Electronic Journal of Statistics

Trimmed Granger causality between two groups of time series

Ying-Chao Hung, Neng-Fang Tseng, and Narayanaswamy Balakrishnan

Full-text: Open access


The identification of causal effects between two groups of time series has been an important topic in a wide range of applications such as economics, engineering, medicine, neuroscience, and biology. In this paper, a simplified causal relationship (called trimmed Granger causality) based on the context of Granger causality and vector autoregressive (VAR) model is introduced. The idea is to characterize a subset of “important variables” for both groups of time series so that the underlying causal structure can be presented based on minimum variable information. When the VAR model is specified, explicit solutions are provided for the identification of important variables. When the parameters of the VAR model are unknown, an efficient statistical hypothesis testing procedure is introduced to estimate the solution. An example representing the stock indices of different countries is used to illustrate the proposed methods. In addition, a simulation study shows that the proposed methods significantly outperform the Lasso-type methods in terms of the accuracy of characterizing the simplified causal relationship.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 1940-1972.

First available in Project Euclid: 29 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F03: Hypothesis testing

Granger causality multivariate time series vector autoregressive process Wald test with constrained parameter spaces multiple hypothesis testing Lasso-penalized VAR approach


Hung, Ying-Chao; Tseng, Neng-Fang; Balakrishnan, Narayanaswamy. Trimmed Granger causality between two groups of time series. Electron. J. Statist. 8 (2014), no. 2, 1940--1972. doi:10.1214/14-EJS940. https://projecteuclid.org/euclid.ejs/1414588183

Export citation


  • [1] Arnold, A., Liu, Y. and Abe, N. (2008). Temporal causal modeling with graphical Granger methods., Proceedings of the 13th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 66–75.
  • [2] Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break Detection in the covariance structure of multivariate time series models., Ann. Statist. 37 4046–4087.
  • [3] Basu, S. and Michailidis, G. (2013). Estimation in high-dimensional vector autoregressive models. Arxiv preprint, arXiv:1311.4175.
  • [4] Berberian, S. K. (1961)., Introduction to Hilbert Space. Oxford University Press, New York.
  • [5] Boudjellaba, H., Dufour, J. N. and Roy, R. (1992). Testing causality between two vectors in multivariate autoregressive moving average models., J. Amer. Statist. Assoc. 87 1082–1090.
  • [6] Copeland, M. and Copeland, T. (1998). Lags, lags, and trading in global markets., Financ. Anal. J. 54 70–80.
  • [7] Davis, R. A., Zang, P. and Zheng, T. (2012). Sparse vector autoregressive modeling. Arxiv preprint, arXiv:1207.0520v1.
  • [8] Dufour, J. M. and Renault, E. (1998). Short-run and long-run causality in time series theory., Econometrica 66 1099–1125.
  • [9] Fujita, A., Sato, J. R., Garay-Malpartida, H. M., Morettin, P. A., Sogayar, M. C. and Ferreira, C. E. (2007). Time-varying modeling of gene expression regulatory networks using the wavelet dynamic vector autoregressive method., Bioinformatics 23 1623–1630.
  • [10] Geweke, J. (1982). Measurement of linear dependence and feedback between multiple time series., J. Amer. Statist. Assoc. 77 304–313.
  • [11] Geweke, J. (1984)., Inference and Causality in Economic Time Series. Handbook of Econometrics, Vol. 2, North-Holland, Amsterdam, 1101–1144.
  • [12] Granger, C. W. J. (1969). Investigating causal relations by econometric models and cross-spectral methods., Econometrica 37 424–438.
  • [13] Hacker, R. S. and Hatemi-J, A. (2006). Tests for causality between integrated variables using asymptotic and bootstrap distributions: Theory and application., Appl. Econ. 38 1489–1500.
  • [14] Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance., Biometrika 75 800–802.
  • [15] Hocking, R. R. (1976). The analysis and selection of variables in linear regression., Biometrics 32 1–49.
  • [16] Holm, S. (1979). A simple sequentially rejective multiple test procedure., Scand. J. Statist. 6 65–70.
  • [17] Hsiao, C. (1980). Autoregressive modeling and causal ordering of econometric variables., J. Econ. Dyn. Control 4 243–259.
  • [18] Jeong, J. (1999). Cross-border transmission of stock price volatility: Evidence from the overlapping trading hours., Global Fianc. J. 10 53–70.
  • [19] Kutner, M. H., Nachtsheim, C. J. and Neter, J. (2004)., Applied Liner Regression Models, 4th ed. MacGraw-Hill, New York.
  • [20] Luenberger, D. G. (1997)., Optimization by Vector Space Methods. Wiley, New York.
  • [21] Lütkepohl, H. (1993)., Testing for Causation Between two Variables in Higher Dimensional VAR Models. In H. Schneeweiss and K. F. Zimmermann (eds), Studies in Applied Econometrics, North-Holland, Amsterdam, 75–91.
  • [22] Lütkepohl, H. and Burda, M. M. (1997). Modified Wald tests under nonregular conditions., J. Econometrics 78 315–332.
  • [23] Lütkepohl, H. (2005)., New Introduction to Multiple Time Series Analysis. Springer-Verlag, Berlin.
  • [24] Miller, A. (2002)., Subset Selection in Regression, 2nd ed., Chapman & Hall/CRC Press, Boca Raton, FL.
  • [25] Miller, R. G. (1981)., Simultaneous Statistical Inference, 2nd ed., Springer-Verlag, New York.
  • [26] Mosconi, R. and Giannine, C. (1992). Non-causality in cointegrated system: Representation, estimation and testing., Oxford Bull. Econ. Statist. 54 399–417.
  • [27] Osborn, D. R. (1984). Causality testing and its implication for dynamic econometric models., Econ. J. 94 82–96.
  • [28] Rapach, D., Strauss, J. and Zhou, G. (2013). International stock return predictability: What is the role of the United States?, J. Finance, DOI: 10.1111/jofi.12041.
  • [29] Roebroech, A., Formisano, E. and Goebel, R. (2005). Mapping directed influence over the brain using Granger causality and fMRI., NeuroImage 25 230–242.
  • [30] Shojaie, A. and Michailidis, G. (2010). Discovering graphical Granger causality using the truncating lasso penalty., Bioinformatics 26 i517–i523.
  • [31] Šidák, Z. (1968). On multivariate normal probabilities of rectangles: Their dependence on correlations., Ann. Math. Statist. 39 1425–1434.
  • [32] Šidák, Z. (1971). On probabilities of rectangles in multivariate Student distributions: Their dependence on correlations., Ann. Math. Statist. 42 169–175.
  • [33] Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance., Biometrika 73 751–754.
  • [34] Sims, C. A. (1980). Macroeconomics and reality., Econometrica 48 1–48.
  • [35] Song, S. and Bickel, P. J. (2011). Large vector auto regressions. Arxiv preprint, arXiv:1106.3915.
  • [36] Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso., J. Roy. Statist. Soc. Ser. B 58 267–288.
  • [37] Worsley, K. J. (1982). An improved Bonferroni inequality and applications., Biometrika 69 297–302.