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PopescuH. White and X. Lu. Granger Causality and Dynamic Structural Systems. Journal ofFinancial Econometrics, 8(2):193, 2010.N. Wiener. The theory of prediction. Modern mathematics for engineers, Series, 1:125–139, 1956.H. O. Wold. A Study in the Analysis of Stationary Time Series. Stockholm: Almqvistand Wiksell., 1938.A. Yatchew. Nonparametric regression techniques in economics. Journal of EconomicLiterature, 36(2):669–721, 1998.G. U. Yule. Why do we sometimes get nonsense correlations between time series?a study in sampling and the nature of time series. Journal of the Royal StatisticalSociety, 89:1–64, 1926.A. Ziehe and K.-R. Mueller. Tdsep- an efficient algorithm for blind separation usingtime structure. ICANN Proceedings, pages 675–680, 1998.S. T. Ziliak and D. N. McCloskey. The Cult of Statistical Significance: How the StandardError Costs Us Jobs, Justice, and Lives. University of Michigan Press, February2008. ISBN 0472050079.Appendix A. Statistical significance tables for Type I and Type III errorsIn order to assist practitioners in evaluating the statistical significance of bivariatecausality testing, tables were prepared for type I and type III error probabilities as definedin (1) for different values of the base statistic. Below tables are provided for boththe jacknifed statistic Ψ/std(Ψ) and for the raw statistic Ψ, which is needed in case thenumber of points is too low to allow a jackknife/cross-validation/bootstrap or computationalspeed is at a premium. The spectral evaluation method is Welch’s method asdescribed in Section 6. There were 2000 simulations for each condition. The tables inthis Appendix differ in one important aspect with respect to those in the main text. Inorder to avoid non-informative comparison of datasets which are, for example, analysesof the same physical process sampled at different sampling rates, the number of pointsis scaled by the ‘effective’ number of points which is essentially the number of samplesrelative to a simple estimate of the observed signal bandwidth:N * = Nτ S /̂︂BŴ︂BW =‖X‖ F‖∆X/∆T‖ FThe values marked with an asterisk have values of both α and γ which are less than5%. Note also that Ψ is non-dimensional index.70
Robust Statistics for CausalityTable 4: α vs. Ψ/std(Ψ)N * → 50 100 200 500 750 1000 1500 2000 50000.125 0.82 0.83 0.86 0.87 0.88 0.90 0.89 0.89 0.890.25 0.67 0.69 0.73 0.77 0.76 0.78 0.78 0.78 0.770.5 0.41 0.44 0.50 0.54 0.55 0.56 0.56 0.59 0.570.75 0.26 0.26 0.32 0.36 0.36 0.37 0.38 0.40 0.391 0.15 0.15 0.20 0.23 0.23 0.25 0.25 0.26 0.261.25 0.09 0.09 0.11 0.13 0.14 0.16 0.14 0.15 0.161.5 0.06 0.05 0.06 0.07 0.08 0.09 0.08 0.10 0.101.75 0.04 0.03 0.04 0.05 0.05 * 0.05 * 0.04 * 0.06 0.062 0.03 0.02 0.03 0.02 * 0.02 * 0.03 * 0.02 * 0.03 * 0.03 *2.5 0.01 0.01 0.01 0.01 * 0.01 * 0.01 * 0.00 * 0.01 * 0.01 *3 0.01 0.00 0.00 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *4 0.00 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *8 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *16 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *Table 5: γ vs. Ψ/std(Ψ)N * → 50 100 200 500 750 1000 1500 2000 50000.125 0.46 0.41 0.35 0.26 0.23 0.21 0.19 0.18 0.160.25 0.46 0.39 0.33 0.24 0.21 0.19 0.18 0.16 0.140.5 0.44 0.36 0.29 0.21 0.18 0.15 0.14 0.13 0.120.75 0.43 0.35 0.28 0.17 0.14 0.12 0.11 0.10 0.091 0.43 0.31 0.23 0.13 0.12 0.09 0.08 0.07 0.061.25 0.42 0.31 0.20 0.09 0.08 0.06 0.05 0.05 0.041.5 0.40 0.26 0.20 0.06 0.05 0.04 0.04 0.04 0.021.75 0.42 0.26 0.16 0.05 0.04 * 0.02 * 0.02 * 0.03 0.012 0.41 0.23 0.11 0.03 * 0.03 * 0.02 * 0.02 * 0.02 * 0.01 *2.5 0.41 0.20 0.06 0.02 * 0.02 * 0.01 * 0.01 * 0.01 * 0.01 *3 0.33 0.09 0.06 0.03 * 0.01 * 0.00 * 0.00 * 0.00 * 0.00 *4 0.33 0.00 * 0.00 * 0.02 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *8 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *16 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 * 0.00 *71
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Causality in Time SeriesVolume 5:Ca
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PrefaceThe following is a print ver
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JMLR: Workshop and Conference Proce
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Linking Granger Causality and the P
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Page 42 and 43: Popescu1. IntroductionCausality is
Page 44 and 45: Popescugeneral terms, if we are pre
Page 46 and 47: PopescuType III error prob. γ = P
Page 48 and 49: Popescuwhich would correspond to a
Page 50 and 51: Popescuvalued vector. A Data Genera
Page 52 and 53: Popescuy i =K∑︁A k y i−k + Bu
Page 54 and 55: Popescuy i =K∑︁A k y i−k + Bu
Page 56 and 57: PopescuFigure 1: SVAR causality and
Page 58 and 59: Popescu6. Spectral methods and phas
Page 60 and 61: PopescuH GCj→i|u = logD i − log
Page 62 and 63: Popescu8.1. The cardinal transform
Page 64 and 65: Popescuy N,i = Bx N,i (37)y = (1
Page 66 and 67: Popescu(a) Unmixed colored noise(b)
Page 68 and 69: PopescuAs we can see in both Figure
Page 70 and 71: Popescuβ > 0 y C,i =x N,i =⎡K∑
Page 72 and 73: Popescupretation of information flo
Page 74 and 75: Popescuit is suggested that a princ
Page 76 and 77: PopescuA. N. Kolmogorov and A. N. S
Page 80 and 81: PopescuTable 6: α vs. ΨN * → 50
Page 82 and 83: Roebroeck Seth Valdes-Sosaincreasin
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Page 86 and 87: Roebroeck Seth Valdes-Sosasignal fr
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Page 90 and 91: Roebroeck Seth Valdes-Sosaat lower
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