dynamicgenerative model: at each time unit the system is affected by unobserved innovationterms, by which, once filtered by the model, the variables come to take the valueswe observe. But, differently from the Cowles Commission approach, and similarly tothe Granger-VAR model, the data generating process is generally enough articulatedso that time series variables are not distinguished a priori between exogenous and endogenous.A linear SVAR model is in principle a VAR model ‘augmented’ by thecontemporaneous structure:Γ 0 Y t = Γ 1 Y t−1 + ... + Γ p Y t−p + ε t . (4)This is easily obtained by pre-multiplying each side of the VAR modelY t = A 1 Y t−1 + ... + A p Y t−p + u t , (5)by a matrix Γ 0 so that Γ i = Γ 0 A i , for i = 1,...,k and ε t = Γ 0 u t . Note, however, that notany matrix Γ 0 will be suitable. The appropriate Γ 0 will be that matrix corresponding107