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White Chalak LuWe then reject CE if T T > ĉ T,n,1−α , where, with n chosen sufficiently large, ĉ T,n,1−α isthe 1 − α percentile of the weighted bootstrap statisticsT T,i := T (ˆθ T,i − ˆθ T ) ′ S ′ 2 S 2 (ˆθ T,i − ˆθ T ) = T (ˆβ 2,0,T,i − ˆβ 2,0,T ) ′ (ˆβ 2,0,T,i − ˆβ 2,0,T ),i = 1,...,n.This procedure is asymptotically valid, even though T T is based on the “unstudentized"statistic S 2 ˆθ T = ˆβ 2,0,T . Alternatively, one can construct a studentized statisticT * T := T ˆθ ′ T S′ 2 [S 2 ĈT,n S ′ 2 ]−1 S 2 ˆθ T ,where Ĉ T,n is an asymptotic covariance estimator constructed from √ T(ˆθ T,i − ˆθ T ), i =1,...,n. The test rejects CE if T * T > c 1−α, where c 1−α is the 1 − α percentile of the chisquareddistribution with dim(β 0,2 ) degrees of freedom. This method is more involvedbut may have better control over the level of the test. WL provide further discussionand methods.Because the given asymptotic distribution is joint for ˆθ 1,T and ˆθ 2,T , the same methodsconveniently apply to testing GN, i.e., β 1,0 = S 1 θ 0 = 0, where S 1 selects β 1,0 fromθ 0 . In this way, GN and CE test statistics can be constructed at the same time.WL discuss three examples, illustrating tests for direct structural non-causalitybased on tests of Granger non-causality and conditional exogeneity. A matlab module,testsn, implementing the methods described here is available at http://ihome.ust.hk/~xunlu/code.htm.7. Summary and Concluding RemarksIn this paper, we explore the relations between direct structural causality in the settablesystems framework and direct causality in the PCM for both recursive and nonrecursivesystems. The close correspondence between these concepts in recursive systemsand the equivalence between direct structural causality and G−causality establishedby WL enable us to show the close linkage between G−causality and PCM notionsof direct causality. We apply WL’s results to provide straightforward practicalmethods for testing direct causality using tests for Granger causality and conditionalexogeneity.The methods and results described here draw largely from work of WC and WL.These papers contain much additional relevant discussion and detail. WC provide furtherexamples contrasting settable systems and the PCM. Chalak, K. and H. White(2010) build on WC, examining not only direct causality in settable systems, but alsonotions of indirect causality, which in turn yield implications for conditional independencerelations, such as those embodied in conditional exogeneity, which plays a keyrole here. WL treat not only the structural VAR case analyzed here, but also the “timeseriesnatural experiment" case, where causal effects of variables D t , absorbed here intoZ t , are explicitly analyzed. The sequence {D t } represents external stimuli, not driven by{Y t }, whose effects on {Y t } are of interest. For example, {D t } could represent passivelyobserved visual or auditory stimuli, and {Y t } could represent measured neural activity.Interest may attach to which stimuli directly or indirectly affect which neurons or28
Linking Granger Causality and the Pearl Causal Model with Settable Systemsgroups of neurons. WL also examine the structural content of classical Granger causalityand a variety of related alternative versions that emerge naturally from differentversions of Assumption A.1.AcknowledgmentsWe express our deep appreciation to Sir Clive W.J. Granger for his encouragement ofthe research underlying the work presented here.ReferencesE. Candès. Ridgelets: Estimating with Ridge Functions. Annals of Statistics, 31:1561–1599, 1999.K. Chalak and H. White. Causality, Conditional Independence, and Graphical Separationin Settable Systems. Technical report, Department of Economics, BostonCollege, 2010.A. P. Dawid. Conditional Independence in Statistical Theory. Journal of the RoyalStatistical Society, Series B, 41:1–31, 1979.A. P. Dawid. Beware of the DAG! Proceedings of the NIPS 2008 Workshop on Causality,Journal of Machine Learning Research Workshop and Conference Proceedings,6:59–86, 2010.M. A. Delgado and W. Gonzalez-Manteiga. Significance Testing in NonparametricRegression Based on the Bootstrap. Annals of Statistics, 29:1469–1507, 2001.M. Eichler. Granger Causality and Path Diagrams for Multivariate Time Series. Journalof Econometrics, 137:334-353, 2007.M. Eichler and V. Didelez. Granger-causality and the Effect of Interventions in TimeSeries. Lifetime Data Analysis, forthcoming.R. Engle, D. Hendry, and J.F. Richard. Exogeneity. Econometrica 51:277–304, 1983.M. Fernandes and R. G. Flores. Tests for Conditional Independence, Markovian Dynamics,and Noncausality. Technical report, European University Institute, 2001.J.P. Florens and D. Fougère. Non-causality in Continuous Time. Econometrica,64:1195–1212, 1996.J.P. Florens and M. Mouchart. A Note on Non-causality. Econometrica, 50:583–591,1982.A.R. Gallant and H. White. On Learning the Derivatives of an Unknown Mapping withMultilayer Feedforward Networks. Neural Networks, 5:129–138, 1992.29
Page 1: Causality in Time SeriesVolume 5:Ca
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Page 42 and 43: Popescu1. IntroductionCausality is
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